Definition of an equivalent functional unit for structural … · 2015. 12. 5. · Giles Dobbelaere...

Giles Dobbelaere

concrete incorporating recycled aggregatesDefinition of an equivalent functional unit for structural

Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: Prof. Marc VanhaelstDepartment of Industrial Technology and Construction

Master of Science in de industriële wetenschappen: bouwkundeMaster's dissertation submitted in order to obtain the academic degree of

Counsellor: dhr. Luis Evangelista (Instituto Superior Técnico)Supervisors: Prof. Patrick Ampe, Prof. Jorge de Brito (Instituto Superior Técnico)

Definition of an equivalent functional unit for structural

concrete incorporating recycled aggregates

Giles Dobbelaere

Dissertation to obtain the Master of Science Degree in

Civil Engineering

Supervisors

Professor Doctor Jorge Manuel Caliço Lopes de Brito

Professor Doctor Luís Manuel da Rocha Evangelista

Examination Committee

Chairperson:

Supervisor:

Member of Committee:

June 2015

I

Acknowledgments

This work is accomplished with the help of a couple of persons to whom I want to express my

gratitude.

First and foremost I would like to take this opportunity to express my sincere gratefulness to my

supervisor, Professor Doctor Jorge Manuel Caliço de Brito, for encouraging me to pursue this

dissertation. Professor de Brito helped me through the various aspects of the dissertation and was

available at any moment of the day to discuss and help me with the occurring problems. He was the

most important source of guidance throughout this project and taught me not only a lot about the

subject, but also about discipline and criticism throughout the dissertation.

Furthermore, I want to thank Professor Doctor Luís Manuel da Rocha Evangelista who was my co-

supervisor in this project. He also gave me advice concerning the calculations according to Eurocode

2 and was always available to discuss aspects of the work.

It would be remiss of me not to thank Mister Rui Vasco Silva. He was an important source of

information for the dissertation. Moreover, he helped me with the calculations and provided the

dissertation with the various aspects of the fundamental parameters and relationships.

I also want to thank the people of Internationalization at Técnico and University Ghent who made the

Erasmus-experience possible. Also Mister Marc Wylaers of campus Schoonmeersen in Ghent was a

great help to complete the administrative issues concerning the dissertation.

Last but not least, a special thank you to my family, particularly to my father, who made this Erasmus

stay possible. His encouragement, guidance and understanding helped me to pursue the dissertation.

Giles Dobbelaere

II

Abstract

Many developers, researchers and engineers are seeking efficient, sustainable building solutions that

conserve non-renewable resources. Owners want to use the research solutions in response to

growing environmental forces and concrete incorporating recycled aggregates is a good choice to

meet these goals. This study intends to determine an equivalent functional unit in concrete with

recycled aggregates to conventional structural concrete in the context of Life Cycle Assessment

analyses. The work aims to contribute to a better understanding and greater confidence in the use of

concrete products with recycled aggregates.

The relationship between recycled aggregates concrete and conventional concrete is expressed by

fundamental parameters α, which describe the relevant equivalent properties of recycled aggregates

concrete in function of the same property of conventional concrete. Using those parameters, the

dissertation performs a thorough analysis according to Eurocode 2: the various compliance checks

with the limit states are performed to obtain the amount of recycled aggregates concrete required to

reach the same functionality as for conventional structural concrete. Conversion criteria for concrete

structures with recycled aggregates (concerning its structural performance) are established and the

conversion formulas are tested in case studies. The method in this dissertation is specifically

developed for slabs and beams, but remarks are made for other structural elements, e.g. columns and

footings. The results show that the method is validated for slabs and beams and that the conversion

formulas yield good results. Further research should improve the conversion formulas and

fundamental parameters.

Key-words

Recycled aggregates concrete, Eurocode 2 (EC2), equivalent functional unit, fundamental parameters,

Life Cycle Assessment (LCA)

III

Abstract

Onderzoekers en ingenieurs zoeken naar efficiënte en duurzame constructie-oplossingen die

hernieuwbare bronnen gebruiken. Eigenaars van bedrijven willen de onderzoeksresultaten in hun

productieproces introduceren zodat voldaan wordt aan de groeiende milieueisen. Beton met

gerecycleerde granulaten is een goede manier om de doelstellingen te bereiken en hiervoor wordt

meer onderzoek uitgevoerd.

Deze masterproef vormt een eerste poging om een gelijkwaardige eenheid in beton met

gerecycleerde granulaten te bepalen. Het is vereist dat deze eenheid dezelfde functionaliteit als een

eenheid conventioneel beton heeft. De resultaten zullen uiteindelijk in het kader van Life Cycle

Assessment kunnen gebruikt worden. Het project heeft als doel om bij te dragen aan een groter

vertrouwen in het gebruik van betonproducten met gerecycleerde granulaten.

Het verband tussen beton met gerecycleerde granulaten en normaal beton wordt uitgedrukt door de

fundamentele parameters α. Deze parameters beschrijven de relevante equivalente eigenschappen in

functie van de corresponderende eigenschap van normaal beton. Met behulp van deze parameters

wordt in de masterproef een diepgaande analyse volgens Eurocode 2 uitgevoerd: er moet voldaan

worden aan duurzaamheid en de verschillende grenstoestanden om de hoeveelheid beton met

gerecycleerde granulaten te bekomen. Deze hoeveelheid is dus nodig om dezelfde functionaliteit als

voor elementen in normaal beton te bereiken. De omzettingscriteria (met betrekking tot structurele

prestaties) voor betonstructuren met gerecycleerde granulaten worden opgesteld en de uiteindelijke

omzettingsformules worden getest in case studies. De methode in de masterproef is vooral ontwikkeld

voor platen en balken, maar er worden ook opmerkingen gegeven met betrekking tot andere

structurele elementen zoals kolommen en funderingen. De resultaten bewijzen dat de methode geldig

is voor platen en balken en dat de conversieformules goedgekozen zijn. Verder onderzoek zou de

conversieformules en de fundamentele parameters kunnen verbeteren.

Sleutelwoorden

Gerecycleerd beton, Eurocode 2 (EC2), gelijkwaardige functionele eenheid, fundamentele parameters,

Life Cycle Assessment (LCA)

IV

Resumo

Diversos projectistas, investigadores e engenheiros estão à procura de novas soluções construtivas

sustentáveis, eficientes e que consigam preservar os recursos não renováveis. Existe uma tendência

gradual para a utilização de soluções de investigação sustentáveis, em resposta aos crescentes

impactos ambientais, sendo uma delas a utilização de betões com agregados reciclados. Este estudo

pretende determinar uma unidade funcional equivalente que consiga relacionar as propriedades de

betões com agregados reciclados com aquelas de betões estruturais convencionais no contexto de

uma Avaliação do Ciclo de Vida. O trabalho visa contribuir para uma melhor compreensão e uma

maior confiança na utilização de produtos de betão com agregados reciclados.

A relação entre betões com agregados reciclados e betões convencionais pode ser representada por

parâmetros fundamentais α, que descrevem as propriedades equivalentes relevantes de betões com

agregados reciclados em função da mesma propriedade de um betão convencional equivalente. Esta

dissertação contém uma análise aprofundada da utilização estes parâmetros, de acordo com as

especificações do Eurocódigo 2. Foram efectuadas diversas verificações dos estados limites, em

conformidade com esta norma, de forma a obter uma quantidade necessária de betão com agregados

reciclados que demonstre o mesmo desempenho de um betão estrutural convencional. Foram

estabelecidos critérios de conversão para estruturas de betão com agregados reciclados (relativos ao

seu desempenho estrutural), cujas fórmulas de conversão foram testadas em casos de estudo.

Embora este método tivesse sido desenvolvido para lajes e vigas nesta dissertação, é possível

adaptá-lo para outros elementos estruturais (e.g. colunas e sapatas).

Os resultados demonstraram que o método é válido para lajes e vigas que as fórmulas de conversão

mostraram bons resultados. Contudo, é necessária investigação adicional de forma a incluir os outros

elementos estruturais e melhorar as fórmulas de conversão e parâmetros fundamentais.

Palavras-chave

Betão com agregados reciclados, Eurocódigo 2 (EC2), unidade funcional equivalente, parâmetros

fundamentais, Avaliação do Ciclo de Vida (ACV)

V

Table of contents

Table of contents ..................................................................................................................................... V

List of tables............................................................................................................................................ XI

List of figures ........................................................................................................................................ XV

List of acronyms ................................................................................................................................. XVII

List of symbols ..................................................................................................................................... XIX

Chapter 1

Introduction .............................................................................................................................................. 1

1.1 Overview ........................................................................................................................................ 1

1.1.1 Compressive strengths fcm ...................................................................................................... 1

1.1.2 Secant moduli of elasticity Ecm and axial tensile strength fctm................................................. 2

1.1.3 Depths of carbonation and chloride penetration .................................................................... 3

1.1.4 Creep ...................................................................................................................................... 4

1.2 Motivation and contents ................................................................................................................. 5

1.3 Structure of the thesis .................................................................................................................... 5

Chapter 2

General data and scope .......................................................................................................................... 7

2.1 Limit states in Eurocode 2 ............................................................................................................. 7

2.1.1 Ultimate Limit States ............................................................................................................... 7

2.1.2 Serviceability Limit States ....................................................................................................... 7

2.1.2.1 Crack control ................................................................................................................... 7

2.1.2.2 Deflection control ............................................................................................................. 8

2.2 Scope of the dissertation ............................................................................................................... 8

2.2.1 Concrete and classes ............................................................................................................. 8

2.2.2 Loads ...................................................................................................................................... 9

2.2.2.1 Slabs ................................................................................................................................ 9

2.2.2.2 Differences and adaptations to beams ............................................................................ 9

2.3 Durability ........................................................................................................................................ 9

2.4 Assumptions and simplifications ................................................................................................. 10

2.4.1 Assumptions ......................................................................................................................... 10

VI

2.4.1.1 Slabs .............................................................................................................................. 10

2.4.1.2 Differences and adaptations to beams .......................................................................... 11

2.4.2 Simplifications ....................................................................................................................... 11

2.4.2.1 Slabs .............................................................................................................................. 11


2.5 Relationship between RC and RAC ............................................................................................ 13

2.5.1 Fundamental parameters ..................................................................................................... 14

2.5.2 Justification of the parameters used ..................................................................................... 14

2.5.2.1 Mean value of the compressive strength ....................................................................... 14

2.5.2.2 Secant modulus of elasticity of concrete ....................................................................... 15

2.5.2.3 Depth of carbonation ..................................................................................................... 15

2.5.2.4 Depth of chlorides .......................................................................................................... 15

2.5.2.5 Shrinkage....................................................................................................................... 15

2.6 Methodology and flowchart.......................................................................................................... 16

2.7 Life cycle assessment ................................................................................................................. 17

2.7.1 Definition of goal and scope ................................................................................................. 18

2.7.2 Life cycle inventory ............................................................................................................... 18

2.7.3 Assessment of the environmental impacts ........................................................................... 18

2.7.4 Interpretation of the results ................................................................................................... 19

Chapter 3

Parametric studies involving the limit states .......................................................................................... 21

3.1 Main purpose ............................................................................................................................... 21

3.2 Design compliance criteria .......................................................................................................... 21

3.2.1 Durability ............................................................................................................................... 21

3.2.2 Deformation serviceability limit state .................................................................................... 22

3.2.3 Bending ultimate limit state................................................................................................... 23

3.2.4 Cracking serviceability limit state ......................................................................................... 24

3.3 Methodology compliance criteria ................................................................................................. 25

3.4 Parametric studies ....................................................................................................................... 25

3.4.1 Durability ............................................................................................................................... 25

3.4.1.1 Methodology .................................................................................................................. 26

3.4.1.2 Results ........................................................................................................................... 26

3.4.1.3 Discussion ..................................................................................................................... 27


VII

3.4.2 Deformation serviceability limit state .................................................................................... 27

3.4.2.1 Methodology and verification formula ............................................................................ 27

3.4.2.2 Results ........................................................................................................................... 28

3.4.2.3 Discussion ..................................................................................................................... 28


3.4.3 Bending ultimate limit state................................................................................................... 30

3.4.3.1 Methodology and verification formula ............................................................................ 30

3.4.3.2 Results ........................................................................................................................... 31

3.4.3.3 Discussion ..................................................................................................................... 31


3.4.4 Cracking serviceability limit state ......................................................................................... 32

3.4.4.1 Methodology .................................................................................................................. 32

3.4.4.1.1 Stress in tension reinforcement .............................................................................. 33

3.4.4.1.2 Bending moment .................................................................................................... 33

3.4.4.1.3 Height of the compressive zone ............................................................................. 33

3.4.4.1.4 Effective cross-section area of concrete in tension ................................................ 34

3.4.4.2 Verification formula ........................................................................................................ 34

3.4.4.3 Results ........................................................................................................................... 35

3.4.4.4 Discussion ..................................................................................................................... 35


3.5 Conclusion of Chapter 3 .............................................................................................................. 38

Chapter 4

Definition of the equivalent functional units ........................................................................................... 39

4.1 Functionality ................................................................................................................................ 39

4.2 K m³ of RAC ................................................................................................................................ 39

4.3 Design compliance criteria .......................................................................................................... 39

4.4 Methodology compliance criteria ................................................................................................. 39

4.5 Calculation of equivalent functional unit ...................................................................................... 40

4.5.1 hRAC/hRC in function of α3 and α4 ............................................................................................ 40

4.5.1.1 Methodology .................................................................................................................. 40

4.5.1.2 Results ........................................................................................................................... 40

4.5.1.3 Discussion ..................................................................................................................... 40


4.5.2 hRAC/hRC in function of α2 and α6 ............................................................................................ 44

4.5.2.1 Methodology .................................................................................................................. 44

VIII

4.5.2.2 Results ........................................................................................................................... 44

4.5.2.3 Discussion ..................................................................................................................... 45


4.5.3 hRAC/hRC in function of α1 ....................................................................................................... 45

4.5.3.1 Methodology .................................................................................................................. 45

4.5.3.2 Results ........................................................................................................................... 45

4.5.3.3 Discussion ..................................................................................................................... 46


4.5.4 hRAC/hRC in function of α5 (including α2 and α6) ...................................................................... 47

4.5.4.1 Methodology .................................................................................................................. 47

4.5.4.2 Results ........................................................................................................................... 47

4.5.4.3 Discussion ..................................................................................................................... 47


4.6 Conclusion of Chapter 4 .............................................................................................................. 49

Chapter 5

Validation of the method using real mixes ............................................................................................. 51

5.1 Scope ........................................................................................................................................... 51

5.2 Design criteria .............................................................................................................................. 52

5.2.1 General ................................................................................................................................. 52

5.2.2 Equivalent in RAC ................................................................................................................ 52

5.3 Missing data ................................................................................................................................ 54

5.4 Structural design .......................................................................................................................... 55

5.4.1 Bending ULS ........................................................................................................................ 55

5.4.2 Deformation SLS .................................................................................................................. 58

5.4.3 Cracking SLS ........................................................................................................................ 59

5.5 Design results and discussion ..................................................................................................... 60

5.6 Over-conservatism ...................................................................................................................... 66

5.7 Limitations of the method ............................................................................................................ 67

5.8 Other structural elements ............................................................................................................ 68

5.9 Conclusions of Chapter 5 ............................................................................................................ 70

Chapter 6

Conclusions and developments ............................................................................................................ 71

6.1 Conclusions ................................................................................................................................. 71

IX

6.2 Recommendations ....................................................................................................................... 74

References ............................................................................................................................................ 75

Annexes

Annex A: Parametric study for the verification of the simplifications (slabs) ........................................ A.1

A.1 Validation.................................................................................................................................... A.1

A.2 Data ............................................................................................................................................ A.1

A.3 Methodology ............................................................................................................................... A.1

A.4 Results ....................................................................................................................................... A.3

A.4.1 Part I .................................................................................................................................... A.3

A.4.2 Part II ................................................................................................................................... A.3

A.4.3 Part III .................................................................................................................................. A.3

A.5 Discussion .................................................................................................................................. A.3

A.5.1 Part I .................................................................................................................................... A.3

A.5.2 Part II ................................................................................................................................. A.11

A.5.3 Part III ................................................................................................................................ A.11

A.5.4 Comparison with other cover increases (∆c = 0.025 m) ................................................... A.12

A.6 Conclusion ............................................................................................................................... A.12

Annex B: Tables with results of the compliance of the bending ultimate limit state (slabs) ............... A.13

Annex C: Tables with results of the compliance of the cracking serviceability limit state (slabs) ...... A.15

Annex D: Results of the equivalent functional unit in RAC, concerning durability (slabs) ................. A.19

Annex E: Results of the equivalent functional unit in RAC, concerning deformation (slabs) ............. A.22

Annex F: Results of the equivalent functional unit in RAC, concerning bending (slabs) ..................... A.24

Annex G: Tables with design results (slabs) ...................................................................................... A.29

Annex H: Parametric study for the verification of the simplifications (beams) ................................... A.35

H.1 Validation ................................................................................................................................. A.35

H.2 Data .......................................................................................................................................... A.35

H.3 Methodology ............................................................................................................................ A.35

H.4 Results ..................................................................................................................................... A.36

H.5 Discussion ................................................................................................................................ A.40

H.5.1 Part I .................................................................................................................................. A.40

H.5.2 Part II ................................................................................................................................. A.40

X

H.5.3 Part III ................................................................................................................................ A.40

H.5.4 Comparison with other cover increases (∆c = 0.020 m) .................................................... A.40

Annex I: Tables with results of the compliance of the deformation serviceability limit state (beams) A.41

Annex J: Tables with results of the compliance of the bending ultimate limit state (beams) ............. A.42

Annex K: Tables with results of the compliance of the cracking serviceability limit state (beams) .... A.44

Annex L: Results of the equivalent functional unit in RAC, concerning durability (beams) ............... A.47

Annex M: Results of the equivalent functional unit in RAC, concerning deformation (beams) ............ A.49

Annex N: Results of the equivalent functional unit in RAC, concerning bending (beams) .................. A.51

Annex O: Tables with design results (beams) .................................................................................... A.53

XI

List of tables

Table 2-1: Live loads for buildings according to EN 1991-1 .....................................................................9

Table 3-1: Compliance criteria in function of structural and exposure classes ..................................... 22

Table 3-2: Values of wmax, according to EC2 ......................................................................................... 25

Table 3-3: α3 in function of structural and exposure class (∆cslabs = 0.015 m and ∆cbeams = 0.020 m) . 26




Table 3-7: Calculated α2/α6 for slabs in function of ∆c and load combinations ..................................... 28

Table 3-8: Adapted table of α2/α6 for slabs in function of ∆c and load combinations ............................ 29

Table 3-9: Calculated α2/α6 for beams (0.50 m * 0.25 m) in function of ∆c and load combinations ...... 29

Table 3-10: α1 for slabs in function of ∆cslabs and load combinations .................................................... 31

Table 3-11: α1 for beams in function of ∆cbeams, γ and load combinations ............................................. 32

Table 3-12: α5 for slabs in function of two cases of ∆c .......................................................................... 36

Table 3-13: α5 in function of α6 (∆c = 0.015 m) ...................................................................................... 37

Table 3-14: Control parameters for α5 .................................................................................................. 37

Table 3-15: α5 for beams in function of two cases of ∆c ....................................................................... 38

Table 5-1: Examples with all parameters available: compliance check (slabs) .................................... 62

Table 5-2: Examples with not all parameters available: compliance check (slabs) .............................. 62

Table 5-3: Assumption that not all fundamental parameters are available: comparison and compliance

check (slabs) ........................................................................................................................................................ 63

Table 5-4: Examples with all parameters available: compliance check (corresponding simply supported

beam) ..................................................................................................................................................... 63

Table 5-5: Examples with all parameters available: compliance check (corresponding continuous

beam) ..................................................................................................................................................... 63

Table 5-6: Examples with not all parameters available: compliance check (corresponding simply

supported beam) ................................................................................................................................... 64

Table 5-7: Examples with not all parameters available: compliance check (corresponding continuous

beam) ..................................................................................................................................................... 64

XII

Table 5-8: Assumption that not all fundamental parameters are available: comparison and compliance

check (beams) ..................................................................................................................................................... 65

Table 5-9: Highest K-values .................................................................................................................. 65

Table 5-10: K-values of the 14 examples analysed ............................................................................. 67

Table 5-11 Relative volume of structural elements in a standard framed building ......................................... 68

Table 5-12 Relative volume of structural elements in a standard framed building - own calculations .......... 69

Table A-1: Part I (∆c = 0.015 m) ........................................................................................................... A.4

Table A-2: Part II (∆c = 0.015 m) .......................................................................................................... A.5

Table A-3: Part III (∆c = 0.015 m) ......................................................................................................... A.6

Table A-4: Part I (∆c = 0.025 m) ........................................................................................................... A.7

Table A-5: Part II (∆c = 0.025 m) .......................................................................................................... A.8

Table A-6: Part III (∆c = 0.025 m) ......................................................................................................... A.9

Table A-7: Influence of the cover in the first section (∆c = 0.015 m) ................................................. A.10

Table A-8: Comparison between load combinations (∆c = 0.015 m) ................................................. A.10

Table A-9: Comparison between different values of µRC (∆c = 0.015 m) ........................................... A.10

Table A-10: Comparison between different concrete strength classes (∆c = 0.015 m) ..................... A.11

Table A-11: Loss in compressive strength (∆c = 0.015 m) ................................................................ A.11

Table A-12: Comparison between ∆c = 0.015 m and ∆c = 0.025 m: general .................................... A.12

Table B-1: Compliance of the bending ULS for slabs (∆c = 0.000 m and 0.010 m) .......................... A.13

Table B-2: Compliance of the bending ULS for slabs (∆c = 0.015 m, 0.025 m and 0.030 m) ........... A.14

Table C-1: Compliance of the cracking SLS for slabs (first 2 groups of columns, section A) ............ A.16

Table C-2: Compliance of the cracking SLS for slabs (third and fourth group of columns, section A) ......

............................................................................................................................................................ A.17

Table C-3: Compliance of the cracking SLS for slabs (last 2 groups of columns, section A) ............ A.18

Table D-1: Equivalent unit in RAC in function of S3, exposure class, α3 and the height in RC, hRC .. A.19

Table D-2: Equivalent unit in RAC in function of S3, exposure class, α4 and the height in RC, hRC .. A.20

Table E-1: Equivalent unit in RAC in function of α6/α2 (∆c = 0.000 m, 0.005 m, 0.010 m, 0.015 m, 0.020 m) ..

........................................................................................................................................................................... A.22

Table E-2: Equivalent unit in RAC in function of α6/α2 (∆c = 0.025 m, 0.030 m, 0.035 m, 0.040 m, 0.045

m, 0.050 m) ................................................................................................................................................A.23

Table F-1: Equivalent unit in RAC in function of α1 for C20/25 (∆c = 0.000 m, 0.005 m, 0.010 m) ........A.24

XIII




Table G-1: Design of slabs when all fundamental parameters are available (fundamental parameters and

data) ...........................................................................................................................................................A.29

Table G-2: Design of slabs when all fundamental parameters are available (bending ULS) .................A.30

Table G-3: Design of slabs when all fundamental parameters are available (deformation and cracking

SLS) ...........................................................................................................................................................A.31

Table G-4: Design of slabs when not all fundamental parameters are available (fundamental parameters

and data) ....................................................................................................................................................A.32

Table G-5: Design of slabs when not all fundamental parameters are available (bending ULS) ...........A.33

Table G-6: Design of slabs when not all fundamental parameters are available (deformation and cracking

SLS) ........................................................................................................................................................................... A.34

Table H-1: Relationship between ∆cslab and ∆cbeam ............................................................................ A.35

Table H-2: Part I (∆c = 0.035 m and 0.5 m * 0.25 m) ................................................................................... A.37

Table H-3: Part II (∆c = 0.035 m and 0.5 m * 0.25 m) .................................................................................. A.38

Table H-4: Part III (∆c = 0.035 m and 0.5 m * 0.25 m) ................................................................................. A.39

Table I-1: Calculated α2/α6 for beams (0.40 m * 0.20 m) in function of ∆c and load combinations ... A.41

Table I-2: Calculated α2/α6 for beams (0.60 m * 0.30 m) in function of ∆c and load combinations ... A.41

Table J-1: Compliance of the bending ULS for beams (∆c = 0.000 m and 0.015 m) ........................ A.42

Table J-2: Compliance of the bending ULS for beams (∆c = 0.020 m, 0.035 m and 0.040 m) ......... A.43

Table K-1: Compliance of the cracking SLS for beams (first 2 groups of columns, section A) .......... A.44

Table K-2: Compliance of the cracking SLS for beams (third and fourth group of columns, section A) ....

............................................................................................................................................................ A.45

Table K-3: Compliance of the cracking SLS for beams (last 2 groups of columns, section A) .......... A.46

Table M-1: Equivalent unit in RAC in function of α6/α2 (∆c = 0.000 m, 0.010 m, 0.015 m) (beams) .. A.49

Table M-2: Equivalent unit in RAC in function of α6/α2 (∆c = 0.020 m, 0.025 m, 0.035 m, 0.040 m)

(beams) ......................................................................................................................................................A.50

Table N-1: Equivalent unit in RAC in function of α1 for C25/30 (∆c = 0.000 m, 0.010 m, 0.015 m) (beams)

....................................................................................................................................................................A.51


....................................................................................................................................................................A.52

XIV

Table O-1: Design of simply supported beams (fundamental parameters and data) .............................A.53

Table O-2: Design of simply supported beams (bending ULS) ...............................................................A.54

Table O-3: Design of simply supported beams (deformation SLS and cracking SLS) ...........................A.55

Table O-4: Design of continuous beams (fundamental parameters and data) .......................................A.56

Table O-5: Design of continuous beams (bending ULS) .........................................................................A.57

Table O-6: Design of continuous beams (deformation SLS and cracking SLS) .....................................A.58

XV

List of figures

Figure 1-1: Ratio in function of the coarse RA content (%) (adaptation of Silva et al., 2014) ..................2

Figure 1-2: Ratio in function of the coarse RA content (%) (adaptation of Silva, 2014c) .........................2

Figure 1-3: Ratio in function of the coarse RA content (%), (adaptation of Silva, 2015)..........................3

Figure 1-4: Ratio in function of the coarse RA content (%) (adaptation of Silva, 2014f)..........................4

Figure 1-5: Ratio in function of the coarse RA content (%), (adaptation of Silva, 2014e)........................4

Figure 1-6: Ratio in function of the coarse RA content (%), (adaptation of Silva, 2014d)........................5

Figure 2-1: Flowchart of the methodology ............................................................................................. 16

Figure 4-1: hRAC/hRC in function of α3 for S3 (slabs) ............................................................................... 41

Figure 4-2: hRAC/hRC in function of α4 for S3 (slabs) ............................................................................... 41

Figure 4-3: hRAC/hRC in function of α3 for S3 - feasible cases only (slabs) ............................................. 42

Figure 4-4: hRAC/hRC in function of α4 for S3 - feasible cases only (slabs) ............................................. 42

Figure 4-5: hRAC/hRC in function of α3 for S4 and slab 15 cm thick (beams) .......................................... 43

Figure 4-6: hRAC/hRC in function of α4 for S4 and slab 15 cm thick (beams) .......................................... 43

Figure 4-7: hRAC/hRC in function of α6/α2 (slabs) ..................................................................................... 44

Figure 4-8: hRAC/hRC in function of α6/α2 (beams) ................................................................................... 45

Figure 4-9: hRAC/hRC in function of α1 (slabs) ........................................................................................ 46

Figure 4-10: hRAC/hRC in function of α1 (beams) ..................................................................................... 47

Figure 4-11: hRAC/hRC in function of α5 (slabs) ....................................................................................... 48

Figure 4-12: hRAC/hRC in function of α5 (beams) ..................................................................................... 48

Figure 4-13: pEd,RAC/pEd,RC in function of the K-value ............................................................................. 49

Figure 4-14: pqp,RAC/pqp,RC in function of the K-value ............................................................................. 49

Figure 5-1: Scatter of the K-value for slabs ........................................................................................... 61

Figure 5-2: Scatter of the K-value for simply supported beams ............................................................ 61

Figure 5-3: hRAC/hRC in function of α3 for S4 (footings) ........................................................................... 69

Figure 5-4: hRAC/hRC in function of α4 for S4 (footings) ........................................................................... 70

Figure D-1: Equivalent unit in RAC in function of S1, exposure class, α3 and the height in RC, hRC . A.20


XVI





Figure L-1: hRAC/hRC in function of α3 for S4 and smallest slab (beams) ............................................ A.47

Figure L-2: hRAC/hRC in function of α3 for S4 and thickest slab (beams) ............................................. A.47

Figure L-3: hRAC/hRC in function of α4 for S4 and smallest slab (beams) ............................................ A.48

Figure L-4: hRAC/hRC in function of α4 for S4 and thickest slab (beams) ............................................. A.48

XVII

List of acronyms

Cx/y Concrete strength class with fck,cyl = x MPa and fck,cube = y MPa

CEM I Portland cement with less than 5 % of other substances

EC2 Eurocode 2

LCA Life cycle assessment

LCI Life cycle inventory

LCIA Life cycle impact assessment

MRA Mixed recycled aggregates

NA Conventional/natural aggregates

RC Conventional concrete

RA Recycled aggregates

RAC Recycled aggregates concrete

S1 Structural class with a service life of 10 years

S2 Structural class with a service life of 10-15 years

S3 Structural class with a service life of 15-30 years

S4 Structural class with a service life of 50 years

S5 Structural class 5 with a service life of 100 years

S6 Structural class 6 with a service life of >100 years

S500 Steel strength class: characteristic yield stress of 500 N/mm²

SLS Serviceability Limit State

ULS Ultimate Limit State

WTCB Wetenschappelijk en Technisch Centrum voor het Bouwbedrijf

X0 Exposure class with no risk of corrosion or attack

XC Exposure class with corrosion induced by carbonation

XC1 Exposure class with corrosion induced by carbonation: dry or permanently wet

XC2 Exposure class with corrosion induced by carbonation: wet, rarely dry

XC3 Exposure class with corrosion induced by carbonation: moderate humidity

XC4 Exposure class with corrosion induced by carbonation: cyclic wet and dry

XVIII

XD Exposure class with corrosion induced by chlorides

XD1 Exposure class with corrosion induced by chlorides: moderate humidity

XD2 Exposure class with corrosion induced by chlorides: wet, rarely dry

XD3 Exposure class with corrosion induced by chlorides: cyclic wet and dry

XS Exposure class with corrosion induced by chlorides from sea water

XS1 Exposure class with corrosion induced by chlorides from sea water: exposed to

airborne salt but not in direct contact with sea water

XS2 Exposure class with corrosion induced by chlorides from sea water: permanently

submerged

XS3 Exposure class with corrosion induced by chlorides from sea water: tidal, splash and

spray zones

XF Exposure class with freeze/thaw attacks

XA Exposure class with chemical attacks

XIX

List of symbols

Latin upper case letters

A Concrete cross-section [cm²]

Ac,eff Effective area of concrete in tension [cm²]

Ac,eff,RAC Effective area of RAC in tension [cm²]

As Cross sectional area of reinforcement in concrete [cm²]

As,RAC Cross sectional area of reinforcement in RAC [cm²]

As,RC Cross sectional area of reinforcement in RC [cm²]

D Diffusion coefficient [m²/s]

Dchl Chloride migration coefficient [m²/s]

DRAC Chloride migration coefficient of RAC [m²/s]

DRC Chloride migration coefficient of RC [m²/s]

Ec,eff Effective modulus of elasticity of concrete [GPa]

Ecm Secant modulus of elasticity of concrete [GPa]

Ecm,RAC Secant modulus of elasticity of RAC [GPa]

Ecm,RC Secant modulus of elasticity of RC [GPa]

Es Design value of modulus of elasticity of reinforcing steel [GPa]

F Action, force [kN]

Fc Resultant of the compressive force of concrete [kN]

Fs Resultant of the tensile force of the reinforcement [kN]

Fs,RAC Resultant of the tensile force of the reinforcement in RAC [kN]

FS,RC Resultant of the tensile force of the reinforcement in RC [kN]

I Moment of inertia of a concrete section [cm4]

II Moment of inertia of a concrete section, assuming an uncracked section [cm4]

III Moment of inertia of a concrete section, assuming a cracked section [cm4]

K Amount of the equivalent weight in RAC [/]

Kcarb Carbonation coefficient [mm/√(years)]

Kcarb,RAC Carbonation coefficient of RAC [mm/√(years)]

XX

Kcarb,RC Carbonation coefficient of RC [mm/√(years)]

L Length, span [m]

M Bending moment [kNm]

Mcr Cracking moment [kNm]

MEd Design value of the bending moment strength [kNm]

MEd,mid-span Design value of the mid-span bending moment strength [kNm]

MEd,support Design value of the support bending moment strength [kNm]

MEd,RAC Design value of the bending moment strength of RAC [kNm]

MEd,RC Design value of the bending moment strength of RC [kNm]

Mqp Bending moment strength in SLS (quasi-permanent combination) [kNm]

Mqp,RAC Bending moment strength in SLS of RAC (quasi-permanent combination) [kNm]

Mqp,RC Bending moment strength in SLS of RC (quasi-permanent combination) [kNm]

N Action, vertical force [kN]

Latin lower case letters

a Coefficient [/]

a∞ Long-term deformation [mm]

a∞,RAC Long-term deformation of a slab in RAC [mm]

a∞,RC Long-term deformation of a slab in RC [mm]

b Overall width of a cross-section, coefficient in quadratic equation [m,/]

c Nominal reinforcement concrete cover, coefficient in quadratic equation [mm,/]

cnom Nominal reinforcement cover [mm]

cRAC Nominal RAC reinforcement cover [mm]

cRC Nominal RC reinforcement cover [mm]

cmin Minimum reinforcement cover [mm]

cmin,RAC Minimum RAC reinforcement cover [mm]

cmin,RC Minimum RC reinforcement cover [mm]

cmin,RC,slabs Minimum RC reinforcement cover of slabs [mm]

cmin,RC,beams Minimum RC reinforcement cover of beams [mm]

cmin,b Minimum reinforcement cover due to bond requirements [mm]

XXI

cmin,dur Minimum reinforcement cover due to environmental conditions [mm]

∆cdur,γ Additive safety element for the concrete cover, provided by the National Annex [mm]

∆cdur,st Reduction of minimum reinforcement cover due to the use of stainless steel [mm]

∆cdur,add Reduction of minimum reinforcement cover due to the use of additional protection

[mm]

∆cdev Allowance in design for deviation [mm]

∆c Difference in reinforcement cover between RAC and RC [mm]

∆cslabs Difference in reinforcement cover between RAC and RC, concerning slabs [mm]

∆cbeams Difference in reinforcement cover between RAC and RC, concerning beams [mm]

d Effective depth of a cross-section of concrete [m]

dRAC Effective depth of a cross-section of RAC [m]

dRC Effective depth of a cross-section of RC [m]

fc Compressive strength of concrete [MPa]

fcd Design value of concrete compressive strength [MPa]

fcd,RAC Design value of RAC compressive strength [MPa]

fcd,RC Design value of RC compressive strength [MPa]

fck Characteristic compressive cylinder strength of concrete at 28 days [MPa]

fck,RAC Characteristic compressive cylinder strength of RAC at 28 days [MPa]

fck,cyl Characteristic compressive cylinder strength of concrete at 28 days [MPa]

fck,cube Characteristic compressive cube strength of concrete at 28 days [MPa]

fcm Mean value of concrete cylinder compressive strength [MPa]

fcm,RAC Mean value of RAC cylinder compressive strength [MPa]

fcm,RC Mean value of RC cylinder compressive strength [MPa]

fct Tensile strength of concrete [MPa]

fct,eff Effective tensile strength of concrete [MPa]

fctm Mean value of axial tensile strength of concrete [MPa]

fctm,RAC Mean value of axial tensile strength of RAC [MPa]

fctm,RC Mean value of axial tensile strength of RC [MPa]

fct,sp Splitting tensile strength of concrete [MPa]

fyd Design value of the tensile strength of reinforcement steel [MPa]

XXII

fyk Characteristic steel reinforcement’s yield tensile strength [MPa]

g Dead weight [kN/m²]

∆g Other permanent loads [kN/m²]

h Total height, overall depth of a cross-section of concrete [m]

hRAC Total height, overall depth of a cross-section of RAC [m]

hRC Total height, overall depth of a cross-section of RC [m]

hmin Minimum height of the RAC examples to comply with the various limit states [m]

hrounded Rounded height of the RAC examples [m]

k1 coefficient that takes into account the bond properties [/]

k2 coefficient that takes into account the distribution of strain [/]

k3 coefficient according to clause 7.3.4(3) of EC2 [/]

k4 coefficient according to clause 7.3.4(3) of EC2 [/]

kt factor dependent on the duration of the load [/]

pbeam,Ed Load, dead weight of the beam in ULS [kN/m²]

pbeam,Ed Load, dead weight of the beam in SLS [kN/m²]

pEd Total load in ULS [kN/m²]

pEd,RAC Total load in ULS, concerning RAC [kN/m²]

pEd,RC Total load in ULS, concerning RC [kN/m²]

pqp Total load in SLS (quasi-permanent combination) [kN/m²]

pqp,RAC Total load in SLS (quasi-permanent combination), concerning RAC [kN/m²]

pqp,RC Total load in SLS (quasi-permanent combination), concerning RC [kN/m²]

pslab,Ed Transferred load of the slab on the beam in ULS [kN/m²]

pslab,qp Transferred load of the slab on the beam in SLS [kN/m²]

q Live loads [kN/m²]

sr,max Maximum crack spacing [mm]

sr,max,RAC Maximum crack spacing in RAC [mm]

t Lifetime, service life, age of concrete [year]

wmax Maximum crack width [mm]

wk Characteristic crack width [mm]

wk,RAC Characteristic crack width in RAC [mm]

XXIII

wk,RC Characteristic crack width in RC [mm]

x Height of compressive zone, carbonation depth [m]

x1 First solution of a quadratic equation [m]

x1,RAC First solution of a quadratic equation, concerning RAC [m]

x1,RC First solution of a quadratic equation, concerning RC [m]

x2 Second solution of a quadratic equation [m]

x2,RAC Second solution of a quadratic equation, concerning RAC [m]

x2,RC Second solution of a quadratic equation, concerning RC [m]

y Distance of the neutral axis from the top of a concrete section [m]

z Lever arm of internal forces in concrete [m]

zRAC Lever arm of internal forces in RAC [m]

zRC Lever arm of internal forces in RC [m]

Greek lower case letters

α1 Ratio between the mean compressive strengths of RAC and RC [/]

α2 Ratio between the effective moduli of elasticity of RAC and RC [/]

α3 Ratio between the carbonation coefficients of RAC and RC [/]

α4 Square root of the ratio between the diffusion coefficients of chlorides of RAC and RC

[/]

α5 Ratio of the mean tensile strengths of RAC and RC [/]

α6 Ratio of the creep coefficients+1 of RAC and RC [/]

β Constant, coefficient presenting boundary conditions, correction factor [/]

γ Partial factor, ratio between pEd,RAC and pEd,RC, ratio between pqp,RAC and pqp,RC [/]

γc Partial factor for concrete [/]

γg Partial factors for dead weight [/]

γq Partial factors for live loads [/]

γs Partial factor for reinforcement steel [/]

δ Deflection, deformation [mm]

∆ Discriminant of quadratic equation [/]

∆RAC Discriminant of quadratic equation, concerning RAC [/]

XXIV

∆RC Discriminant of quadratic equation, concerning RC [/]

εcm Mean strain in the concrete between the cracks [/]

εsm Mean strain in the reinforcement under the relevant combination of loads, including the

effect of imposed deformations and taking into account the effects of tension stiffening

[/]

ξ Distribution coefficient [/]

ρp,eff Ratio between the cross-section of reinforcement, As, and the effective cross-section

area of concrete in tension, Ac,eff [/]

µ Dimensionless value of the moment [/]

µRAC Dimensionless value of the moment of RAC [/]

µRC Dimensionless value of the moment of RC [/]

σadm admissible stress [MPa]

σc Compressive stress in the concrete [MPa]

σs Stress in the tension reinforcement in concrete [MPa]

σs,RAC Stress in the tension reinforcement in RAC [MPa]

σs,RAC Stress in the tension reinforcement in RAC [MPa]

σs,RC Stress in the tension reinforcement in RC [MPa]

Ø Diameter of a reinforcing bar [mm]

Østirb Diameter of the shear reinforcement [mm]

ω Reinforcement area ratio [/]

ωRAC Reinforcement area ratio of RAC [/]

ωRC Reinforcement area ratio of RC [/]

ψ2 Combination coefficient [/]

φ(∞,t0) Final value of the creep coefficient [/]

φ(∞,t0)RAC Final value of the creep coefficient of RAC [/]

φ(∞,t0)RC Final value of the creep coefficient of RC [/]

1

Chapter 1

Introduction

This chapter introduces the reader to the concept of concrete incorporating recycled aggregates. The

motivation and objectives are presented, along with the content and outline of the thesis structure.

1.1 Overview

At present, many developers, researchers and engineers are seeking efficient, sustainable building

solutions that conserve non-renewable resources. Owners want to use the research solutions in

response to growing environmental forces. Considering the environmental impact of materials, recycled

aggregates concrete is an excellent choice. Unfortunately, not many products are reintroduced into the

construction sector to be used as recycled aggregates (RA) in the production of concrete. An important

cause for this trend is the lack or conservative stance of regulations, which does not allow designers to

use the RA in the concrete production. (Bravo et al., 2015a)

Already several researches have been executed to evaluate the use of different types of RA with several

replacement ratios in concrete. The problem mostly occurring is that scarce studies thoroughly focus all

the properties of RA through the analysis of their composition and physical/chemical tests. Recently

performed research (Silva et al., 2014a), (Silva et al., 2014b), (Silva et al., 2014c), (Silva et al., 2014d),

(Silva et al., 2014e), (Silva et al., 2014f), (Silva et al., 2015) collected data from an extensive number of

RAC studies developed during the last decades that allowed concluding that the use of RA worsens, at

varying levels, most of the durability and mechanical properties tested.

1.1.1 Compressive strengths fcm

The mechanical performance of RA is found to be mainly influenced by the recycling procedure used and

the quality of the original materials. Compressive strength usually allows good correlation with the other

mechanical and durability-related properties of concrete (i.e. these normally improve as the compressive

strength increases). Several factors related to the use of RA significantly affect the compressive strength:

as the replacement level increases, the compressive strength of concrete decreases. The degree of this

loss, however, was found to be mainly dependent on the aggregate type, size and quality. A

comprehensive literature review demonstrated that the mean compressive strength of full replacement

RAC ranges from 0.56 to 1.17 relative to that of RC (average value = 0.89) (Silva et al., 2014b). Figure 1-

1 shows the ratio between the compressive strengths in function of the relative coarse recycled

aggregates content. The previous values were collected from the following references: (Amorim et al.,

2012), (Dhir and Paine, 2007), (Ferreira et al., 2011), (Gómez-Soberón, 2002), (Yang et al., 2008),

(Limbachiya et al., 2012).

2

Figure 1-1: Ratio in function of the coarse RA content (%) (adaptation of Silva et al., 2014)

1.1.2 Secant moduli of elasticity Ecm and axial tensile strength fctm

The modulus of elasticity, Ecm, and axial tensile strength, fctm, are known to be influenced by the cement

paste, the aggregate’s nature, the replacement level of RA, the aggregates’ size and quality, the mixing

procedure, the curing conditions, the chemical admixtures and additions content, the concrete age and the

compacity of concrete. Silva et al. show that the range of the ratio between the secant moduli of elasticity of

full replacement RAC and that of RC is [0.44 - 0.96] when all factors are taken into account (average value

= 0.80). The ratio between the respective tensile strengths varies between 0.40 and 1.14 (average value =

0.88) (Silva et al., 2015). Figures 1-2 and 1-3 show the ratios in function of the relative coarse recycled

aggregates content.

Figure 1-2: Ratio in function of the coarse RA content (%) (adaptation of Silva, 2014c)

The following references were used to obtain the values of Ecm: (Akbarnezhad et al., 2011), (Amorim

et al., 2011), (Cachim, 2009), (Casuccio et al., 2008), (Chen et al., 2003), (Choi and Yun, 2012),

(Corinaldesi ,2010), (Dapena et al., 2011), (Dhir and Paine, 2007), (Etxeberria et al., 2007), (Ferreira

et al., 2011), (Gómez-Soberón, 2002), (González and Etxeberria, 2014), (Juan and Gutiérrez, 2004),

(Kou et al., 2007), (Koulouris et al., 2004), (Limbachiya et al., 2012), (Manzi et al., 2013), (Park, 1999),

(Poon and Kou, 2010), (Rahal, 2007), (Rao et al., 2010), (Razaqpur et al., 2010), (Salem et al., 2003),

(Thomas et al., 2013), (Vieira et al., 2011) (Waleed and Canisius, 2007), (Yang et al., 2008).

y = -0.0011x + 1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0 10 20 30 40 50 60 70 80 90 100

fcm,RAC/

fcm,RC

Coarse RA content (%)

y = -0.0020x + 1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 10 20 30 40 50 60 70 80 90 100

Ecm,RAC/

Ecm,RC


3

Figure 1-3: Ratio in function of the coarse RA content (%), (adaptation of Silva, 2015)

In Figure 1-3, the ratio between the tensile splitting strengths, fct,sp, is used instead of the ratio between

pure tensile strengths, fct. The next formula shows that the results are the same.

f�� = 0.90 ∗ f��,� (Equation 1-1)

The following references were used: (Fonseca et al., 2011), (Evangelista and de Brito, 2007), (Pedro

et al., 2014a), (Duan and Poon, 2014), (González and Etxeberria, 2014), (Kim et al., 2013), (Kou et al.,

2007), (Kou and Poon, 2009a (Kou et al., 2004), (Kou et al., 2008), (Kou et al., 2012), (Kou and Poon,

2013), (Pedro et al., 2014b), (Vaishali and Rao, 2012), (Evangelista, 2014), (Ajdukiewicz and

Kliszczewicz, 2002), (Arezoumandi et al., 2014), (González-Fonteboa et al., 2011), (Çakir, 2014),

(Thomas et al., 2013), (Folino and Xargay, 2014), (Matias et al., 2013), (Manzi et al., 2013), (Schubert

et al., 2012), (Pereira et al., 2012).

1.1.3 Depths of carbonation and chloride penetration

RA generally have bigger porosity than NA. As a result, the carbonation and chlorides penetration depths

normally increase in RAC: the carbonation coefficient of full replacement RAC, Kcarb,RAC, ranges from 0.82

to 2.47 relative to that of RC (average value = 1.46) and the respective relative chloride diffusion coefficient,

D, from 0.90 to 1.72 (average value = 1.10) (Silva et al., 2014e), (Silva et al., 2014f). The lower 95%-

certainty limits are normally not used because, due to their bigger porosity, it is quite unlikely to obtain a

better resistance against chlorides penetration in concrete with recycled aggregates. Figure 1-4 shows the

ratio between the carbonation coefficients in function of the relative coarse recycled aggregates

content whilst Figure 1-5 does this for the diffusion coefficients.

A set of seven references was used to obtain the values of carbonation (Razaqpur et al., 2010), (Buyle-

Bodin et al., 2002), (Amorim et al., 2011), (Bravo et al., 2015b), (Katz, 2003), (Kou and Poon, 2012), (Pedro

et al., 2014) and eight references were considered to determine the values of chloride penetration:

(Amorim et al., 2011), (Bravo et al., 2015b), (Cartuxo, 2013), (Evangelista and de Brito, 2010), (Pedro

et al., 2014a), (Vieira, 2013), (Pedro et al., 2014b), (Evangelista, 2014).

y = -0.0012x + 1

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0 10 20 30 40 50 60 70 80 90 100

fctm,sp,RAC /

fctm,sp,RC


4

Figure 1-4: Ratio in function of the coarse RA content (%) (adaptation of Silva, 2014f)

Figure 1-5: Ratio in function of the coarse RA content (%), (adaptation of Silva, 2014e)

1.1.4 Creep

Creep of concrete is a complex phenomenon, which is influenced by many factors including the mix design

(i.e. replacement level, size and type of aggregates, quality of the original material, mixing procedure, etc.)

and environmental conditions. Creep affects the long-term deformation and the effective modulus of

elasticity can be obtained as follows, according to Eurocode 2 (EC2) (Silva et al., 2014d):

E�. �� = ��(�,��) (Equation 1-2)

Where Ec.eff is the effective modulus of elasticity, Ecm - secant modulus of elasticity and φ(∞, t0) - creep

coefficient for a given time period and load. Previous research suggests that the range of the

denominator of equation 1 for full replacement RAC relative to RC falls between 1.05 and 1.40.

(average value = 1.17).

6 references provided data to obtain Figure 1-5: (Gomez-Soberon et al., 2002), (Domingo et al., 2010),

(Kou et al., 2007), (Bravo et al., 2015b), (Manzi et al., 2013), (Limbachiya et al., 2000).

y = 0.0046x + 1

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0 10 20 30 40 50 60 70 80 90 100

Kcarb,RAC/

Kcarb,RC


y = 0,0010x + 1

0.80

0.90

1.00

1.10

1.20

1.30

1.40

0 10 20 30 40 50 60 70 80 90 100

√(DRAC / DRC)


5

Figure 1-6: Ratio in function of the coarse RA content (%), (adaptation of Silva, 2014d)

1.2 Motivation and contents

This project intends to better understand the possibility of defining an equivalent functional unit in

recycled aggregates concrete (RAC) to conventional structural concrete (RC) for Life Cycle Analysis

(LCA) purposes (not for structural design purposes). The goal is to obtain the minimum volume of RAC

that complies with all the limit states as 1 m3 of RC concrete does. To achieve this, a number of

parametrical studies has been executed. It is necessary to make some simplifications because not

everything can be taken into account in this theoretical exercise. If too many parameters are

considered simultaneously, the method will become too complex.

The previous section showed the relationships between the properties of RAC and RC. Consequently,

it is possible to introduce equations and parameters, which show these relationships.These are called

the fundamental parameters α that will be introduced in the next chapter.

In practical terms, the element in RAC will have a bigger height and consequently a bigger volume

than the one in RC. It has to be stated that it is one aim of this study to limit this increase. If the

magnification is not too big and the RAC example complies with the various limit states as the

conventional concrete does according to Eurocode 2 (EC2), designers and developers can be

encouraged to consider the use of RA, namely by allowing comparative LCA studies.

1.3 Structure of the thesis

This dissertation is composed of several main chapters. Two types of structural elements, slabs and

bemas, are considered. The project is mainly presented for slabs and differences and adaptations for

beams are described in the corresponding sections. In short, the calculations for beams form an

extrapolation of those of slabs.

y = 0.0017x + 1

0.80

1.00

1.20

1.40

1.60

0 10 20 30 40 50 60 70 80 90 100

(φRAC +1)/

(φRC +1)


6

In Chapter 1, an introduction in the research of RA is presented. The motivation and content of the

dissertation are also described.

Chapter 2 consists of the general data, conditions and scope of the dissertation. Furthermore, various

assumptions are made and some simplifications are developed in order to make the equations (and as

a result the project) not overly long. Some assumptions and simplifications are required for the

calculations in Chapters 3, 4 and 5. Chapter 2 also provides the definition of the fundamental

parameters. The limit states in Eurocode 2 are summarised in section 2.5. All the previous aspects

lead to the methodology of the dissertation.

In Chapters 3, 4 and 5, the actual analyses are performed. Chapter 3 consists of parametric studies to

check the compliance with the various limit states. Four aspects are considered - durability, bending

ULS, deformation SLS and cracking SLS - and each aspect is eventually described with a verification

formula. The conditions for the compliance controls and the methodology compliance criteria lead to

the accomplishment and results of the various parametric studies in this chapter.

The compliance checks with the limit states can be used in Chapter 4, which provides the definition of

the equivalent functional units concerning the various limit states. The meaning and consequences of

the previous term are explained. Furthermore, the design and methodology compliance criteria are

introduced. They lead to the final result of Chapter 4: the K-value, the most conditioning of the

equivalent functional units obtained for the various limit states.

Chapter 5 provides the validation of the method proposed when real mixes of RC and RAC are

produced. The scope and design criteria for the calculations are described as well as the relationships

between the fundamental parameters when there is lack of data. The structural design leads to the

results of Chapter 5.

Finally, Chapter 6 presents the conclusions about the project to demonstrate which goals of the

dissertation are achieved and which limitations should be taken into account. Chapter 6 also provides

the recommendations to improve the method and to eliminate some of these limitations.

The annexes show the tables and figures concerning all the calculations to complete the dissertation.

There are already some abbreviated tables in the text, but the complete tables, concerning the various

parts of the dissertation, can be seen in the annexes.

7

Chapter 2

General data and scope

This chapter provides the data and scope of the dissertation. The properties, parameters and

assumptions used, are introduced and the various influencing factors and methods are covered.

Several aspects are not straightforward and require a further explanation.

2.1 Limit states in Eurocode 2

The dissertation performs an analysis according to EC2, which sets the limit states related to design

situations, for which compliance is required. Relevant design situations are selected taking into account

the circumstances under which the structure is required to fulfil its function. EC2 makes a distinction

between ultimate limit states (ULS) and serviceability limit state (SLS). ULS concern the safety of people

and/or the safety of the structure under normal use. The comfort of people and the appearance of the

construction are considered in SLS.

2.1.1 Ultimate Limit States

The ULS are bending with or without axial force, shear, torsion and punching. This dissertation only

considers bending without axial force. It applies to undisturbed regions of slabs, beams and other

similar types of members for which sections remain approximately plane before and after loading.

Several assumptions are made (EC2):

- Plane sections remain plane;

- The strain in bonded reinforcement is the same as that in the surrounding concrete;

- The tensile strength of the concrete is ignored;

- The stresses in the compressive zone are derived from design stress/strain relationships (EC2);

- The stresses in the reinforcement are derived from design curves (EC2).

With the assumptions made, it is possible to partly design the slab and more specifically to calculate

the cross-section of reinforcement.

2.1.2 Serviceability Limit States

The SLS are stress limitation, crack control and deflection control. There can be other limit states that may

be of importance in particular structures but those are not covered by Eurocode 2. The stress limitation is

not specifically implemented throughout the dissertation. The crack and deflection control limitations are

checked to determine whether compliance with the SLS is obtained for the equivalent in RAC.

2.1.2.1 Crack control

Cracks are normal in reinforced concrete structures subjected to bending, shear or torsion that result

from either direct loading or restraint or imposed deformations. Cracking needs to be limited to an

8

extent that will not impair the proper functioning or durability of the structure or cause its appearance to

be unacceptable. This limit state will generally not be conditioning and there is the choice between

cracking control with or without direct calculation. It is decided to opt for the second choice due to the

greater preciseness of the method. Chapter 3 gives more information about the calculations.

2.1.2.2 Deflection control

Deformation of a member or structure needs to be limited in such a way that it does not affect its

proper functioning or appearance. Appropriate limiting values of deflection, taking into account the

nature of the structure, the partition walls and the function of the structure, are required. The limiting

value of the deflection of a beam or slab, subjected to quasi-permanent loads, is span/250.

Due to the quasi-permanent loads, deformation in the long-term needs to be considered. Total deflection

including creep may be calculated by using the effective modulus of elasticity for concrete instead of the

secant modulus of elasticity.

2.2 Scope of the dissertation

The main goal of the dissertation is to determine an equivalent functional unit in RAC to RC with regards to

its environmental impact. The philosophy throughout the project is that the beam’s equivalent functional unit

in RAC depends on the corresponding slab’s equivalent functional unit in RAC. The extra loads of the RAC

slabs need to be taken into account for the calculations of the beam’s equivalent functional unit in RAC.

The main reason for the dependency of the beams is that RA will generally be used in the weaker structural

elements of a building: as slabs form weaker structural elements than beams, RA will surely be

implemented in slabs if they are used in beams. Another reason is that the casting of beams and slabs of a

standard framed building normally always takes place at the same time.

2.2.1 Concrete and classes

The project only considers slabs used in standard framed buildings. Exposure classes XC, XD and XS will

be considered in order to take into account the influence of carbonation and chlorides penetration. The

study does not take into account harsh environmental conditions, such as freeze-thaw and chemical attack.

It is also possible to classify concrete into structural classes (EC2 provides minimum covers in function of

the exposure and structural classes). These classes represent the target lifetime of a structural element,

ranging from S1 to S6. The relevant structural class for the dissertation is S4, corresponding to a lifetime

of 50 years and used for standard framed buildings.

The concrete strength class of current slabs does normally not exceed C40/50 or go below C16/20.

The classes C20/25, C25/30 and C30/37 will be used during the compliance checks of the limit states

and the definition of the equivalent functional unit. In the validation of the method for real mixes, higher

strength classes are also used due to lack of data. As will become clear later, not all the compliance

checks of the limit states depend on the strength classes.

9

2.2.2 Loads

2.2.2.1 Slabs

The total loads on standard framed building slabs consist of three types: the dead weight g, the other

permanent loads ∆g and the live loads q. The thickness of the slabs considered falls within a specific

range, based on design experience. Besides the type of construction, the height also depends on the

compressive strength of the concrete and the amount of reinforcement in the element. The thickness

of solid slabs in standard framed buildings normally ranges from 12 to 18 cm. Solid slabs may be

thicker than 20 cm, but waffle plates will become more efficient in that case. A height lower than 10 cm is

also possible, but the nominal cover will become too big in relative terms and the effective height will

suffer an excessive decrease. RC has a dead weight of approximately 25 kN/m³. The same can be used

for RAC (even though a small reduction is expected in most cases). This means that the dead weight of

the slabs analysed ranges between 3.0 kN/m² and 4.5 kN/m². The other permanent loads depend on

the floors, walls or furniture that are placed on the slab. The limits are based on design experience:

1.0 kN m�⁄ ≤ ∆g ≤ 3.5 kN m�⁄ (Equation 2-1)

The live loads are provided by Eurocode 1 (EN 1991-1) (EC1, 2004) in function of the use of the

spaces. Not all values are feasible for standard framed building slabs, e.g. an archive needs bigger

live loads but these slabs should be thicker and are not commonly used. Table 2-1 contains the values

of the live loads used.

Table 2-1: Live loads for buildings according to EN 1991-1

Load case Effective vertical load (kN/m²)

A - Rooms, kitchens, hotel rooms,… (floors) 1.5 or 2.0

B and C1 - Offices, restaurants, dining rooms, reading rooms,… 3.0

C2 - Areas with fixed seats: conference rooms, waiting rooms,… 4.0

2.2.2.2 Differences and adaptations to beams

A beam used in standard framed buildings has a thickness which is normally compatible with its span

(height = span/12). Heights ranging from 0.40 to 0.60 m are considered in the compliance checks of

the limit states and the definition of the equivalent functional units. The width of the beams considered

is usually equal to the half of the height, 0.20 m – 0.60 m, which results in a dead weight, g, varying

between 2 kN/m and 4.5 kN/m. Live loads, q, and other permanent loads, ∆g, are not defined for

beams as those are already included in the total loads of the slab, which are transferred to the beams.

2.3 Durability

Durability affects the cover of the slabs. The cover, defined as the minimum distance between the

envelope of the reinforcement and the concrete surface, protects the reinforcement against carbonation

10

and chlorides penetration. The greater value of the minimum cover, cmin,RC, provided by EC2, satisfying

the requirements for both bond and environmental conditions must be used:

c&'( = max+c&'(,,; c&'(,./0 +∆c./0,2 − ∆c./0,� −∆c./0,4..; 10mm5 (Equation 2-2)

cmin,b is the minimum cover due to bond requirements and cmin,dur is the one due to environmental

conditions. The other parameters are: ∆cdur,y - additive safety element; ∆cdur,st and ∆cdur,add - reductions

of the minimum cover due to the use of stainless steel and additional protection, respectively.

The minimum cover must be increased by a deviation, ∆cdev, to take into account execution errors, to

obtain the nominal cover that is specified in the drawings:

c(6& = c&'( +∆c. 7 (Equation 2-3)

A deviation of 0.5 cm, corresponding to high-quality control casting conditions, is used throughout the

dissertation. The dissertation includes various covers for RC in the parametric study and other calculations:

c89 = 1.0, 2.0or3.0cm (Equation 2-4)

The biggest cover will be the most relevant for durability purposes. A thinner slab cannot be combined

with a high cover as the effective height will suffer a much too big decrease.

The cover, cRC, used in the calculations concerning the beams is 5 mm bigger than that for slabs: 1.5,

2.5 or 3.5 cm. This is because of the bigger minimum cover, cmin,RC, provided by EC2. The reason for

the difference is because of the structural classes, which is explained in section 3.2.1.

2.4 Assumptions and simplifications

In research work, it is often necessary to make simplifications and assume a number of issues

(otherwise, the conversion formulas become too long). It is not the intention of the dissertation to

comprehensively cover of all aspects of concrete design.

2.4.1 Assumptions

2.4.1.1 Slabs

The dissertation focuses on the environmental impact of RAC and not on that of the reinforcement in the

concrete. The cross-section of the reinforcement is always expected to be the same. If a different cross-

section of steel were used, the environmental impact of this material would also have to be taken into

account. The steel strength class is considered to be S500, corresponding to a characteristic yield

stress of 500 N/mm². The bars have a diameter Ø of 8 mm, 10 mm or 12 mm, according to the slabs’

thickness. These values are used in solid slabs for standard framed buildings.

There is always the same amount of cement in the compared mixes of RC and RAC. Cement affects

strength and the hardening process. If different amounts of cement were used, it would not be possible

to directly compare the environmental impact of RC and RAC.

11

Furthermore, the composition of the concrete mixes remains the same in each comparison, except for

the aggregates themselves and the water content. A fraction of the normal aggregates (NA) will be

replaced by RA. The replacement level varies between 0% and 100%. RA have a bigger porosity than

NA and need more effective water (water content of fresh concrete minus the water absorbed by the

aggregates) as a result (Pedro et al. 2014).

Another assumption has to be taken into account, considering the dimensionless value of the moment

of RC, µRC. This value is assumed to be 0.18 for slabs in standard framed buildings and expresses the

optimum balance between concrete and steel consumption, based on design experience of slabs. If

the value is lower, e.g. 0.12, the maximum strains in concrete fall below the maximum allowed values

and as a result, the slab design is not economical. On the other hand, if the µRC increases, e.g. 0.24,

more steel is used in the slab, which means that the design is not economical as well. These trends

increase as the distance between µRC and 0.18 increases.


It must be noted that other bar diameters, Ø, are used for slabs: 16 mm, 20 mm or 25 mm, according

to the beams’ thickness. Also the optimal dimensionless value of the moment, µRC, for beams is not

the same as that for slabs. The optimal value varies around 0.25 and the limits used in the

parametrical study are 0.20 and 0.30.

Furthermore, if RA are used in the concrete of slabs, the height increase and the capacity of the

beams needs to be higher in order to take into account the extra loads. Nevertheless, the width of the

beams, b, remains unchanged.

2.4.2 Simplifications

2.4.2.1 Slabs

Simplifications will be used in all the compliance checks of the limit states and in the calculations of the ratio

hRAC/hRC. The simplifications are verified by a parametric study (see Annex A) to prove that they can be

used in the further parts of the dissertation.

The total and effective heights of the RAC slabs are defined as follows:

h8>9 = h89 + 2 ∗ (c8>9 −c89) = h89 + 2 ∗ Δc (Equation 2-5)

d8>9 = d89 + (c8>9 −c89) = d89 + Δc (Equation 2-6)

Where hRAC and hRC are the total heights of the slabs in RAC and RC, respectively. The difference

between covers in RAC and RC (cRAC and cRC) is ∆c and dRAC and dRC represent the effective heights

of the slabs in RAC and RC, respectively. The expressions have to be considered as educated

guesses, based on design experience.

At first sight, it is expected that the total height hRC needs to rise with ∆c instead of 2*∆c. If that were

the case, the loads would be higher but the effective height dRC would remain unchanged. This would

result in a higher cross-section of reinforcement to take into account the increase of the bending

12

moment. As the compressive strength of RAC is also expected to decline, the cross-section of

reinforcement would need to increase even more. This does not comply with the initial assumption of

constant reinforcement. Therefore, it is necessary to increase the effective height as well.

The purpose of these simplifications is to make the formulas in the deformation service limit state

(SLS) and the bending ultimate limit state (ULS) not more complex than strictly necessary. The cover

changes in function of durability, i.e. the heights change as well. Deformation strongly depends on the

height of the structural element, which means that Equation 2-5 and equation 2-6 need to be used or

the deformation will also depend on durability, unnecessarily complicating the deformation SLS check.

Another simplification is introduced to calculate the cross-section of reinforcement in RAC and RC:

A,89 = ω89 ∗ ��C∗,∗.DE�FC (Equation 2-7)

Where b is the width of the element (equal to 1 m for slabs), fcd - the design value of the compressive

strength of concrete, which is equal to the characteristic compressive strength of the concrete, fck,

divided by the partial safety factor for concrete. fyd - the design value of the tensile strength of steel

and ωRC (reinforcement area ratio) is the simplification in the formula. This parameter is expressed in

function of µRC. Normally, it is necessary to search the value of ωRC in tables but a conservative

simplification (Equation 2-8) is used instead. Equation 2-8 will apply if the value of µRC is relatively

small (the case in the dissertation):

ω89 =μ89 ∗ (1 + μ89) (Equation 2-8)


The simplifications for the total and effective heights, given by Equations 2-5 and 2-6, are not valid for

beams. These structural elements need a simplification formula that is more conservative because the

parametric study, concerning the verification of those simplifications, led to unsafe results for beams. A

simplification formula that applies for beams can be obtained by stating that the cross-section of

reinforcement, As, is the same in RAC and RC:

A,89 =μ89 ∗ (1 + μ89) ∗ ��C∗,∗.DE�FC =μ8>9 ∗ (1 + μ8>9) ∗ ��C∗,∗.DHE�FC =A,8>9 (Equation 2-9)

Where As,RC and As,RAC are the cross-sections of reinforcement in RC and RAC, respectively, µRC and

µRAC - the dimensionless values of the moment in RC and RAC, respectively, fcd - the design value of

the compressive strength of concrete, b - the width of the beam, dRC and dRAC - the effective heights of

the beam in RC and RAC, respectively, and fyd - the design value of the tensile strength of

reinforcement. Including Equation 2-10, which expresses the relationship between µRAC and µRC for

slabs, leads to equation 2-14:

μ8>9 = μ89 ∗ I .DE.DHEJ� ∗ �KC,DHE�KC,DE = μ89 ∗ I .DE.DHEJ

� ∗ γ (Equation 2-10)

Where pEd,RAC and pEd,RC are the total design loads in RAC and RC, respectively, and γ is the ratio of

between the two previous parameters. The reason that this philosophy is used for beams is that the

13

use of RA in concrete of slabs leads to an increase of the vertical loads, so also for the bending

moments. This is demonstrated by Equation 2-10 and γ. The height of beams used in standard framed

buildings in Portugal is normally increased for seismic reasons. Resulting from this, the loads are also

higher. The seismic coefficient depends on the frequency of vibrations, stiffness of the building, type of

soil, type of structure, etc. γ is used following the same philosophy as for seismic activity, but it is

slightly smaller than the seismic parameter because RAC elements have higher stiffness (even though

their modulus of elasticity is lower, which in principle leads to a slightly higher seismic coefficient).

→ μ89 ∗ (1 + μ89) ∗ d89 =μ89 ∗ I .DE.DHEJ� ∗ γ ∗ N1 + μ89 ∗ I .DE.DHEJ

� ∗ γO ∗ d8>9 (Equation 2-11)

↔ (1 + μ89) = .DE.DHE ∗ γ ∗ N1 + μ89 ∗ I .DE.DHEJ� ∗ γO (Equation 2-12)

The following simplification is accepted, for simplification purposes:

(1 + μ89) ≃ N1 + μ89 ∗ I .DE.DHEJ� ∗ γO (Equation 2-13)

This means that dRAC can be expressed in function of dRC and the design loads in RC and RAC, pEd,RC

and pEd,RAC:

d8>9 = d89 ∗ γ (Equation 2-14)

Equation 2-14 ensures that the cross-section of reinforcement remains the same in RAC and RC but

the compressive strength needs to stay constant. As the latter is normally lower for RAC, it is

necessary to take into account an extra margin like it is done for Equation 2-5. This can be done by

including a power for γ, which is empirically obtained: 1.2. Equation 2-15 presents this:

d8>9 = d89 ∗ γ�.� (Equation 2-15)

As a consequent, the total height of the beam in RAC, hRAC, is as follows:

h8>9 = d89 ∗ γ�.� + h89 −d89 + ∆c (Equation 2-16)

The parametric study concerning Equations 2-15 and 2-16 gives more information about their

applicability (see Annex H), basically showing that they can be applied in the whole range of valid values

of the parameters used in this study.

2.5 Relationship between RC and RAC

As stated in the introduction, it is not possible to design RAC using the properties of RC. RC is

designed according to Eurocode 2 (EC2, 2008). Unlike RC, in which all the relevant properties for

design of reinforced concrete elements are determined based solely on the strength class (i.e. the

compressive strength), RAC does not have the same relationships between its various properties and

compressive strength. Therefore, it is not enough to know the compressive strength of RAC to deduct

the other RAC properties. The fundamental parameters express the equivalent properties of RAC in

function of the same property of RC. The properties used in the dissertation are limited as some of the

14

general properties are irrelevant for the purpose of the dissertation and structural design. Using the

fundamental parameters, it is possible to obtain the equivalent functional unit and design the structural

elements according to EC2.

2.5.1 Fundamental parameters

The following list defines the fundamental parameters that are considered in the dissertation. The

ranges of earlier research (section 1.1) are used throughout the dissertation:

1. α1 is the ratio between the average compressive strengths of RAC and RC. The strengths are

expressed in MPa and this ratio is used in the compliance check of the bending ULS:

α� = ��,DHE��,DE (Equation 2-17)

2. α2 is the ratio between the moduli of elasticity of RAC and RC. The moduli of elasticity are

expressed in GPa:

α� = ��,DHE��,DE (Equation 2-18)

3. α3 is the ratio between the carbonation coefficients of RAC and RC. The carbonation

coefficients are expressed in mm/√(years):

αS =T�UVW,DHET�UVW,DE (Equation 2-19)

4. α4 is the ratio between the chlorides diffusion coefficients of RAC and RC. The diffusion coefficients

are expressed in m²/s and the reason for using a square root is given in section 2.5.2.4:

αX =YZDHEZDE (Equation 2-20)

5. α5 is the ratio between the average tensile strengths of RAC and RC. The strengths are

expressed in N/mm²:

α[ = ��\�,DHE��\�,DE (Equation 2-21)

6. α6 is the ratio between the creep coefficients plus 1 of RAC and RC. The reason for this ratio

is already explained in section 1.1.4:

α] = (^(�,_�)`ab��)(^(�,_�)`b��) (Equation 2-22)

2.5.2 Justification of the parameters used

Several fundamental parameters are not straightforward and require a further explanation.

2.5.2.1 Mean value of the compressive strength

The ratio α1 expresses the relationship between the average values of the compressive strengths of

RAC and RC. Although design values of the concrete compressive strengths fcd are required, namely

for the bending ULS, it is more practical to express the relationship between average values because

available research normally only provides average values. Using the ratio between the design values

of the compressive strengths would lead to more complicated formulas. The compressive strengths of

15

RAC and RC are the essential properties to use in the method. In some cases, there will be missing

parameters in the comparison between RAC and RC, namely in the validation of the method when real

mixes are produced (Chapter 5). It is possible to express other fundamental parameters in function of α1

but, if α1 is unknown, the equations are useless.

2.5.2.2 Secant modulus of elasticity of concrete

The effective modulus of elasticity, Ec,eff, includes the effect of creep and is in the long-term more

relevant than the secant modulus of elasticity, Ecm. The ratio between the effective moduli of elasticity

would be more relevant in this dissertation, but there is not enough available research concerning that

parameter. This is the reason why the ratio between the effective moduli of elasticity is divided into the

parameters α2 and α6: α2 concerns the secant moduli of elasticity of RAC and RC and α6 involves the

creep coefficients increased by 1.

2.5.2.3 Depth of carbonation

The carbonation depth, x, is expressed in function of the concrete age, t, and the carbonation coefficient,

Kcarb, which depends on the quality of the concrete and the exposure conditions (WTCB, 2007):

x = K�40, ∗ √t (Equation 2-23)

The target service life of RAC and RC is assumed to be the same (50 years), which means that it can

be omitted if the ratio between the carbonation depths in RAC and RC is considered.

2.5.2.4 Depth of chlorides

Fick’s laws of diffusion explain the square root used in ratio α4. Equation 2-24 shows a simplified version

of Fick’s law: the diffusion length (i.e. chlorides penetration depth) depends on the diffusion coefficient,

D, the age of concrete, t, and a constant, which represents the diffusion conditions:

diffusionlength = constant ∗ √D ∗ t (Equation 2-24)

As the target service life of RAC and RC is 50 years in the dissertation, it can be omitted in the ratio.

The same diffusion conditions must be considered in RAC and RC, which means that this factor can

be left out as well. The simplifications result in the square root of the ratio between the diffusion

coefficients in RAC and RC.

2.5.2.5 Shrinkage

Besides creep, shrinkage is also a physical property of concrete. This phenomenon is one of the factors

that contribute to the cracks in floors and slabs. Although shrinkage is an important part of the total change

of volume of concrete, this parameter is not used. The limit states that are checked do not directly depend

on shrinkage and the dissertation does not need to go further into that phenomenon.

16

2.6 Methodology and flowchart

In previous sections, the data, assumptions and simplifications were described. These are required to

describe the methodology of the dissertation.

It is noted that the method proposed in this dissertation is not for structural design purposes. This

dissertation works with average values of fundamental parameters while structural designers should

use, at least until further confidence on recycled aggregates concrete is acquired, 95%-certainty

(characteristic) values. This method solely aims to demonstrate that it is possible to obtain an

equivalent in RAC to RC concerning various aspects: the limit states for which compliance is required.

Figure 2-1 provides a flowchart of the methodology of the study. The different steps are exemplified below.

Figure 2-1: Flowchart of the methodology

RA generally have a bigger porosity than NA. Consequently, the depths of carbonation and chlorides

penetration in RAC increase and the fundamental parameters α3 and α4 become bigger than 1. This leads

to the conclusion that RAC needs a bigger cover than RC, resulting in the first step of the flowchart. The

definition of a difference in cover ∆c between RAC and RC is necessary to take into account the bigger

porosity of the RA but it has to be limited for slabs and beams (∆cslabs = 0 - 2.5 cm and ∆cbeams = 0 - 3.5

cm). After the definition of the difference in reinforcement cover and height of the elements in RAC, it

is possible to perform the parametric studies, which check whether the hRAC value adopted indeed

always leads to α values that approximately fall between the imposed limits by previous research.

(Section 1.1) If this is not the case, new hRAC and other conditions must be defined. The flowchart does

not provide the option for the compliance with the cracking SLS because this limit state is never

expected to be conditioning (as will be shown). Different load combinations, covers cRC, differences in

cover ∆c, concrete strength classes and dimensionless values of the moment µ are tested. Every

compliance check is executed as a parametric study and its relevant conditions and parameters are

described in section 3.2.

17

When all compliance checks are positive, the equivalent functional units, concerning the various limit

states, can be calculated in function of the fundamental parameters. The parametric studies of

Chapter 3 can be used in Chapter 4, due to the fact that the calculated range of fundamental

parameters mostly corresponds to the limits proposed. The most conditioning equivalent functional

unit ratio represents the K-value.

The sequence of checking the various limit states is the same in both chapters. The validation of the

method using real mixes, treated in Chapter 5, is slightly different because it first considers the

bending ULS. This is the usual practice in design situations due to the need of knowing the cross-

section of reinforcement in the deformation control and crack control calculations. Various examples of

RAC and RC with the respective fundamental parameters for slabs and beams are provided in the

assessment of the method. The slabs are designed first and their loads are used in the design of the

beams. The calculations are done according to the available data (namely concerning the RA

properties) and their purpose is that the equivalent slab or beam in RAC complies with the limit states

just like a similar slab or beam in RC. If this succeeds in (almost) every case, the method proposed is

considered validated, i.e. the results can be used for LCA purposes.

2.7 Life cycle assessment

As the dissertation and final results will be used for life cycle assessment (LCA) purposes, it is

important to know what this entails.

LCA, according to ISO 14040-14043 (ISO, 2006), is the most acknowledged and standardized

methodology for environmental assessment. Concrete is one of the most widely used building

materials in structures. Because of that global extensive use, it is imperative to evaluate the

environmental impact of RC and RAC correctly. The general goal of LCA is to compare the full range

of environmental effects assignable to products and services by quantifying all inputs and outputs of

material flows and assessing how these material flows impact the environment. It is required that the

environmental impact of products and processes is assessed from cradle-to-grave. This involves raw

materials acquisition, material production and construction, use phase and end-of-life phase. It is not

possible to implement a simple cradle-to-gate analysis when the environmental benefits of potential

RACs are evaluated. On that level, only the influencing parameters workability and strength can be

considered, not durability. Cradle-to-grave on the other hand, looks at the material’s impact over its

entire life cycle (Marinković et al., 2013 and Van den Heede et al.,2012).

LCA of reinforced concrete with RA is an example of LCA that seeks to identify the environmental

consequences of a proposed change in a system under study, which means that market and

economic implications of a decision may have to be taken into account.

Based on the analysis of up-to-date experimental evidence, it can be concluded that the use of RA for

low-to-middle strength structural concrete and non-aggressive exposure conditions is technically

feasible. Results of earlier research show that the impact of aggregates and cement production

phases is slightly bigger for RAC than for RC. This is because more cement is used in those case-

18

studies; this is not the case in this dissertation, which means that the impact of the cement remains the

same. The total environmental impacts depend on the natural and recycled aggregates transport

distances and on transport types. If the transport distance is limited, environmental impacts of RAC

can be equal or even lower than those of its corresponding RC (Marinković et al., 2010).

Four steps need to be executed during LCA: definition of goal and scope, life cycle inventory,

assessment of the environmental impacts and interpretation of the results. It can be concluded that

small changes in the first three steps may induce important differences in the environmental score

eventually obtained in the interpretation phase.

2.7.1 Definition of goal and scope

This includes the functional unit in RC. It defines what is studied and quantifies the service delivered

by the concrete systems because it provides a reference to which the inputs and outputs can be

related. The environmental impacts of the production of two types of ready-mixed concrete are

compared: RC with NA and RAC made with natural fine and recycled coarse aggregates. Normally,

fine RA are not recommended because of their high water absorption and high cohesion, which make

the concrete quality control very difficult (DIN, 2002 and BSI, 2006).

Regardless of the scope of LCA, system boundaries must be described clearly using a flow diagram or

process tree. The construction process and the use phase of RC and ARC are assumed to be

comparable and are therefore omitted from the analysis. Consequently, the functional unit is 1 m³ of

RC with a specific strength at the construction site.

It has to be noted that the adopted functional unit for which the environmental impact is calculated

influences the outcomes significantly. This unit should incorporate differences in strength, durability

and service life when different concrete compositions are compared. Even for strength and durability

related functional units, it often remains difficult to decide what and what not to include in the system

(Van den Heede et al.,2012).

Finally, the necessary criteria regarding the quality of the data used in the LCA need to be set. Time-

related coverage, geographical coverage, technology coverage, precision, completeness, etc. are data

requirements, which should be addressed (ISO, 2006).

2.7.2 Life cycle inventory

Life cycle inventory (LCI) is an inventory of flows from and to the environment of the concrete system.

The flows include inputs of water, energy, raw materials, releases to air, etc. and data on inputs and

outputs gathered must be related to the functional unit defined in the goal and scope. Flow models are

illustrated with flow charts that include the activities that are going to be assessed.

2.7.3 Assessment of the environmental impacts

The significance of potential environmental impacts of the RAC examples is evaluated in this step,

based on the life cycle inventory results. This step consists of three mandatory aspects: selection of

the impact categories, classification which means assignment of LCI results to the chosen impact

19

categories, and characterization in which the converted LCI results are aggregated into an indicator

result, the final result of a LCIA. The assessment can be done with various endpoint methods.

Ecoindicator 99 and Ecological Scarcity 2006 are examples of damage oriented LCIA: those methods

focus on the actual damage effect; they try to model the cause-effect chain up to the endpoint, or the

actual environmental damage. This has sometimes high uncertainties. CML 2002 is an example of a

problem oriented LCIA: it limits uncertainties and groups LCI results related to a given environmental

problem, into midpoint indicators (e.g. radiation, climate change, ecotoxity, fossil fuels, etc.) (Knoeri et

al., 2013). According to Benetto et al. (Benetto et al., 2004), the problem related approach provides

reliable results, although it is sometimes difficult to compare them with each other. On the other hand,

a damage oriented impact analysis, allows a much easier interpretation of the LCA output, but is

considered to be not so reliable.

2.7.4 Interpretation of the results

This part consists of a systematic technique to identify, quantify, check and evaluate the results of the

second and third step. Links between the various phases can also be revealed. The outcome is a set

of conclusions and recommendations concerning the RAC examples.

21

Chapter 3

Parametric studies involving the limit states

This chapter consists of the various parametric studies, which demonstrate that the calculated

fundamental parameters vary between the limits provided in Chapter 2. The simplifications, expressed

by Equations 2-5 and 2-6 for slabs and Equations 2-15 and 2-16 for beams, form the basis of Chapter 3.

They are verified by other parametric studies (Annex A for slabs and Annex H for beams).

3.1 Main purpose

The main goal of Chapter 3 is to calculate fundamental parameters α, using conditions that are safe

enough but not too conservative. Those conditions, e.g. different ∆c’s, concrete strength classes,

cross-sections of reinforcement, etc. are necessary to validate the method proposed: if most of the

calculated values fall between the limits provided in Chapter 2, the conditions are well determined and

furthermore, it is possible to define a functional equivalent with the obtained range of fundamental

parameters.

3.2 Design compliance criteria

The conditions for the compliance control of durability and the various limit states form an important

part of the parametric studies. The majority of the data, used throughout the several parametric

studies of Chapter 3, are defined in sections 2.2, 2.3 and 2.4. Those are the elements used in every

parametric study, but some of the criteria, e.g. the difference in cover, differ according to the

parametric study in which they are used.

3.2.1 Durability

The fundamental parameters affecting durability are α3 and α4 (Equations 2-19 and 2-20). Earlier

research shows that the effect of RA on carbonation will be greater than the one on chlorides

penetration. On the other hand, chlorides penetration is more life-limiting to reinforced concrete than

carbonation. This can be demonstrated with Table 3-1, which describes the environmental

requirement for cmin,dur. The table also sets the design compliance criteria for the parametric study in

function of the structural and exposure classes as the values form the basis to calculate the maximum

values of α3 and α4. Equation 2-3 shows that the nominal cover of concrete consists of the minimum

cover and an allowance in design, ∆cdev (irrelevant for the sensitivity analysis as it will be kept

constant). cmin is, according to Equation 2-2, the maximum of several values. Considering durability,

only cmin,dur needs to be taken into account (Table 3-1). ∆cdur,γ, ∆cdur,st and ∆cdur,add are recommended

to be 0, according to the national annexes.

22

Table 3-1: Compliance criteria in function of structural and exposure classes

Environmental requirement for cmin,dur (mm)

Structural class

Exposure class according to Table 4.1 EC2

X0 XCS XC2 / XC3 XC4 XD1 / XS1 XD2 / XS2 XD3 / XS3

S1 10 10 10 15 20 25 30

S2 10 10 15 20 25 30 35

S3 10 10 20 25 30 35 40

S4 10 15 25 30 35 40 45

S5 15 20 30 35 40 45 50

S6 20 25 35 40 45 50 55

As section 2.2.1 stated, the relevant structural class in the dissertation is S4, corresponding to a target

lifetime of 50 years, but EC2 allows modifications to the structural class in Table 4.3N: if calculations

are performed for members with a slab geometry, it is allowed to reduce the structural class by 1 and

use the corresponding minimum cover, cmin,dur. This means that the cmin,RC values of S3 will be used for

further calculations of the slabs.

The parametric study of slabs is performed with two differences in cover: ∆cslabs = 0.015 m and ∆cslabs

= 0.025 m. Higher values are not considered because they will result in an excessive thickness

increase of the slab in RAC.

Durability is also affected by α3 and α4 when beams are examined. Consequently, the same

philosophy can be applied to these structural elements but as a beam does not have a slab geometry,

cmin,RC values of S4 need to be used for further calculations. The parametric study of beams is

performed with ∆cbeams = 0.020 m and ∆cbeams = 0.035 m, which correspond to the differences in cover

of 0.015 m and 0.025 m used for slabs.

3.2.2 Deformation serviceability limit state

The deformation SLS depends on the fundamental parameters α2 and α6 (Equations 2-18 and 2-22).

Quasi-permanent loads, pqp, are used in these limit states: the live loads, q, must be multiplied by

combination coefficients, Ѱ2, according to EC2:

pn� = (g + Δg) +Ѱ� ∗ q (Equation 3-1)

Where g is the dead weight and ∆g the other permanent loads.

The differences in cover used range from 0.000 m to 0.050 m. In practice, it is not economical to use

such a high increase of the total height of the slab, but this still will lead to complying results.

It does not matter which concrete strength class is used for the compliance check of the deformation

SLS because the results are the same. The parametric study is executed for C20/25.

If the deformation SLS is considered during the design of concrete slabs, the deflection needs to be

limited to particular values according to EC2 (see section 2.1.2.2). Compliance with the limit state

demands that the deflection in long-term of a slab in RAC, a∞,RAC, is smaller than (or almost the same

as) the equivalent deflection in RC a∞,RC:

a�,8>9 ≤a�,89 (Equation 3-2)

23

This condition will eventually lead to the verification formula (Equation 3-3) of the fundamental

parameters, derived in section 3.4.2.1:

qrqs ≥ �uv,DHE(wDHEwDE )x∗�uv,DE (Equation 3-3)

Equations 3-2 and 3-3 for slabs can also be used when beams are examined. The loads, pqp, do not

need to be calculated for beams because the ratio between the quasi-permanent loads of the slabs is

used (instead of the ratio between the loads of the beams) in the verification formula. The reason for

this is provided in section 2.4.2.2. The parametric study is performed for various geometry conditions

that lead to different results: beams with the following dimensions (hRC * b) are considered: 0.40 m *

0.20 m, 0.50 m * 0.25 m and 0.60 m * 0.30 m.

3.2.3 Bending ultimate limit state

The compliance check of the bending ULS concerns fundamental parameter α1 (Equation 2-17).

Because this is an ULS, it is necessary to multiply the loads (see section 2.1.2) by the partial safety

factors, γg (= 1.35) and γq (= 1.5), according to EC2 to obtain the total design loads, pEd,RC:

p�.,89 = 1.35 ∗ (g + Δg) + 1.5 ∗ q (Equation 3-4)

Where g is the permanent load, ∆g - the other permanent loads and q - the live loads. The parametric

study is executed for concrete strength class C25/30. Other concrete strength classes lead to the

same results for the calculated fundamental parameter α1. Various differences in cover, ∆cslabs, are

included: 0.000 m, 0.010 m, 0.015 m, 0.025 m and 0.030 m. Higher values are not used because

those cases are not useful for practical purposes.

Section 2.1.1 stated that the dissertation solely considers pure bending without axial forces, which

means that there is a balance of forces (resultant of the compressive force of concrete, Fc, equal to the

resultant of the tensile force of the reinforcement, Fs)::

F =A ∗ fz. =F� (Equation 3-5)

Where As is the cross-section of reinforcement and fyd is the design value of the tensile strength of

steel. As is assumed to be the same in RAC and RC (see section 2.4.1). Therefore, As,RC is used in the

calculations (instead of As,RAC), which is accepted due to the small difference between the cross-

sections (between -5 % and 15 %). As stated above, the whole idea of comparing the environmental

impacts of RAC and RC is that steel remains unchanged.

This parametric study uses another simplification, in which the real compressive zone of concrete is

replaced by the compressive zone according to EC2. The compressive zone has a height, x, and a

simplified compressive zone can be used with a height of 0.8*x and constant stress, equal to the

maximum capacity of concrete. Consequently, the resultant of the compressive strength of concrete,

Fc, can be expressed as follows:

F� = 1 ∗ f�. ∗ 0.8 ∗ x ∗ b (Equation 3-6)

Where b is the width of the structural element (equal to 1 m for slabs) and fcd presents the design value

24

of the compressive strength of concrete.

Compliance with the bending ULS requires that the ultimate bending moment strength of RAC, MEd,RAC,

is bigger or approximately the same as that of RC, MEd,RC, taking into account the corresponding loads:

M�.,8>9 ≥ �KC,DHE�KC,DE ∗ M�.,89 (Equation 3-7)

Beams take the same conditions into account and Equations 3-5, 3-6 and 3-7 apply for these

structural elements as well. Nevertheless, the absolute values of the total design loads do not need to

be calculated for beams because the increase of the total design loads of the slabs is used to describe

the loads of the beams (ratio pEd,RAC/pEd,RC) (section 2.4.2.2). The parametric study is performed for

various heights of the beams (0.40 m, 0.50 m and 0.60 m), but they lead to the same results.

Corresponding differences in cover, ∆cbeams, to those of slabs are taken into account to calculate the

parameters: 0.000 m, 0.015 m, 0.020 m and 0.035 m.

3.2.4 Cracking serviceability limit state

The cracking SLS depends particularly on fundamental parameter α5 (Equation 2-21). Nevertheless,

fundamental parameters α2 and α6 (Equations 2-18 and 2-22) affect the calculated values of α5.

Consequently, it is necessary to include the ranges of α2 and α6 as conditions in this parametric study:

- α2 in the case of a coarse recycled aggregates content of 100%: [0.44; 0.96];

- α6 in the case of a coarse recycled aggregates content of 100%: [1.05; 1.40].

The concrete strength class used in this parametric study is C25/30. The lower the concrete strength

class goes, the more conditioning the calculated results for α5 become. The range C20/25 to C50/60

(concrete strength classes used in Chapter 5) is considered and this leads to values of α5 that are

lower than 1.14 (characteristic value). As a result it is sufficient to solely include C25/30. Two values of

∆cslabs are included in the parametric study: 0.015 m and 0.025 m. Lower differences in cover will

never impose a problem. It is possible to go until ∆c = 0.030 m and fall between the limits but these

cases are less useful for slabs, as already explained.

The expected basis for the verification formula of the cracking SLS would be as follows:

w�,8>9 ≤w�,89 (Equation 3-8)

Where wk,RAC and wk,RC are the characteristic crack widths in RAC and RC, respectively. This

philosophy, coherent with that of the previous sections, was firstly followed but led to useless results

for the purpose of the dissertation: all results were bigger than 1, which is highly unlikely in practice.

As a result, another condition needed to be determined: EC2 recommends values of wmax, the

maximum crack width, which need to be used in function of the exposure class (Table 3-2). It is

assumed that the characteristic crack width in RAC can be bigger than the one in RC but it has to be

restricted to 0.3 mm for all exposure classes. This is the basis for the verification formula.

w�,8>9 ≤ 0.3mm (Equation 3-9)

25

Table 3-2: Values of wmax, according to EC2

Recommended values of wmax (mm)

Exposure Class

Reinforced members and prestressed members with unbonded tendons

Quasi-permanent load combination

XO, XC1 0.4*

XC2, XC3, XC4 0.3

XD1, XD2, XS1, XS2, XS3

Note *: For X0, XC1 exposure classes, crack width has no influence on durability and this limit is set to guarantee acceptable appearance.

Equation 3-9 is also used in the parametric study for the beams and the same philosophy is

implemented. The parametric study is executed for ∆cbeams = 0.020 m and 0.035 m, which correspond

to the values of ∆cslabs.

3.3 Methodology compliance criteria

The methodology compliance criteria describe the results that are expected in the parametric studies.

The criteria for each parametric study consist of a range in which the calculated fundamental

parameters must fall. These characteristic ranges are provided by earlier research (section 1.1):






- α6 in the case of a coarse recycled aggregates content of 100%: [1.05; 1.40].

Durability concerns the range of α3 and α4, deformation - the range of α2/α6 ([0.31; 0.91]), bending - the

range of α1 and cracking - the range of α5.

3.4 Parametric studies

This section describes how the fundamental parameters are calculated. Furthermore, it provides the

results and the discussion of the results for each compliance check of the limit states.

3.4.1 Durability

The parametric studies for slabs and beams, concerning durability, consist of the methodology for α3 and

α4, the outcomes and the discussion. The three parts are presented for slabs, but Tables 3-3 to 3-6 show

the results for both structural elements. The differences for beams are explained in section 3.4.1.4.

26

3.4.1.1 Methodology

The methodology is the same for carbonation (α3) and chlorides penetration (α4), but only the

equations concerning α3 are explained.

The minimum RAC cover, cmin,RAC, is expressed in function of the minimum RC cover, cmin,RC, and α3:

c&'(,8>9 ≥ αS ∗ c&'(,89 (Equation 3-10)

If ∆c is introduced, it is possible to express α3 in function of this parameter and cmin,RC:

∆c = c&'(,8>9 − c&'(,89 ≥ (αS − 1) ∗ c&'(,89 (Equation 3-11)

→ αS ≤ ∆��,DE + 1 (Equation 3-12)

The maximum cmin,RC leads to the most conditioning case and, as seen in Table 3-1, this occurs for the

combination of S3 and XC4. The same equations can be obtained if chlorides penetration is

considered. In this case, α3 needs to be replaced by α4 in Equations 3-10, 3-11 and 3-12. The same

conditions are taken into account and the most conditioning case occurs for S3 and XD3/XS3.

3.4.1.2 Results

The tables beneath show the values obtained for α3 and α4 in function of ∆cslabs = 0.015 m and ∆cbeams =

0.020 m or ∆cslabs = 0.025 m and ∆cbeams = 0.035 m.

Table 3-3: α3 in function of structural and exposure class (∆cslabs = 0.015 m and ∆cbeams = 0.020 m)

Values for α3 and ∆c = 0,015 m Values for α3 and ∆c = 0.020 m

Structural class


Structural class


X0 XCS XC2 / XC3 XC4

X0 XCS XC2 / XC3 XC4

S1 2.50 2.50 2.50 2.00 S1 3.00 3.00 3.00 2.33

S2 2.50 2.50 2.00 1.75 S2 3.00 3.00 2.33 2.00

S3 2.50 2.50 1.75 1.60 S3 3.00 3.00 2.00 1.80

S4 2.50 2.00 1.60 1.50 S4 3.00 2.33 1.80 1.67


Values for α3 and ∆c = 0,025 m Values for α3 and ∆c = 0,035 m

Structural class

Exposure class according to Table 4.1 EC2 Structural

class


X0 XCS XC2 / XC3 XC4 X0 XCS XC2 / XC3 XC4

S1 3.50 3.50 3.50 2.67 S1 4.50 4.50 4.50 3.33

S2 3.50 3.50 2.67 2.25 S2 4.50 4.50 3.33 2.75

S3 3.50 3.50 2.25 2.00 S3 4.50 4.50 2.75 2.40

S4 3.50 2.67 2.00 1.83 S4 4.50 3.33 2.40 2.17

27



Structural class

Exposure class according to Table 4.1 EC2 Structural

class


XD1 / XS1 XD2 / XS2 XD3 / XS3 XD1 / XS1 XD2 / XS2 XD3 / XS3

S1 1.75 1.60 1.50 S1 2.00 1.80 1.67

S2 1.60 1.50 1.43 S2 1.80 1.67 1.57

S3 1.50 1.43 1.38 S3 1.67 1.57 1.50

S4 1.43 1.38 1.33 S4 1.57 1.50 1.44



Structural class


Structural class


XD1 / XS1 XD2 / XS2 XD3 / XS3 XD1 / XS1 XD2 / XS2 XD3 / XS3

S1 2.25 2.00 1.83 S1 2.75 2.40 2.17

S2 2.00 1.83 1.71 S2 2.40 2.17 2.00

S3 1.83 1.71 1.63 S3 2.17 2.00 1.88

S4 1.71 1.63 1.56 S4 2.00 1.88 1.78

3.4.1.3 Discussion

If the results are compared with the criteria of the parametric study for durability, it can be concluded

that most part of the α3 values fall outside the characteristic limits. Furthermore, all the values of α4 fall

outside the provided limits as well. The fact that they do not vary between the limits is still acceptable

because the values are not below 1. In other words, the values of α3 and α4 are too conservative. For

the given extra covers, they could go up to limits that are not useful, but on the safe side.


The same methodology is followed when beams are considered and the outcomes are shown in section

3.4.1.2. It can be concluded that the more ∆c increases (the case for beams if they are compared to slabs),

the more the parameters fall outside the provided limits.

3.4.2 Deformation serviceability limit state

This section treats the methodology of the parametric study, concerning deformation. Furthermore, the

verification formula is obtained and the results and discussion are described. The various parts are

presented for slabs, but differences and adaptations for beams are described in section 3.4.2.4.

3.4.2.1 Methodology and verification formula

The verification formula concerning α2/α6 is obtained according to EC2. Deformation in the long-term,

a∞, can be expressed in function of the quasi-permanent loads, pqp, the effective modulus of elasticity

Ec,eff, the height, h, and a constant, β, presenting the geometric/boundary conditions of the slab (which

are the same for slabs in RAC and RC):

a� = β ∗ �uv��,��∗�³ (Equation 3-13)

28

The effective modulus of elasticity, Ec,eff, depends on the secant modulus of elasticity of concrete, Ecm,

and the creep coefficient, φ(∞,t0):

E�, �� = ��(�,��) (Equation 3-14)

Introducing Equations 3-13 and 3-14 in Equation 3-2 leads to the verification formula (Equation 3-18):

β ∗ �uv,DHEK��,DHE��(�,\�)DHE∗�DHE³≤ β ∗ �uv,DEK��,DE��(�,\�)DE∗�DE³

(Equation 3-15)

↔ �uv,DHE�r∗K��,DE�s∗(��(�,\�)DE)∗(�DHE)³≤ �uv,DEK��,DE��(�,\�)DE∗�DE³

(Equation 3-16)

↔ �uv,DHE�r�s∗(�DHE)³ ≤�uv,DE�DE³ (Equation 3-17)

↔ �uv,DHE�uv,DE∗(wDHEwDE )³ ≤ qrqs (Equation 3-18)

3.4.2.2 Results

The complete table concerning the compliance of the deformation SLS consists of the combination of

Table A-1 and Table 3-7. The results of the verification formula are shown in Table 3-7.

Table 3-7: Calculated α2/α6 for slabs in function of ∆c and load combinations

Load comb.

hRC (m)

pqp,RC

(kN/m²) cRC (m)

∆cslabs (m)

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

min (α2/α6)

1 0.120 4.450

0.010 1.000 0.831 0.700 0.598 0.517 0.451 0.396 0.351 0.313 0.281 0.253

0.020 1.000 0.831 0.700 0.598 0.517 0.451 0.396 0.351 0.313 0.281 0.253

0.030 1.000 0.831 0.700 0.598 0.517 0.451 0.396 0.351 0.313 0.281 0.253

2 0.180 10.400

0.010 1.000 0.871 0.764 0.675 0.600 0.537 0.483 0.436 0.396 0.360 0.330

0.020 1.000 0.871 0.764 0.675 0.600 0.537 0.483 0.436 0.396 0.360 0.330

0.030 1.000 0.871 0.764 0.675 0.600 0.537 0.483 0.436 0.396 0.360 0.330

3 0.150 6.900

0.010 1.000 0.854 0.737 0.642 0.563 0.498 0.444 0.397 0.358 0.324 0.294

0.020 1.000 0.854 0.737 0.642 0.563 0.498 0.444 0.397 0.358 0.324 0.294

0.030 1.000 0.854 0.737 0.642 0.563 0.498 0.444 0.397 0.358 0.324 0.294

Table 3-7 must be restricted to the cases useful for practical purposes, as described in section 3.3.

Consequently, the cases for ∆c = 0.000 m and ∆c = 0.050 m are excluded in Table 3-8. Also the

lowest load combination, combined with a high ∆c (0.040 m or 0.045 m) must be excluded due to its

irrelevancy.

3.4.2.3 Discussion

Table 3-8 contains the calculated fundamental parameters α2/α6 in function of the various ∆c’s and

load combinations. It shows that the higher ∆c is, the lower the minimum value of the ratio α2/α6

becomes (this makes sense as ∆c is in the denominator of Equation 3-18). On the other hand, the

29

minimum ratio α2/α6 increases with a higher load combination: a thicker slab in RC is less sensitive to

the difference in cover so the factor �`b��∗∆��`b declines. As a result, the ratio α2/α6 rises and the case

becomes more conditioning if ∆c remains unchanged. Table 3-8 also demonstrates that a different

cover cRC does not affect the results.

Table 3-8: Adapted table of α2/α6 for slabs in function of ∆c and load combinations

Load comb.

hRC (m)

pqp,RC

(kN/m²) cRC

(m)

∆cslabs (m)

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 min (α2/α6)

1 0.120 4.450 0.010 0.831 0.700 0.598 0.517 0.451 0.396 0.351 / / 0.020 0.831 0.700 0.598 0.517 0.451 0.396 0.351 / / 0.030 0.831 0.700 0.598 0.517 0.451 0.396 0.351 / /

2 0.180 10.400 0.010 0.871 0.764 0.675 0.600 0.537 0.483 0.436 0.396 0.360 0.020 0.871 0.764 0.675 0.600 0.537 0.483 0.436 0.396 0.360 0.030 0.871 0.764 0.675 0.600 0.537 0.483 0.436 0.396 0.360

3 0.150 6.900 0.010 0.854 0.737 0.642 0.563 0.498 0.444 0.397 0.358 0.324 0.020 0.854 0.737 0.642 0.563 0.498 0.444 0.397 0.358 0.324 0.030 0.854 0.737 0.642 0.563 0.498 0.444 0.397 0.358 0.324


The methodology of the slabs can be implemented for beams as well. It has to be stressed again that the

ratio pqp,RAC/pqp,RC of slabs is used in the calculations and not that of beams (see section 2.4.2.2).

The results for the intermediate beam (0.50 m * 0.25 m) are shown below and the results for other

beams are presented in Annex I. The complete table concerning the deformation SLS is a combination

of Tables A-4, H-2 and 3-9. This last table demonstrates the results of the verification formula. The

cases that are not feasible (∆cslab = 0.000 m and ∆cslab ≥ 0.045 m) are excluded from the table.

Table 3-9: Calculated α2/α6 for beams (0.50 m * 0.25 m) in function of ∆c and load combinations

Load combination

(slabs)

dRC (m)

(pqp,RAC/ pqp,RC)slabs

∆cslabs

0.005 0.005 0.010 0.015 0.020 0.020 0.025 0.030 0.035 0.035 0.040

∆cbeams

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055

min (α2/α6)

1

0.467 1.169 0.854 0.831 0.718 0.625 0.548 0.536 0.474 0.421 0.377 0.369 0.332

0.457 1.169 0.857 0.834 0.723 0.631 0.555 0.543 0.481 0.429 0.384 0.376 0.339

0.447 1.169 0.861 0.837 0.728 0.638 0.562 0.550 0.488 0.436 0.391 0.384 0.346

2

0.467 1.072 0.918 0.892 0.823 0.760 0.704 0.687 0.638 0.595 0.555 0.542 0.507

0.457 1.072 0.920 0.894 0.825 0.764 0.709 0.691 0.643 0.600 0.560 0.547 0.513

0.447 1.072 0.921 0.895 0.828 0.767 0.713 0.695 0.648 0.605 0.566 0.553 0.518

3

0.467 1.109 0.893 0.868 0.780 0.705 0.639 0.623 0.568 0.519 0.476 0.466 0.429

0.457 1.109 0.895 0.870 0.784 0.709 0.644 0.629 0.574 0.525 0.482 0.472 0.435

0.447 1.109 0.898 0.872 0.788 0.714 0.650 0.634 0.580 0.532 0.489 0.479 0.442

The results demonstrate the same trends as Table 3-8 but not the same reasons are the cause of

these trends. The fact that a lower minimum value is obtained with a bigger ∆c can also be explained

by the parameter in the denominator, but the higher minimum value of the ratio for higher load

combinations can not be explained by the sensitivity of the height of the beam (fixed height). A higher

slab load combination leads to a smaller pqp,RAC/pqp,RC because thicker slabs are less sensitive to extra

30

loads. The ratio between the loads is in the numerator of the verification formula but it is also included

in hRAC, which is raised to the third power in the denominator. The latter factor is the reason for the

increasing minimum value of α2/α6 when higher load combinations are examined. It must also be noted

that different covers in a load combination lead to different ratios for beams.

The minimum value for the ratio α2/α6 is more conditioning for beams than for slabs as it is slightly

higher. This leads to a smaller margin for the range of the fundamental parameters.

3.4.3 Bending ultimate limit state

This section contains the same parts as for previous limit states but now for the bending ULS. The

calculations, results and discussion are presented for slabs, but the differences for beams are

described subsequently.

3.4.3.1 Methodology and verification formula

Equation 3-7 is the basis for the calculations in this section. The ultimate bending moment strength,

MEd, is equal to the multiplication of the resultant of the tensile force of the reinforcement, Fs, and the

lever arm, z. Introducing this in Equation 3-7 leads to Equation 3-19:

F,8>9 ∗ z8>9 ≥ �KC,DHE�KC,DE ∗ F,89 ∗ z89 (Equation 3-19)

The simplification in section 3.2.3 (Equation 3-6) ensures that the lever arm, z, normally given by

Equation 3-20, can be expressed in function of Fs and fcd. Fs,RAC is assumed to be equal to Fs,RC (section

3.2.3). Consequently, they can be omitted in Equation 3-19:

z = d − 0.4 ∗ x (Equation 3-20)

z = d − ��∗,∗��C (Equation 3-21)

z8>9 ≥ �KC,DHE�KC,DE ∗ z89 (Equation 3-22)

If Equations 2-6 and 3-21 are inserted in Equation 3-22, it is possible to derive the verification formula:

↔ .DHE� ��,DHEr∗W∗��C,DHE.DE� ��r∗W∗��C,DE

≥�KC,DHE�KC,DE (Equation 3-23)

↔ .DE�∆�� ,DHEr∗W∗��C,DHE.DE� ��,DEr∗W∗��C,DE

≥ �KC,DE��∗�[∗�.S[∗∆��KC,DE (Equation 3-24)

↔ (.DE�∆�)∗�∗,∗��C,DHE��,DHE.DE∗�∗,��C,DE��,DE ∗ ��C,DE��C,DHE ≥ �KC,DE�]�.[∗∆��KC,DE (Equation 3-25)

↔ (.DE�∆�)∗�∗∝�∗,∗��C,DE��,DHE.DE∗�∗,∗��C,DE��,DE ≥∝�∗ �KC,DE�]�.[∗∆��KC,DE (Equation 3-26)

↔ (d89 + ∆c) ∗ 2 ∗∝�∗ b ∗ f�.,89 − F,8>9 ≥∝�∗ (�KC,DE�]�.[∗∆�)∗(.DE∗�∗,∗��C,DE��,DE)�KC,DE (Equation 3-27)

↔ (d89 + ∆c) ∗ 2 ∗∝�∗ b ∗ f�.,89 −∝�∗ (�KC,DE�]�.[∗∆�)∗(.DE∗�∗,∗��C,DE��,DE)�KC,DE ≥ F,8>9 (Equation 3-28)

31

↔ p�.,89 ∗ (d89 + ∆c) ∗ 2 ∗∝�∗ b ∗ f�.,89 −∝� �p�.,89 + 67.5 ∗ ∆c�(d89 ∗ 2 ∗ b ∗ f�.,89 − F,89) ≥ p�.,89 ∗ F,8>9 (Equation 3-29)

↔∝� �p�.,89 ∗ (d89 + ∆c) ∗ 2 ∗ b ∗ f�.,89 − �p�.,89 + 67.5 ∗ ∆c�(d89 ∗ 2 ∗ b ∗ f�.,89 − F,89)� ≥ p�.,89 ∗ F,8>9

(Equation 3-30)

↔∝�≥ �KC,DE∗��,DHE��KC,DE∗(.DE�∆�)∗�∗,∗��C,DE��KC,DE�]�.[∗∆��(.DE∗�∗,∗��C,DE��,DE)� (Equation 3-31)

Where pEd,RC is the total load in ULS, dRC - the effective height of the slab in RC, ∆c - the difference in cover,

b - the width of the slab (equal to 1 m), fcd,RC - the design value of the compressive strength of RC and Fs,RC

and Fs,RAC - the resultants of the tensile forces in the reinforcement in RC and RAC, respectively.

3.4.3.2 Results

The results can be seen in Annex B. An example is provided in section 3.4.3.3.

3.4.3.3 Discussion

The difference in cover of ∆cslabs = 0.000 m leads to α1=1. The higher ∆cslabs is, the more α1 declines.

This is because ∆cslabs is in the denominator of the verification formula. Furthermore, Table 3-10

proves that a given combination with a smaller calculated α1 than the minimum can be solved by rising

∆c. The table also demonstrates that α1 declines (bigger margin for losses) if a higher cover is used.

On the other hand, it is necessary to limit the nominal cover of the concrete slab because its thickness

will already increase a lot (Equation 2-5). A higher load combination will not necessarily cause a lower

α1. This can be explained by the non-linearity of the verification formula.

Table 3-10: α1 for slabs in function of ∆cslabs and load combinations

Load combination pEd,RC (kN/m²) cRC (m) ∆cslabs (m) Calculated

minimum α1

4 7.650 0.010 0.015 0.787 7.650 0.020 0.015 0.704 7.650 0.030 0.015 0.622

4 7.650 0.010 0.025 0.689 7.650 0.020 0.025 0.587 7.650 0.030 0.025 0.496

5 16.800 0.010 0.015 0.721 16.800 0.020 0.015 0.691 16.800 0.030 0.015 0.660

5 16.800 0.010 0.025 0.608 16.800 0.020 0.025 0.573 16.800 0.030 0.025 0.538

The range of the calculated fundamental parameters approximately complies with the provided range

if ∆cslabs ranges from 0.000 m to 0.025 m. From the moment that ∆c = 0.030 m, too many cases fall

outside the characteristic limits of Chapter 2 (but still on the safe side). In the worst case, a loss of

slightly more than 20% in the compressive strength is possible with ∆c = 0.015 m. With ∆c = 0.025 m,

it is possible to go further to 31%.

32


The calculations for beams are also derived from Equation 3-19 but the verification formula needs to

be reformed (Equation 3-32) because the absolute loads of the beams are not available. Furthermore,

the effective height of the beams, dRAC, is obtained in another way than for slabs (Equation 2-15

instead of Equation 2-6):

↔∝�≥ ��,DHE�(.DHE)∗�∗,∗��C,DE�2(.DE∗�∗,∗��C,DE��,DE)� (Equation 3-32)

The results of the bending SLS can be seen in Annex J. Table 3-11 shows the example of beams

corresponding to Table 3-10. Most of the trends are the same as for slabs, but different covers in a

load combination lead to the same results of α1. Moreover, a higher load combination of the slabs leads

to higher values of α1. This can be explained by the lower value of γ (=pEd,RAC / pEd,RC) that is in the

denominator of the verification formula. The range of the calculated minimum α1 lies between the

provided 95%-certainty limits in Chapter 2 if ∆cbeams ranges from 0.000 m to 0.035 m. It is possible to go

even further and still comply (not the case for slabs). In some cases, the results are less conditioning

(load combination 1) but other cases lead to more conditioning values (load combination 2) if beams

are compared with slabs.

Table 3-11: α1 for beams in function of ∆cbeams, γ and load combinations

Load combination

dRC (m) cRC (m) pEd,RAC / pEd,RC ∆cbeams

(m) Calculated

minimum α1

1

0.467 0.015 1.132 0.020 0.761

0.457 0.025 1.132 0.020 0.761

0.447 0.035 1.132 0.020 0.761

1

0.467 0.015 1.221 0.035 0.650

0.457 0.025 1.221 0.035 0.650

0.447 0.035 1.221 0.035 0.650

2

0.467 0.015 1.060 0.020 0.877

0.457 0.025 1.060 0.020 0.877

0.447 0.035 1.060 0.020 0.877

2

0.467 0.015 1.100 0.035 0.809

0.457 0.025 1.100 0.035 0.809

0.447 0.035 1.100 0.035 0.809

3.4.4 Cracking serviceability limit state

This section provides the parameters necessary to obtain a verification formula for slabs and beams.

Furthermore, the realization of the verification formula is described. Lastly, it provides for slabs the

results and discussion of the results. The differences and adaptations to beams are described

subsequently.

3.4.4.1 Methodology

The various parameters to obtain the verification formula need to be described.

33

3.4.4.1.1 Stress in tension reinforcement

The first factor highly affecting the crack width is the stress in the tension reinforcement, σs, assuming

a cracked section. This parameter depends on the bending moment, M, and the height of the

compressive zone, x. Considering the stress diagram in SLS, according to EC2, the balance of forces

and moments (Equations 3-33 and 3-34) can be defined:

��∗,∗�� = σ ∗ A (Equation 3-33)

��∗,∗�� ∗ �∗�S + σ ∗ A ∗ (d − x) = M (Equation 3-34)

Where As is the cross-section of reinforcement, σs - the stress in the tension reinforcement, b - the

width of the slab, d - the effective height, x - the height of the compressive zone and M - the bending

moment. The stress σs can be obtained by substituting the two balances:

σ ∗ A ∗ �∗�S + σ ∗ A ∗ (d − x) = M (Equation 3-35)

↔ σ = �>�∗r∗ x �>�∗(.��) (Equation 3-36)

Equation 3-36 is calculated for RC and RAC due to the different M and x. Both parameters are

determined in the following sections.

3.4.4.1.2 Bending moment

The bending moment in SLS, Mqp, is obtained by the fact that bending moments always depend on the

total loads and geometric/boundary conditions. The latter are the same in ULS and SLS, which means

that Mqp can be expressed as follows:

Mn� = �uv�KC ∗ M�. (Equation 3-37)

Where pqp and pEd are the total loads in SLS and ULS, respectively.

3.4.4.1.3 Height of the compressive zone

The height of the compressive zone, x, obtained by Equation 3-6 in ULS, cannot be used in SLS. Equation

3-33 and the following formula (which represents the relationship between the compressive stress in

concrete, σc, and the stress in the tension reinforcement, σs) lead to Equation 3-39:

σ� = σ ��,�� ∗ �.�� (Equation 3-38)

Where Es is the design value of the modulus of elasticity of reinforcement steel and Ec,eff is the

effective modulus of elasticity of concrete.

↔ A ∗ ��,�� ∗ (d − x) − ,∗�r� = 0 (Equation 3-39)

The latter equation can be changed if Equation 3-14 is included:

↔ A ∗ �� ∗ (1 + φ(∞, t£)) ∗ (d − x) − ,∗�r� = 0 (Equation 3-40)

34

↔ A ∗ �� ∗ (1 + φ(∞, t£)) ∗ d − A ∗ �� ∗ (1 + φ(∞, t£)) ∗ x − ,∗�r� = 0 (Equation 3-41)

Where Ecm is the secant modulus of elasticity of concrete, b is the width (1 m for slabs) and φ(∞,to) is

the creep coefficient of concrete. x can be solved out of the quadratic equation above (c + bx + ax² =

0) with Equation 3-42:

x = �,±¥,r�X4��4 (Equation 3-42)

x can be calculated for RC but the fundamental parameters α2 and α6 have to be included to obtain the

value of x in RAC (Ecm,RAC and (1+φ)RAC are unknown):

A,8>9 ∗ EE�&,8>9 ∗ (1 + φ(∞, t£)8>9) ∗ d − A,8>9 ∗ EE�&,8>9 ∗ (1 + φ(∞, t£)8>9) ∗ x − b ∗ x�2 = 0

(Equation 3-43)

As,RAC is assumed to be equal to AS,RC because of the comparison of the environmental impact of RC

and RAC (see section 2.4.1):

↔ A,89 ∗ ��qr∗��,DE ∗ α] ∗ (1 + φ(∞, t£)89) ∗ d − A,89 ∗ ��qr∗��,DE ∗ α] ∗ (1 + φ(∞, t£)89) ∗ x − ,∗�r� = 0

(Equation 3-44)

It is possible to solve this in the way explained above, including the range of α2 and α6.

3.4.4.1.4 Effective cross-section area of concrete in tension

The effective area of concrete in tension, Ac,eff, surrounds the reinforcement. This cross-seciton can be

calculated according to EC2 if the height of the compressive zone, x, the width of the slab, b, and the

total height, h, are known:

A�, �� = b ∗ min ¦2.5 ∗ (h − d); (��)S ; ��§ (Equation 3-45)

3.4.4.2 Verification formula

Equation 3-9 will be developed, according to EC2. The standard describes the characteristic crack

width, wk, as follows:

w� = s0,&4� ∗ (ε& −ε�&) (Equation 3-46)

Where sr,max is defined as the maximum crack spacing. This parameter can be calculated with the

following formula:

s0,&4� =kS ∗ c + ��∗�r∗�©∗∅«v,�� (Equation 3-47)

Where c is the cover of the concrete slab, k1 = 0.8 (coefficient that takes into account the bond properties:

bars with high bond), k2 = 0.5 (coefficient that takes into account the strain diagram within the cross

section), k3 = 3.4 (coefficient according to clause 7.3.4(3) of EC2), k4 = 0.425 (coefficient according to

clause 7.3.4(3) of EC2), Ø - the bar diameter, ρp,eff - the ratio between the cross-section of reinforcement,

35

As, and the effective cross-section area of concrete in tension, Ac,eff (see section 3.4.4.1.4).

εsm is the mean strain in the reinforcement under the relevant combination of loads, including the effect

of imposed deformations and taking into account the effects of tension stiffening. Only the additional

tensile strain beyond the state of zero strain of the concrete at the same level is considered. εcm is the

mean strain in the concrete between the cracks. EC2 provides the equation to calculate εsm-εcm:

ε& −ε�& = ��\∗��\,��¬v,��∗(��q�∗«v,��)�� (Equation 3-48)

Where σs is the stress in the tension reinforcement, assuming a cracked section, kt = 0.4 (factor

dependent on the duration of the load: long term loading), fct,eff is the effective tensile strength of

concrete (at the time when the cracks may first be expected to occur). fct,eff is assumed to be equal to

the mean value of axial tensile strength of concrete, fctm. ρp,eff is defined in the previous paragraph and

αe is the ratio between the design value of modulus of elasticity of reinforcing steel, Es, and the secant

modulus of elasticity of concrete, Ecm. Equation 3-9 leads to the verification formula:

s0,&4�,8>9 ∗ (ε& −ε�&)8>9 ≤ 0.3mm (Equation 3-49)

↔ kS(c89 + ∆c) + ��r�©∅H�,DHEH�,��,DHE® ∗ °̄°

±��,DHE��\ ��\�,DHEH�,DHEH�,��,DHE(�� K�K��,DHE

H�,DHEH�,��,DHE)�� ²³

³́ ≤ 0.3mm (Equation 3-50)

↔ kS(c89 + ∆c) + ��r�©∅H�,DEH�,��,DHE® ∗ °̄°

±��,DHE��\�µ��\�,DEH�,DEH�,��,DHE(�� K��rK��,DE

H�,DEH�,��,DHE)�� ²³

³́ ≤ 0.3mm (Equation 3-51)

↔ σ,8>9 −k� qµ��\�,DEH�,DEH�,��,DHE(1 + ��qr��,DE >�,DE>�,��,DHE) ≤ £.S&&∗��

�x(�DE�∆�)� ¶�¶r¶©∅H�,DEH�,��,DHE (Equation 3-52)

↔ N��,DHE� �.x��∗K��V,�U ,DHEO>�,DE�\��\�,DE>�,��,DHE(�� K��rK��,DE

H�,DEH�,��,DHE)≤ α[ (Equation 3-53)

The latter expression is the verification formula for the compliance check of the cracking SLS, which is

calculated in function of the various conditions and parameters.

3.4.4.3 Results

The tables can be seen in Annex C.

3.4.4.4 Discussion

This section discusses the results of α5. When the cases with different ∆c values are compared, it can

be concluded that the bigger the ∆c is, the more conditioning the value of α5 becomes. The table

below shows the values of α5 for α2 = 0.44 and α6 = 1.40, which are the most conditioning. All cases

fall below the highest 95%-certainty limit, which means that compliance is guaranteed. The results are

in most cases smaller than 0, which proves that the cracking SLS never imposes problems.

36

Table 3-12: α5 for slabs in function of two cases of ∆c

Load combination hRC (m) ∆c (m) α2 α6 α5

4 0.120 0.015 0.440 1.400 -4.610 0.120 0.015 0.440 1.400 -1.838 0.120 0.015 0.440 1.400 -0.436

5 0.180 0.015 0.440 1.400 -4.049 0.180 0.015 0.440 1.400 -1.503 0.180 0.015 0.440 1.400 -0.093

6

0.150 0.015 0.440 1.400 -4.956

0.150 0.015 0.440 1.400 -2.223

0.150 0.015 0.440 1.400 -0.741

4

0.120 0.025 0.440 1.400 -1.746

0.120 0.025 0.440 1.400 -0.406

0.120 0.025 0.440 1.400 0.285

5 0.180 0.025 0.440 1.400 -1.512 0.180 0.025 0.440 1.400 -0.153 0.180 0.025 0.440 1.400 0.634

6 0.150 0.025 0.440 1.400 -2.181 0.150 0.025 0.440 1.400 -0.758 0.150 0.025 0.440 1.400 0.051

There is also a trend in function of the cover used. A bigger sr,max is obtained if the cover of the

concrete rises. The trends of sr,max leads to a numerator that becomes less negative as sr,max grows.

The denominator of the verification formula increases for higher cRC, but there are contrasting trends in

this part. If the numerator is less negative and the denominator is bigger, the calculated minimum

value of α5 gets bigger (less negative).

The whole range of α2 and α6 is included, which means that the calculated values of α5 vary in function

of the values of the other fundamental parameters. It is not possible to specifically describe the trends

of α5 in function of the other parameters (because of contrasting trends), but all cases lead to results

that are smaller than 1.

If there is a comparison between the same cases of α2, the effect of α2 in the verification formula itself

remains unchanged (only the indirect effect of α2/α6 on α5). The influence is bigger as the ratio is smaller

and that is why the values of α5 are even smaller for a higher α6, which can be demonstrated by Table 3-

13. The cases are calculated for ∆c = 0.015 m, but the same can be concluded for other ∆c’s.

The last columns of the tables in Annex C consist of parameters, which check whether the calculated

α5 always leads to a characteristic crack width smaller than 0.3 mm. To obtain the results, the factor

(εcm – εsm) is calculated. This factor depends on α5 but it also has to be bigger than a specific limit,

provided by EC2:

(ε�& − ε&) ≥ 0.6 ∗ ��,DHE�� (Equation 3-54)

Table 3-14 shows the control parameters for various load combinations. In this case is ∆c = 0.025 m,

α2 = 0.8 and α6 = 1.2. The results show that cracking is conditioning with a difference in cover, ∆c, up

to 0.025 m and in the worst cases of α2 and α6.

37

Table 3-13: α5 in function of α6 (∆c = 0.015 m)

Load comb

hRC (m)

α2 α6 x1RAC (m)

σsRAC (kN/m²)

Minimum [2.5*(h-d) ;

(h-x)/3 ; h/2] RAC (m)

srmax Numerator Denominator α5

4 0.120 0.900 1.050 0.049 255552 0.024 0.073 -507.672 31.032 -16.360 0.120 0.900 1.050 0.044 255552 0.025 0.114 -223.239 32.068 -6.961 0.120 0.900 1.050 0.040 255552 0.027 0.157 -97.864 33.104 -2.956

5 0.180 0.900 1.050 0.076 271960 0.035 0.089 -561.854 46.030 -12.206 0.180 0.900 1.050 0.071 271960 0.036 0.129 -259.293 47.066 -5.509 0.180 0.900 1.050 0.067 271960 0.038 0.171 -109.147 48.102 -2.269

6

0.150 0.900 1.050 0.063 240580 0.029 0.081 -570.979 38.531 -14.819

0.150 0.900 1.050 0.058 240580 0.031 0.121 -271.695 39.567 -6.867

0.150 0.900 1.050 0.053 240580 0.032 0.163 -129.891 40.603 -3.199

4 0.120 0.900 1.200 0.051 257821 0.023 0.071 -518.227 30.249 -17.132 0.120 0.900 1.200 0.046 257821 0.025 0.113 -226.086 31.362 -7.209 0.120 0.900 1.200 0.041 257821 0.026 0.155 -98.481 32.474 -3.033

5 0.180 0.900 1.200 0.080 274375 0.033 0.087 -578.142 44.817 -12.900 0.180 0.900 1.200 0.075 274375 0.035 0.127 -264.909 45.929 -5.768 0.180 0.900 1.200 0.070 274375 0.037 0.169 -110.982 47.042 -2.359

6

0.150 0.900 1.200 0.066 242716 0.028 0.079 -584.800 37.533 -15.581

0.150 0.900 1.200 0.061 242716 0.030 0.120 -276.128 38.646 -7.145

0.150 0.900 1.200 0.056 242716 0.031 0.162 -131.271 39.758 -3.302

4 0.120 0.900 1.400 0.054 260532 0.022 0.070 -531.111 29.331 -18.108 0.120 0.900 1.400 0.049 260532 0.024 0.111 -229.513 30.534 -7.517 0.120 0.900 1.400 0.044 260532 0.025 0.154 -99.214 31.736 -3.126

5 0.180 0.900 1.400 0.084 277259 0.032 0.085 -598.234 43.395 -13.786 0.180 0.900 1.400 0.079 277259 0.034 0.125 -271.744 44.597 -6.093 0.180 0.900 1.400 0.073 277259 0.036 0.167 -113.198 45.800 -2.472

6 0.150 0.900 1.400 0.069 245268 0.027 0.077 -601.759 36.363 -16.549 0.150 0.900 1.400 0.064 245268 0.029 0.118 -281.492 37.566 -7.493 0.150 0.900 1.400 0.058 245268 0.031 0.160 -132.924 38.768 -3.429

Table 3-14: Control parameters for α5

Load combination

hRC (m)

cRC (m)

sr,max

(m) α5

εcm-εsm (respectiv

e α5)

0.6*σsRAC

/Es Control wk (mm)

εcm-εsm ( α5=1)

Control wk (mm) (α5=1)

4

0.120 0.010 0.070 -17.391 0.004 0.0007 0.300 0.001 0.075

0.120 0.020 0.111 -7.269 0.003 0.0007 0.300 0.001 0.116

0.120 0.030 0.154 -3.041 0.002 0.0007 0.300 0.001 0.155

5

0.180 0.010 0.085 -13.189 0.004 0.0008 0.300 0.001 0.098

0.180 0.020 0.126 -5.860 0.002 0.0008 0.300 0.001 0.143

0.180 0.030 0.167 -2.387 0.002 0.0008 0.300 0.001 0.187

6

0.150 0.010 0.078 -15.863 0.004 0.0007 0.300 0.001 0.077

0.150 0.020 0.119 -7.226 0.003 0.0007 0.300 0.001 0.116

0.150 0.030 0.160 -3.323 0.002 0.0007 0.300 0.001 0.153


The calculations and equations concerning the cracking SLS can be applied when beams are

considered. The complete tables of the results can be seen in Annex K but abbreviated versions are

shown below. Most of the conclusions are equal to those in slabs. Table 3-15 is the corresponding

table for beams of Table 3-12. When cases with different ∆c values are compared, it can be concluded

that the bigger the ∆c is, the more conditioning the value of α5 becomes. All cases fall below the

38

highest 95%-certainty limit if ∆cbeams = 0.020 m, which means that compliance is guaranteed. It must to

be noted that not all the cases comply with the cracking SLS when ∆cbeams = 0.035 m. Some of the

results are bigger than 1.14, which is not acceptable. The reason for this is that a lot of cases are not

economical in practice when ∆cbeams = 0.035 m is used.

Table 3-15: α5 for beams in function of two cases of ∆c

hRC (m) dRC (m) ∆cbeams (m) α2 α6 α5

0.500 0.467 0.020 0.440 1.400 -2.126 0.500 0.457 0.020 0.440 1.400 -0.812 0.500 0.447 0.020 0.440 1.400 0.115 0.500 0.467 0.020 0.440 1.400 -2.063 0.500 0.457 0.020 0.440 1.400 -0.554 0.500 0.447 0.020 0.440 1.400 0.494 0.500 0.467 0.020 0.440 1.400 -2.723 0.500 0.457 0.020 0.440 1.400 -1.279 0.500 0.447 0.020 0.440 1.400 -0.267

0.500 0.467 0.035 0.440 1.400 -0.221 0.500 0.457 0.035 0.440 1.400 0.466 0.500 0.447 0.035 0.440 1.400 0.983 0.500 0.467 0.035 0.440 1.400 0.029 0.500 0.457 0.035 0.440 1.400 0.867 0.500 0.447 0.035 0.440 1.400 1.484 0.500 0.467 0.035 0.440 1.400 -0.655 0.500 0.457 0.035 0.440 1.400 0.132 0.500 0.447 0.035 0.440 1.400 0.719

If the same cases of α2, with different cases of α6 are compared, it is not possible to make specific

conclusions as it is done for slabs because of contrasting trends. The other trends are the same as for

slabs and are not repeated in this section. The complete table can be seen in Annex K.

3.5 Conclusion of Chapter 3

Chapter 3 demonstrated that the ranges of the calculated fundamental parameters for slabs vary

between 95%-certainty limits provided in Chapter 2 if all the respective conditions and parameters are

taken into account. For some limit states, it was possible to go even further than required.

The same can be concluded for beams, except for the cracking SLS. There were some cases that led

to a non-compliance with this limit state but this can be explained by their non-feasibilty in practice.

The compliance checks of the other limit states demonstrated that the results for beams generally vary

in the same area as those for slabs.

In short, the calculated ranges can be used in Chapter 4 to determine the equivalent functional unit in

RAC for slabs and beams.

39

Chapter 4

Definition of the equivalent functional units

Chapter 4 explains how the equivalent unit in RAC can be obtained for slabs and beams. 1 m³ of RC must

lead to a particular amount, K m³, of RAC. This equivalent weight of RAC must have the same functionality

as 1 m³ of RC. Durability and each limit state will lead to a result for hRAC/hRC. As a result, the equivalent

functional unit, K m³, will be the most conditioning of the ratio’s obtained in the various limit states.

4.1 Functionality

As stated above, it is required that the equivalent weight of RAC has the same functionality as 1 m³ of

RC. Firstly the service life needs to be the same in both cases. It is assumed that the service life is 50

years, corresponding to structural class S4. During that time, the equivalent in RAC needs to resist

and chlorides penetration. Another aspect of functionality is that the deformation of the equivalent in

RAC is equal, or slightly smaller, than that of the example in RC. Furthermore, the cross-section of

reinforcement in RAC must be approximately equal to that in RC. The difference in design cross-

section may range from -5% to 15% if RAC is considered (underlying concept of comparing

environmental impacts of RAC and RC). Lastly, the crack width, wk, of the equivalent slab in RAC

needs to be limited as well. The cracking SLS will normally never pose a problem under normal

circumstances, but wk must be smaller than 0.3 mm.

4.2 K m³ of RAC

The equivalent weight of RAC is expressed by constant K. It must be noted that the difference in thickness

between the examples in RAC and RC must not be excessive to make sense for practical purposes

(economic and structural). K is normally bigger than 1 but it is also possible that the amount of RAC used is

smaller than that of RC. This can be the case when RA with a higher quality than NA are employed.

4.3 Design compliance criteria

The ranges, described as methodology compliance criteria in section 3.3, form the conditions for the

determination of K. The other data, used throughout the calculations in this chapter, are defined in

sections 2.2, 2.3 and 2.4. Moreover, most of the parametric studies involving the limit states are also

used for the calculations of this chapter.

4.4 Methodology compliance criteria

As previously explained, these criteria define the target results of the various calculations: ranges in

40

which the various hRAC/hRC must vary. It is expected that the ratios will not be lower than 1 (although this

is possible in some cases). hRAC/hRC for slabs must be limited to around 1.5 because higher ratios will not

be economical/feasible for practical purposes. The corresponding limit of the ratio for beams is 1.30.

4.5 Calculation of equivalent functional unit

This section contains the calculations to determine K for slabs and beams. Durability handles α3 and

α4, deformation - α2 and α6, bending - α1 and cracking - α5. The definition of the equivalent functional

unit is made independently for each limit state and in the end, the most conditioning of the various

ratios is selected.

4.5.1 hRAC/hRC in function of α3 and α4

The methodology, results and discussion of the results, concerning durability, are described in this section.

The three parts are described for slabs but the differences for beams are explained in section 4.5.1.4.

4.5.1.1 Methodology

As previously stated, EC2 provides values for cmin,RC and Table 3-1 forms the basis for the

calculations. The minimum cover in RAC, cmin,RAC, can be described in function of that in RC if α3 (or

α4) is included. This is demonstrated by Equation 3-10. Consequently, the difference in cover between

RAC and RC, ∆c, is expressed by Equation 3-11. If Equation 2-5 is introduced, hRAC/hRC can be

calculated as follows:

�`ab�`b = �`b��∗∆��`b = �`b��∗(·x��)∗�¸¹º,`b�`b (Equation 4-1)

The same equation can be obtained if chlorides penetration is considered but α3 needs to be replaced

by α4. The calculations are executed in function of the fundamental parameter, structural and exposure

class. Various RC slab heights, hRC, have to be taken into account as well: 0.010 m, 0.012 m, 0.015 m

and 0.018 m.

4.5.1.2 Results

The complete tables can be seen in Annex D. Each structural class represents a diagram in function of

the other parameters. Figures 4-1 and 4-2 show the values of the ratio in function of α3 or α4 for S3.

This structural class is shown instead of S4 because EC2 allows to reduce the structural class by 1 if

cmin,RC needs to be determined for slabs.

4.5.1.3 Discussion

Some cases in Figures 4-1 and 4-2 result in very high ratios (e.g. in Figure 4-1: a ratio of 1.75 is

obtained if α3 = 2.5). This is not useful for practical purposes because the example corresponds to hRC

= 10 cm and XC4. Resulting from this, dRC would be 5.7 cm. The ratio between the effective height,

dRC, and the total height, hRC, would be too low and consequently not economical in design situations.

Therefore, several cases need to be excluded from the figures if the previous reasoning is considered.

41

Based on experience in design of concrete slabs, a practical limit for dRC/hRC can be defined if

carbonation is considered:

.DE�DE ≥ 0.75 (Equation 4-2)

Figure 4-1: hRAC/hRC in function of α3 for S3 (slabs)

Figure 4-2: hRAC/hRC in function of α4 for S3 (slabs)

Figure 4-3 shows the adaptation of Figure 4-1, taking Equation 4-2 into account. The higher the

structural class is, the more cases are excluded. This is because of the higher cmin,RC that leads to a

lower dRC (e.g. the condition is almost never satisfied for XC4: a bigger slab is required to comply with

this case).

The general trends demonstrate that hRAC/hRC grows with a higher α3. All the cases (even characteristic

values of α3) comply with the criteria described in section 4.4. hRAC/hRC is equal to 1.14 when the

average value of α3 (=1.50) is considered. A higher exposure class leads to higher results because

cmin,RC becomes bigger in Equation 4-1. Comparing the same exposure class for slabs with a different

hRC shows that the ratio declines for thicker slabs.

If chlorides penetration (fundamental parameter α4) is considered, no result for S3 complies with

0.900

1.000

1.100

1.200

1.300

1.400

1.500

1.600

1.700

1.8000

.80

0

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=10cm

XC1 (hRC=10cm)

XC2/XC3 (hRC=10cm)

XC4 (hRC=10cm)

X0 (hRC=12cm)

XC1 (hRC=12cm)

XC2/XC3 (hRC=12cm)

XC4 (hRC=12cm)

X0 (hRC=15cm)

XC1 (hRC=15cm)

XC2/XC3 (hRC=15cm)

XC4 (hRC=15cm)

X0 (hRC=18cm)

XC1 (hRC=18cm)

XC2/XC3 (hRC=18cm)

XC4 (hRC=18cm)

1.000

1.050

1.100

1.150

1.200

1.250

1.300

1.0

00

1.0

50

1.1

00

1.1

50

1.2

00

1.2

50

1.3

00

1.3

50

hRAC/hRC

α4

XD1/XS1 (hRC=10cm)

XD2/XS2 (hRC=10cm)

XD3/XS3 (hRC=10cm)

XD1/XS1 (hRC=12cm)

XD2/XS2 (hRC=12cm)

XD3/XS3 (hRC=12cm)

XD1/XS1 (hRC=15cm)

XD2/XS2 (hRC=15cm)

XD3/XS3 (hRC=15cm)

XD1/XS1 (hRC=18cm)

XD2/XS2 (hRC=18cm)

XD3/XS3 (hRC=18cm)

42

Equation 4-2. In this case it is necessary to lower the criteria:

.DE�DE ≥ �S (Equation 4-3)

Figure 4-3: hRAC/hRC in function of α3 for S3 - feasible cases only (slabs)

Figure 4-4 shows the complying cases for S3. The same conclusions as for α3 can be derived. The

criteria of section 4.4 are satisfied for all values and the average value of α4 for full replacement is

1.10, which leads to hRAC/hRC = 1.05. Furthermore, the values of cmin,RC given by EC2 are conservative.

A normal slab in a standard framed building will normally never be exposed to such conditions that a

minimum cover of 30 mm to 40 mm is needed. Only slabs for balconies will be exposed to those

exposure classes.

Figure 4-4: hRAC/hRC in function of α4 for S3 - feasible cases only (slabs)


The methodology of the slabs cannot be applied for beams because the simplification concerning the

total height (Equation 2-5) is only for slabs. Beams follow another philosophy that uses the

simplifications given by Equations 2-15 and 2-16. Equation 3-10, which expresses the relationship

between cmin,RAC and cmin,RC, is included and this leads to the following formula for hRAC/hRC:

0.900

1.000

1.100

1.200

1.300

1.400

1.500

0.8

00

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=10cmXC1 (hRC=10cm)X0 (hRC=12cm)XC1 (hRC=12cm)X0 (hRC=15cm)XC1 (hRC=15cm)XC2/XC3 (hRC=15cm)X0 (hRC=18cm)XC1 (hRC=18cm)XC2/XC3 (hRC=18cm)XC4 (hRC=18cm)

1.000

1.020

1.040

1.060

1.080

1.100

1.120

1.140

1.160

1.180

1.200

1.0

00

1.0

50

1.1

00

1.1

50

1.2

00

1.2

50

1.3

00

1.3

50

hRAC/hRC

α4

XD1/XS1 (hRC=12cm)

XD1/XS1 (hRC=15cm)

XD2/XS2 (hRC=15cm)

XD3/XS3 (hRC=15cm)

XD1/XS1 (hRC=18cm)

XD2/XS2 (hRC=18cm)

XD3/XS3 (hRC=18cm)

43

�DHE�DE = .DE∗γ�.r��DE�.DE�∆�W�U��DE = .DE∗»vuv,DE�µ�∗(αx¼�)∗��,DE,�½UW�vuv,DE ¾�.r��DE�.DE�(αx��)∗��,DE,W�U��

�DE

(Equation 4-4)

Where dRC is the effective height of the beam, γ - the ratio between the quasi-permanent slab loads in

RAC and RC, pqp,RAC and pqp,RC. hRC is the total height of the beam, cmin,RC,slabs and cmin,RC,beams are the

minimum covers of slabs and beams (EC2). The equation is determined for various heights of the

beams (0.40 m, 0.50 m and 0.60 m) and as the loads of the slabs affect the height of the beam, it is

necessary to include the various load combinations of the slabs as well (heights 0.12 m, 0.15 m and

0.18 m). This means that 3 different diagrams for 1 structural class are obtained. The same equation

can be obtained if chlorides penetration is considered but α3 needs to be replaced by α4. The

calculations are done in function of the fundamental parameter, structural class S4 and the exposure

class (for each case of height of beam and slab). The figures showing the results for various cases are

presented in Annex L, but Figures 4-5 and 4-6 show the ratio in function of α3 or α4 for the intermediate

load combination of the slabs.

Figure 4-5: hRAC/hRC in function of α3 for S4 and slab 15 cm thick (beams)

Figure 4-6: hRAC/hRC in function of α4 for S4 and slab 15 cm thick (beams)

1.000

1.050

1.100

1.150

1.200

1.250

1.300

1.350

1.400

1.450

0.8

00

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=40cm)

XC1 (hRC=40cm)

XC2/XC3 (hRC=40cm)

XC4 (hRC=40cm)

X0 (hRC=50cm)

XC1 (hRC=50cm)

XC2/XC3 (hRC=50cm)

XC4 (hRC=50cm)

X0 (hRC=60cm)

XC1 (hRC=60cm)

XC2/XC3 (hRC=60cm)

XC4 (hRC=60cm)

1.000

1.020

1.040

1.060

1.080

1.100

1.120

1.140

1.160

1.000 1.050 1.100 1.150 1.200 1.250 1.300 1.350

hRAC/hRC

α4

XD1/XS1 (hRC=40cm)

XD2/XS2 (hRC=40cm)

XD3/XS3 (hRC=40cm)

XD1/XS1 (hRC=50cm)

XD2/XS2 (hRC=50cm)

XD3/XS3 (hRC=50cm)

XD1/XS1 (hRC=60cm)

XD2/XS2 (hRC=60cm)

XD3/XS3 (hRC=60cm)

44

The ratio between the effective height, dRC, and the total height, hRC is for all cases bigger than the

practical limit (0.75), which means that none of the examples needs to be excluded from the diagrams.

The other trends for of hRAC/hRC are the same as those described in section 4.5.1.3 and all cases comply

with the criteria of section 4.4 when average values of the fundamental parameters are considered. The

results of the maximum hRAC/hRC are then 1.11 and 1.04, for α3 and α4 respectively.The highest load

combinations of the slabs lead to the lowest result of hRAC/hRC. This makes sense as γ becomes smaller

in the verification formula (thicker slabs are less sensitive to ∆cslabs).

4.5.2 hRAC/hRC in function of α2 and α6

This section presents the slabs’ methodology to obtain the equivalent unit in RAC if deformation is

considered. Also the results and their discussion are provided. The differences for beams are

explained thereafter.

4.5.2.1 Methodology

In this case, it is possible to use the parametric study of Chapter 3 to obtain the equivalent functional

unit in RAC for slabs: Equation 4-5 is calculated for various load combinations:

qsqr ≤I�DHE�DE JS ∗ �uv,DE�uv,DHE (Equation 4-5)

Where hRAC and hRC are the total slab heights in RAC and RC, respectively. pqp,RAC and pqp,RC represent

the corresponding quasi-permanent loads. It was expected that hRAC/hRC would be in one part of the

equation but heights affect the loads as well: α6/α2 is set on one side of the expression and the other

parameters affected by the heights on the other side.

4.5.2.2 Results

The complete table can be seen in Annex E. With the results and ratio hRAC/hRC, it is possible to

express the equivalent unit in RAC in function of the relevant range of α6/α2 and determine an average

trend line. The summarizing diagram is shown in Figure 4-7.

Figure 4-7: hRAC/hRC in function of α6/α2 (slabs)

y = 0.9983x0.4154

R² = 0.9916

1.000

1.100

1.200

1.300

1.400

1.500

1.600

1.700

1.000 1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800 3.000 3.200 3.400α6/α2

hRAC/hRC

45

4.5.2.3 Discussion

The calculations for the deformation SLS result in average trend lines in function of α6/α2. This is

demonstrated by Figure 4-7 for slabs and Figure 4-8 for beams. Various conditions are included and it

can be seen that the results of hRAC/hRC vary within the target limits when (full replacement) average

values of fundamental parameters are used: mean values of α6 (= 1.17) and α2 (= 0.8) lead to α6/α2 =

1.46, which results in hRAC/hRC equal to 1.17 for slabs.


Equation 4-5 is used for beams as well. The complete tables are shown in Annex M, but the diagram that

takes all the cases (concerning the height of the beam) into account is presented below.

Figure 4-8: hRAC/hRC in function of α6/α2 (beams)

The ratio α6/α2 only ranges from 1.09 to 2.44, which means that not the whole range is included. The

obtained relationship can be used for higher ratios as well. Mean values of α6 (= 1.17) and α2 (= 0.8)

lead to α6/α2 = 1.46, which results in hRAC/hRC equal to 1.18 for beams.

4.5.3 hRAC/hRC in function of α1

The methodology, results and discussion are handled for slabs and differences or adaptations to

beams are described in section 4.5.3.4.

4.5.3.1 Methodology

As it is not possible to specifically express hRAC/hRC in function of α1, this part is performed as for the

deformation SLS. The parametric study concerning the bending ULS and the calculated range of α1

form the basis to determine K. Figure 4-9 depicts hRAC/hRC of the various combinations in function of

the results of Equation 3-31. The calculations are executed for various concrete strength classes

(C20/25, C25/30 and C30/37) but the same results are reached (demonstrated in Annex F).

Differences in cover, varying between 0.000 m and 0.025 m are used.

4.5.3.2 Results

The complete tables can be seen in Annex F. Figure 4-9 shows the summarized diagram of the relevant

y = 0.9956x0.446

R² = 0.9977

1.000

1.100

1.200

1.300

1.400

1.500

1.600

1.000 1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600

hRAC/hRC

α6/α2

46

results with trend lines for the minimum values and the maximum values, respectively.

Figure 4-9: hRAC/hRC in function of α1 (slabs)

4.5.3.3 Discussion

The ratio hRAC/hRC can be achieved out of three values of α1 because of the different cover, cRC, in each

load combination. A higher cover leads to a lower value of α1 for the same ratio hRAC/hRC. This makes

sense because the cover affects dRAC, which is in the denominator of the verification formula. If a

higher cover is used, dRC is smaller and dRAC changes relatively more because of ∆c. The cases with

the lower α1 (bigger cRC) for the same hRAC/hRC give the best results: even if the compressive strength

of RAC is that much smaller than the one in RC, the same hRAC/hRC is reached.

All cases of RAC, even with poor quality aggregates, comply with the criteria with 95% probability if α1 is

equal or bigger than 0.56. The average value of α1 is approximately 0.9 in the case of 100% replacement.

In that case, the ratio hRAC/hRC is maximum 1.08, which is a feasible result for practical purposes.

Although some cases lead to the same ratio hRAC/hRC, they have a different value of α1. It needs to be

noted that the case with the lower value for µRAC will result in the lower value of α1. This is because a

lower µRAC leads to a lower ωRAC and a lower cross-section of reinforcement in RAC. Furthermore, a

lower µRAC is also caused by a bigger dRAC, which is in the denominator of the verification formula

(Equation 3-31).


The same methodology as in section 4.5.3.1 is applied, but calculations are only done for C25/30 (the

same results are obtained for the other classes). The equivalent functional unit is determined for the

three heights of the beams: 0.40 m, 0.50 m and 0.60 m. Figure 4-10 shows the outcomes for the three

cases together. The complete table for h = 0.50 m can be seen in Annex N.

The value of α1 does not change in function of the cover, cRC, which means that there is just one result

of α1 that leads to a particular height. The beams with the biggest size lead to the smallest results of

hRAC/hRC. This can be explained by the smaller cross-section of reinforcement in those beams if the

same load combination is considered. A smaller cross-section of reinforcement leads to a smaller

1.000

1.050

1.100

1.150

1.200

1.250

1.300

1.350

1.400

1.450

0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100

hRAC/hRC

α1

all values

minimum values

maximum values

line minimum values

line maximum values

y = x-0,356

y = x-0,932

47

Fs,RAC in the verification formula (Equation 3-32). RAC complies with the criteria if (full replacement)

average value of α1 (= 0.9) are considered. In that case, the ratio hRAC/hRC is maximum 1.08, which is a

feasible result for practical purposes.

Figure 4-10: hRAC/hRC in function of α1 (beams)

4.5.4 hRAC/hRC in function of α5 (including α2 and α6)

The methodology, results and discussion of the results, concerning cracking, are described in this

section. It is impossible to define an equivalent functional unit in RAC in function of only α5 because α2

and α6 affect the parameters in the verification formula. The various parts are presented for slabs, but

the differences for beams are shown in section 4.5.4.4.

4.5.4.1 Methodology

As in sections 4.5.2 and 4.5.3, the parametric study of the compliance check of the limit state forms

the basis for the results of the equivalent unit. Equation 3-53 is calculated for various load

combinations and differences in cover (0.010 m, 0.015 m, 0.020 m and 0.025 m). The ranges of

parameters α2 and α6 are included as well. Ratio hRAC/hRC is for every load combination depicted in

function of α5 (Figure 4-11). The calculations are only performed for C25/30 (results obtained with

other strength classes are not conditioning as well).

4.5.4.2 Results

The tables are based on those in Annex C (exactly the same sequence). Only the parameter hRAC/hRC

is included for every case. The tables concerning cracking consider differences in cover, ∆c, ranging

from 0.010 m to 0.025 m (whilst only the cases of 0.015 m and 0.025 m were tested in Annex C). The

summarizing figure is given below.

4.5.4.3 Discussion

Most of the cases result in a negative fundamental parameter. These cases do not make sense for

practical purposes. If solely cases with α1 between the 95%-certainty limits are included, few results

are obtained. Moreover, these cases will lead to higher equivalent units if other limit states are

considered. Therefore, it can be concluded that the cracking SLS does not need to be taken into

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

0.5 0.6 0.7 0.8 0.9 1 1.1

hRAC/hRC

α1

all values (0.40 m * 0.20 m)

all values (0.60 m * 0.30 m)

all values (0.50 m * 0.25 m)

line minimum values

line maximum values

y=x-0,58

y=x-0,86

48

account in the determination of the final result of the equivalent functional unit in RAC.

Figure 4-11: hRAC/hRC in function of α5 (slabs)


The same methodology and Equation 3-54 can be used for beams, but the differences in cover need

to be adapted to the corresponding ones for beams. The calculations are also only made for concrete

strength class C25/30 and the various sizes of the beams are included as they lead to different results.

The tables for beams are based on those in Annexes K and C but hRAC/hRC is included for every load

combination. The tables include differences in cover, ∆c, ranging from 0.015 m to 0.035 m (whilst only

the cases of 0.020 m and 0.035 m were tested in Annex K). Figure 4-12 shows the results in function

of α5. Most of the values of α5 are negative; the figure only shows the examples until -7.000 but the

range goes until -18.000. Only the cases with α5 between 0 and 1 make sense, but it is not possible to

determine a formula with those results. As cracking is expected to be never conditioning for beams, it

does not need to be taken into account in the determination of the equivalent functional unit.

Figure 4-12: hRAC/hRC in function of α5 (beams)

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

-8.000 -6.000 -4.000 -2.000 0.000 2.000

hRAC/hRC

α5

delta c=0.010m

delta c=0.015m

delta c=0.020m

delta c=0.025m

1.000

1.050

1.100

1.150

1.200

1.250

1.300

1.350

1.400

1.450

-7.000 -6.000 -5.000 -4.000 -3.000 -2.000 -1.000 0.000 1.000 2.000

hRAC/hRC

α5

delta c=0.015m (0.50m*0.25m)

delta c=0.015m (0.40m*0.20m)

delta c=0.015m (0.60m*0.30m)

delta c=0.020m (0.50m*0.25m)

delta c=0.020m (0.60m*0.30m)

delta c=0.020m (0.40m*0.20m)

delta c=0.025m (0.50m*0.25m)

delta c=0.025m (0.60m*0.30m)

delta c=0.025m (0.40m*0.20m)

delta c=0.035m (0.50m*0.25m)

delta c=0.035m (0.60m*0.30m)

delta c=0.035m (0.40m*0.20m)

49

4.6 Conclusion of Chapter 4

Each limit state, except the cracking SLS, led to an equivalent functional unit in RAC.. Some cases

needed to be excluded, due to their irrelevancy for practical purposes and therefore this dissertation.

The final result of the equivalent functional unit in RAC for slabs or beams is the most conditioning of

the various ones obtained in the limit states (Equation 4-6):

K = �DHE�DE = max ¿�DHE�DEq� ; �DHE�DE

qx ; �DHE�DEq© ; �DHE�DE

qs qrÀ Á (Equation 4-6)

The ratio between the loads in ULS and SLS can also be described in function of the K-values of the

slabs, which is demonstrated by Figures 4-13 and 4-14. These ratios show the increase of the loads

when a particular K-value is chosen. This is interesting for designers and owners who want to know

the effect of RA on the loads of the slabs. The ratios are also required to perform the design for slabs,

as stated in section 2.4.2.2.

Figure 4-13: pEd,RAC/pEd,RC in function of the K-value Figure 4-14: pqp,RAC/pqp,RC in function of the K-value

y = 0.3425x + 0.6595

R² = 0.8868

0.980

1.000

1.020

1.040

1.060

1.080

1.100

1.120

1.140

1.000 1.100 1.200 1.300 1.400

pEd,RAC/

pEd,RC

Kslab

y = 0.4384x + 0.565

R² = 0.8616

0.980

1.000

1.020

1.040

1.060

1.080

1.100

1.120

1.140

1.160

1.180

1.000 1.100 1.200 1.300 1.400

pqp,RAC/

pqp,RC

Kslab

51

Chapter 5

Validation of the method using real mixes

This chapter demonstrates that the method proposed works when real concrete mixes are produced.

Cases of slabs with various geometry/boundary conditions are examined. For each case of slabs,

there are two cases of beams examined. The equivalent functional units in RAC are obtained via the

factor K and their properties can be expressed in function of the examples in RC via the fundamental

parameters. It is thus possible to design the slabs and beams using RAC similarly to RC slabs and

beams. Cases in which some of the fundamental parameters are not available need to be examined

as well. The missing parameters can be expressed in function of α1, the only fundamental parameter

always required if cases in RAC are compared with those in RC.

5.1 Scope

The scopes of slabs and beams are mostly the same but there are some differences for beams that

require a further explanation.

The relevant conditions for the calculations are selected:

1. The thickness of the slabs in RC ranges from 0.12 m to 0.18 m (minimum height that complies

with all the limit states) and consequently the dead weight loads range from 3.0 kN/m² to 4.5

kN/m². As previously stated, this range represents the vast majority of solid slabs in standard

framed buildings;

2. Other permanent loads, ranging from 2.0 kN/m² to 3.0 kN/m², are included;

3. The live loads used vary between 2.0 kN/m² and 4.0 kN/m²;

4. Environmental classes XC2, XC3, XD1 (airborne chlorides) and XS1 (chlorides from sea

water) are included;

5. The concrete strength classes used range from C20/25 to C50/60. The latter strength is

normally too high to be used in slabs in standard framed buildings, but they are used because

those are the available research results;

6. Only structural class S4 is considered throughout the calculations;

7. The bar diameters of the reinforcement are restricted to 8 mm, 10 mm or 12 mm, which

represent the steel bars commonly used. Furthermore, the bar spacing is also applied according

to commonly used values in design situations: 0.075 m, 0.100 m, 0.125 m, 0.150 m and 0.20 m;

8. The creep coefficient for RC can be derived from regulations by EC2, but Chapter 5 always

uses φ(∞,t0) = 2.5. This value is a good approach for all the cases of RC in current design

situations;

9. The design value of the modulus of elasticity of reinforcing steel is assumed to be 200 GPa (EC2).

52

The relevant differences for the calculations of beams are selected:

1. Beams supporting slabs do not need their own live loads or other permanent loads as those

are already included in the slabs;

2. The height of the beam, h, is the minimum height that complies with all limit states;

3. The bar diameters of the reinforcement are restricted to 16 mm, 20 mm or 25 mm,

corresponding to the height of the beam;

4. The span is similar to that of the slabs when one-way slabs are considered while the span is

the same when two-way slabs are considered;

5.2 Design criteria

5.2.1 General

Various geometry/boundary conditions, one- and two-way slabs and differences in the supports of the

slabs (simply supported or continuous) are included. The calculations for the bending moments, MEd and

Mqp, and deformation of the slabs will depend on those conditions. Furthermore, Bares coefficients are

used in the case of two-way slabs to calculate the deflection and bending moment of those cases (Bares,

1981) When continuous slabs are considered (on one or more borders), the support bending moments will

be used for further calculations instead of the mid-span bending moment. This will lead to more

conditioning cases. The span of the slabs examined ranges from 4.0 m to 6.0 m. When two-way slabs

are considered, it is assumed that the slabs are square. This is done for simplification purposes.

It is also necessary to define the geometry/boundary conditions of the beams. Each slab designed

results in two beams with different supports: simply supported or continuous. The calculation of the

loads, bending moments and deformation is affected by this aspect. When the border of the slab is

simply supported, it is assumed that the beams designed represent a border beam, i.e. it receives

loads from only one side. Continuous borders of the slabs lead to the assumption that middle beams in

the building are designed, i.e. it receives loads from two sides. Furthermore, it is assumed that two

slabs supported by the middle beam are exactly the same (simplification purposes).

When RC is considered, the mean compressive strength obtained in the tests, needs to be compared

with EC2. Section 5.2.2 explains how the compressive strength used in further calculations is

calculated. The other properties (e.g. Ecm and fctm) of the RC are based on the value provided by EC2.

5.2.2 Equivalent in RAC

RAC is related to RC by the fundamental parameters and this leads to the definition of the equivalent

unit, K. Equations 5-1 to 5-4 show the formulas of the ratios of the slabs, obtained in Chapter 4.

Equations 5-5 to 5-8 show the corresponding formulas for the beams:

�DHE�DE ∝� =∝��£.S[] (Equation 5-1)

53

�DHE�DE ∝x =�DE��∗(qx��)∗��,DE�DE (Equation 5-2)

�DHE�DE ∝© =�DE��∗(q©��)∗��,DE�DE (Equation 5-3)

�DHE�DE ∝s ∝rÀ = 0.9983 ∗ I∝s∝rJ£.X�[X

(Equation 5-4)

�DHE�DE ∝� =∝��£.[Â (Equation 5-5)

�DHE�DE ∝x =.DE∗»vuv,DE�µ�∗(�x¼�)∗��,DE,�½UW�vuv,DE ¾�.r��DE�.DE�(qx��)∗��,DE,W�U��

�DE (Equation 5-6)

�DHE�DE ∝© =.DE∗»vuv,DE�µ�∗(�©¼�)∗��,DE,�½UW�vuv,DE ¾�.r��DE�.DE�(q©��)∗��,DE,W�U��


�DHE�DE ∝s ∝rÀ = 0.9956 ∗ I∝s∝rJ£.XX]

(Equation 5-8)

Where hRAC and hRC are the total heights of the structural elements in RAC and RC, respectively, dRC -

the effective height of the beam, γ - the ratio between the quasi-permanent loads in RAC and RC,

pqp,RAC and pqp,RC, respectively, and cmin,RC,slabs and cmin,RC,beams - the minimum covers of slabs and

beams (EC2). In various cases of real mixes, the values of the fundamental parameters fall outside the

characteristic limits. Nevertheless, it is more relevant to use the values obtained in previous research

instead of restricting them to the 95%-certainty limits. Various research results are provided (Amorim

et al., 2012), (Bravo et al., 2015b), (Cakir, 2014), (Cartuxo, 2013), (Evangelista, 2014), (González-

Fonteboa, 2011), (Pedro et al., 2014b). The same references are used for both structural elements.

In most cases, each property of RAC can be achieved by multiplying the same property of the

respective RC by the corresponding fundamental parameter. This is not the case for the mean

compressive strength of RAC, fcm,RAC. In practice, it is not possible to know the exact characteristic

compressive strength of RC based solely on its strength class. Knowing the strength class, e.g.

C30/37, only ensures that fck,cyl ranges between 30 and 35 MPa and fck,cube between 37 and 45 MPa.

The upper limits concern the lower limits of the next class, in this case C35/45. The characteristic

compressive strength for cubes is used in the calculations due to the fact that fundamental parameters

obtained in research are mostly based on cubes. fck,RC is equal to the average value of the limits, in

this example 41 MPa. fck,RAC can be calculated by multiplying fck,RC by α1. Resulting from this, the mean

compressive strength in RAC, fcm,RAC, is calculated with Equation 5-9:

f�&,8>9 = f��,8>9 + 8 (Equation 5-9)

Expressing the property in this way is a slight inaccuracy because α1 expresses the ratio between the mean

compressive strengths in RAC and RC, respectively. However, this is acceptable for design purposes. The

height of the slab or beam in RAC is equal to the height of the corresponding example in RC,

multiplied by K. This result must be rounded to the nearest number of centimetres. The minimum

cover of the slab or beam in RAC, cmin,RAC, is equal to the maximum of the minimum covers of the slab

54

or beam in RC, cmin,RC, respectively multiplied by α3 or α4. In practice, it is necessary to round the result

as well, but as this chapter is a theoretical exercise, it is better to keep the value obtained via the


5.3 Missing data

Research, concerning the subject of this dissertation, often does not provide all the fundamental

parameters. In this case, it is necessary to express the missing fundamental parameters in function of

α1, which must always be available to execute the design.

EC2 provides Equation 5-10 for the relationship between the secant modulus of elasticity and mean

compressive strength. This formula can also be used for examples in RAC:

E�& = β ∙ 22 ∙ I��£ J£.S (Equation 5-10)

Where Ecm is the 28-day secant modulus of elasticity, fcm - the 28-day mean compressive strength and β

- the correction factor depending on the nature of the aggregates used (1-2 - basalt; 1.0 - quartzite; 0.9 -

limestone; 0.7 - sandstone). A previous comprehensive statistical analysis with information provided by

Silva (Silva et al., 2014c) led to several trend lines. The value of β, for which the highest coefficient of

determination, R², is obtained, is used to express the average relationship between α2 and α1:

∝�= 0.871 ∗∝�£.S (Equation 5-11)

Concerning carbonation and chlorides penetration, results of earlier studies (Hasaba et al., 1981)

suggest that it is possible to correlate the accelerated carbonation coefficient and chloride migration

coefficient with the mean compressive strength of concrete. The relationship is not affected by the

replacement level, type or size of RA, which can be demonstrated by a comprehensive statistical

analysis (Silva et al., 2015). The relationship between RAC and RC with similar mix design may be

calculated using the following equations:

∝S= T�UVW,àbT�UVW,Db = N ÄÅ¸,DEÄÅ¸,àbO�,� = I �

∝�J�,�

(Equation 5-12)

∝X= YÆàbÆDb = ¥Ç�£,£�S(ÄÅ¸,àb�ÄÅ¸,Db) = ¥Ç�£.£�S∗ÄÅ¸,`b∗(∝��) (Equation 5-13)

Where Kcarb,RAC and Kcarb,RC are the coefficients of accelerated carbonation in RAC and RC,

respectively, DRAC and DRC - the respective 28-day chloride migration coefficients and fcm,RAC and fcm,RC

their 28-day mean compressive strength.

Equation 5-14 is also provided by EC2. This formula proves that the tensile strength varies

proportionally to the compressive strength:

È�_É = 0.30 ∙ È�Ê�/S (Equation 5-14)

Where fck is the target characteristic compressive strength. This formula applies to concrete strength

classes ≤C50/60. A comprehensive statistical analysis of Silva (Silva et al., 2015) showed that there is

55

no significant difference between the actual and expected fctm, when considering a given compressive

strength. This can be concluded for all RAC mixes, regardless of the replacement level and type of RA.

In other words, the relationship between fctm and fck may also be used in RAC mixes. Knowing this, it is

possible to express α5 in function of α1 (Equation 5-15).

∝[=∝�� SÀ (Equation 5-15)

The relationship between the fundamental parameters α6 and α1 is extremely complex and requires a

creep prediction model by ACI. (Silva et al., 2014d) The model requires other properties, which are

also not analysed in this dissertation. That is why ACI is not used to obtain the relationship. The

average relationship (Equation 5-16) between α6 and α1 is less comprehensive, but the most simplified

equation between the two properties. It is obtained by using the average trend lines of the respective

fundamental parameters in Figures 1-1 and 1-6. The replacement ratio (x) of the properties can be

isolated in the expressions and rearranging both formulae leads to Equation 5-16.

∝]= 2.55 − 1.55 ∗ ∝� (Equation 5-16)

This equation is only useful for a specific range of α1: [0.0 – 1.0]. If a higher value is introduced in

Equation 5-16, α6 goes below 1.0., which is does not make sense for practical purposes.

Four cases with missing data are used to check whether the method proposed can also be applied to

those examples:

1. Only α1 is available;

2. α1, α2 and α5 are available;

3. α1, α3 and α4 are available;

4. All fundamental parameters are available, except α6.

In case 1, a random example of RAC is considered, but it is also useful to use an example for which it

is assumed that the fundamental parameters are not available. In this way it is possible to compare the

results obtained considering two different philosophies.

5.4 Structural design

The examples in RAC must be designed in the same way as in RC. Bending is considered before

deformation and cracking, due to the need of knowing the cross-section of reinforcement in the

deformation control and crack control calculations. The calculations are explained for slabs;

differences for beams are described at the end of each section.

5.4.1 Bending ULS

The design compressive strength of concrete, fcd, and design steel yield tensile strength, fyd are

obtained as follows:

f�. = ��¶2E (Equation 5-17)

56

fz. = �F¶2Ì (Equation 5-18)

Where γc is concrete’s partial safety factor (= 1.50), fck - the characteristic compressive strength. γs -

steel reinforcement’s partial safety factor (= 1.15) and fyk is the steel reinforcement’s characteristic

yield tensile strength.

The loads chosen, g, ∆g and q, lead to the total loads, pEd and pqp, taking into account the partial

safety factors, γ, and combination coefficients, Ѱ2 :

p�. = 1.35 ∗ (g + Δg)+ 1.5 ∗ q (Equation 5-19)

pn� = (g + Δg) +Ѱ� ∗ q (Equation 5-20)

The slab’s mid-span and support design bending moments are calculated according to the

geometry/boundary conditions of the slab and EC2. There is a distinction between one- and two-way slabs.

One-way slabs can be simply supported or continuous. Simply supported slab’s mid-span design bending

moment is calculated as follows:

M�.,&'.��4( = ÍKC×ÏrÂ (Equation 5-21)

Where L is the span of the slab. If the slab is continuous on one side and simply supported on the

other side, the bending moments can be calculated as follows:

M�.,&'.��4( = Ð∗ÍKC∗Ïr��Â (Equation 5-22)

M�.,/��60� = − ÍKC∗ÏrÂ (Equation 5-23)

It is also possible that the slab is continuous on both sides. The bending moments for this case are

calculated as follows:

M�.,&'.��4( = ÍKC∗Ïr�X (Equation 5-24)

M�.,/��60� = − ÍKC∗Ïr�� (Equation 5-25)

A cantilevered slab only has a support bending moment, which can be calculated as follows:

M�.,/��60� = − ÍKC∗Ïr� (Equation 5-26)

Two-way slabs require the consideration of the Bares coefficients, which need to be introduced in the

calculations of the bending moments and deformation (Bares, 1981) Simply supported square slabs’

mid-span bending moment is calculated with Equation 5-27:

M�.,&'.��4( = 0.0423 ∗ p�. ∗ L� (Equation 5-27)

If a continuous square slab is considered, there is also a support bending moment. Both are

calculated with the following equations:

M�.,&'.��4( = 0.0202 ∗ p�. ∗ L� (Equation 5-28)

57

M�.,/��60� = −0.0515 ∗ p�. ∗ L� (Equation 5-29)

Equations 5-21 to 5-29 must be calculated for SLS as well. In this case, it is required to replace pEd by

pqp. The results for the bending moments in SLS, Mqp, need to be used in sections 5.4.2 and 5.4.3.

With the values of the cover, c, and the bar diameter, Ø, it is possible to calculate the effective height,

d, of the slab:

d = h − c − ∅ (Equation 5-30)

The simplification for the reinforcement area ratio, which is introduced in Chapter 2, given by Equation

2-8, is used to calculate the cross-section of the reinforcement in the slab:

A,89 = ω89 ∗ ��C∗,∗.DE�FC (Equation 5-31)

Where b is the width (equal to 1 m for slabs), fcd - the design value of the compressive strength of

concrete, fyd - the design value of the tensile strength of steel, and ωRC (reinforcement area ratio) - the

simplification in the formula. The results lead to the theoretical cross-section of reinforcement, which is

replaced by the practical one found in the tables. In practical design, the cross-section actually used is

normally higher than the theoretical one. Lower practical sections can be accepted, but the difference

between the theoretical and practical real cross-section must be restricted to 3%. Shear reinforcement

design is not considered because this was not analysed in the dissertation and the effect of the use of

RA on shear is still unclear in the literature. Furthermore, this limit state is never conditioning in well-

designed solid slabs.

Beams are designed in the same way as slabs, but the parameters used are slightly different. The live

loads, q, and the other permanent loads, ∆g, are already included in the slabs’ load. Only the dead

weight is introduced, which can be calculated by multiplying the cross-section of the beam by 25

kN/m³. The loads of the slabs are transferred to the beams and are multiplied by a factor according to

the boundary conditions of the slabs.

- One-way slab, simply supported: pÒ4,,�. = p�. ∗ Ï� ∗ 1 (Equation 5-32)

- One-way slab, continuous on both borders: pÒ4,,�. = p�. ∗ Ï� ∗ 2 (Equation 5-33)

- One-way slab, continuous on 1border: pÒ4,,�. = p�. ∗ [∗ÏÂ ∗ 2 (Equation 5-34)

- Two-way slab, simply supported: Equation 5-32;

- Two-way slab, continuous on all borders: Equation 5-33.

Where pslab,Ed is the transferred design load on the beam and pEd the total design loads of the slabs.

The total load of the beam can be obtained by adding the design load of the beam, pbeam,Ed, to the

transferred load.

The design bending moments of the beams depend on the loads and the geometry/boundary

conditions of the slabs. One-way slabs lead to rectangular transferred loads that can be summed to

the dead weight of the beam to calculate the bending moments. In this case, Equations 5-21, 5-24 and

5-25 apply to simply supported and continuous beams, respectively. Two-way slabs are more

complicated as the transferred loads are triangular. Resulting from this, it is not possible to take the

58

transferred loads and the dead weight together (they lead to other bending moments). Simply

supported beams’ mid-span design bending moment is calculated as follows:

M�.,&'.��4( = �W�U�,KC×ÏrÂ +��½UW�,KC×Ïr�� (Equation 5-35)

The bending moments of continuous beams are in that case obtained with Equations 5-36 and 5-37:

M�.,&'.��4( = ÍW�U�,KC∗Ïr�X + Í�½UW,KC∗ÏrS� (Equation 5-36)

M�.,/��60� = − ÍW�U�,KC∗Ïr�� − Í�½UW,KC∗Ïr�Ð.� (Equation 5-37)

The moments need to be calculated in SLS as well (for the deformation and cracking SLS), which

means that the design loads need to be replaced by the quasi-permanent loads.

The effective height, d, is calculated in another way than for slabs because the bar diameter of shear

reinforcement, presented by Østirb, is included. Equation 5-38 demonstrates this:

d = h − c − ∅� − ∅�'0, (Equation 5-38)

Where h is the total height of the beam, Ø – the bar diameter and c – the cover of the beam. The

cross-section of reinforcement in beams is obtained as explained above (Equation 5-31).

5.4.2 Deformation SLS

The moments of inertia of the cracked and uncracked section need to be calculated to obtain the final

moment of inertia. To calculate both, the location of the neutral axis, y, is required. If the effective

reinforcement is included, the following equation is used:

y = W∗w²r �>�∗» K�K�,��¾∗.,∗��>�∗( K�K�,��)

(Equation 5-39)

Where b is the width of the structural element, h is the total height, As is the cross-section of

reinforcement, Es is the design value of modulus of elasticity of reinforcement steel, Ec,eff is the effective

modulus of elasticity of concrete and d is the effective height. The moments of inertia, assuming an

uncracked and cracked section, are calculated as follows (using the Huygens-Steiner theorem):

IÖ = ,∗�³�� + b ∗ h ∗ Iy − ��J� +A ∗ N ��,�� − 1O ∗ (d − y)² (Equation 5-40)

IÖÖ = ,∗�³�� + b ∗ x ∗ I��J� +A ∗ ��,�� ∗ (d − x)² (Equation 5-41)

Where x is the height of the compressive zone of the cracked section, which is equal to the location of

the neutral axis. EC2 states that it is possible to adequately predict the behaviour of a concrete

member that is expected to crack, (but may not be fully cracked) which will behave in an intermediate

manner between the uncracked and fully cracked conditions. Therefore, it is necessary to calculate the

equivalent moment of inertia with the following equation:

I = ξ ∗ IÖÖ + (1 − ξ) ∗ IÖ (Equation 5-42)

59

Where I is the moment of inertia, II and III - the values of the moment of inertia of the uncracked and

cracked section, respectively, and ξ - the distribution coefficient, given by the following equation:

ξ = 1 − β ∗ I��V� J� (Equation 5-43)

Where β is a coefficient taking into account the influence of the loading duration or repeated loading

on the average strain (equal to 0.5 for sustained loads), M is the bending moment in SLS and Mcr

represents the cracking moment calculated on the basis of a cracked section under the loading

conditions causing first cracking. If the final moment of inertia is calculated, it is possible to obtain the

value of the deformation of the slab:

- One-way slab, simply supported: δ = [SÂX ∗ �uv∗Ï©��,��∗Ö (Equation 5-44)

- One-way slab, continuous on one side: δ = �SÂX ∗ �uv∗Ï©��,��∗Ö (Equation 5-45)

- One-way slab, continuous on both sides: δ = �SÂX ∗ �uv∗Ï©��,��∗Ö (Equation 5-46)

- One-way slab, cantilevered: δ = �Â ∗ �uv∗Ï©��,��∗Ö (Equation 5-47)

- Two-way slab, simply supported on all sides: δ = 0.00397 ∗ �uv∗Ï©��,��∗Ö (Equation 5-48)

- Two-way slab, continuous on all sides: δ = 0.00124 ∗ �uv∗Ï©��,��∗Ö (Equation 5-49)

The methodology for the slabs and Equations 5-39 to 5-43 can also be used for the determination of

the deformation of beams. There are also several cases, depending on the consideration of a one- or

two-way slab and the boundary conditions of the beam:

- One-way slab combined with a simply supported beam: δ = [SÂX ∗ �uv∗Ï©��,��∗Ö (Equation 5-50)

- One-way slab combined with a continuous beam: δ = �SÂX ∗ �uv∗Ï©��,��∗Ö (Equation 5-51)

- Two-way slab combined with a simply supported beam:δ = [SÂX ∗ �W�U�,uv∗Ï©��,��∗Ö + �

��Â ∗ ��½UW,uv∗Ï©��,��∗Ö

(Equation 5-52)

- Two-way slab combined with a continuous beam:δ = �SÂX ∗ �W�U�,uv∗Ï©��,��∗Ö + �

[XÂ.[� ∗ ��½UW,uv∗Ï©��,��∗Ö

(Equation 5-53)

Where pslab,qp is the transferred quasi-permanent load on the beam, pbeam,qp is the quasi-permanent

load of the beam (dead weight) and pqp the total quasi-permanent loads of the beam. L is the span of

the beam, I is the final moment of inertia and Ec,eff is the effective modulus of elasticity.

The calculated deflections need to be compared with the limit (span/250), provided by EC2.

5.4.3 Cracking SLS

EC2 provides regulations and formulas to calculate the crack width, wk, in order to comply with the

cracking SLS:

60

w� = s0,&4� ∗ (ε& − ε�&) (Equation 5-54)

Where sr,max is the maximum crack spacing, εsm is the mean strain in the reinforcement and εcm is the

mean strain in the concrete between the cracks. The difference between εsm and εcm may be

calculated using the following equation:

ε& − ε�& = ��\∙��\,��¬v,��∙��q�∙«v,�� (Equation 5-55)

Where σs is the tension in the reinforcement assuming a cracked section, αe - the ratio between the

steel’s modulus of elasticity, Es, and that of concrete, Ecm. ρp,eff - the ratio between the area of steel, As,

and the effective area of concrete in tension surrounding the reinforcement, Ac,eff. fct,eff is the effective

tensile strength of concrete (at the time when cracking is first expected to occur). It is assumed to be

equal to the mean value of axial tensile strength of concrete, fctm. kt is a factor dependent on the

duration of loading (equal to 0.4 for long-term loading). The maximum crack spacing (sr,max) may be

calculated with the following equation:

s0,&4� = kS ∙ c + ��∙�r∙�©∙Ø«v,�� (Equation 5-56)

Where Ø is the bar diameter, c is the cover, k1 is a coefficient that takes into account the bond

properties of the bonded reinforcement (equal to 0.8 for high bond bars), k2 is a coefficient that takes

into account the strain distribution (equal to 0.5 for bending) and the values of k3 and k4 are equal to

3.4 and 0.425, respectively, according to clause 7.3.4(3) of EC2.

The methodology and equations are used for slabs and beams. If the characteristic crack width is

obtained, it needs to be smaller than 0.3 mm, as per table 7.1N in EC2.

5.5 Design results and discussion

Tables with calculations of the slabs and beams can be seen in Annex G and Annex O, respectively,

but abbreviated versions are used in this section. 41 examples are considered for the design of slabs

and the design of beams handles 74 cases.

Table 5-1 demonstrates that all slabs cases, in which the fundamental parameters are available,

comply with the various limit states. Three references with geometry/boundary conditions are

considered and each of them leads to feasible results. All examples in RAC have a smaller theoretical

cross-section of reinforcement, As, than that of the corresponding RC. Therefore, the same or even

lower practical cross-section of reinforcement is used for the examples in RAC. This table also

demonstrates that the deformation of the examples in RAC is not always smaller than that of the

corresponding example in RC. The main reason for this is that K is sometimes not much bigger than 1

and consequently, because of the rounding, the same thickness for the slab in RAC is used. The main

purpose, comply with the limit state, is still reached because the deformation is always smaller than

span/250. The crack width, wk, is always smaller than 0.3 mm.

The other examples, in which not all the fundamental parameters were available, are shown in Table 5-

61

2, which leads to other conceptions and conclusions. When not all fundamental parameters are

available, some examples have a higher theoretical cross-section of reinforcement, As, in RAC than the

corresponding example in RC. This is accepted if the theoretical As,RAC is smaller than the theoretical

As,RC, multiplied by 1.05 (see section 3.2.3). Furhermore, the same can be concluded as in Table 5-1.

Section 5.3 stated that it is also useful to assume that the fundamental parameters are not available in

order to check whether the formulas for the missing parameters lead to feasible results (the results do

not differ too much from those in which the values of the fundamental parameters are known). The

comparison, shown in Table 5-3, is done for the example provided by Bravo (Bravo et al., 2015b) and

this proves that the formulas used to obtain the missing parameters lead to complying results: the

cross-section of reinforcement, As, is in both examples smaller in RAC than in RC. The same can be

said about the deformation. Furthermore, cracking is always smaller than 0.3 mm and not conditioning.

Figure 5-1 provides the scatter of the K-values for slabs and Figure 5-2 does this for beams: the K-

value is not always the same for simply supported and continuous beams. This is the case when the

most conditioning K-value is the result of durability and if different heights for the two sorts of beams

are used. Nevertheless, the same amount of examples fall between the described limits of Figure 5-2

and consequently, only 29 cases are examined.

Figure 5-1: Scatter of the K-value for slabs

Figure 5-2: Scatter of the K-value for simply supported beams

Figure 5-1 above shows that the spread of the K-value for slabs falls between 1.0 and 1.37:

- 13 examples have a K-value below 1.1;

- 9 examples range between 1.1 and 1.2;


- 4 examples have a K-value above 1.3.

1.000

1.100

1.200

1.300

1.400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

K-value

RAC mixes

1.000

1.100

1.200

1.300

1.400

1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829

K-value

RAC mixes

62

The K-value for the corresponding beams is generally lower and falls between 1.0 and 1.30:

- 13 examples have a K-value below 1.1;


- 11 examples range between 1.2 and 1.3.

Table 5-1: Examples with all parameters available: compliance check (slabs)

Reference Boundary/geometry

conditions Mix Total height h (m)

As (cm²/m)

Real As Real As (cm²/m)

L/250 (mm)

δ (mm)

wk

(mm)

Bravo et al. (2015b)

One-way slab, Continuous on both borders, span L = 6

m

RC 0.140 9.876 Ø10//0,075 10.470 24.000 20.898 0.133

MRA 100% 0.190 9.016 Ø10//0,075 10.470 24.000 18.733 0.203

MRA 100% 0.180 8.598 Ø10//0,075 10.470 24.000 17.600 0.174

MRA 10% 0.150 9.592 Ø10//0,075 10.470 24.000 18.886 0.154

MRA 10% 0.150 9.421 Ø10//0,075 10.470 24.000 17.544 0.140

MRA 50% 0.160 9.342 Ø10//0,075 10.470 24.000 19.522 0.166

Pedro et al. (2014)

One-way slab, Simply supported on both borders, span L

= 5 m

RC 0.180 7.978 Ø12//0,125 9.050 20.000 18.340 0.161

RAC 100% 0.230 6.483 Ø12//0,150 7.540 20.000 11.714 0.173

RAC 100% 0.210 6.937 Ø12//0,150 7.540 20.000 16.359 0.192

Cartuxo (2013)

One-way slab, 1 border simply

supported and 1 border continuous,

span L = 5 m

RC 0.170 11.935 Ø12//0,075 15.080 24.000 20.353 0.113

RAC 100% 0.240 11.388 Ø12//0,100 11.310 24.000 19.104 0.288

RAC 100% 0.230 11.007 Ø12//0,100 11.310 24.000 17.531 0.263

RAC 100% 0.180 11.093 Ø12//0,100 11.310 24.000 20.575 0.166

Table 5-2: Examples with not all parameters available: compliance check (slabs)


conditions

Available fundamental parameters

Mix Total

height h (m)

As (cm²/m)


L/250 (mm)

δ (mm)

wk

(mm)

Evangelista (2014)

Two-way slab, continuous on all

borders, span L = 5 m

α1, α2, α3, α4, α5

RC 0.155 4.320 Ø10//0,150 5.240 24.000 18.272 0.158

RAC 10% 0.170 4.117 Ø10//0,150 5.240 24.000 17.295 0.163

RAC 30% 0.170 4.149 Ø10//0,150 5.240 24.000 16.312 0.163

RAC 50% 0.180 3.946 Ø10//0,200 3.930 24.000 14.039 0.235

RAC 100% 0.180 4.091 Ø10//0,150 5.240 24.000 17.679 0.185

González-Fonteboa et al. (2011)

Two-way slab, simply supported on all


α1, α2, α5

RC 0.120 5.427 Ø8//0,075 6.700 20.000 10.771 0.124

RAC 20% 0.120 5.459 Ø8//0,075 6.700 20.000 11.956 0.131

RAC 50% 0.120 5.365 Ø8//0,075 6.700 20.000 11.736 0.124

RAC 100% 0.130 5.126 Ø8//0,100 5.030 20.000 12.065 0.197

Cakir (2014)

Two-way slab, simply supported on all


α1

RC 0.120 2.665 Ø8//0,150 3.350 16.000 8.611 0.123

RAC 25% 0.140 2.417 Ø8//0,200 2.510 16.000 6.277 0.194

RAC 50% 0.160 2.414 Ø8//0,200 2.510 16.000 5.562 0.214

RAC 75% 0.170 2.350 Ø8//0,200 2.510 16.000 4.512 0.214

RAC 100% 0.180 2.429 Ø8//0,200 2.510 16.000 4.861 0.247

Amorim et al. (2012)

One-way slab, continuous on all


α1, α3, α4

RC 0.140 8.532 Ø10//0,075 10.470 24.000 19.693 0.151

RAC 20% 0.150 7.924 Ø10//0,100 7.850 24.000 19.422 0.213

RAC 50% 0.150 7.778 Ø10//0,100 7.850 24.000 19.293 0.202

RAC 100% 0.150 7.764 Ø10//0,100 7.850 24.000 19.729 0.201

Kou et al. (2007)

Cantilevered slab, span L =1.5 m

α1, α2, α3, α5

RC 0.150 4.037 Ø10//0,200 3.930 6.000 4.810 0.270

RAC 20% 0.160 4.031 Ø10//0,200 3.930 6.000 5.334 0.276

RAC 50% 0.170 3.522 Ø10//0,200 3.930 6.000 3.816 0.248

RAC 100% 0.200 3.103 Ø10//0,200 3.930 6.000 2.708 0.217

63

Table 5-3: Assumption that not all fundamental parameters are available: comparison and compliance check (slabs)


conditions


Mix Total

height h (m)

As (cm²/m)


L/250 (mm)

δ (mm)

wk

(mm)


One-way slab, Continuous on both borders, span L = 6

m

α1, α2, α3, α4, α5, α6

RC 0.140 9.876 Ø10//0,075 10.470 24.000 20.898 0.133

MRA 100% 0.190 9.016 Ø10//0,075 10.470 24.000 18.733 0.203

MRA 100% 0.180 8.598 Ø10//0,075 10.470 24.000 17.600 0.174

MRA 10% 0.150 9.592 Ø10//0,075 10.470 24.000 18.886 0.154

MRA 10% 0.150 9.421 Ø10//0,075 10.470 24.000 17.544 0.140

MRA 50% 0.160 9.342 Ø10//0,075 10.470 24.000 19.522 0.166


One-way slab, Continuous on both ends, span L = 6 m

α1

RC 0.140 9.876 Ø10//0,075 10.470 24.000 20.898 0.133

MRA 100% 0.180 8.942 Ø10//0,075 10.470 24.000 17.112 0.193

MRA 100% 0.170 9.203 Ø10//0,075 10.470 24.000 18.721 0.183

MRA 10% 0.150 9.182 Ø10//0,075 10.470 24.000 18.555 0.132

MRA 10% 0.150 9.103 Ø10//0,075 10.470 24.000 18.064 0.124

MRA 50% 0.160 9.249 Ø10//0,075 10.470 24.000 19.161 0.161

Tables 5-4 to 5-8 form the corresponding examples for beams to Tables 5-1 to 5-3.

Table 5-4: Examples with all parameters available: compliance check (corresponding simply supported beam)


conditions Mix

Total height h (m)

b (m) As

(cm²/m) Real As

Real As (cm²/m)

L/250 (mm)

δ (mm)

wk

(mm)


Simply supported beam, span L = 6 m

RC 0.510 0.260 23.138 8Ø20 25.130 24.000 22.617 0.136

MRA 100% 0.640 0.260 20.518 7Ø20 21.990 24.000 21.465 0.248

MRA 100% 0.660 0.260 19.231 7Ø20 21.990 24.000 19.865 0.214

MRA 10% 0.530 0.260 22.944 8Ø20 25.130 24.000 22.083 0.159

MRA 10% 0.530 0.260 22.763 8Ø20 25.130 24.000 21.048 0.151

MRA 50% 0.590 0.260 20.833 7Ø20 21.990 24.000 21.494 0.196

Pedro et al. (2014)


RC 0.380 0.200 9.060 5Ø16 10.050 20.000 19.501 0.182

RAC 100% 0.480 0.200 7.624 4Ø16 8.040 20.000 16.839 0.235

RAC 100% 0.460 0.200 7.698 4Ø16 8.040 20.000 17.003 0.221

Cartuxo (2013)


RC 0.570 0.280 25.788 8Ø20 25.130 24.000 20.691 0.161

RAC 100% 0.710 0.280 23.841 8Ø20 25.130 24.000 17.452 0.296

RAC 100% 0.700 0.280 23.596 8Ø20 25.130 24.000 18.342 0.283

RAC 100% 0.600 0.280 23.716 8Ø20 25.130 24.000 19.609 0.165

Table 5-5: Examples with all parameters available: compliance check (corresponding continuous beam)


conditions Mix

Total height h (m)

b (m) As

(cm²/m) Real As

Real As (cm²/m)

L/250 (mm)

δ (mm)

wk

(mm)


Continuous beam, span L = 6 m

RC 0.450 0.220 17.625 6Ø20 18.850 24.000 7.918 0.137

MRA 100% 0.570 0.220 15.440 5Ø20 15.710 24.000 7.533 0.263

MRA 100% 0.580 0.220 14.708 5Ø20 15.710 24.000 7.202 0.230

MRA 10% 0.470 0.220 17.372 6Ø20 18.850 24.000 7.647 0.160

MRA 10% 0.470 0.220 17.220 6Ø20 18.850 24.000 7.277 0.151

MRA 50% 0.520 0.220 15.888 5Ø20 15.710 24.000 7.740 0.209

Pedro et al. (2014)


RC 0.240 0.200 11.543 6Ø16 12.060 20.000 14.559 0.170

RAC 100% 0.310 0.200 8.878 5Ø16 10.050 20.000 11.172 0.193

RAC 100% 0.290 0.200 9.375 5Ø16 10.050 20.000 11.972 0.186

Cartuxo (2013)


RC 0.460 0.230 22.601 7Ø20 21.990 24.000 8.668 0.146

RAC 100% 0.580 0.230 20.665 7Ø20 21.990 24.000 7.265 0.275

RAC 100% 0.570 0.230 20.447 7Ø20 21.990 24.000 7.737 0.264

RAC 100% 0.480 0.230 20.765 7Ø20 21.990 24.000 8.455 0.151

64

Table 5-6: Examples with not all parameters available: compliance check (corresponding simply supported beam)

Reference Boundary/ geometry conditions


Mix Total

height h (m)

b (m) As

(cm²/m) Real As

Real As (cm²/m)

L/250 (mm)

δ (mm)

wk

(mm)

Evangelista (2014)

Simply supported beam, span L = 5

m

α1, α2, α3, α4, α5

RC 0.420 0.210 9.733 5Ø16 10.050 24.000 23.351 0.195 RAC 10% 0.450 0.210 9.187 5Ø16 10.050 24.000 21.000 0.199 RAC 30% 0.450 0.210 9.193 5Ø16 10.050 24.000 20.746 0.216 RAC 50% 0.460 0.210 9.177 5Ø16 10.050 24.000 20.921 0.211 RAC 100% 0.470 0.210 9.129 5Ø16 10.050 24.000 21.611 0.241

González-Fonteboa

et al. (2011)


m α1, α2, α5

RC 0.420 0.200 12.236 6Ø16 12.060 20.000 18.284 0.157 RAC 20% 0.430 0.200 11.833 6Ø16 12.060 20.000 17.731 0.158 RAC 50% 0.440 0.200 11.320 6Ø16 12.060 20.000 16.797 0.148 RAC 100% 0.470 0.200 10.863 6Ø16 12.060 20.000 16.264 0.164

Cakir (2014)


m α1

RC 0.300 0.200 4.128 2Ø16 4.020 16.000 13.824 0.223 RAC 25% 0.350 0.200 3.691 2Ø16 4.020 16.000 11.410 0.247 RAC 50% 0.360 0.200 3.802 2Ø16 4.020 16.000 12.261 0.295 RAC 75% 0.370 0.200 3.854 2Ø16 4.020 16.000 12.507 0.328 RAC 100% 0.380 0.200 3.783 2Ø16 4.020 16.000 12.041 0.340


Simply supported beam, span L = 5 m (different span)

α1, α3, α4

RC 0.450 0.220 14.499 5Ø20 15.710 20.000 18.450 0.192 RAC 20% 0.480 0.220 13.712 5Ø20 15.710 20.000 17.278 0.189 RAC 50% 0.490 0.220 13.262 5Ø20 15.710 20.000 16.363 0.177 RAC 100% 0.490 0.220 13.293 5Ø20 15.710 20.000 16.645 0.176

Table 5-7: Examples with not all parameters available: compliance check (corresponding continuous beam)

Reference Boundary/ geometry conditions


Mix Total

height h (m)

b (m) As

(cm²/m) Real As

Real As (cm²/m)

L/250 (mm)

δ (mm) wk

(mm)

Evangelista (2014)

Continuous beam, span L

= 5 m

α1, α2, α3, α4, α5

RC 0.320 0.200 8.487 5Ø16 10.050 24.000 11.790 0.156

RAC 10% 0.340 0.200 8.068 4Ø16 8.040 24.000 12.123 0.209

RAC 30% 0.340 0.200 8.082 4Ø16 8.040 24.000 12.003 0.214

RAC 50% 0.350 0.200 7.979 4Ø16 8.040 24.000 11.890 0.221

RAC 100% 0.360 0.200 7.889 4Ø16 8.040 24.000 12.135 0.238

González-Fonteboa et al. (2011)


= 5 m α1, α2, α5

RC 0.340 0.200 9.646 5Ø16 10.050 20.000 8.329 0.148

RAC 20% 0.350 0.200 9.239 5Ø16 10.050 20.000 7.945 0.147

RAC 50% 0.350 0.200 9.141 5Ø16 10.050 20.000 7.995 0.141

RAC 100% 0.380 0.200 8.562 5Ø16 10.050 20.000 7.418 0.154

Cakir (2014) Continuous

beam, span L = 4 m

α1

RC 0.210 0.200 4.099 2Ø16 4.020 16.000 8.990 0.190

RAC 25% 0.240 0.200 3.762 2Ø16 4.020 16.000 8.103 0.219

RAC 50% 0.260 0.200 3.643 2Ø16 4.020 16.000 7.608 0.253

RAC 75% 0.260 0.200 3.916 2Ø16 4.020 16.000 8.727 0.298

RAC 100% 0.270 0.200 3.784 2Ø16 4.020 16.000 8.158 0.306



= 5 m (different

span)

α1, α3, α4

RC 0.400 0.200 10.812 6Ø16 12.060 20.000 6.044 0.179

RAC 20% 0.430 0.200 10.107 5Ø16 10.050 20.000 6.076 0.220

RAC 50% 0.430 0.200 10.061 5Ø16 10.050 20.000 6.085 0.210

RAC 100% 0.440 0.200 9.752 5Ø16 10.050 20.000 5.789 0.203

Tables 5-4 and 5-5 correspond to Table 5-1: for each slab, a simply supported beam and a continuous

beam are analysed. All the examples in RAC have a theoretical cross-section of reinforcement that is

smaller than that in RC. Resulting from this, the practical cross-section of the examples in RAC is

always equal to or smaller than that in RC. Furthermore, the deformation of the examples in RAC is

always smaller than that in RC and the crack width never poses problems.

The same conclusions can be drawn for Tables 5-6 and 5-7 that correspond to Table 5-2, but it must

be noted that there are two simply supported and one continuous example (Cakir, 2014) for which the

method is not valid. The reason for this is the non-compliance with the cracking SLS. However, the

examples with the non-compliance are still accepted, because they have exaggerated covers that

should not be used in most practical instances. Nevertheless, if the heights of the beams are

respectively increased by 2, 3 and 1 cm, the compliance with the cracking SLS is reached.

65

Table 5-8: Assumption that not all fundamental parameters are available: comparison and compliance check (beams)

Reference Boundary/geo

metry conditions


Mix Total

height h (m)

b (m) As

(cm²/m) Real As

Real As (cm²/m)

L/250 (mm)

δ (mm)

wk

(mm)


Simply supported

beam, span L = 6 m

α1, α2, α3, α4, α5, α6

RC 0.510 0.260 23.138 8Ø20 25.130 24.000 22.617 0.136 MRA 100% 0.640 0.260 20.518 7Ø20 21.990 24.000 21.465 0.248 MRA 100% 0.660 0.260 19.231 7Ø20 21.990 24.000 19.865 0.214 MRA 10% 0.530 0.260 22.944 8Ø20 25.130 24.000 22.083 0.159 MRA 10% 0.530 0.260 22.763 8Ø20 25.130 24.000 21.048 0.151 MRA 50% 0.590 0.260 20.833 7Ø20 21.990 24.000 21.494 0.196


Continuous beam, span L =

6 m

α1, α2, α3, α4, α5, α6



Simply supported

beam, span L = 6 m

α1



Continuous beam, span L =

6 m α1


Table 5-8 checks whether the formulas for the missing data make sense if beams are considered. The

results of Table 5-8 lead to the same conclusions as for the corresponding Table 5-3.

The feasibility of all the cases needs to be considered as well. This is done by the K-values: concrete mixes

with RA that have a higher K-value will normally not be used if common sense in design situations is

considered (i.e. the advantages of using RA are in those cases offset by the disadvantage of having to use

much more material and increasing the dead weight of the structure). The replacement ratio and the quality

of the aggregates of the 32 slab and 58 beam examples are checked whether they cause the higher values

of the equivalent functional unit. Table 5-9 demonstrates that the higher K-values almost always

correspond to a replacement ratio of 100%. In the examples provided by Cakir (2014), even a replacement

ratio of 75% leads to a higher K-value (because of very poor quality of the RA).

Table 5-9: Highest K-values

K-value slabs K-value simple

beams K-value

continuous beams Mix Reference

1.268 1.261 1.261 MRA (100%) Bravo et al., 2015b

1.278 1.298 1.298 MRA (100%)

1.255 1.272 1.272 RAC (100% ) Pedro et al., 2014

1.315 1.254 1.264 RAC (100% ) Cartuxo, 2013

1.285 1.230 1.239 RAC (100% )

1.322 1.249 1.249 RAC (75%) Cakir, 2014

1.373 1.267 1.267 RAC (100% )

1.256 1.240 1.240 MRA (100%) Bravo et al., 2015b

1.358 / / RAC (100%) Kou et al., 2007

66

5.6 Over-conservatism

As seen in the results, some of the cases in RAC lead to results that are considerably smaller than in

RC. It is necessary to define criteria to show which cases are too conservative in this study: over-

conservatism occurs when a lower cross-section in RAC (lower thickness of the slab or beam) still

complies with all the limit states.

32 RAC mixes for slabs, which correspond to 9 RC examples, are examined and this leads to the

following results:

- 17 RAC mixes comply in the case of a decrease in total height, hRAC, of 1.0 cm;

- 1 RAC mix complies in the case of a decrease in total height, hRAC, of 2.0 cm;

- 1 RAC mix complies in the case of a decrease in total height, hRAC, of 3.0 cm.

The results are examined in order to understand why the outliers with the highest decreases do not

make sense; overly high covers in some of the examples are the reason for this. The calculations are

done for all the examples, but some of them are not feasible in practice. This is the case if the total

cover in RAC, cRAC, exceeds 4.0 cm. One mix in RAC of Kou et. al (2007), two mixes of Bravo (2014),

one of Evangelista (2014) and one of Cakir (2014) are examples of this. It can be concluded that the

method yields exaggerated heights usually only if poor-quality recycled aggregates, high replacement

ratios and harsh environmental conditions are combined.

The accuracy of the method needs to be checked for the other 14 cases. This is possible with

Equation 5-57, which calculates the difference (in percentage) between the lowest possible value of

the height, hmin, and the rounded height used, hrounded:

Errorinheight = �VÚÛ�C�C��VÚÛ�C�C ∗ 100 (Equation 5-57)

The smaller the error in height, the more accurate the method; the analysis showed that the 14

examples have an error in height of less than 10%. The lower the hRC, the higher the relative error in

height becomes when a decrease in total height is possible. In short, it can be stated that the method

proposed is accurate for these concrete mixes.

As done in section 5.5 for all the RAC cases, it is possible to express these 14 cases’ practical

feasibility in terms of the K-values. It is expected that the highest K-values will occur in these cases.

Table 5-10 demonstrates that this aspect does not lead to higher K-values, which means that some of

the cases show that even feasible examples in practical design can have slightly lower cross-sections

and still comply with the various limit states.

The same control is executed for the 58 RAC beam examples. As the height of a beam is considerably

higher than that of a slab, it is expected that it will be possible to decrease the height of the beams

more than that of the slabs. The following results are obtained:



- 4 RAC mixes comply in the case of a decrease in total height, hRAC, of 4.0 cm.

67

Table 5-10: K-values of the 14 examples analysed

Mix K-value Error in height (%)

MRA (100%) 1.278 5.556

MRA (10%) 1.069 6.667

MRA (10%) 1.048 6.667

RAC (100%) 1.194 4.762

RAC (10%) 1.085 5.882

RAC (30%) 1.086 5.882

RAC (20%) 1.023 8.333

RAC (50%) 1.038 8.333

MRA (100%) 1.199 5.882

MRA (10%) 1.069 6.667

MRA (10%) 1.059 6.667

MRA (50%) 1.149 6.250

RAC (50%) 1.116 5.882

RAC (100%) 1.255 8.696

The same reason as for slabs can be the cause of the compliance with high decreases in height in some

cases: the high total cover in RAC (≥ 4.5 cm). This is the case for 1 mix that complies with all the limit

states if a decrease of 4.0 cm is taken into account, for 8 mixes if a decrease of 2.0 cm is taken into

account and for 6 mixes if a decrease of 1.0 cm is taken into account. These examples are mostly not

economical for practical purposes. The unfeasible combination of poor-quality RA, high replacement

ratios and harsh environmental conditions are the cause of that phenomenon.

If the unfeasible cases are excluded, still 28 cases reach a compliance with lower cross-sections. The

accuracy of the method for those beams is checked with Equation 5-57 and it is expected that the

errors in height will be smaller than those of slabs because higher structural elements are considered.

The analysis of the 28 examples gave the following results:

- 21 examples have an error in height of less than 5%;

- 6 examples have an error in height between 5% and 10%;

- 1 example has an error in height higher than 10%.

The results show that the method proposed is accurate for the concrete mixes as the error in heights

are mostly smaller than 5%.

5.7 Limitations of the method

Although the method proposed is validated, it must be noted that there are some limitations.

It must be stressed again that the method is only developed for slabs and beams, not for other

structural concrete elements, e.g. columns and footings. More specifically, the method is only valid for

solid slabs and beams in standard framed buildings. Only slabs with thicknesses ranging from 12 cm

to 18 cm and beams with thicknesses ranging from 20 cm to 55 cm have been tested. Also waffle

plates are not considered in the method; those structural elements become more efficient when spans

of more than 6-10 m are used. The method is solely tested for slab and beams spans up to 6 m.

Moreover, the spans of the beams considered is never too dissimilar to that of the slabs (one-way

68

slabs) Furthermore, only one-way and square two-way slabs are used to validate the method, in this

case for simplification purposes only.

Another limitation of the method is that no extremely harsh environmental conditions were used in its

validation. Freeze-thaw and chemical attacks were not considered. The validation of the method is

done for environmental classes up to XC2/XC3 (carbonation) and XD1/XS1 (chlorides penetration).

The structural class used for the calculations throughout the dissertation is S4. It is expected that

lower structural classes will not pose any problems, but higher structural classes possibly will.

Examples provided by other research must have fundamental parameters that approximately fall between

specific limits. Otherwise, the method will for most cases not be valid. In some of them, the method was

validated but the examples are slightly uneconomical/unfeasible for practical purposes.

Finally, the validation was made for concrete strength classes ranging between C20/25 and C50/60,

which means that the method cannot be used for special structures that require higher concrete

strength classes without validation for that purpose.

5.8 Other structural elements

The dissertation proves that it is possible to define an equivalent functional unit in RAC and consequently

that the method proposed works well for slabs and beams. Other structural elements, i.e. columns and

footings, are also affected by the elements they support, e.g. if RAC is used in slabs, it will also have an

influence on the structural elements that support the slabs.

Three small standard framed structures are considered to check the relative importance of the slabs,

beams, columns and footings in terms of concrete volume. Moreover, a fictional simple structure is

designed to control this as well. The outcomes of Tables 5-11 and 5-12 show that the percentage of slabs

is approximately 60% in a normal building. Beams represent approximately 11% of the volume of all

structural elements while more or less 23% of the concrete is used for footings. Columns (5%) represent

the smallest part of the various structural elements. As the method is valid for slabs and beams, it is

satisfied for the biggest part (71%) of the various structural parts in a standard framed building.

Table 5-11 Relative volume of structural elements in a standard framed building

Structure 1 (m³)

Structure 1 (%)

Structure 2 (m³)

Structure 2 (%)

Structure 3 (m³)

Structure 3 (%)

Average (%)

Standard deviation

(%)

Slabs 196 55.21 176 56.77 119 60.10 57.36 2.039

Beams 32 9.01 30 9.68 23 11.62 10.10 1.104

Columns 17 4.79 15 4.84 14 7.07 5.57 1.064

Footings (and foundation beams)

110 30.99 89 28.71 42 21.21 26.97 4.176

Total 355 100.00 310 100.00 198 100.00 100.00 0.000

69

Table 5-12 Relative volume of structural elements in a standard framed building - own calculations

Part infinite structure (5*5*2.5

m³), own calculations (m³) Part infinite structure (5*5 *2.5

m³), own calculations (%)

Slab (h = 0.16 m) 4 60.61

Beams (0.25 * 0.50 m²) 0.85 12.88

Columns (0.40 * 0.40 m²) 0.4 6.06

Footing (1.50 * 1.50 * 0.60 m³) 1.35 20.45

Total 6.6 100.00

Footings represent also a relatively big part of the concrete used for structural elements. It must be

noted that it is impossible to validate the method for footings if RA are used in the other structural

elements, i.e. slabs, beams and columns. This is because extra loads need to be taken into account in

the calculations for footings. The soil under the footings has a maximum admissible stress. The

concrete cross-section (and resulting the cross-section of reinforcement) depends on that stress,

which is demonstrated by Equation 5-58.

A = Ü�UC� (Equation 5-58)

Where A is the concrete cross-section, N - the vertical force and σadm - the maximum admissible

stress. If RA are used in the slabs, beams or columns, N increases because of the extra loads.

Resulting from this, the concrete area in plan and the cross-section of reinforcement increase as well.

This goes against the assumption that steel remains unchanged.

On the other hand, it is interesting to analyse and examine footings in RAC when there is no use of RA

in the other structural elements. The method is valid when RA are only implemented in footings because

a loss in compressive strength hardly affects the design of the footings, since concrete’s compressive

strength is not conditioning. The method only depends on durability and consequently, the K-value is the

most conditioning of the ones obtained in function of α3 and α4. The effective height, d, remains

unchanged (because the compressive strength is not conditioning); only the total height increases in

RAC because of the higher cover. Resulting from this, the ratio hRAC/hRC is obtained as follows:

�DHE�DE = .DE�Ø�qx∗��,DE�£.££[&.DE�Ø��,DE�£.££[& (Equation 5-59)

Where dRC is the effective height, Ø - the bar diameter (according to dRC), cmin,RC - the minimum RC

cover and 0.005 m - the deviation corresponding to high-quality control casting conditions. Figures 5-3

and 5-4 represent the K-values obtained for carbonation and chlorides penetration, respectively.

Figure 5-3: hRAC/hRC in function of α3 for S4 (footings)

0.980

1.000

1.020

1.040

1.060

1.080

1.100

0.8

00

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=40cm

XC1 (hRC=40cm)XC2/XC3 (hRC=40cm)XC4 (hRC=40cm)X0 (hRC=60cm)

XC1 (hRC=60cm)XC2/XC3 (hRC=60cm)XC4 (hRC=60cm)

70

Figure 5-4: hRAC/hRC in function of α4 for S4 (footings)

As can be seen, the K-value is always almost equal to 1, which means that the use of RA does not

have a big influence on the design of footings. In short, footings regardless of slabs, beams and

columns in a building form the least demanding elements to validate the method.

The method proposed in this dissertation is not validated for columns due to the complexity of the

calculations. The ULS considered for slabs and beams is the bending ULS without axial force. This is

impossible for columns because they support the vertical loads of the other structural elements.

Consequently, it is required to implement the bending ULS with axial force when columns are

analysed. On the other hand, the method is validated for slabs (60%) and beams (11%), which can

imply that the method proposed will also work for columns, representing a much smaller percentage

(5%) than the other elements. As sometimes happens, comprehensiveness does not always go

together with efficiency; therefore, it is not worth adapting the method to columns because this would

make the calculations too extensive and complicated.

5.9 Conclusions of Chapter 5

This chapter demonstrated that the methodology proposed works for real mixes and the validation of

the method is thus accomplished for slabs and beams. All RAC examples of the slabs comply with the

various limit states like a given example in RC. Three RAC examples of the beams did not reach a

compliance with the various limit states like its corresponding example in RC does. This is because a

non-compliance with the cracking SLS. As the examples represent only 5% of all cases of the beams,

this is negligible. Moreover, the examples have exaggerated covers, which are normally not used for

most practical purposes.

Another conclusion of Chapter 5 is that if there is missing data, it is also possible to express the

properties in RAC in function of those in RC and the resulting element complies with all the limit states.

Furthermore, it can be concluded that some of the examples lead to results too conservative for the

purpose of the dissertation. Those cases are shown in section 5.6: the over-conservatism is

acceptable since the most conservative examples are not feasible or economical for practical

purposes. Also a lot of examples of beams are not feasible in practice, but more cases are over-

conservative. Nevertheless, the accuracy of the method is higher than for slabs because smaller

errors in height are obtained.

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.0

00

1.0

50

1.1

00

1.1

50

1.2

00

1.2

50

1.3

00

1.3

50

hRAC/hRC

α4

XD1/XS1 (hRC=40cm)

XD2/XS2 (hRC=40cm)XD3/XS3 (hRC=40cm)

XD1/XS1 (hRC=60cm)

XD2/XS2 (hRC=60cm)XD3/XS3 (hRC=60cm)

XD1/XS1 (hRC=80cm)

XD2/XS2 (hRC=80cm)

71

Chapter 6

Conclusions and developments

Chapter 6 describes the most pertinent results of the previous chapters. Furthermore, conclusions are

drawn and recommendations for future research and developments are made.

6.1 Conclusions

The main goal of the master’s dissertation is to search and define an equivalent functional unit in recycled

aggregates concrete to conventional structural concrete with regards to its environmental impact. A specific

volume of RAC, K m³, has to comply with the various limit states as 1 m3 of conventional structural concrete

does and this can be used for LCA purposes. It can be concluded that the method proposed for slabs and

beams in this dissertation leads to the accomplishment of the main purpose. It must be noted that the

method is not developed for structural design purposes, but for LCA purposes. It demonstrates that a

functional equivalent unit in RAC can be defined, regarding the compliance with the various limit

states. Average values and relationships between the properties of RAC and RC, based on an

extensive literature review, are used throughout the dissertation and not values with 95 % probability

(the latter ones would/could be used for structural design purposes).

It was required to make a couple of assumptions and simplifications to make the calculations simpler.

Thus, the method becomes more straightforward to understand. The most interesting simplifications

for slabs, given by Equations 6-1 and 6-2, consider the total height of the slab in RAC, hRAC, and the

corresponding effective height, dRAC.

h8>9 = h89 + 2 ∗ (c8>9 −c89) = h89 + 2 ∗ Δc (Equation 6-1)

d8>9 = d89 + (c8>9 −c89) = d89 + Δc (Equation 6-2)

Where cRAC and cRC are the nominal RAC and RC covers, respectively. As a first iteration, it is expected

that the total height hRC needs to increase by only ∆c as for dRC. If that were the case, the loads would

be higher but the effective height dRC would remain unchanged. Consequently, a bigger cross-section

of reinforcement would be required to take into account the higher bending moment. Furthermore, the

compressive strength of RAC is expected to decrease, which means that the cross-section of

reinforcement would need to increase even more. This goes against the assumption that the cross-

section of reinforcement is the same in RAC and RC. Increasing hRC by 2*∆c solves the previous

problem. A parametric study showed that the simplifications lead to good results and consequently

that they can be used for the purpose of the dissertation.

Equations 6-1 and 6-2 do not apply to beams. As a result, other simplifications are developed.

Equations 6-3 and 6-4 represent the main simplifications for beams and handle the total and effective

height in RAC, described by hRAC and dRAC.

72

h8>9 = d89 ∗ γ�.� + h89 −d89 + ∆c (Equation 6-3)

d8>9 = d89 ∗ γ�.� (Equation 6-4)

Where γ is the ratio between pEd,RAC and pEd,RC, the total design loads of the corresponding slab in RAC

and RC, respectively. ∆c is the difference in cover between RAC and RC. The loads of the slabs are

used in the calculations of the beams because the use of RA in the concrete of the slabs leads to an

increase of the vertical loads and bending moments. As beams are always designed after slabs, the

extra vertical loads of the slabs need to be taken into account in the beams in order to obtain a higher

effective height that takes into account the increase of the bending moment. Equation 6-3 without the

power 1.2 results from the assumption that the cross-section of reinforcement must be the same in

RAC and RC. This is accepted because the concept of comparing the environmental impacts of RAC

and RC (motivation of the dissertation) demands that steel remains unchanged. If the power 1.2 would

not be used, the cross-section of reinforcement would always be the same in RAC and RC in the

parametric study. On the other hand, the compressive strength of RAC is generally expected to

decrease. This means that the cross-section needs to increase, which goes against the previously

stated assumption. The power is empirically determined and is included to obtain a margin for losses

in compressive strength. The total height is obtained by adding the effective height by the cover (hRC -

dRC) and the difference in cover. A parametric study showed that also these simplifications can be

used for the purpose of the dissertation.

The most important parameters of the dissertation are the K-value and the fundamental parameters, α1 to

α6. The latter ones are used to express the equivalent properties of RAC (α1 - compressive strength, α2 -

modulus of elasticity, α3 - carbonation, α4 - chloride penetration, α5 - axial tensile strength and α6 -

creep) in function of the same property in RC. They are primordial to execute the study and their

ranges are provided by previous research. Chapter 3 consists of parametric studies that result in

calculated ranges of the fundamental parameters when the compliance with the various limit states are

met. Several conditions (e.g. load combinations, differences in cover, covers, etc.) need to be taken

into account for the calculations. It can be concluded that the calculated fundamental parameters vary

between the limits, provided by earlier research, if all the respective conditions and parameters are

considered. Even more demanding conditions lead to reasonable results in some of the limit states.

The compliance checks with durability and the limit states (i.e. bending ULS, deformation SLS and cracking

SLS) lead to results for hRAC/hRC if the respective conditions and calculated ranges of the fundamental

parameters are taken into account. This is demonstrated by Equations 6-5 to 6-8 for slabs and 6-9 to 6-

12 for beams. The final K-value, as referred above, defines the functional equivalent unit in RAC,

considering the various limit states; it is the most conditioning of the various values, obtained in the limit

states (Equation 6-13):

�DHE�DE ∝� =∝��£.S[] (Equation 6-5)

�DHE�DE ∝x =�DE��∗(qx��)∗��,DE�DE (Equation 6-6)

73

�DHE�DE ∝© =�DE��∗(q©��)∗��,DE�DE (Equation 6-7)

�DHE�DE ∝s ∝rÀ = 0.9983 ∗ I∝s∝rJ£.X�[X

(Equation 6-8)

�DHE�DE ∝� =∝��£.[Â (Equation 6-9)

�DHE�DE ∝x =.DE∗»vuv,DE�µ�∗(�x¼�)∗��,DE,�½UW�vuv,DE ¾�.r��DE�.DE�(qx��)∗��,DE,W�U��


�DHE�DE ∝© =.DE∗»vuv,DE�µ�∗(�©¼�)∗��,DE,�½UW�vuv,DE ¾�.r��DE�.DE�(q©��)∗��,DE,W�U��


�DHE�DE ∝s ∝rÀ = 0.9956 ∗ I∝s∝rJ£.XX]

(Equation 6-12)

K = �DHE�DE = max ¿�DHE�DEq� ; �DHE�DE

qx ; �DHE�DEq© ; �DHE�DE

qs qrÀ Á (Equation 6-13)

As can be seen, there is no equation concerning the cracking SLS for slabs and beams (α5 - axial tensile

strength). Examining the results, it can be concluded that the cracking SLS is almost never conditioning.

Therefore, it is not necessary to include this limit state in the determination of the final result of the

equivalent unit in RAC.

The final step is the validation of the method proposed when real mixes are produced. 9 slab cases in

RC and 32 in RAC with various geometry/boundary conditions are analysed, using K and the

fundamental parameters α. The design of the corresponding beams handles 8 cases in RC and 58 in

RAC (with simple or continuous supports). Research concerning the subject of this dissertation does

often not provide all the required fundamental parameters. Cases with missing data are also

considered and therefore, it is required to develop relationships between the fundamental parameters.

It is essential that α1 is always available to perform the design. The equations below provide the

relationships between the missing fundamental parameter and α1, also based on an extensive

literature review. (Silva et al., 2014a), (Silva et al., 2014b), (Silva et al., 2014c), (Silva et al., 2014e),

(Silva et al., 2014f), (Silva et al., 2015)

∝�= 0.871 ∗∝�£.S (Equation 6-14)

∝S= T�UVW,àbT�UVW,Db = N ÄÅ¸,DEÄÅ¸,àbO�,� = I �

∝�J�,�

(Equation 6-15)

∝X= YÆàbÆDb = ¥Ç�£,£�S(ÄÅ¸,àb�ÄÅ¸,Db) = ¥Ç�£.£�S∗ÄÅ¸,`b∗(∝��) (Equation 6-16)

∝[=∝�� SÀ (Equation 6-17)

∝]= 2.55 − 1.55 ∗ ∝� (Equation 6-18)

The results, when data are not available, are approximately the same as those for the cases with the

fundamental parameters available, which means that the expressions are generally correct and,

74

furthermore, they can be used in the calculations of the validation of the method.

Moreover, the results showed that 100% of the slab cases in RAC and 95% of the beam cases comply

with the various limit states just like the corresponding case in RC. This leads to the conclusion that the

assumptions, simplifications and method proposed in the dissertation work well for slabs and beams.

19 of the 32 RAC slab cases examined are slightly too conservative. 5 cases can be excluded

because the method proposed led to exaggerated values of some parameters, i.e. they are not

feasible in practice. Although the remaining 14 cases are slightly uneconomical for practical purposes,

they are still acceptable in the validation of the method. The calculations of the 58 beams are also

checked to determine whether they lead to over-conservative results. This led to the conclusion that

eventually 28 of the 58 cases are slightly too conservative. Since the method is not to be used for

structural design purposes but only as a tool in comparative Life Cycle Analysis studies, this is not

relevant for practical purposes.

6.2 Recommendations

The statements above showed that the method proposed in the dissertation is validated and works

well for the described conditions and parameters. It must be noted that the method also has some

limitations, which means that it is necessary to conduct further research on this subject.

A first remark is that the fundamental parameters provided by earlier research are only valid for mixes

produced with CEM I (Portland cement with less than 5 % of other substances). This means that

further investigations could include other types of cement (CEM II to CEM V).

The equations used when some of the data are missing (Equations 6-14 to 6-18), showed that the

results approximately correspond to those when the fundamental parameters are available. It has to

be remarked that some of the expressions are not comprehensive enough. Namely the expression

concerning α6 and creep should be improved in future developments. It is now little comprehensive and the

simplest equation between α6 and α1.

The scope in the beginning of the project showed that only solid slabs in standard framed building

(concerning thickness, type of element, span, environmental class, etc.) are considered throughout the

dissertation. The method is valid for this type of elements but it can be useful to extend the method also for

other slabs. Another limitation of the method is that no extremely harsh environmental conditions are

implemented to execute the validation. Harsher environmental classes than XC2/XC3 and XD1/XS1

also need a validation within the method. Finally, the validation was made for concrete strength

classes ranging between C20/25 and C50/60. All the previous limitations can be developed in further

research in order to include other buildings and structures as well.

The target life time in the dissertation was set as 50 years (structural class S4). It can be useful to validate

the method for other structural classes as well.

75

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Waleed N.; Canisius T. D. G. (2007) - Engineering properties of concrete containing recycled

aggregates. Banbury, Oxon, UK, Waste & Resources Action Program, 104 p.

Yang K.; Chung H.; Ashour A. (2008) - Influence of type and replacement level of recycled aggregates

on concrete properties. ACI Materials Journal 105(3), pp. 289-296.

Websites

Wetenschappelijk en Technisch Centrum voor het Bouwbedrijf. (2007). - Wapeningscorossie door

carbonatatie van beton voorkomen. Obtained at 12/03/2015 on

<<http://www.wtcb.be/homepage/download.cfm?dtype=publ&doc=wtcb_artonline_2007_3_nr2.pdf&lang=n

l>>

A.1

Annexes

Annex A: Parametric study for the verification of

the simplifications (slabs)

A.1 Validation

The parametric study needs to prove that the given simplifications lead to elements that are on the

safe side but not too conservative. The height, described by the simplification is used in the various

other limit states. If the simplification is accepted, it is possible to make the height independent of α3

and α4, which makes the compliance checks easier to perform. The simplifications, Equations 2-5 and

2-6, need to be verified by comparing the sections of reinforcement steel in RC and RAC. The cross-

section of reinforcement will be smaller in RAC, due to the bigger effective height, dRAC, of the slab.

The criteria for the parametric study are described as follows: the decrease in the design cross-section

of the reinforcement needs to be smaller than 15% or the simplifications are too conservative. The

design cross-section of the reinforcement in RAC can also be bigger than that of the RC, which can be

the case if the compressive strength of RAC decreases. An increase of the section of reinforcement in

RAC is acceptable if it stays below 5%.

A.2 Data

Most of the data (loads, load combinations, concrete strength classes, steel strength class, cover, etc.)

used throughout the parametric study are provided in sections 2.2, 2.3 and 2.4. The other parameters,

necessary to execute the study, are described in this section: it is important to define a range for the

difference in cover, ∆c, because this parameter affects the total and the effective height of the slab.

The parametric study was made with ∆c = 0.015 m and ∆c = 0.025 m. A bigger difference in cover is

not used because slabs whose thickness increases by more than 5 cm relative to that of RAC will

generally not be used in practice. A smaller difference is possible and ∆c = 0.000 m is included, which

leads to the same cross-section of reinforcement in RAC as in RC. There are also different values for

the dimensionless value of the bending moment, µRC, used in order to check if all cases comply.

A.3 Methodology

This part contains the explanation of how the various parameters are obtained. To calculate the cross-

section of reinforcement in RAC, the study is made for an ULS. This means that the loads (defined in

A.2

section 2.2.2) must be multiplied by partial safety factors, γg (= 1.35) and γq (= 1.5), according to EC2 to

obtain the total load combination in RC, pEd,RC:

p�.,89 = 1.35 ∗ (g + Δg) + 1.5 ∗ q (Equation A-1)Where g is the permanent load, ∆g - the other permanent loads and q - the live loads.

The total height, hRC, the bar diameter, Ø, and the nominal cover, cRC, defined in section 2.2 and 2.3,

lead to a value of the effective height in RC, dRC, as follows:

d89 = h89–c89– Ø (Equation A-2)With the effective height and the dimensionless value of the bending moment in RC, µRC, determined

in section 2.4.1, it is possible to calculate the ultimate bending moment strength MEd,RC:

M�.,89 = μ89 ∗ d89² ∗ f�. (Equation A-3)

Where fcd is the design value of the compressive strength of the concrete. The next step is the

calculation of the cross-section of the reinforcement steel. This is done with the simplification,

introduced in section 2.4.2 by Equation 2-7. Once the cross-section is obtained, all parameters of RC

are calculated in order to proceed with RAC. In this part, ∆c and Equations 2-5 and 2-6 are used to

obtain the total and effective height in RAC. The equivalent total load, pEd,RC, rises too and pEd,RAC can

be obtained as follows:

p�.,8>9 = p�.,89 + 2 ∗ Δc ∗ 25 (Equation A-4)

Where 25 represents the weight of concrete per m³. If the total design load is known, the ultimate

bending moment strength of the RAC slab, MEd,RAC, can be calculated, which depends on the loads

and the loading conditions. The loading conditions are assumed to be the same for RAC and RC,

which results in the fact that MEd,RAC solely depends on MEd,RC, pEd,RC and pEd,RAC:

M�.,8>9 = �KC,DHE�KC,DE ∗ M�.,89 (Equation A-5)

With MEd,RAC, it is possible to determine the dimensionless value of the bending moment in RAC, µRAC,

and the corresponding ωRAC:

μ8>9 = �KC,DHE.²DHE∗��C (Equation A-6)

ω8>9 =μ8>9 ∗ (1 + μ8>9) (Equation A-7)

In the final step, the cross-section of reinforcement in RAC, As,RAC, is calculated:

A,8>9 = ω8>9 ∗ ��C∗,∗.DHE�FC (Equation A-8)

Where fyd is the design yield strength of reinforcement. b is equal to 1 m for slabs. The loss in section

is calculated with Equation A-9. As explained in the validation, acceptable differences in design cross-

section can range from -5% to 15%. If most of the cases comply with the criteria, the simplifications

can be used in Chapters 3 and 4.

differenceindesigncross − section = >�,DE�>�,DHE>�,DE ∗ 100 (Equation A-9)

A.3

A.4 Results

The results can be seen in the tables below. There are several cases examined, depending on the

possible values of the parameters. The study contains two main cases: one with ∆c = 0.015 m and one

with ∆c = 0.025 m. The spread sheets are divided into three groups.

A.4.1 Part I

In the first four sections, concrete strength class C20/25 is considered. µRC is kept constant (= 0.18) in the

first two blocks, but there is a different ∆c. In each section, a low, high and medium load combination is

considered. Every combination is divided in three cases, in which the cover of RC changes. The next two

sections of part I show a difference in the value of µRC.

A.4.2 Part II

Concrete strength class is changed to C30/37 in order to show that the same results are obtained for

various strength classes. Only µRC = 0.18 is considered.

A.4.3 Part III

Concrete strength class C30/37 is used, but a loss in the compressive strength of RAC is introduced

as this can be expected in practice. Only µRC = 0.18 is considered and the cases in which ∆c = 0.000

m are omitted, due to the fact that the differences in design cross-section will always be 0 in those

cases. The differences have to be taken into account in the equations for µRAC and As,RAC.

A.5 Discussion

This section discusses the results obtained in the parametric study. The discussion is focused on the

cases where ∆c = 0.015 m, but the same conclusions can be reached for the cases where ∆c = 0.025 m.

A.5.1 Part I

Table A-7 shows a small part of the first section. The case with the biggest difference is, for each load

combination, the one with the highest cover, cRC. This makes sense because, if a higher cover is used,

∆c is in relative terms a bigger addition to dRC and the ratio between dRAC and dRC increases. That is

why µRAC and ωRAC decrease and the difference between the cross-sections increases.

A.4

Table A-1: Part I (∆c = 0.015 m)

Islabs Load

combination hRC (m)

Dead weight

g (kN/m²)

Other permanent loads ∆g (kN/m²)

Life loads q (kN/m²)

pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC

(m) MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

Difference As,RC -

As,RAC (%)

C20/25, µ is

0.18, ∆c =0.015

1

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.015 0.150 8.663 0.117 28.274 0.155 0.179 6.416 3.381

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.015 0.150 8.663 0.107 23.002 0.151 0.173 5.687 5.058

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.015 0.150 8.663 0.097 18.273 0.146 0.167 4.962 7.061

2

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.015 0.210 17.813 0.173 63.524 0.159 0.185 9.785 4.874

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.015 0.210 17.813 0.163 55.738 0.157 0.182 9.098 5.579

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.015 0.210 17.813 0.153 48.460 0.155 0.179 8.412 6.373

3

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.015 0.180 13.613 0.145 43.819 0.156 0.181 8.033 5.085

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.015 0.180 13.613 0.135 37.337 0.154 0.177 7.335 6.113

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.015 0.180 13.613 0.125 31.374 0.151 0.173 6.639 7.298

C20/25, µ is

0.18, ∆c =0.000

4

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.000 0.120 7.650 0.102 24.970 0.180 0.212 6.641 0.000

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.000 0.120 7.650 0.092 20.314 0.180 0.212 5.990 0.000

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.000 0.120 7.650 0.082 16.138 0.180 0.212 5.338 0.000

5

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.000 0.180 16.800 0.158 59.914 0.180 0.212 10.286 0.000

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.000 0.180 16.800 0.148 52.570 0.180 0.212 9.635 0.000

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.000 0.180 16.800 0.138 45.706 0.180 0.212 8.984 0.000

6

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.000 0.150 12.600 0.130 40.560 0.180 0.212 8.463 0.000

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.000 0.150 12.600 0.120 34.560 0.180 0.212 7.812 0.000

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.000 0.150 12.600 0.110 29.040 0.180 0.212 7.161 0.000

C20/25, µ is

0.12, ∆c =0.015

7

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.120 16.646 0.134 4.202 0.015 0.150 8.663 0.117 18.850 0.103 0.114 4.086 2.756

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.120 13.542 0.134 3.790 0.015 0.150 8.663 0.107 15.335 0.100 0.111 3.626 4.338

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.120 10.758 0.134 3.378 0.015 0.150 8.663 0.097 12.182 0.097 0.107 3.168 6.232

8

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.120 39.942 0.134 6.509 0.015 0.210 17.813 0.173 42.350 0.106 0.117 6.225 4.366

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.120 35.046 0.134 6.097 0.015 0.210 17.813 0.163 37.159 0.105 0.116 5.790 5.029

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.120 30.470 0.134 5.685 0.015 0.210 17.813 0.153 32.307 0.104 0.114 5.357 5.776

9

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.120 27.040 0.134 5.355 0.015 0.180 13.613 0.145 29.213 0.104 0.115 5.114 4.506

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.120 23.040 0.134 4.943 0.015 0.180 13.613 0.135 24.891 0.102 0.113 4.673 5.474

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.120 19.360 0.134 4.531 0.015 0.180 13.613 0.125 20.916 0.100 0.110 4.233 6.593

C20/25, µ is

0.24, ∆c =0.015

10

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.240 33.293 0.298 9.304 0.015 0.150 8.663 0.117 37.699 0.207 0.249 8.937 3.945

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.240 27.085 0.298 8.392 0.015 0.150 8.663 0.107 30.670 0.201 0.241 7.913 5.708

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.240 21.517 0.298 7.480 0.015 0.150 8.663 0.097 24.365 0.194 0.232 6.896 7.810

11

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.240 79.885 0.298 14.413 0.015 0.210 17.813 0.173 84.699 0.212 0.257 13.644 5.333

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.240 70.093 0.298 13.500 0.015 0.210 17.813 0.163 74.317 0.210 0.254 12.680 6.076

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.240 60.941 0.298 12.588 0.015 0.210 17.813 0.153 64.614 0.207 0.250 11.718 6.912

12

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.240 54.080 0.298 11.858 0.015 0.180 13.613 0.145 58.426 0.208 0.252 11.193 5.608

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.240 46.080 0.298 10.946 0.015 0.180 13.613 0.135 49.783 0.205 0.247 10.214 6.689

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.240 38.720 0.298 10.034 0.015 0.180 13.613 0.125 41.831 0.201 0.241 9.238 7.935

A.5

Table A-2: Part II (∆c = 0.015 m)

IIslabs Load

combination hRC (m)

Dead weight

g (kN/m²)



pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC

(m) MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

Difference As,RC -

As,RAC (%)

C30/37, µ is

0.18, ∆c

=0.015

13

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.015 0.150 8.663 0.117 42.412 0.155 0.179 9.624 3.381

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.015 0.150 8.663 0.107 34.503 0.151 0.173 8.530 5.058

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.015 0.150 8.663 0.097 27.410 0.146 0.167 7.442 7.061

14

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.015 0.210 17.813 0.173 95.287 0.159 0.185 14.677 4.874

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.015 0.210 17.813 0.163 83.607 0.157 0.182 13.647 5.579

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.015 0.210 17.813 0.153 72.690 0.155 0.179 12.618 6.373

15

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.015 0.180 13.613 0.145 65.729 0.156 0.181 12.050 5.085

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.015 0.180 13.613 0.135 56.006 0.154 0.177 11.002 6.113

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.015 0.180 13.613 0.125 47.060 0.151 0.173 9.958 7.298

C30/37, µ is

0.18, ∆c

=0.000

16

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.000 0.120 7.650 0.102 37.454 0.180 0.212 9.961 0.000

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.000 0.120 7.650 0.092 30.470 0.180 0.212 8.984 0.000

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.000 0.120 7.650 0.082 24.206 0.180 0.212 8.008 0.000

17

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.000 0.180 16.800 0.158 89.870 0.180 0.212 15.430 0.000

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.000 0.180 16.800 0.148 78.854 0.180 0.212 14.453 0.000

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.000 0.180 16.800 0.138 68.558 0.180 0.212 13.476 0.000

18

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.000 0.150 12.600 0.130 60.840 0.180 0.212 12.695 0.000

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.000 0.150 12.600 0.120 51.840 0.180 0.212 11.719 0.000

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.000 0.150 12.600 0.110 43.560 0.180 0.212 10.742 0.000

A.6

Table A-3: Part III (∆c = 0.015 m)

IIIslabs Load

combination hRC (m)

Dead weight

g (kN/m²)



pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC

(m) MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

Difference As,RC -

As,RAC (%)

C30/37, µ is 0.18, ∆c =0.015, fcd changes

19

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.015 0.150 8.663 0.117 42.412 0.172 0.202 9.767 1.941

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.015 0.150 8.663 0.107 34.503 0.167 0.195 8.654 3.676

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.015 0.150 8.663 0.097 27.410 0.162 0.188 7.547 5.748

20

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.015 0.210 17.813 0.173 95.287 0.177 0.208 14.901 3.423

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.015 0.210 17.813 0.163 83.607 0.175 0.205 13.853 4.153

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.015 0.210 17.813 0.153 72.690 0.173 0.202 12.806 4.975

21

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.015 0.180 13.613 0.145 65.729 0.174 0.204 12.231 3.659

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.015 0.180 13.613 0.135 56.006 0.171 0.200 11.165 4.723

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.015 0.180 13.613 0.125 47.060 0.167 0.195 10.103 5.950


22

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.015 0.150 8.663 0.117 42.412 0.194 0.231 9.947 0.141

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.015 0.150 8.663 0.107 34.503 0.188 0.224 8.809 1.950

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.015 0.150 8.663 0.097 27.410 0.182 0.215 7.679 4.107

23

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.015 0.210 17.813 0.173 95.287 0.199 0.239 15.181 1.608

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.015 0.210 17.813 0.163 83.607 0.197 0.235 14.110 2.370

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.015 0.210 17.813 0.153 72.690 0.194 0.232 13.042 3.227

24

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.015 0.180 13.613 0.145 65.729 0.195 0.234 12.457 1.877

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.015 0.180 13.613 0.135 56.006 0.192 0.229 11.369 2.986

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.015 0.180 13.613 0.125 47.060 0.188 0.224 10.284 4.265


25

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.015 0.150 8.663 0.117 42.412 0.221 0.270 10.177 -2.173

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.015 0.150 8.663 0.107 34.503 0.215 0.262 9.009 -0.271

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.015 0.150 8.663 0.097 27.410 0.208 0.251 7.848 1.997

26

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.015 0.210 17.813 0.173 95.287 0.227 0.279 15.541 -0.724

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.015 0.210 17.813 0.163 83.607 0.225 0.275 14.442 0.078

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.015 0.210 17.813 0.153 72.690 0.222 0.271 13.344 0.980

27

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.015 0.180 13.613 0.145 65.729 0.223 0.273 12.748 -0.414

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.015 0.180 13.613 0.135 56.006 0.220 0.268 11.630 0.754

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.015 0.180 13.613 0.125 47.060 0.215 0.261 10.517 2.098


28

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.015 0.150 8.663 0.117 42.412 0.258 0.325 10.485 -5.259

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.015 0.150 8.663 0.107 34.503 0.251 0.314 9.275 -3.231

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.015 0.150 8.663 0.097 27.410 0.243 0.302 8.073 -0.816

29

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.015 0.210 17.813 0.173 95.287 0.265 0.336 16.021 -3.835

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.015 0.210 17.813 0.163 83.607 0.262 0.331 14.883 -2.979

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.015 0.210 17.813 0.153 72.690 0.259 0.326 13.748 -2.016

30

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.015 0.180 13.613 0.145 65.729 0.261 0.328 13.136 -3.469

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.015 0.180 13.613 0.135 56.006 0.256 0.322 11.979 -2.224

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.015 0.180 13.613 0.125 47.060 0.251 0.314 10.827 -0.791

A.7

Table A-4: Part I (∆c = 0.025 m)

Islabs Load

combination hRC (m)

Dead weight

g (kN/m²)



pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC

(m) MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

Difference As,RC -

As,RAC (%)

C20/25, µ is 0.18, ∆c

=0.025

1

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.025 0.170 9.338 0.127 30.478 0.142 0.162 6.299 5.149

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.025 0.170 9.338 0.117 24.795 0.136 0.154 5.534 7.614

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.025 0.170 9.338 0.107 19.697 0.129 0.146 4.778 10.500

2

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.025 0.230 18.488 0.183 65.932 0.148 0.169 9.505 7.593

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.025 0.230 18.488 0.173 57.850 0.145 0.166 8.802 8.653

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.025 0.230 18.488 0.163 50.297 0.142 0.162 8.101 9.835

3

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.025 0.200 14.288 0.155 45.992 0.144 0.164 7.801 7.832

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.025 0.200 14.288 0.145 39.189 0.140 0.159 7.082 9.355

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.025 0.200 14.288 0.135 32.929 0.136 0.154 6.367 11.089

C20/25, µ is 0.18, ∆c

=0.000

4

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.000 0.120 7.650 0.102 24.970 0.180 0.212 6.641 0.000

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.000 0.120 7.650 0.092 20.314 0.180 0.212 5.990 0.000

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.000 0.120 7.650 0.082 16.138 0.180 0.212 5.338 0.000

5

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.000 0.180 16.800 0.158 59.914 0.180 0.212 10.286 0.000

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.000 0.180 16.800 0.148 52.570 0.180 0.212 9.635 0.000

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.000 0.180 16.800 0.138 45.706 0.180 0.212 8.984 0.000

6

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.000 0.150 12.600 0.130 40.560 0.180 0.212 8.463 0.000

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.000 0.150 12.600 0.120 34.560 0.180 0.212 7.812 0.000

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.000 0.150 12.600 0.110 29.040 0.180 0.212 7.161 0.000

C20/25, µ is 0.12, ∆c

=0.025

7

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.120 16.646 0.134 4.202 0.025 0.170 9.338 0.127 20.318 0.094 0.103 4.025 4.202

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.120 13.542 0.134 3.790 0.025 0.170 9.338 0.117 16.530 0.091 0.099 3.542 6.545

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.120 10.758 0.134 3.378 0.025 0.170 9.338 0.107 13.132 0.086 0.093 3.064 9.297

8

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.120 39.942 0.134 6.509 0.025 0.230 18.488 0.183 43.954 0.098 0.108 6.065 6.818

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.120 35.046 0.134 6.097 0.025 0.230 18.488 0.173 38.567 0.097 0.106 5.620 7.821

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.120 30.470 0.134 5.685 0.025 0.230 18.488 0.163 33.531 0.095 0.104 5.177 8.942

9

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.120 27.040 0.134 5.355 0.025 0.200 14.288 0.155 30.661 0.096 0.105 4.983 6.958

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.120 23.040 0.134 4.943 0.025 0.200 14.288 0.145 26.126 0.093 0.102 4.528 8.404

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.120 19.360 0.134 4.531 0.025 0.200 14.288 0.135 21.953 0.090 0.099 4.076 10.053

C20/25, µ is 0.24, ∆c

=0.025

10

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.240 33.293 0.298 9.304 0.025 0.170 9.338 0.127 40.637 0.189 0.225 8.746 6.003

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.240 27.085 0.298 8.392 0.025 0.170 9.338 0.117 33.059 0.181 0.214 7.672 8.579

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.240 21.517 0.298 7.480 0.025 0.170 9.338 0.107 26.263 0.172 0.202 6.613 11.586

11

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.240 79.885 0.298 14.413 0.025 0.230 18.488 0.183 87.909 0.197 0.236 13.217 8.293

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.240 70.093 0.298 13.500 0.025 0.230 18.488 0.173 77.133 0.193 0.231 12.231 9.404

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.240 60.941 0.298 12.588 0.025 0.230 18.488 0.163 67.062 0.189 0.225 11.248 10.642

12

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.240 54.080 0.298 11.858 0.025 0.200 14.288 0.155 61.323 0.191 0.228 10.836 8.621

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.240 46.080 0.298 10.946 0.025 0.200 14.288 0.145 52.251 0.186 0.221 9.828 10.215

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.240 38.720 0.298 10.034 0.025 0.200 14.288 0.135 43.906 0.181 0.213 8.827 12.026

A.8

Table A-5: Part II (∆c = 0.025 m)

IIslabs Load

combination hRC (m)

Dead weight

g (kN/m²)



pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC

(m) MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

Difference As,RC -

As,RAC (%)

C30/37, µ is 0.18, ∆c

=0.025

13

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.025 0.170 9.338 0.127 45.716 0.142 0.162 9.448 5.149

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.025 0.170 9.338 0.117 37.192 0.136 0.154 8.300 7.614

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.025 0.170 9.338 0.107 29.546 0.129 0.146 7.167 10.500

14

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.025 0.230 18.488 0.183 98.898 0.148 0.169 14.258 7.593

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.025 0.230 18.488 0.173 86.775 0.145 0.166 13.202 8.653

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.025 0.230 18.488 0.163 75.445 0.142 0.162 12.151 9.835

15

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.025 0.200 14.288 0.155 68.988 0.144 0.164 11.701 7.832

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.025 0.200 14.288 0.145 58.783 0.140 0.159 10.622 9.355

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.025 0.200 14.288 0.135 49.394 0.136 0.154 9.551 11.089

C30/37, µ is 0.18, ∆c

=0.000

16

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.000 0.120 7.650 0.102 37.454 0.180 0.212 9.961 0.000

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.000 0.120 7.650 0.092 30.470 0.180 0.212 8.984 0.000

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.000 0.120 7.650 0.082 24.206 0.180 0.212 8.008 0.000

17

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.000 0.180 16.800 0.158 89.870 0.180 0.212 15.430 0.000

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.000 0.180 16.800 0.148 78.854 0.180 0.212 14.453 0.000

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.000 0.180 16.800 0.138 68.558 0.180 0.212 13.476 0.000

18

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.000 0.150 12.600 0.130 60.840 0.180 0.212 12.695 0.000

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.000 0.150 12.600 0.120 51.840 0.180 0.212 11.719 0.000

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.000 0.150 12.600 0.110 43.560 0.180 0.212 10.742 0.000

A.9

Table A-6: Part III (∆c = 0.025 m)

IIIslabs Load

combination hRC (m)

Dead weight

g (kN/m²)



pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC

(m) MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

Difference As,RC -

As,RAC (%)


19

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.025 0.170 9.338 0.127 45.716 0.157 0.182 9.578 3.840

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.025 0.170 9.338 0.117 37.192 0.151 0.174 8.411 6.386

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.025 0.170 9.338 0.107 29.546 0.143 0.164 7.258 9.363

20

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.025 0.230 18.488 0.183 98.898 0.164 0.191 14.462 6.272

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.025 0.230 18.488 0.173 86.775 0.161 0.187 13.388 7.368

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.025 0.230 18.488 0.163 75.445 0.158 0.183 12.319 8.590

21

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.025 0.200 14.288 0.155 68.988 0.160 0.185 11.864 6.546

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.025 0.200 14.288 0.145 58.783 0.155 0.179 10.767 8.120

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.025 0.200 14.288 0.135 49.394 0.151 0.173 9.677 9.910


22

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.025 0.170 9.338 0.127 45.716 0.177 0.209 9.741 2.205

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.025 0.170 9.338 0.117 37.192 0.170 0.199 8.548 4.851

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.025 0.170 9.338 0.107 29.546 0.161 0.187 7.372 7.943

23

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.025 0.230 18.488 0.183 98.898 0.185 0.219 14.717 4.621

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.025 0.230 18.488 0.173 86.775 0.181 0.214 13.620 5.761

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.025 0.230 18.488 0.163 75.445 0.177 0.209 12.529 7.033

24

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.025 0.200 14.288 0.155 68.988 0.179 0.212 12.068 4.939

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.025 0.200 14.288 0.145 58.783 0.175 0.205 10.948 6.576

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.025 0.200 14.288 0.135 49.394 0.169 0.198 9.836 8.437


25

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.025 0.170 9.338 0.127 45.716 0.202 0.243 9.951 0.103

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.025 0.170 9.338 0.117 37.192 0.194 0.232 8.726 2.878

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.025 0.170 9.338 0.107 29.546 0.184 0.218 7.518 6.116

26

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.025 0.230 18.488 0.183 98.898 0.211 0.255 15.044 2.498

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.025 0.230 18.488 0.173 86.775 0.207 0.250 13.919 3.696

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.025 0.230 18.488 0.163 75.445 0.203 0.244 12.798 5.031

27

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.025 0.200 14.288 0.155 68.988 0.205 0.247 12.330 2.873

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.025 0.200 14.288 0.145 58.783 0.200 0.240 11.181 4.591

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.025 0.200 14.288 0.135 49.394 0.194 0.231 10.039 6.542


28

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 37.454 0.212 9.961 0.025 0.170 9.338 0.127 45.716 0.236 0.292 10.230 -2.701

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 30.470 0.212 8.984 0.025 0.170 9.338 0.117 37.192 0.226 0.278 8.962 0.247

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 24.206 0.212 8.008 0.025 0.170 9.338 0.107 29.546 0.215 0.261 7.713 3.681

29

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 89.870 0.212 15.430 0.025 0.230 18.488 0.183 98.898 0.246 0.307 15.481 -0.333

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 78.854 0.212 14.453 0.025 0.230 18.488 0.173 86.775 0.242 0.300 14.317 0.942

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 68.558 0.212 13.476 0.025 0.230 18.488 0.163 75.445 0.237 0.293 13.158 2.362

30

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 60.840 0.212 12.695 0.025 0.200 14.288 0.155 68.988 0.239 0.297 12.680 0.118

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 51.840 0.212 11.719 0.025 0.200 14.288 0.145 58.783 0.233 0.287 11.491 1.944

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 43.560 0.212 10.742 0.025 0.200 14.288 0.135 49.394 0.226 0.277 10.311 4.016

A.10

Table A-7: Influence of the cover in the first section (∆c = 0.015 m)

The comparison of the load combinations (Table A-8) in each block shows that the absolute values of

the differences in design cross-section of the reinforcement steel are in the same area, but it is not

possible to specifically compare them. This is because there are two opposite trends: on the one hand,

the influence of ∆c decreases in a thicker slab, but on the other hand MEd,RC increases because a

constant value of µRC is imposed. The only conclusion is that all the load combinations are safe and

not too conservative.

Table A-8: Comparison between load combinations (∆c = 0.015 m)

Load combination hRC (m) cRC (m) MEd,RC (kNm/m) Difference As,RC - As,RAC (%)

1 0.12 0.010 24.970 3.381 0.020 20.314 5.058 0.030 16.138 7.061

2 0.18 0.010 59.914 4.874 0.020 52.570 5.579 0.030 45.706 6.373

3 0.15 0.010 40.560 5.085 0.020 34.560 6.113 0.030 29.040 7.298

The second section uses the same load combinations but ∆c = 0.000 m. As a consequence, the loss

in section will be 0. The first two sections of the table considered an optimal value of µRC (0.18). In

practice, designers will try to pursue this value for most of the slabs in a building. If there are a lot of

different spans and loading conditions, it is logical to take a µRC that is slightly higher for the most

demanding slabs and the optimal value for most other slabs. Other possibilities are also used but this

statement basically shows that more acceptable values of µRC need to be considered in the parametric

study. Table A-9 shows that the differences in design cross-section remain between the target limits

for the lower (= 0.12) and higher (= 0.24) values of µRC.

Table A-9: Comparison between different values of µRC (∆c = 0.015 m)

Load combination

µRC Difference As,RC - As,RAC (%)

1

0.180 3.381

0.180 5.058

0.180 7.061

7

0.120 2.756

0.120 4.338

0.120 6.232

10

0.240 3.945

0.240 5.708

0.240 7.810

Load combination

cRC (m) Difference As,RC - As,RAC (%)

1

0.01 3.381

0.02 5.058

0.03 7.061

A.11

A.5.2 Part II

The second part describes the study for concrete strength class C30/37. Table A-10 demonstrates that

the same results are obtained if a different strength class is considered. This is because µRAC is

eventually the same for different strength classes and the compressive strength can be omitted in the

formula of the differences in design cross-section if two strength classes are compared.

Table A-10: Comparison between different concrete strength classes (∆c = 0.015 m)

Strength class

Load combination

MEd,RC

(kNm/m) ωRC

MEd,RAC

(kNm/m) ωRAC

Difference As,RC - As,RAC (%)

C20/25 1

24.970 0.212 28.274 0.179 3.381

20.314 0.212 23.002 0.173 5.058

16.138 0.212 18.273 0.167 7.061

C30/37 13

37.454 0.212 42.412 0.179 3.381

30.470 0.212 34.503 0.173 5.058

24.206 0.212 27.410 0.167 7.061

A.5.3 Part III

Table A-11 shows the maximum loss in compressive strength of RAC to stay on the safe side

(increase in cross-section of reinforcement in RAC smaller than 5%). Strength class C30/37 is used

and all the other parameters are kept the same as in part II. The low load combinations are the most

conditioning because they have a smaller margin for the differences in design cross-section of the

reinforcement steel.

The parameters µRAC and As,RAC increase, which leads to a smaller difference between the design

cross-sections because the design cross-section of reinforcement in RC remains unchanged. It is

possible to use a RAC that has a loss in compressive strength of almost 40%.

Table A-11: Loss in compressive strength (∆c = 0.015 m)

Load combination

fcd,RAC / fcd,RC µRAC As,RAC (cm²/m) Difference As,RC - As,RAC (%)

13 1 0.155 9.624 3.381 0.151 8.530 5.058 0.146 7.442 7.061

19 0.9 0.172 9.767 1.941 0.167 8.654 3.676 0.162 7.547 5.748

22 0.8 0.194 9.947 0.141 0.188 8.809 1.950 0.182 7.679 4.107

25 0.7 0.221 10.177 -2.173 0.215 9.009 -0.271 0.208 7.848 1.997

28 0.6 0.258 10.485 -5.259 0.251 9.275 -3.231 0.243 8.073 -0.816

A.12

A.5.4 Comparison with other cover increases (∆c = 0.025 m)

The results with a bigger ∆c are generally more conservative. This is because a higher ∆c has a

greater influence on the effective height, dRC, of the slab. A bigger dRAC leads to a bigger MEd,RAC and a

smaller As,RAC. As a result, the differences between the design cross-sections increase. Table A-12

demonstrates this statement. This also means that it is possible to have a higher loss in compressive

strength if ∆c = 0.025 m is used. The limit on the conservative side of the difference As,RC - As,RAC (%) is

assumed to be 15%. This is satisfied for ∆c = 0.025 m. The biggest difference is 12.026% in load

combination 12 (Table A-4).

Table A-12: Comparison between ∆c = 0.015 m and ∆c = 0.025 m: general

Load combination

hRC (m) ∆c (m) cRC (m) Difference As,RC - As,RAC (%)

1 0.120 0.015 0.010 3.381 0.020 5.058 0.030 7.061

1 0.120 0.025 0.010 5.149 0.020 7.614 0.030 10.500

3 0.150 0.015 0.010 5.085 0.020 6.113 0.030 7.298

3 0.150 0.025 0.010 7.832 0.020 9.355 0.030 11.089

A.6 Conclusion

The outcomes with both values of ∆c are on the safe side and not too conservative. The simplifications

can be used in the compliance checks of the limit states, as expected.

A.13

Annex B: Tables with results of the compliance of the bending ultimate limit

state (slabs)

The tables of the compliance of the bending ultimate limit state with differences in cover ranging from 0.000 m to 0.030 m are shown below.

Table B-1: Compliance of the bending ULS for slabs (∆c = 0.000 m and 0.010 m)

Load

combination hRC (m)

Dead weight

g (kN/m²)


Live loads q (kN/m²)

pEd,RC (kN/m²)

pqp,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) pqp,RAC

(kN/m²) dRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

As,RC - As,RAC

(%) α1

C25/30, µ is 0.18, ∆c

=0.000

1

0.120 3.000 1.000 1.500 7.650 4.450 0.010 0.102 0.180 31.212 0.212 8.301 0.000 0.120 7.650 4.450 0.102 31.212 0.180 0.212 8.301 0.000 1.000

0.120 3.000 1.000 1.500 7.650 4.450 0.020 0.092 0.180 25.392 0.212 7.487 0.000 0.120 7.650 4.450 0.092 25.392 0.180 0.212 7.487 0.000 1.000

0.120 3.000 1.000 1.500 7.650 4.450 0.030 0.082 0.180 20.172 0.212 6.673 0.000 0.120 7.650 4.450 0.082 20.172 0.180 0.212 6.673 0.000 1.000

2

0.180 4.500 3.500 4.000 16.800 10.400 0.010 0.158 0.180 74.892 0.212 12.858 0.000 0.180 16.800 10.400 0.158 74.892 0.180 0.212 12.858 0.000 1.000

0.180 4.500 3.500 4.000 16.800 10.400 0.020 0.148 0.180 65.712 0.212 12.044 0.000 0.180 16.800 10.400 0.148 65.712 0.180 0.212 12.044 0.000 1.000

0.180 4.500 3.500 4.000 16.800 10.400 0.030 0.138 0.180 57.132 0.212 11.230 0.000 0.180 16.800 10.400 0.138 57.132 0.180 0.212 11.230 0.000 1.000

3

0.150 3.750 2.250 3.000 12.600 6.900 0.010 0.130 0.180 50.700 0.212 10.579 0.000 0.150 12.600 6.900 0.130 50.700 0.180 0.212 10.579 0.000 1.000

0.150 3.750 2.250 3.000 12.600 6.900 0.020 0.120 0.180 43.200 0.212 9.766 0.000 0.150 12.600 6.900 0.120 43.200 0.180 0.212 9.766 0.000 1.000

0.150 3.750 2.250 3.000 12.600 6.900 0.030 0.110 0.180 36.300 0.212 8.952 0.000 0.150 12.600 6.900 0.110 36.300 0.180 0.212 8.952 0.000 1.000

C25/30, µ is 0.18, ∆c

=0.010

4

0.120 3.000 1.000 1.500 7.650 4.450 0.010 0.102 0.180 31.212 0.212 8.301 0.010 0.140 8.325 4.950 0.112 33.966 0.162 0.189 8.104 2.366 0.847

0.120 3.000 1.000 1.500 7.650 4.450 0.020 0.092 0.180 25.392 0.212 7.487 0.010 0.140 8.325 4.950 0.102 27.632 0.159 0.185 7.220 3.563 0.781

0.120 3.000 1.000 1.500 7.650 4.450 0.030 0.082 0.180 20.172 0.212 6.673 0.010 0.140 8.325 4.950 0.092 21.952 0.156 0.180 6.339 5.010 0.711

5

0.180 4.500 3.500 4.000 16.800 10.400 0.010 0.158 0.180 74.892 0.212 12.858 0.010 0.200 17.475 10.900 0.168 77.901 0.166 0.193 12.425 3.367 0.795

0.180 4.500 3.500 4.000 16.800 10.400 0.020 0.148 0.180 65.712 0.212 12.044 0.010 0.200 17.475 10.900 0.158 68.352 0.164 0.191 11.579 3.863 0.770

0.180 4.500 3.500 4.000 16.800 10.400 0.030 0.138 0.180 57.132 0.212 11.230 0.010 0.200 17.475 10.900 0.148 59.427 0.163 0.189 10.733 4.425 0.744

6

0.150 3.750 2.250 3.000 12.600 6.900 0.010 0.130 0.180 50.700 0.212 10.579 0.010 0.170 13.275 7.400 0.140 53.416 0.164 0.190 10.205 3.535 0.785

0.150 3.750 2.250 3.000 12.600 6.900 0.020 0.120 0.180 43.200 0.212 9.766 0.010 0.170 13.275 7.400 0.130 45.514 0.162 0.188 9.349 4.265 0.750

0.150 3.750 2.250 3.000 12.600 6.900 0.030 0.110 0.180 36.300 0.212 8.952 0.010 0.170 13.275 7.400 0.120 38.245 0.159 0.185 8.494 5.113 0.712

A.14

Table B-2: Compliance of the bending ULS for slabs (∆c = 0.015 m, 0.025 m and 0.030 m)

Load

combination hRC (m)

Dead weight

g (kN/m²)


Live loads q (kN/m²)

pEd,RC (kN/m²)

pqp,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) pqp,RAC

(kN/m²) dRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

As,RC - As,RAC

(%) α1

C25/30, µ is

0.18, ∆c

=0.015

7

0.120 3.000 1.000 1.500 7.650 4.450 0.010 0.102 0.180 31.212 0.212 8.301 0.015 0.150 8.663 5.200 0.117 35.343 0.155 0.179 8.020 3.381 0.787

0.120 3.000 1.000 1.500 7.650 4.450 0.020 0.092 0.180 25.392 0.212 7.487 0.015 0.150 8.663 5.200 0.107 28.753 0.151 0.173 7.108 5.058 0.704

0.120 3.000 1.000 1.500 7.650 4.450 0.030 0.082 0.180 20.172 0.212 6.673 0.015 0.150 8.663 5.200 0.097 22.842 0.146 0.167 6.202 7.061 0.622

8

0.180 4.500 3.500 4.000 16.800 10.400 0.010 0.158 0.180 74.892 0.212 12.858 0.015 0.210 17.813 11.150 0.173 79.406 0.159 0.185 12.231 4.874 0.721

0.180 4.500 3.500 4.000 16.800 10.400 0.020 0.148 0.180 65.712 0.212 12.044 0.015 0.210 17.813 11.150 0.163 69.672 0.157 0.182 11.372 5.579 0.691

0.180 4.500 3.500 4.000 16.800 10.400 0.030 0.138 0.180 57.132 0.212 11.230 0.015 0.210 17.813 11.150 0.153 60.575 0.155 0.179 10.515 6.373 0.660

9

0.150 3.750 2.250 3.000 12.600 6.900 0.010 0.130 0.180 50.700 0.212 10.579 0.015 0.180 13.613 7.650 0.145 54.774 0.156 0.181 10.041 5.085 0.709

0.150 3.750 2.250 3.000 12.600 6.900 0.020 0.120 0.180 43.200 0.212 9.766 0.015 0.180 13.613 7.650 0.135 46.671 0.154 0.177 9.169 6.113 0.666

0.150 3.750 2.250 3.000 12.600 6.900 0.030 0.110 0.180 36.300 0.212 8.952 0.015 0.180 13.613 7.650 0.125 39.217 0.151 0.173 8.298 7.298 0.622

C25/30, µ is

0.18, ∆c

=0.025

10

0.120 3.000 1.000 1.500 7.650 4.450 0.010 0.102 0.180 31.212 0.212 8.301 0.025 0.170 9.338 5.700 0.127 38.097 0.142 0.162 7.873 5.149 0.689

0.120 3.000 1.000 1.500 7.650 4.450 0.020 0.092 0.180 25.392 0.212 7.487 0.025 0.170 9.338 5.700 0.117 30.993 0.136 0.154 6.917 7.614 0.587

0.120 3.000 1.000 1.500 7.650 4.450 0.030 0.082 0.180 20.172 0.212 6.673 0.025 0.170 9.338 5.700 0.107 24.622 0.129 0.146 5.972 10.500 0.496

11

0.180 4.500 3.500 4.000 16.800 10.400 0.010 0.158 0.180 74.892 0.212 12.858 0.025 0.230 18.488 11.650 0.183 82.415 0.148 0.169 11.882 7.593 0.608

0.180 4.500 3.500 4.000 16.800 10.400 0.020 0.148 0.180 65.712 0.212 12.044 0.025 0.230 18.488 11.650 0.173 72.313 0.145 0.166 11.002 8.653 0.573

0.180 4.500 3.500 4.000 16.800 10.400 0.030 0.138 0.180 57.132 0.212 11.230 0.025 0.230 18.488 11.650 0.163 62.871 0.142 0.162 10.126 9.835 0.538

12

0.150 3.750 2.250 3.000 12.600 6.900 0.010 0.130 0.180 50.700 0.212 10.579 0.025 0.200 14.288 8.150 0.155 57.490 0.144 0.164 9.751 7.832 0.594

0.150 3.750 2.250 3.000 12.600 6.900 0.020 0.120 0.180 43.200 0.212 9.766 0.025 0.200 14.288 8.150 0.145 48.986 0.140 0.159 8.852 9.355 0.545

0.150 3.750 2.250 3.000 12.600 6.900 0.030 0.110 0.180 36.300 0.212 8.952 0.025 0.200 14.288 8.150 0.135 41.162 0.136 0.154 7.959 11.089 0.497

C25/30, µ is

0.18, ∆c

=0.030

13

0.120 3.000 1.000 1.500 7.650 4.450 0.010 0.102 0.180 31.212 0.212 8.301 0.030 0.180 9.675 5.950 0.132 39.474 0.136 0.154 7.809 5.923 0.649

0.120 3.000 1.000 1.500 7.650 4.450 0.020 0.092 0.180 25.392 0.212 7.487 0.030 0.180 9.675 5.950 0.122 32.113 0.129 0.146 6.834 8.714 0.543

0.120 3.000 1.000 1.500 7.650 4.450 0.030 0.082 0.180 20.172 0.212 6.673 0.030 0.180 9.675 5.950 0.112 25.512 0.122 0.137 5.875 11.955 0.451

14

0.180 4.500 3.500 4.000 16.800 10.400 0.010 0.158 0.180 74.892 0.212 12.858 0.030 0.240 18.825 11.900 0.188 83.919 0.142 0.163 11.723 8.823 0.564

0.180 4.500 3.500 4.000 16.800 10.400 0.020 0.148 0.180 65.712 0.212 12.044 0.030 0.240 18.825 11.900 0.178 73.633 0.139 0.159 10.836 10.034 0.528

0.180 4.500 3.500 4.000 16.800 10.400 0.030 0.138 0.180 57.132 0.212 11.230 0.030 0.240 18.825 11.900 0.168 64.018 0.136 0.155 9.952 11.381 0.492

15

0.150 3.750 2.250 3.000 12.600 6.900 0.010 0.130 0.180 50.700 0.212 10.579 0.030 0.210 14.625 8.400 0.160 58.848 0.138 0.157 9.621 9.055 0.549

0.150 3.750 2.250 3.000 12.600 6.900 0.020 0.120 0.180 43.200 0.212 9.766 0.030 0.210 14.625 8.400 0.150 50.143 0.134 0.152 8.712 10.785 0.500

0.150 3.750 2.250 3.000 12.600 6.900 0.030 0.110 0.180 36.300 0.212 8.952 0.030 0.210 14.625 8.400 0.140 42.134 0.129 0.146 7.811 12.744 0.451

A.15

Annex C: Tables with results of the compliance of

the cracking serviceability limit state (slabs)

The spreadsheet is divided in groups of columns:

- The first two groups of columns are the ones that are determined in the parametric study in

Annex A (Table C-1);

- The third group of columns uses the first two to calculate the height of the compression zone,

x, and the stress, σs (Table C-2);

- The fourth group of columns tests the different equations to become Ac,eff (Table C-2);

- In the four columns of group five, the different factors of the verification formula are calculated

to lead to a minimum value of α5 (Table C-3);

- The last columns form a control part (Table C-3).

The spreadsheet can also be divided in four sections of rows (A, B, C and D):

- Section A: the calculations are executed for ∆c = 0.000 m and ∆c = 0.015 m (or ∆c = 0.025

m). This block forms the basis for the other blocks and here are the ratios α2 and α6 equal to

respectively 0.96 and 1.05 (best case scenario);

- Section B: the six load combinations of section A are repeated for smaller values of α2: 0.9,

0.8, 0.7, 0.6, 0.5 and 0.44. α6 remains 1.05;

- Section C: this section is the same as section B but α6 = 1.20. This means that the ratio

between α2 and α6 gets smaller and this has an immediate influence on the value of xRAC;

- Section D: section B is repeated again but α6 = 1.40. The ratio between α2 and α6 is even

smaller and the case where α2 = 0.44 and α6 = 1.40 is the worst case scenario for the load

combinations.

Only Section A is provided in the dissertation by Tables C-1, C-2 and C-3. The other Sections have

the same sequence and consist of the same calculations. Only other parameters are used.

A.16

Table C-1: Compliance of the cracking SLS for slabs (first 2 groups of columns, section A)

Load combination

hRC (m)

g (kN/m²)

∆g (kN/m²)

q (kN/m²)

pEd,RC (kN/m²)

pqp,RC

(kN/m²) cRC (m)

dRC (m) µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) pqp,RAC

(kN/m²) dRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m)

As,RC

- As,RAC

(%)

1

0.120 3.000 1.000 1.500 7.650 4.450 0.010 0.102 0.180 31.212 0.212 8.301 0.000 0.120 7.650 4.450 0.102 31.212 0.180 0.212 8.301 0.000

0.120 3.000 1.000 1.500 7.650 4.450 0.020 0.092 0.180 25.392 0.212 7.487 0.000 0.120 7.650 4.450 0.092 25.392 0.180 0.212 7.487 0.000

0.120 3.000 1.000 1.500 7.650 4.450 0.030 0.082 0.180 20.172 0.212 6.673 0.000 0.120 7.650 4.450 0.082 20.172 0.180 0.212 6.673 0.000

2

0.180 4.500 3.500 4.000 16.800 10.400 0.010 0.158 0.180 74.892 0.212 12.858 0.000 0.180 16.800 10.400 0.158 74.892 0.180 0.212 12.858 0.000

0.180 4.500 3.500 4.000 16.800 10.400 0.020 0.148 0.180 65.712 0.212 12.044 0.000 0.180 16.800 10.400 0.148 65.712 0.180 0.212 12.044 0.000

0.180 4.500 3.500 4.000 16.800 10.400 0.030 0.138 0.180 57.132 0.212 11.230 0.000 0.180 16.800 10.400 0.138 57.132 0.180 0.212 11.230 0.000

3

0.150 3.750 2.250 3.000 12.600 6.900 0.010 0.130 0.180 50.700 0.212 10.579 0.000 0.150 12.600 6.900 0.130 50.700 0.180 0.212 10.579 0.000

0.150 3.750 2.250 3.000 12.600 6.900 0.020 0.120 0.180 43.200 0.212 9.766 0.000 0.150 12.600 6.900 0.120 43.200 0.180 0.212 9.766 0.000

0.150 3.750 2.250 3.000 12.600 6.900 0.030 0.110 0.180 36.300 0.212 8.952 0.000 0.150 12.600 6.900 0.110 36.300 0.180 0.212 8.952 0.000

4

0.120 3.000 1.000 1.500 7.650 4.450 0.010 0.102 0.180 31.212 0.212 8.301 0.015 0.150 8.663 5.200 0.117 35.343 0.155 0.179 8.020 3.381

0.120 3.000 1.000 1.500 7.650 4.450 0.020 0.092 0.180 25.392 0.212 7.487 0.015 0.150 8.663 5.200 0.107 28.753 0.151 0.173 7.108 5.058

0.120 3.000 1.000 1.500 7.650 4.450 0.030 0.082 0.180 20.172 0.212 6.673 0.015 0.150 8.663 5.200 0.097 22.842 0.146 0.167 6.202 7.061

5

0.180 4.500 3.500 4.000 16.800 10.400 0.010 0.158 0.180 74.892 0.212 12.858 0.015 0.210 17.813 11.150 0.173 79.406 0.159 0.185 12.231 4.874

0.180 4.500 3.500 4.000 16.800 10.400 0.020 0.148 0.180 65.712 0.212 12.044 0.015 0.210 17.813 11.150 0.163 69.672 0.157 0.182 11.372 5.579

0.180 4.500 3.500 4.000 16.800 10.400 0.030 0.138 0.180 57.132 0.212 11.230 0.015 0.210 17.813 11.150 0.153 60.575 0.155 0.179 10.515 6.373

6

0.150 3.750 2.250 3.000 12.600 6.900 0.010 0.130 0.180 50.700 0.212 10.579 0.015 0.180 13.613 7.650 0.145 54.774 0.156 0.181 10.041 5.085

0.150 3.750 2.250 3.000 12.600 6.900 0.020 0.120 0.180 43.200 0.212 9.766 0.015 0.180 13.613 7.650 0.135 46.671 0.154 0.177 9.169 6.113

0.150 3.750 2.250 3.000 12.600 6.900 0.030 0.110 0.180 36.300 0.212 8.952 0.015 0.180 13.613 7.650 0.125 39.217 0.151 0.173 8.298 7.298

A.17

Table C-2: Compliance of the cracking SLS for slabs (third and fourth group of columns, section A)

Mqp,RC

(kNm/m) Mqp,RAC

(kNm/m) α2/α6 α2 α6 ∆RC

x1RC

(m) x2RC (m)

∆RAC x1RAC (m)

x2RAC (m)

σs,RC (kN/m²)

σs,RAC

(kN/m²) Ø

(m)

2,5*(h-d)RC (m)

2,5*(h-d)RAC (m)

(h-x)/3

RC (m)

(h-x)/3

RAC (m)

h/2

RC (m)

h/2

RAC (m)

Minimum [2,5*(h-d) ; (h-x)/3 ; h/2] RC

(m)

Minimum [2,5*(h-d) ;

(h-x)/3 ; h/2] RAC

(m)

18.156 18.156 0.914 0.960 1.050 0.004 0.047 -0.086 0.005 0.048 -0.091 253030 254483 0.008 0.045 0.045 0.024 0.024 0.060 0.060 0.024 0.024

14.771 14.771 0.914 0.960 1.050 0.004 0.042 -0.078 0.004 0.043 -0.082 253030 254483 0.008 0.070 0.070 0.026 0.026 0.060 0.060 0.026 0.026

11.734 11.734 0.914 0.960 1.050 0.003 0.038 -0.069 0.003 0.039 -0.073 253030 254483 0.008 0.095 0.095 0.027 0.027 0.060 0.060 0.027 0.027

46.362 46.362 0.914 0.960 1.050 0.011 0.072 -0.133 0.012 0.075 -0.141 269276 270822 0.012 0.055 0.055 0.036 0.035 0.090 0.090 0.036 0.035

40.679 40.679 0.914 0.960 1.050 0.009 0.068 -0.125 0.010 0.070 -0.132 269276 270822 0.012 0.080 0.080 0.037 0.037 0.090 0.090 0.037 0.037

35.367 35.367 0.914 0.960 1.050 0.008 0.063 -0.116 0.009 0.065 -0.123 269276 270822 0.012 0.105 0.105 0.039 0.038 0.090 0.090 0.039 0.038

27.764 27.764 0.914 0.960 1.050 0.007 0.059 -0.110 0.008 0.061 -0.116 238205 239574 0.010 0.050 0.050 0.030 0.030 0.075 0.075 0.030 0.030

23.657 23.657 0.914 0.960 1.050 0.006 0.055 -0.101 0.007 0.057 -0.107 238205 239574 0.010 0.075 0.075 0.032 0.031 0.075 0.075 0.032 0.031

19.879 19.879 0.914 0.960 1.050 0.005 0.050 -0.093 0.006 0.052 -0.098 238205 239574 0.010 0.100 0.100 0.033 0.033 0.075 0.075 0.033 0.033

18.156 21.216 0.914 0.960 1.050 0.004 0.047 -0.086 0.006 0.053 -0.096 253030 256997 0.008 0.045 0.083 0.024 0.032 0.060 0.075 0.024 0.032

14.771 17.260 0.914 0.960 1.050 0.004 0.042 -0.078 0.005 0.048 -0.087 253030 253246 0.008 0.070 0.108 0.026 0.034 0.060 0.075 0.026 0.034

11.734 13.712 0.914 0.960 1.050 0.003 0.038 -0.069 0.004 0.043 -0.078 253030 248726 0.008 0.095 0.133 0.027 0.036 0.060 0.075 0.027 0.036

46.362 49.705 0.914 0.960 1.050 0.011 0.072 -0.133 0.013 0.079 -0.146 269276 263645 0.012 0.055 0.093 0.036 0.044 0.090 0.105 0.036 0.044

40.679 43.612 0.914 0.960 1.050 0.009 0.068 -0.125 0.011 0.074 -0.137 269276 262013 0.012 0.080 0.118 0.037 0.045 0.090 0.105 0.037 0.045

35.367 37.918 0.914 0.960 1.050 0.008 0.063 -0.116 0.010 0.070 -0.128 269276 260168 0.012 0.105 0.143 0.039 0.047 0.090 0.105 0.039 0.047

27.764 30.782 0.914 0.960 1.050 0.007 0.059 -0.110 0.009 0.066 -0.121 238205 236484 0.010 0.050 0.088 0.030 0.038 0.075 0.090 0.030 0.038

23.657 26.229 0.914 0.960 1.050 0.006 0.055 -0.101 0.007 0.061 -0.112 238205 234336 0.010 0.075 0.113 0.032 0.040 0.075 0.090 0.032 0.040

19.879 22.039 0.914 0.960 1.050 0.005 0.050 -0.093 0.006 0.056 -0.103 238205 231847 0.010 0.100 0.138 0.033 0.041 0.075 0.090 0.033 0.041

A.18

Table C-3: Compliance of the cracking SLS for slabs (last 2 groups of columns, section A)

srmax (m) Numerator Denominator α5 εcm-εsm

(respective α5)

0,6*σsRAC/Es control wk

(mm) εcm-εsm (α5=1)


0.073 -502.770 31.000 -16.218 0.004 0.001 0.300 0.001 0.076 0.114 -221.904 32.039 -6.926 0.003 0.001 0.300 0.001 0.115 0.157 -97.572 33.078 -2.950 0.002 0.001 0.300 0.001 0.153 0.090 -554.342 45.980 -12.056 0.003 0.001 0.300 0.001 0.100 0.130 -256.680 47.019 -5.459 0.002 0.001 0.300 0.001 0.144 0.172 -108.290 48.059 -2.253 0.002 0.001 0.300 0.001 0.186 0.081 -564.583 38.490 -14.668 0.004 0.001 0.300 0.001 0.079 0.122 -269.624 39.529 -6.821 0.002 0.001 0.300 0.001 0.116 0.164 -129.243 40.569 -3.186 0.002 0.001 0.300 0.001 0.152

0.138 -165.145 39.844 -4.145 0.002 0.001 0.300 0.001 0.138 0.181 -71.251 40.888 -1.743 0.002 0.001 0.300 0.001 0.171 0.226 -20.391 41.933 -0.486 0.001 0.001 0.300 0.001 0.200 0.154 -186.279 54.807 -3.399 0.002 0.001 0.300 0.001 0.162 0.196 -72.431 55.848 -1.297 0.002 0.001 0.300 0.001 0.201 0.238 -5.133 56.890 -0.090 0.001 0.001 0.300 0.001 0.237 0.146 -205.934 47.323 -4.352 0.002 0.001 0.300 0.001 0.133 0.188 -98.474 48.366 -2.036 0.002 0.001 0.300 0.001 0.165 0.231 -36.357 49.409 -0.736 0.001 0.001 0.300 0.001 0.195

A.19

Annex D: Results of the equivalent functional unit in RAC, concerning durability

(slabs)

The complete tables of the equivalent unit in RAC in function of the various conditions and α3 or α4 can be seen below.

Table D-1: Equivalent unit in RAC in function of S3, exposure class, α3 and the height in RC, hRC

S3 hRAC/hRC with hRC = 10cm hRAC/hRC with hRC = 12cm hRAC/hRC with hRC = 15cm hRAC/hRC with hRC = 18cm

α3 X0 XC1 XC2/XC3 XC4 X0 XC1 XC2/XC3 XC4 X0 XC1 XC2/XC3 XC4 X0 XC1 XC2/XC3 XC4

0.800 0.960 0.960 0.920 0.900 0.967 0.967 0.933 0.917 0.973 0.973 0.947 0.933 0.978 0.978 0.956 0.944

0.850 0.970 0.970 0.940 0.925 0.975 0.975 0.950 0.938 0.980 0.980 0.960 0.950 0.983 0.983 0.967 0.958

0.900 0.980 0.980 0.960 0.950 0.983 0.983 0.967 0.958 0.987 0.987 0.973 0.967 0.989 0.989 0.978 0.972

0.950 0.990 0.990 0.980 0.975 0.992 0.992 0.983 0.979 0.993 0.993 0.987 0.983 0.994 0.994 0.989 0.986

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.050 1.010 1.010 1.020 1.025 1.008 1.008 1.017 1.021 1.007 1.007 1.013 1.017 1.006 1.006 1.011 1.014

1.100 1.020 1.020 1.040 1.050 1.017 1.017 1.033 1.042 1.013 1.013 1.027 1.033 1.011 1.011 1.022 1.028

1.150 1.030 1.030 1.060 1.075 1.025 1.025 1.050 1.063 1.020 1.020 1.040 1.050 1.017 1.017 1.033 1.042

1.200 1.040 1.040 1.080 1.100 1.033 1.033 1.067 1.083 1.027 1.027 1.053 1.067 1.022 1.022 1.044 1.056

1.250 1.050 1.050 1.100 1.125 1.042 1.042 1.083 1.104 1.033 1.033 1.067 1.083 1.028 1.028 1.056 1.069

1.300 1.060 1.060 1.120 1.150 1.050 1.050 1.100 1.125 1.040 1.040 1.080 1.100 1.033 1.033 1.067 1.083

1.350 1.070 1.070 1.140 1.175 1.058 1.058 1.117 1.146 1.047 1.047 1.093 1.117 1.039 1.039 1.078 1.097

1.400 1.080 1.080 1.160 1.200 1.067 1.067 1.133 1.167 1.053 1.053 1.107 1.133 1.044 1.044 1.089 1.111

1.450 1.090 1.090 1.180 1.225 1.075 1.075 1.150 1.188 1.060 1.060 1.120 1.150 1.050 1.050 1.100 1.125

1.500 1.100 1.100 1.200 1.250 1.083 1.083 1.167 1.208 1.067 1.067 1.133 1.167 1.056 1.056 1.111 1.139

1.550 1.110 1.110 1.220 1.275 1.092 1.092 1.183 1.229 1.073 1.073 1.147 1.183 1.061 1.061 1.122 1.153

1.600 1.120 1.120 1.240 1.300 1.100 1.100 1.200 1.250 1.080 1.080 1.160 1.200 1.067 1.067 1.133 1.167

1.650 1.130 1.130 1.260 1.325 1.108 1.108 1.217 1.271 1.087 1.087 1.173 1.217 1.072 1.072 1.144 1.181

1.700 1.140 1.140 1.280 1.350 1.117 1.117 1.233 1.292 1.093 1.093 1.187 1.233 1.078 1.078 1.156 1.194

1.750 1.150 1.150 1.300 1.375 1.125 1.125 1.250 1.313 1.100 1.100 1.200 1.250 1.083 1.083 1.167 1.208

1.800 1.160 1.160 1.320 1.400 1.133 1.133 1.267 1.333 1.107 1.107 1.213 1.267 1.089 1.089 1.178 1.222

1.850 1.170 1.170 1.340 1.425 1.142 1.142 1.283 1.354 1.113 1.113 1.227 1.283 1.094 1.094 1.189 1.236

1.900 1.180 1.180 1.360 1.450 1.150 1.150 1.300 1.375 1.120 1.120 1.240 1.300 1.100 1.100 1.200 1.250

1.950 1.190 1.190 1.380 1.475 1.158 1.158 1.317 1.396 1.127 1.127 1.253 1.317 1.106 1.106 1.211 1.264

2.000 1.200 1.200 1.400 1.500 1.167 1.167 1.333 1.417 1.133 1.133 1.267 1.333 1.111 1.111 1.222 1.278

2.050 1.210 1.210 1.420 1.525 1.175 1.175 1.350 1.438 1.140 1.140 1.280 1.350 1.117 1.117 1.233 1.292

2.100 1.220 1.220 1.440 1.550 1.183 1.183 1.367 1.458 1.147 1.147 1.293 1.367 1.122 1.122 1.244 1.306

2.150 1.230 1.230 1.460 1.575 1.192 1.192 1.383 1.479 1.153 1.153 1.307 1.383 1.128 1.128 1.256 1.319

2.200 1.240 1.240 1.480 1.600 1.200 1.200 1.400 1.500 1.160 1.160 1.320 1.400 1.133 1.133 1.267 1.333

2.250 1.250 1.250 1.500 1.625 1.208 1.208 1.417 1.521 1.167 1.167 1.333 1.417 1.139 1.139 1.278 1.347

2.300 1.260 1.260 1.520 1.650 1.217 1.217 1.433 1.542 1.173 1.173 1.347 1.433 1.144 1.144 1.289 1.361

2.350 1.270 1.270 1.540 1.675 1.225 1.225 1.450 1.563 1.180 1.180 1.360 1.450 1.150 1.150 1.300 1.375

2.400 1.280 1.280 1.560 1.700 1.233 1.233 1.467 1.583 1.187 1.187 1.373 1.467 1.156 1.156 1.311 1.389

2.450 1.290 1.290 1.580 1.725 1.242 1.242 1.483 1.604 1.193 1.193 1.387 1.483 1.161 1.161 1.322 1.403

2.500 1.300 1.300 1.600 1.750 1.250 1.250 1.500 1.625 1.200 1.200 1.400 1.500 1.167 1.167 1.333 1.417

d (cm) 7.700 7.700 6.700 6.200 9.700 9.700 8.700 8.200 12.500 12.500 11.500 11.000 15.300 15.300 14.300 13.800

d/h 0.770 0.770 0.670 0.620 0.808 0.808 0.725 0.683 0.833 0.833 0.767 0.733 0.850 0.850 0.794 0.767

A.20

Table D-2: Equivalent unit in RAC in function of S3, exposure class, α4 and the height in RC, hRC

S3 hRAC/hRC with hRC = 10cm hRAC/hRC with hRC = 12cm hRAC/hRC with hRC = 15cm hRAC/hRC with hRC = 18cm

α4 XD1/XS1 XD2/XS2 XD3/XS3 XD1/XS1 XD2/XS2 XD3/XS3 XD1/XS1 XD2/XS2 XD3/XS3 XD1/XS1 XD2/XS2 XD3/XS3

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.050 1.030 1.035 1.040 1.025 1.029 1.033 1.020 1.023 1.027 1.017 1.019 1.022

1.100 1.060 1.070 1.080 1.050 1.058 1.067 1.040 1.047 1.053 1.033 1.039 1.044

1.150 1.090 1.105 1.120 1.075 1.088 1.100 1.060 1.070 1.080 1.050 1.058 1.067

1.200 1.120 1.140 1.160 1.100 1.117 1.133 1.080 1.093 1.107 1.067 1.078 1.089

1.250 1.150 1.175 1.200 1.125 1.146 1.167 1.100 1.117 1.133 1.083 1.097 1.111

1.300 1.180 1.210 1.240 1.150 1.175 1.200 1.120 1.140 1.160 1.100 1.117 1.133

1.350 1.210 1.245 1.280 1.175 1.204 1.233 1.140 1.163 1.187 1.117 1.136 1.156

d (cm) 5.700 5.200 4.700 7.700 7.200 6.700 10.500 1000 9.500 13.300 12.800 12.300

d/h 0.570 0.520 0.470 0.641 0.600 0.558 0.700 0.667 0.633 0.739 0.711 0.683

The figures of the equivalent unit in RAC in function of the various conditions and α3 or α4 concerning

the other structural classes are shown below.

Figure D-1: Equivalent unit in RAC in function of S1, exposure class, α3 and the height in RC, hRC


0.900

1.000

1.100

1.200

1.300

1.400

1.500

0.8

00

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=10cm)XC1 (hRC=10cm)XC2/XC3 (hRC=10cm)XC4 (hRC=10cm)X0 (hRC=12cm)XC1 (hRC=12cm)XC2/XC3 (hRC=12cm)XC4 (hRC=12cm)XO (hRC=15cm)XC1 (hRC=15cm)XC2/XC3 (hRC=15cm)XC4 (hRC=15cm)X0 (hRC=18cm)XC1 (hRC=18cm)XC2/XC3 (hRC=18cm)XC4 (hRC=18cm)

0.900

1.000

1.100

1.200

1.300

1.400

1.500

1.600

1.700

0.8

00

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=10cm)XC1 (hRC=10cm)

XC2/XC3 (hRC=10cm)

XC4 (hRC=10cm)X0 (hRC=12cm)

XC1 (hRC=12cm)

XC2/XC3 (hRC=12cm)XC4 (hRC=12cm)

X0 (hRC=15cm)XC1 (hRC=15cm)

XC2/XC3 (hRC=15cm)

XC4 (hRC=15cm)X0 (hRC=18cm)

XC1 (hRC=18cm)

XC2/XC3 (hRC=18cm)XC4 (hRC=18cm)

A.21





0.900

1.100

1.300

1.500

1.700

1.900

2.100

0.8

00

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=10cmXC1 (hRC=10cm)XC2/XC3 (hRC=10cm)XC4 (hRC=10cm)X0 (hRC=12cm)XC1 (hRC=12cm)XC2/XC3 (hRC=12cm)XC4 (hRC=12cm)X0 (hRC=15cm)XC1 (hRC=15cm)XC2/XC3 (hRC=15cm)XC4 (hRC=15cm)X0 (hRC=18cm)XC1 (hRC=18cm)XC2/XC3 (hRC=18cm)XC4 (hRC=18cm)

1.000

1.050

1.100

1.150

1.200

1.250

1.0

00

1.0

50

1.1

00

1.1

50

1.2

00

1.2

50

1.3

00

1.3

50

hRAC/hRC

α4

XD1/XS1 (hRC=10cm)XD2/XS2 (hRC=10cm)XD3/XS3 (hRC=10cm)XD1/XS1 (hRC=12cm)XD2/XS2 (hRC=12cm)XD3/XS3 (hRC=12cm)XD1/XS1 (hRC=15cm)XD2/XS2 (hRC=15cm)XD3/XS3 (hRC=15cm)

1.000

1.050

1.100

1.150

1.200

1.250

1.300

1.0

00

1.0

50

1.1

00

1.1

50

1.2

00

1.2

50

1.3

00

1.3

50

hRAC/hRC

α4

XD1/XS1 (hRC=10cm)XD2/XS2 (hRC=10cm)XD3/XS3 (hRC=10cm)XD1/XS1 (hRC=12cm)XD2/XS2 (hRC=12cm)XD3/XS3 (hRC=12cm)XD1/XS1 (hRC=15cm)XD2/XS2 (hRC=15cm)XD3/XS3 (hRC=15cm)

1.000

1.050

1.100

1.150

1.200

1.250

1.300

1.350

1.0

00

1.0

50

1.1

00

1.1

50

1.2

00

1.2

50

1.3

00

1.3

50

hRAC/hRC

α4

XD1/XS1 (hRC=10cm)

XD2/XS2 (hRC=10cm)

XD3/XS3 (hRC=10cm)

XD1/XS1 (hRC=12cm)

XD2/XS2 (hRC=12cm)

XD3/XS3 (hRC=12cm)

XD1/XS1 (hRC=15cm)

XD2/XS2 (hRC=15cm)

XD3/XS3 (hRC=15cm)

XD1/XS1 (hRC=18cm)

XD2/XS2 (hRC=18cm)

XD3/XS3 (hRC=18cm)

A.22

Annex E: Results of the equivalent functional unit

in RAC, concerning deformation (slabs)

The complete tables of the equivalent unit in RAC in function of the various conditions and α6/α2 can

be seen below.

Table E-1: Equivalent unit in RAC in function of α6/α2 (∆c = 0.000 m, 0.005 m, 0.010 m, 0.015 m, 0.020 m)

Load combination

hRC (m)

Dead weight

g (kN/m²)



pqp,RC

(kN/m²) cRC (m)

dRC (m)

∆c (m)

hRAC (m)

pqp,RAC

(kN/m²) dRAC (m)

hRAC/hRC α6/α2

1

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.000 0.120 4.450 0.102 1.000 1.000

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.000 0.120 4.450 0.092 1.000 1.000

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.000 0.120 4.450 0.082 1.000 1.000

2 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.000 0.180 10.400 0.158 1.000 1.000 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.000 0.180 10.400 0.148 1.000 1.000 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.000 0.180 10.400 0.138 1.000 1.000

3

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.000 0.150 6.900 0.130 1.000 1.000

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.000 0.150 6.900 0.120 1.000 1.000

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.000 0.150 6.900 0.110 1.000 1.000

4

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.005 0.130 4.700 0.107 1.083 1.204

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.005 0.130 4.700 0.097 1.083 1.204

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.005 0.130 4.700 0.087 1.083 1.204

5 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.005 0.190 10.650 0.163 1.056 1.148 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.005 0.190 10.650 0.153 1.056 1.148 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.005 0.190 10.650 0.143 1.056 1.148

6

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.005 0.160 7.150 0.135 1.067 1.171

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.005 0.160 7.150 0.125 1.067 1.171

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.005 0.160 7.150 0.115 1.067 1.171

7

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.010 0.140 4.950 0.112 1.167 1.428

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.010 0.140 4.950 0.102 1.167 1.428

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.010 0.140 4.950 0.092 1.167 1.428

8 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.010 0.200 10.900 0.168 1.111 1.309 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.010 0.200 10.900 0.158 1.111 1.309 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.010 0.200 10.900 0.148 1.111 1.309

9

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.010 0.170 7.400 0.140 1.133 1.357

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.010 0.170 7.400 0.130 1.133 1.357

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.010 0.170 7.400 0.120 1.133 1.357

10

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.015 0.150 5.200 0.117 1.250 1.671

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.015 0.150 5.200 0.107 1.250 1.671

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.015 0.150 5.200 0.097 1.250 1.671

11 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.015 0.210 11.150 0.173 1.167 1.481 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.015 0.210 11.150 0.163 1.167 1.481 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.015 0.210 11.150 0.153 1.167 1.481

12

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.015 0.180 7.650 0.145 1.200 1.559

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.015 0.180 7.650 0.135 1.200 1.559

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.015 0.180 7.650 0.125 1.200 1.559

13

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.020 0.160 5.450 0.122 1.333 1.935

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.020 0.160 5.450 0.112 1.333 1.935

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.020 0.160 5.450 0.102 1.333 1.935

14 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.020 0.220 11.400 0.178 1.222 1.666 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.020 0.220 11.400 0.168 1.222 1.666 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.020 0.220 11.400 0.158 1.222 1.666

15

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.020 0.190 7.900 0.150 1.267 1.775

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.020 0.190 7.900 0.140 1.267 1.775

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.020 0.190 7.900 0.130 1.267 1.775

A.23

Table E-2: Equivalent unit in RAC in function of α6/α2 (∆c = 0.025 m, 0.030 m, 0.035 m, 0.040 m, 0.045 m, 0.050 m)

Load combination

hRC (m)

Dead weight

g (kN/m²)


Life loads

q (kN/m²)

pqp,RC

(kN/m²) cRC (m)

dRC (m)

∆c (m)

hRAC (m)

pqp,RAC

(kN/m²) dRAC (m)

hRAC/hRC α6/α2

16

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.025 0.170 5.700 0.127 1.417 2.220

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.025 0.170 5.700 0.117 1.417 2.220

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.025 0.170 5.700 0.107 1.417 2.220

17 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.025 0.230 11.650 0.183 1.278 1.862 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.025 0.230 11.650 0.173 1.278 1.862 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.025 0.230 11.650 0.163 1.278 1.862

18

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.025 0.200 8.150 0.155 1.333 2.007

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.025 0.200 8.150 0.145 1.333 2.007

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.025 0.200 8.150 0.135 1.333 2.007

19

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.030 0.180 5.950 0.132 1.500 2.524

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.030 0.180 5.950 0.122 1.500 2.524

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.030 0.180 5.950 0.112 1.500 2.524

20 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.030 0.240 11.900 0.188 1.333 2.072 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.030 0.240 11.900 0.178 1.333 2.072 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.030 0.240 11.900 0.168 1.333 2.072

21

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.030 0.210 8.400 0.160 1.400 2.254

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.030 0.210 8.400 0.150 1.400 2.254

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.030 0.210 8.400 0.140 1.400 2.254

22

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.035 0.190 6.200 0.137 1.583 2.849

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.035 0.190 6.200 0.127 1.583 2.849

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.035 0.190 6.200 0.117 1.583 2.849

23 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.035 0.250 12.150 0.193 1.389 2.293 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.035 0.250 12.150 0.183 1.389 2.293 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.035 0.250 12.150 0.173 1.389 2.293

24

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.035 0.220 8.650 0.165 1.467 2.517

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.035 0.220 8.650 0.155 1.467 2.517

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.035 0.220 8.650 0.145 1.467 2.517

25

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.040 0.200 6.450 0.142 1.667 3.194

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.040 0.200 6.450 0.132 1.667 3.194

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.040 0.200 6.450 0.122 1.667 3.194

26 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.040 0.260 12.400 0.198 1.444 2.528 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.040 0.260 12.400 0.188 1.444 2.528 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.040 0.260 12.400 0.178 1.444 2.528

27

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.040 0.230 8.900 0.170 1.533 2.795

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.040 0.230 8.900 0.160 1.533 2.795

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.040 0.230 8.900 0.150 1.533 2.795

28

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.045 0.210 6.700 0.147 1.750 3.560

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.045 0.210 6.700 0.137 1.750 3.560

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.045 0.210 6.700 0.127 1.750 3.560

29 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.045 0.270 12.650 0.203 1.500 2.775 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.045 0.270 12.650 0.193 1.500 2.775 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.045 0.270 12.650 0.183 1.500 2.775

30

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.045 0.240 9.150 0.175 1.600 3.089

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.045 0.240 9.150 0.165 1.600 3.089

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.045 0.240 9.150 0.155 1.600 3.089

31

0.120 3.000 1.000 1.500 4.450 0.010 0.102 0.050 0.220 6.950 0.152 1.833 3.945

0.120 3.000 1.000 1.500 4.450 0.020 0.092 0.050 0.220 6.950 0.142 1.833 3.945

0.120 3.000 1.000 1.500 4.450 0.030 0.082 0.050 0.220 6.950 0.132 1.833 3.945

32 0.180 4.500 3.500 4.000 10.400 0.010 0.158 0.050 0.280 12.900 0.208 1.556 3.035 0.180 4.500 3.500 4.000 10.400 0.020 0.148 0.050 0.280 12.900 0.198 1.556 3.035 0.180 4.500 3.500 4.000 10.400 0.030 0.138 0.050 0.280 12.900 0.188 1.556 3.035

33

0.150 3.750 2.250 3.000 6.900 0.010 0.130 0.050 0.250 9.400 0.180 1.667 3.398

0.150 3.750 2.250 3.000 6.900 0.020 0.120 0.050 0.250 9.400 0.170 1.667 3.398

0.150 3.750 2.250 3.000 6.900 0.030 0.110 0.050 0.250 9.400 0.160 1.667 3.398

A.24

Annex F: Results of the equivalent functional unit in RAC, concerning bending

(slabs)

The complete tables of the equivalent unit in RAC in function of the various conditions and α1can be seen below.

Table F-1: Equivalent unit in RAC in function of α1 for C20/25 (∆c = 0.000 m, 0.005 m, 0.010 m)

Load combination

hRC (m)

g (kN/m²)

∆g (kN/m²)

q (kN/m²)

pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC (cm²/m)

As,RC - As,RAC

(%) α1 hRAC/hRC

1 0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.000 0.120 7.650 0.102 24.970 0.180 0.212 6.641 0.000 1.000 1.000 0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.000 0.120 7.650 0.092 20.314 0.180 0.212 5.990 0.000 1.000 1.000 0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.000 0.120 7.650 0.082 16.138 0.180 0.212 5.338 0.000 1.000 1.000

2 0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.000 0.180 16.800 0.158 59.914 0.180 0.212 10.286 0.000 1.000 1.000 0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.000 0.180 16.800 0.148 52.570 0.180 0.212 9.635 0.000 1.000 1.000 0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.000 0.180 16.800 0.138 45.706 0.180 0.212 8.984 0.000 1.000 1.000

3 0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.000 0.150 12.600 0.130 40.560 0.180 0.212 8.463 0.000 1.000 1.000 0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.000 0.150 12.600 0.120 34.560 0.180 0.212 7.812 0.000 1.000 1.000 0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.000 0.150 12.600 0.110 29.040 0.180 0.212 7.161 0.000 1.000 1.000

4 0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.005 0.130 7.988 0.107 26.071 0.171 0.200 6.558 1.244 0.917 1.083 0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.005 0.130 7.988 0.097 21.210 0.169 0.198 5.876 1.888 0.877 1.083 0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.005 0.130 7.988 0.087 16.850 0.167 0.195 5.196 2.676 0.831 1.083

5 0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.005 0.190 17.138 0.163 61.117 0.173 0.202 10.107 1.747 0.886 1.056 0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.005 0.190 17.138 0.153 53.626 0.172 0.201 9.442 2.009 0.870 1.056 0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.005 0.190 17.138 0.143 46.624 0.171 0.200 8.777 2.309 0.853 1.056

6 0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.005 0.160 12.938 0.135 41.646 0.171 0.201 8.307 1.846 0.880 1.067 0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.005 0.160 12.938 0.125 35.486 0.170 0.199 7.638 2.236 0.857 1.067 0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.005 0.160 12.938 0.115 29.818 0.169 0.198 6.969 2.693 0.832 1.067

7 0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.010 0.140 8.325 0.112 27.173 0.162 0.189 6.483 2.366 0.847 1.167 0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.010 0.140 8.325 0.102 22.106 0.159 0.185 5.776 3.563 0.781 1.167 0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.010 0.140 8.325 0.092 17.562 0.156 0.180 5.071 5.010 0.711 1.167

8 0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.010 0.200 17.475 0.168 62.321 0.166 0.193 9.940 3.367 0.795 1.111 0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.010 0.200 17.475 0.158 54.682 0.164 0.191 9.263 3.863 0.770 1.111 0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.010 0.200 17.475 0.148 47.542 0.163 0.189 8.587 4.425 0.744 1.111

9 0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.010 0.170 13.275 0.140 42.733 0.164 0.190 8.164 3.535 0.785 1.133 0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.010 0.170 13.275 0.130 36.411 0.162 0.188 7.479 4.265 0.750 1.133 0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.010 0.170 13.275 0.120 30.596 0.159 0.185 6.795 5.113 0.712 1.133

A.25


Load combination

hRC (m)

g (kN/m²)

∆g (kN/m²)

q (kN/m²)

pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC (cm²/m)

As,RC - As,RAC

(%) α1 hRAC/hRC

10

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.015 0.150 8.663 0.117 28.274 0.155 0.179 6.416 3.381 0.787 1.250

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.015 0.150 8.663 0.107 23.002 0.151 0.173 5.687 5.058 0.704 1.250

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.015 0.150 8.663 0.097 18.273 0.146 0.167 4.962 7.061 0.622 1.250

11

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.015 0.210 17.813 0.173 63.524 0.159 0.185 9.785 4.874 0.721 1.167

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.015 0.210 17.813 0.163 55.738 0.157 0.182 9.098 5.579 0.691 1.167

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.015 0.210 17.813 0.153 48.460 0.155 0.179 8.412 6.373 0.660 1.167

12

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.015 0.180 13.613 0.145 43.819 0.156 0.181 8.033 5.085 0.709 1.200

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.015 0.180 13.613 0.135 37.337 0.154 0.177 7.335 6.113 0.666 1.200

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.015 0.180 13.613 0.125 31.374 0.151 0.173 6.639 7.298 0.622 1.200

13

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.020 0.160 9.000 0.122 29.376 0.148 0.170 6.355 4.305 0.735 1.333

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.020 0.160 9.000 0.112 23.898 0.143 0.163 5.606 6.401 0.640 1.333

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.020 0.160 9.000 0.102 18.985 0.137 0.156 4.864 8.879 0.552 1.333

14

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.020 0.220 18.150 0.178 64.728 0.153 0.177 9.640 6.280 0.660 1.222

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.020 0.220 18.150 0.168 56.794 0.151 0.174 8.944 7.171 0.627 1.222

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.020 0.220 18.150 0.158 49.378 0.148 0.170 8.250 8.171 0.592 1.222

15

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.020 0.190 13.950 0.150 44.906 0.150 0.172 7.912 6.513 0.646 1.267

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.020 0.190 13.950 0.140 38.263 0.146 0.168 7.203 7.803 0.600 1.267

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.020 0.190 13.950 0.130 32.151 0.143 0.163 6.497 9.281 0.552 1.267

16

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 24.970 0.212 6.641 0.025 0.170 9.338 0.127 30.478 0.142 0.162 6.299 5.149 0.689 1.417

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 20.314 0.212 5.990 0.025 0.170 9.338 0.117 24.795 0.136 0.154 5.534 7.614 0.587 1.417

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 16.138 0.212 5.338 0.025 0.170 9.338 0.107 19.697 0.129 0.146 4.778 10.500 0.496 1.417

17

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 59.914 0.212 10.286 0.025 0.230 18.488 0.183 65.932 0.148 0.169 9.505 7.593 0.608 1.278

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 52.570 0.212 9.635 0.025 0.230 18.488 0.173 57.850 0.145 0.166 8.802 8.653 0.573 1.278

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 45.706 0.212 8.984 0.025 0.230 18.488 0.163 50.297 0.142 0.162 8.101 9.835 0.538 1.278

18

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 40.560 0.212 8.463 0.025 0.200 14.288 0.155 45.992 0.144 0.164 7.801 7.832 0.594 1.333

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 34.560 0.212 7.812 0.025 0.200 14.288 0.145 39.189 0.140 0.159 7.082 9.355 0.545 1.333

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 29.040 0.212 7.161 0.025 0.200 14.288 0.135 32.929 0.136 0.154 6.367 11.089 0.497 1.333

A.26


Load combination

hRC (m)

g (kN/m²)

∆g (kN/m²)

q (kN/m²)

pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC (cm²/m)

As,RC - As,RAC

(%) α1 hRAC/hRC

1

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 31.212 0.212 8.301 0.000 0.120 7.650 0.102 31.212 0.180 0.212 8.301 0.000 1.000 1.000

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 25.392 0.212 7.487 0.000 0.120 7.650 0.092 25.392 0.180 0.212 7.487 0.000 1.000 1.000

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 20.172 0.212 6.673 0.000 0.120 7.650 0.082 20.172 0.180 0.212 6.673 0.000 1.000 1.000

2

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 74.892 0.212 12.858 0.000 0.180 16.800 0.158 74.892 0.180 0.212 12.858 0.000 1.000 1.000

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 65.712 0.212 12.044 0.000 0.180 16.800 0.148 65.712 0.180 0.212 12.044 0.000 1.000 1.000

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 57.132 0.212 11.230 0.000 0.180 16.800 0.138 57.132 0.180 0.212 11.230 0.000 1.000 1.000

3

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 50.700 0.212 10.579 0.000 0.150 12.600 0.130 50.700 0.180 0.212 10.579 0.000 1.000 1.000

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 43.200 0.212 9.766 0.000 0.150 12.600 0.120 43.200 0.180 0.212 9.766 0.000 1.000 1.000

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 36.300 0.212 8.952 0.000 0.150 12.600 0.110 36.300 0.180 0.212 8.952 0.000 1.000 1.000

4

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 31.212 0.212 8.301 0.005 0.130 7.988 0.107 32.589 0.171 0.200 8.197 1.244 0.917 1.083

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 25.392 0.212 7.487 0.005 0.130 7.988 0.097 26.512 0.169 0.198 7.346 1.888 0.877 1.083

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 20.172 0.212 6.673 0.005 0.130 7.988 0.087 21.062 0.167 0.195 6.494 2.676 0.831 1.083

5

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 74.892 0.212 12.858 0.005 0.190 17.138 0.163 76.397 0.173 0.202 12.633 1.747 0.886 1.056

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 65.712 0.212 12.044 0.005 0.190 17.138 0.153 67.032 0.172 0.201 11.802 2.009 0.870 1.056

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 57.132 0.212 11.230 0.005 0.190 17.138 0.143 58.280 0.171 0.200 10.971 2.309 0.853 1.056

6

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 50.700 0.212 10.579 0.005 0.160 12.938 0.135 52.058 0.171 0.201 10.384 1.846 0.880 1.067

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 43.200 0.212 9.766 0.005 0.160 12.938 0.125 44.357 0.170 0.199 9.547 2.236 0.857 1.067

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 36.300 0.212 8.952 0.005 0.160 12.938 0.115 37.272 0.169 0.198 8.711 2.693 0.832 1.067

7

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 31.212 0.212 8.301 0.010 0.140 8.325 0.112 33.966 0.162 0.189 8.104 2.366 0.847 1.167

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 25.392 0.212 7.487 0.010 0.140 8.325 0.102 27.632 0.159 0.185 7.220 3.563 0.781 1.167

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 20.172 0.212 6.673 0.010 0.140 8.325 0.092 21.952 0.156 0.180 6.339 5.010 0.711 1.167

8

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 74.892 0.212 12.858 0.010 0.200 17.475 0.168 77.901 0.166 0.193 12.425 3.367 0.795 1.111

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 65.712 0.212 12.044 0.010 0.200 17.475 0.158 68.352 0.164 0.191 11.579 3.863 0.770 1.111

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 57.132 0.212 11.230 0.010 0.200 17.475 0.148 59.427 0.163 0.189 10.733 4.425 0.744 1.111

9

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 50.700 0.212 10.579 0.010 0.170 13.275 0.140 53.416 0.164 0.190 10.205 3.535 0.785 1.133

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 43.200 0.212 9.766 0.010 0.170 13.275 0.130 45.514 0.162 0.188 9.349 4.265 0.750 1.133

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 36.300 0.212 8.952 0.010 0.170 13.275 0.120 38.245 0.159 0.185 8.494 5.113 0.712 1.133

A.27


Load combination

hRC (m)

g (kN/m²)

∆g (kN/m²)

q (kN/m²)

pEd,RC

(kN/m²) cRC (m)

dRC (m)

µRC MEd,RC

(kNm/m) ωRC

As,RC

(cm²/m) ∆c (m)

hRAC (m)

pEd,RAC

(kN/m²) dRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC (cm²/m)

As,RC - As,RAC

(%) α1 hRAC/hRC

10

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 31.212 0.212 8.301 0.015 0.150 8.663 0.117 35.343 0.155 0.179 8.020 3.381 0.787 1.250

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 25.392 0.212 7.487 0.015 0.150 8.663 0.107 28.753 0.151 0.173 7.108 5.058 0.704 1.250

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 20.172 0.212 6.673 0.015 0.150 8.663 0.097 22.842 0.146 0.167 6.202 7.061 0.622 1.250

11

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 74.892 0.212 12.858 0.015 0.210 17.813 0.173 79.406 0.159 0.185 12.231 4.874 0.721 1.167

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 65.712 0.212 12.044 0.015 0.210 17.813 0.163 69.672 0.157 0.182 11.372 5.579 0.691 1.167

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 57.132 0.212 11.230 0.015 0.210 17.813 0.153 60.575 0.155 0.179 10.515 6.373 0.660 1.167

12

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 50.700 0.212 10.579 0.015 0.180 13.613 0.145 54.774 0.156 0.181 10.041 5.085 0.709 1.200

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 43.200 0.212 9.766 0.015 0.180 13.613 0.135 46.671 0.154 0.177 9.169 6.113 0.666 1.200

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 36.300 0.212 8.952 0.015 0.180 13.613 0.125 39.217 0.151 0.173 8.298 7.298 0.622 1.200

13

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 31.212 0.212 8.301 0.020 0.160 9.000 0.122 36.720 0.148 0.170 7.943 4.305 0.735 1.333

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 25.392 0.212 7.487 0.020 0.160 9.000 0.112 29.873 0.143 0.163 7.008 6.401 0.640 1.333

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 20.172 0.212 6.673 0.020 0.160 9.000 0.102 23.732 0.137 0.156 6.081 8.879 0.552 1.333

14

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 74.892 0.212 12.858 0.020 0.220 18.150 0.178 80.910 0.153 0.177 12.051 6.280 0.660 1.222

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 65.712 0.212 12.044 0.020 0.220 18.150 0.168 70.992 0.151 0.174 11.180 7.171 0.627 1.222

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 57.132 0.212 11.230 0.020 0.220 18.150 0.158 61.723 0.148 0.170 10.313 8.171 0.592 1.222

15

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 50.700 0.212 10.579 0.020 0.190 13.950 0.150 56.132 0.150 0.172 9.890 6.513 0.646 1.267

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 43.200 0.212 9.766 0.020 0.190 13.950 0.140 47.829 0.146 0.168 9.004 7.803 0.600 1.267

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 36.300 0.212 8.952 0.020 0.190 13.950 0.130 40.189 0.143 0.163 8.121 9.281 0.552 1.267

16

0.120 3.000 1.000 1.500 7.650 0.010 0.102 0.180 31.212 0.212 8.301 0.025 0.170 9.338 0.127 38.097 0.142 0.162 7.873 5.149 0.689 1.417

0.120 3.000 1.000 1.500 7.650 0.020 0.092 0.180 25.392 0.212 7.487 0.025 0.170 9.338 0.117 30.993 0.136 0.154 6.917 7.614 0.587 1.417

0.120 3.000 1.000 1.500 7.650 0.030 0.082 0.180 20.172 0.212 6.673 0.025 0.170 9.338 0.107 24.622 0.129 0.146 5.972 10.500 0.496 1.417

17

0.180 4.500 3.500 4.000 16.800 0.010 0.158 0.180 74.892 0.212 12.858 0.025 0.230 18.488 0.183 82.415 0.148 0.169 11.882 7.593 0.608 1.278

0.180 4.500 3.500 4.000 16.800 0.020 0.148 0.180 65.712 0.212 12.044 0.025 0.230 18.488 0.173 72.313 0.145 0.166 11.002 8.653 0.573 1.278

0.180 4.500 3.500 4.000 16.800 0.030 0.138 0.180 57.132 0.212 11.230 0.025 0.230 18.488 0.163 62.871 0.142 0.162 10.126 9.835 0.538 1.278

18

0.150 3.750 2.250 3.000 12.600 0.010 0.130 0.180 50.700 0.212 10.579 0.025 0.200 14.288 0.155 57.490 0.144 0.164 9.751 7.832 0.594 1.333

0.150 3.750 2.250 3.000 12.600 0.020 0.120 0.180 43.200 0.212 9.766 0.025 0.200 14.288 0.145 48.986 0.140 0.159 8.852 9.355 0.545 1.333

0.150 3.750 2.250 3.000 12.600 0.030 0.110 0.180 36.300 0.212 8.952 0.025 0.200 14.288 0.135 41.162 0.136 0.154 7.959 11.089 0.497 1.333

A.28

The graphs and tables demonstrate that the results are the same for various concrete strength

classes, but this can be demonstrated analytically as well:

∝�= �KC,DE∗��,DE��KC,DE∗(.DE�∆�)∗�∗ÄÅÞ,`b��KC,DE�]�.[∗∆��(.DE∗�∗ÄÅÞ,`b��,DE)� (Equation F-1)

Where pEd,RC is the total design load, dRC - the effective height of the slab in RC, ∆c - the difference in

cover, fcd,RC - the design value of the compressive strength of RC and Fs,RC - the resultant of the tensile

force in the reinforcement.

Introducing Equation F-2 leads to equation F-4:

x = ��C∗£.Â = ��C∗£.Â = >�∗�FC��C∗£.Â ↔ F = ��C∗£.Â� (Equation F-2)

Where x is the height of the compressive zone, Fc is the resultant of the compressive force of

concrete, AS is the cross-section of reinforcement and fyd is the design value of the yield strength of the

reinforcement.

→∝�= �KC,DE∗��C,DE∗�.ß DE à�KC,DE∗.DHE∗�∗��C,DE��KC,DHE∗(.DE∗�∗��C,DE��C,DE∗�.ß DE )á (Equation F-3)

↔∝�= �KC,DE∗ �.ßâ`bà�KC,DE∗.DHE∗��KC,DHE∗(.DE∗�� .ßâ`b)á

(Equation F-4)

The tables show that most parameters remain the same in this last formula. x is not calculated so it is

necessary to compare this parameter in the two cases. Load combination 1 is considered in the case

of ∆c = 0.010 m for C20/25 and C25/30.

6.641 cm�m ∗ 435000kN/m²13333 ∗ 0.8 = 8.301 cm�m ∗ 435000kN/m²

16667 ∗ 0.8

4.981 = 4.981

This calculation is done for several cases and the results are always the same. As a consequence, it

can be concluded that the results for α1 do not depend on the strength class.

A.29

Annex G: Tables with design results (slabs)

Not all the examples will be included in the dissertation as this would become too extensive. Tables G-

1, G-2 and G-3 show an RC-example with its corresponding RAC-examples when the fundamental

parameters are available. Tables G-4, G-5 and G-6 demonstrate the same but with missing


Table G-1: Design of slabs when all fundamental parameters are available (fundamental parameters and data)

One-way slab, continuous on both borders, L = 6m, Bravo et al. (2015b)

Mixture RC MRA

(100%)

MRA

(100%)

MRA

(10%)

MRA

(10%)

MRA

(50%)

Fundamental parameters

α1 1 0.789 0.822 0.985 0.998 0.885

α2 1 0.649 0.521 0.931 0.965 0.777

α3 1 1.939 1.712 1.242 1.166 1.439

α4 1 1.208 1.165 1.062 1.043 1.113

α5 1 0.725 0.775 0.975 1.000 0.900

α6 1 1.103 0.945 0.965 0.928 1.076

Data

fcm, cylinder (MPa) 38.240 40.349 41.709 48.400 48.914 44.282

fck (MPa) research 30.240 32.349 33.709 40.400 40.914 36.282

Strength class C30/37 / / / / /

fck used for calculations 41.000 32.349 33.709 40.400 40.914 36.282

Ecm (GPa) 33 21.417 17.193 30.727 31.859 25.626

(creep coefficient + 1) * α6 3.500 3.860 3.307 3.379 3.248 3.767

Ec,eff (GPa)= Ecm/ ((creep

coefficient+1)*α6) 9.429 5.548 5.199 9.094 9.810 6.802

Es (GPa) 200 200 200 200 200 200

fctm (MPa) 2.9 2.1025 2.2475 2.8275 2.9 2.61

Exposure class XC2/XC3 XC2/XC3 XC2/XC3 XC2/XC3 XC2/XC3 XC2/XC3

Structural class S3 S3 S3 S3 S3 S3

Minimum cover cmin (m) in function of

XC2/XC3 0.020 0.020 0.020 0.020 0.020 0.020

hRAC/hRC because of α1 1 1.088 1.072 1.005 1.001 1.044

hRAC/hRC because of α3 1 1.268 1.203 1.069 1.048 1.125

hRAC/hRC because of α4 / / / / / /

hRAC/hRC because of α6/α2 1 1.244 1.278 1.013 0.982 1.143

K-value 1 1.268 1.203 1.069 1.048 1.125

A.30

Table G-2: Design of slabs when all fundamental parameters are available (bending ULS)

BENDING ULS

fcd (MPa) 27.333 21.566 22.473 26.933 27.276 24.188

fyk (MPa) 500.000 500.000 500.000 500.000 500.000 500.000

fyd (MPa) 434.783 434.783 434.783 434.783 434.783 434.783

h (m) 0.140 0.178 0.179 0.150 0.147 0.160

real h (m) 0.140 0.180 0.180 0.150 0.150 0.160

b (m) 1.000 1.000 1.000 1.000 1.000 1.000

L (m) 6.000 6.000 6.000 6.000 6.000 6.000

g (kN/m²) 3.500 4.500 4.500 3.750 3.750 4.000

∆g (kN/m²) 3.000 3.000 3.000 3.000 3.000 3.000

q (kN/m²) 3.000 3.000 3.000 3.000 3.000 3.000

pEd (kN/m²) 13.275 14.625 14.625 13.613 13.613 13.950

pqp (kN/m²) 7.400 8.400 8.400 7.650 7.650 7.900

MEd,support = pEdl²/12 (kNm) 39.825 43.875 43.875 40.838 40.838 41.850

MEd,midspan = pEdl²/24 (kNm) 19.913 21.938 21.938 20.419 20.419 20.925

Mqp,support =p*l²/12 (kNm) 22.200 25.200 25.200 22.950 22.950 23.700

Mqp,midspan =p*l²/24 (kNm) 11.100 12.600 12.600 11.475 11.475 11.850

Minimum cover cmin (m) in function of exposure class

0.020 0.039 0.034 0.025 0.023 0.029

Cover c (m) 0.025 0.044 0.039 0.030 0.028 0.034

Ø (m) 0.010 0.010 0.010 0.010 0.010 0.010

d = h - c - Ø (m) 0.105 0.126 0.131 0.110 0.112 0.116

µd = MEd,support/(fcd*b*d²) 0.132 0.128 0.114 0.125 0.120 0.128

ω = µd * (1+µd) 0.150 0.144 0.127 0.141 0.134 0.144

As = ω*fcd*b*d/fyd (cm²/m) 9.876 9.016 8.598 9.592 9.421 9.342

REAL As Ø10//0,075 Ø10//0,075 Ø10//0,075 Ø10//0,075 Ø10//0,075 Ø10//0,075

As (cm²/m) 10.470 10.470 10.470 10.470 10.470 10.470

A.31

Table G-3: Design of slabs when all fundamental parameters are available (deformation and cracking SLS)

DEFORMATION SLS

y (neutral axis) (m) 0.075 0.096 0.097 0.079 0.079 0.086

II (m4) 0.000 0.001 0.001 0.000 0.000 0.000

II (cm4) 25118.622 52598.923 53952.619 30494.817 30529.375 37424.057

As*(d-x)*Es/(Ec,eff)-b*x2/2=0 0.000 0.000 0.000 0.000 0.000 0.000

x (m) 0.050 0.067 0.070 0.052 0.051 0.059

III (m4) 0.000 0.000 0.000 0.000 0.000 0.000

III (cm4) 10883.790 23262.060 26305.552 12475.576 12278.971 16926.442

w=II/(h-y) (m³) 0.004 0.006 0.007 0.004 0.004 0.005

Mcr = fctm * w (kNm) 11.138 13.186 14.662 12.229 12.535 13.142

β 0.500 0.500 0.500 0.500 0.500 0.500

Mqp,support (kNm) 22.200 25.200 25.200 22.950 22.950 23.700

ξ = 1-β*(Mcr/M)² 0.874 0.863 0.831 0.858 0.851 0.846

I = ξ*III+(1-ξ)*II (m4) 0.000 0.000 0.000 0.000 0.000 0.000

I (cm4) 12675.192 27278.426 30984.815 15033.812 15001.313 20077.953

δ (m) 0.021 0.019 0.018 0.019 0.018 0.020

L/250 (m) 0.024 0.024 0.024 0.024 0.024 0.024

L/250 (mm) 24.000 24.000 24.000 24.000 24.000 24.000

δ (mm) 20.898 18.733 17.600 18.886 17.544 19.522

TEST (<L/250) OK! OK! OK! OK! OK! OK!

CRACKING SLS

φ (m) 0.010 0.010 0.010 0.010 0.010 0.010

As*(d-x)*Es/(Ec,eff)-b*x2/2=0 0.000 0.000 0.000 0.000 0.000 0.000

x (m) 0.050 0.067 0.065 0.063 0.050 0.065

Ac,eff (m²) 0.030 0.038 0.038 0.029 0.033 0.032

ρp.eff = As/Ac,eff 0.035 0.028 0.027 0.036 0.031 0.033

k1 0.800 0.800 0.800 0.800 0.800 0.800

k2 0.500 0.500 0.500 0.500 0.500 0.500

k3 3.400 3.400 3.400 3.400 3.400 3.400

k4 0.425 0.425 0.425 0.425 0.425 0.425

Mqp,support (kNm) 22.200 25.200 25.200 22.950 22.950 23.700

σs (kN/m²) 239680.967 231616.750 220888.317 246110.034 230766.671 239524.908

kt 0.400 0.400 0.400 0.400 0.400 0.400

αe 6.061 9.338 11.633 6.509 6.278 7.805

fct,eff=fctm (MPa) 2.900 2.103 2.248 2.828 2.900 2.610

wmax (m) 0.000 0.000 0.000 0.000 0.000 0.000

wmax (mm) 0.300 0.300 0.300 0.300 0.300 0.300

Sr,max=k3*c+k1*k2*k4*φ/ρp.eff (m) 0.134 0.210 0.195 0.148 0.150 0.166

εsm-εcm 0.001 0.001 0.001 0.001 0.001 0.001

wk=Sr,max*(εsm-εcm) (m) 0.000 0.000 0.000 0.000 0.000 0.000

wk (mm) 0.133 0.203 0.174 0.154 0.140 0.166

TEST (<wmax) OK! OK! OK! OK! OK! OK!

A.32

Table G-4: Design of slabs when not all fundamental parameters are available (fundamental parameters and data)

One-way slab, continuous on both borders, L = 6m, Amorim et al. (2012)

Mixture RC RAC (20%) RAC (50% ) RAC (100%)


α1 (available) 1 0.994 0.977 0.955

α2 1 0.869 0.865 0.859

α3 (available) 1 1.007 1.163 1.248

α4 (available) 1 1.033 0.974 0.962

α5 1 0.996 0.984 0.970

α6 1 1.009 1.036 1.069

Data

fcm, cylinder (MPa) 41.280 48.762 48.047 47.172

fck (MPa) research 33.280 40.762 40.047 39.172

strength class C30/37 / / /

fck used for calculations 41.000 40.762 40.047 39.172

Ecm (GPa) 33.000 28.693 28.541 28.352

(creep coefficient + 1) * α6 3.500 3.532 3.626 3.742

Ec,eff (GPa)= Ecm/((creep coefficient+1)*α6) 9.429 8.125 7.871 7.577

Es (GPa) 200.000 200.000 200.000 200.000

fctm (MPa) 2.900 2.889 2.855 2.813

Exposure class XC2/XC3 and

XD1/XS1 XC2/XC3 and

XD1/XS1 XC2/XC3 and

XD1/XS1 XC2/XC3 and

XD1/XS1 Minimum cover cmin (m) in function of XC2/XC3

0.020 0.020 0.020 0.020

Minimum cover cmin (m) in function of XD1/XS1

0.030 0.030 0.030 0.030

Structural class S3 S3 S3 S3

hRAC/hRC because of α1 1 1.002 1.008 1.016



hRAC/hRC because of α6/α2 1 1.062 1.076 1.093

K-value 1 1.062 1.076 1.093

A.33

Table G-5: Design of slabs when not all fundamental parameters are available (bending ULS)

Bending ULS

fcd (MPa) 27.333 27.174 26.698 26.115

fyk (MPa) 500.000 500.000 500.000 500.000

fyd (MPa) 434.783 434.783 434.783 434.783

h (m) 0.140 0.149 0.151 0.153

real h (m) 0.140 0.150 0.150 0.150

b (m) 1.000 1.000 1.000 1.000

L (m) 6.000 6.000 6.000 6.000

g (kN/m²) 3.500 3.750 3.750 3.750

∆g (kN/m²) 2.000 2.000 2.000 2.000

q (kN/m²) 2.000 2.000 2.000 2.000

pEd (kN/m²) 10.425 10.763 10.763 10.763

pqp (kN/m²) 6.100 6.350 6.350 6.350

MEd,support = pEdl²/12 (kNm) 31.275 32.288 32.288 32.288

MEd,midspan = pEdl²/24 (kNm) 15.638 16.144 16.144 16.144

Mqp,support =p*l²/12 (kNm) 18.300 19.050 19.050 19.050

Mqp,midspan =p*l²/24 (kNm) 9.150 9.525 9.525 9.525


0.030 0.031 0.029 0.029

Cover c (m) 0.035 0.036 0.034 0.034

Ø (m) 0.010 0.010 0.010 0.010

d = h - c - Ø (m) 0.095 0.104 0.106 0.106

µd = MEd,support/(fcd*b*d²) 0.127 0.110 0.108 0.110

ω = µd * (1+µd) 0.143 0.122 0.120 0.122

As = ω*fcd*b*d/fyd (cm²/m) 8.532 7.924 7.778 7.764

REAL As Ø10//0,075 Ø10//0,100 Ø10//0,100 Ø10//0,100

As (cm²/m) 10.470 7.850 7.850 7.850

A.34

Table G-6: Design of slabs when not all fundamental parameters are available (deformation and cracking SLS)

Deformation SLS y (neutral axis) (m) 0.073 0.078 0.078 0.079

II (m4) 0.000 0.000 0.000 0.000

II (cm4) 24015.624 29513.464 29736.265 29831.449

As*(d-x)*Es/(Ec,eff)-b*x2/2=0 0.000 0.000 0.000 0.000

x (m) 0.046 0.047 0.048 0.049

III (m4) 0.000 0.000 0.000 0.000

III (cm4) 8575.598 9741.004 10349.057 10686.550

w=II/(h-y) (m³) 0.004 0.004 0.004 0.004

Mcr = fctm * w (kNm) 10.439 11.873 11.871 11.762

β 0.500 0.500 0.500 0.500

Mqp,support (kNm) 18.300 19.050 19.050 19.050

ξ = 1-β*(Mcr/M)² 0.837 0.806 0.806 0.809

I = ξ*III+(1-ξ)*II (m4) 0.000 0.000 0.000 0.000

I (cm4) 11087.618 13581.096 14113.353 14335.975

δ (m) 0.020 0.019 0.019 0.020

L/250 (m) 0.024 0.024 0.024 0.024

L/250 (mm) 24.000 24.000 24.000 24.000

δ (mm) 19.693 19.422 19.293 19.729

TEST (<L/250) OK! OK! OK! OK!

Cracking SLS

φ (m) 0.010 0.010 0.010 0.010

As*(d-x)*Es/(Ec,eff)-b*x2/2=0 0.000 0.000 0.000 0.000

x (m) 0.046 0.047 0.048 0.049

Ac,eff (m²) 0.031 0.034 0.034 0.034

ρp.eff = As/Ac,eff 0.034 0.023 0.023 0.023

k1 0.800 0.800 0.800 0.800

k2 0.500 0.500 0.500 0.500

k3 3.400 3.400 3.400 3.400

k4 0.425 0.425 0.425 0.425

Mqp,support (kNm) 18.300 19.050 19.050 19.050

σs (kN/m²) 219802.011 274653.970 270269.585 269978.160

kt 0.400 0.400 0.400 0.400

αe 6.061 6.970 7.008 7.054

fct,eff=fctm (MPa) 2.900 2.889 2.855 2.813

wmax (m) 0.000 0.000 0.000 0.000

wmax (mm) 0.300 0.300 0.300 0.300

Sr,max=k3*c+k1*k2*k4*φ/ρp.eff (m) 0.170 0.197 0.190 0.188

εsm-εcm 0.001 0.001 0.001 0.001

wk=Sr,max*(εsm-εcm) (m) 0.000 0.000 0.000 0.000

wk (mm) 0.151 0.213 0.202 0.201

TEST (<wmax) OK! OK! OK! OK!

A.35

Annex H: Parametric study for the verification of

the simplifications (beams)

H.1 Validation

The simplifications for the effective and total height of the beam, dRAC and hRAC, respectively,

(Equations 2-15 and 2-16) need to be verified with a parametric study like it is done for slabs. The

same reasoning as in Annex A can be used for this parametric studies: if the simplifications lead to

feasible results in the study, the height can be made independent of α3 and α4.

The cross-sections of reinforcement in RAC and RC are compared with each other and the difference

between the two needs to vary between – 5% and 15% (as it is the case for slabs). The cross-sections

of reinforcement will not exactly be the same because of the included empirical power (=1.2)

(explained in section 2.4.2.2).

H.2 Data

The same data as for slabs (section A.2) is used. Only the differences in cover, ∆c, for beams differ

from those for slabs. If ∆c = 0.015 m is used for slabs, then ∆c = 0.020 m is used for beams. The ratio

between the two is approximately 1.33 and this factor is used to determine the various differences in

cover for beams. Table H-1 demonstrates this. The parametric study is performed with ∆cbeams = 0.020

m or ∆cbeams = 0.035 m. Higher cases do not make sense in practice. Both cases are performed for a

lower slab (0.40 m * 0.20 m), a higher slab (0.60 m * 0.30 m) and an intermediate slab (0.50 m * 0.25

m). This means that in total 6 cases are handled but different heights lead to the same results.

Table H-1: Relationship between ∆cslab and ∆cbeam

∆cslab

(m) 0.000 0.005 0.010 0.015 0.020 0.020 0.025 0.030 0.035 0.035 0.040 0.045 0.050

∆cbeam

(m) 0.000 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065

H.3 Methodology

Most of the parameters for beams are obtained in the same way as for slabs, which is described in

section A.3. Only the differences for beams are explained in this part.

The parametric study for the beams is developed starting form the parametric study for the slabs. This

is because beams are in practice never designed before the slabs. That is why the load combinations

of the slabs correspond to load combinations chosen for the beams.

The absolute loads of the beams are not calculated because they are not necessary to perform the

parametric study. Only the ratio between the total design loads of the slabs, pEd,RAC and pEd,RC, are

A.36

necessary to determine dRAC (Equation 2-15) and MEd,RAC (Equation A-5). The other parameters to

eventually calculate the difference in design cross-section of reinforcement in beams are calculated in

the same way as for slabs.

H.4 Results

The parametric study concerning the beams is presented in the same way as for the slabs: Part I,

Part II and Part III, with corresponding values of the parameters for beams, are included for the two

cases of difference in cover (∆cbeams = 0.020 m and ∆cbeams = 0.035 m). The tables below show the

calculations for a beam of 0.50 m * 0.25 m and ∆c = 0.035 m, which corresponds to a ∆c = 0.025 m for

slabs. This means that Tables H-1, H-2 and H-3 present the calculations for beams, starting from the

calculations for slabs in Tables A-4, A-5 and A-6. The other difference in cover for beams is not presented

as this is exactly the same; only the difference in design cross-section of reinforcement is a little bit smaller,

which results in a smaller margin for losses in compressive strength, fcd.

A.37

Table H-2: Part I (∆c = 0.035 m and 0.5 m * 0.25 m)

Ibeams hRC (m)

b (m) cRC

(m) Ø

(m) dRC (m)

MEd,RC

(kNm/m) µRC ωRC As,RC (cm²)

∆c (m)

(pEd,RAC/ pEd,RAC)slabs

dRAC (m) hRAC (m) MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC (cm²/m)

As,RC - As,RAC (%)

C20/25, µ is

0,25, ∆c =0,035

0.500 0.250 0.015 0.020 0.467 181.741 0.250 0.313 11.189 0.035 1.221 0.593 0.661 221.831 0.189 0.225 10.228 8.588

0.500 0.250 0.025 0.020 0.457 174.041 0.250 0.313 10.949 0.035 1.221 0.580 0.658 212.432 0.189 0.225 10.009 8.588

0.500 0.250 0.035 0.020 0.447 166.508 0.250 0.313 10.709 0.035 1.221 0.568 0.656 203.237 0.189 0.225 9.790 8.588

0.500 0.250 0.015 0.020 0.467 181.741 0.250 0.313 11.189 0.035 1.100 0.524 0.592 199.996 0.219 0.266 10.701 4.357

0.500 0.250 0.025 0.020 0.457 174.041 0.250 0.313 10.949 0.035 1.100 0.513 0.591 191.523 0.219 0.266 10.472 4.357

0.500 0.250 0.035 0.020 0.447 166.508 0.250 0.313 10.709 0.035 1.100 0.501 0.589 183.233 0.219 0.266 10.243 4.357

0.500 0.250 0.015 0.020 0.467 181.741 0.250 0.313 11.189 0.035 1.134 0.543 0.611 206.081 0.210 0.254 10.559 5.629

0.500 0.250 0.025 0.020 0.457 174.041 0.250 0.313 10.949 0.035 1.134 0.531 0.609 197.350 0.210 0.254 10.333 5.629

0.500 0.250 0.035 0.020 0.447 166.508 0.250 0.313 10.709 0.035 1.134 0.520 0.608 188.808 0.210 0.254 10.107 5.629

C20/25, µ is

0,25, ∆c =0

0.500 0.250 0.015 0.020 0.467 181.741 0.250 0.313 11.189 0.000 1.000 0.467 0.500 181.741 0.250 0.313 11.189 0.000

0.500 0.250 0.025 0.020 0.457 174.041 0.250 0.313 10.949 0.000 1.000 0.457 0.500 174.041 0.250 0.313 10.949 0.000

0.500 0.250 0.035 0.020 0.447 166.508 0.250 0.313 10.709 0.000 1.000 0.447 0.500 166.508 0.250 0.313 10.709 0.000

0.500 0.250 0.015 0.020 0.467 181.741 0.250 0.313 11.189 0.000 1.000 0.467 0.500 181.741 0.250 0.313 11.189 0.000

0.500 0.250 0.025 0.020 0.457 174.041 0.250 0.313 10.949 0.000 1.000 0.457 0.500 174.041 0.250 0.313 10.949 0.000

0.500 0.250 0.035 0.020 0.447 166.508 0.250 0.313 10.709 0.000 1.000 0.447 0.500 166.508 0.250 0.313 10.709 0.000

0.500 0.250 0.015 0.020 0.467 181.741 0.250 0.313 11.189 0.000 1.000 0.467 0.500 181.741 0.250 0.313 11.189 0.000

0.500 0.250 0.025 0.020 0.457 174.041 0.250 0.313 10.949 0.000 1.000 0.457 0.500 174.041 0.250 0.313 10.949 0.000

0.500 0.250 0.035 0.020 0.447 166.508 0.250 0.313 10.709 0.000 1.000 0.447 0.500 166.508 0.250 0.313 10.709 0.000

C20/25, µ is

0,20, ∆c =0,035

0.500 0.250 0.015 0.020 0.467 145.393 0.200 0.240 8.593 0.035 1.221 0.593 0.661 177.465 0.151 0.174 7.922 7.808

0.500 0.250 0.025 0.020 0.457 139.233 0.200 0.240 8.409 0.035 1.221 0.580 0.658 169.946 0.151 0.174 7.752 7.808

0.500 0.250 0.035 0.020 0.447 133.206 0.200 0.240 8.225 0.035 1.221 0.568 0.656 162.590 0.151 0.174 7.583 7.808

0.500 0.250 0.015 0.020 0.467 145.393 0.200 0.240 8.593 0.035 1.100 0.524 0.592 159.997 0.175 0.206 8.254 3.947

0.500 0.250 0.025 0.020 0.457 139.233 0.200 0.240 8.409 0.035 1.100 0.513 0.591 153.218 0.175 0.206 8.077 3.947

0.500 0.250 0.035 0.020 0.447 133.206 0.200 0.240 8.225 0.035 1.100 0.501 0.589 146.586 0.175 0.206 7.900 3.947

0.500 0.250 0.015 0.020 0.467 145.393 0.200 0.240 8.593 0.035 1.134 0.543 0.611 164.865 0.168 0.196 8.154 5.105

0.500 0.250 0.025 0.020 0.457 139.233 0.200 0.240 8.409 0.035 1.134 0.531 0.609 157.880 0.168 0.196 7.980 5.105

0.500 0.250 0.035 0.020 0.447 133.206 0.200 0.240 8.225 0.035 1.134 0.520 0.608 151.046 0.168 0.196 7.805 5.105

C20/25, µ is

0,30, ∆c =0,035

0.500 0.250 0.015 0.020 0.467 218.089 0.300 0.390 13.963 0.035 1.221 0.593 0.661 266.197 0.227 0.278 12.664 9.308

0.500 0.250 0.025 0.020 0.457 208.849 0.300 0.390 13.664 0.035 1.221 0.580 0.658 254.919 0.227 0.278 12.392 9.308

0.500 0.250 0.035 0.020 0.447 199.809 0.300 0.390 13.365 0.035 1.221 0.568 0.656 243.885 0.227 0.278 12.121 9.308

0.500 0.250 0.015 0.020 0.467 218.089 0.300 0.390 13.963 0.035 1.100 0.524 0.592 239.995 0.262 0.331 13.302 4.735

0.500 0.250 0.025 0.020 0.457 208.849 0.300 0.390 13.664 0.035 1.100 0.513 0.591 229.827 0.262 0.331 13.017 4.735

0.500 0.250 0.035 0.020 0.447 199.809 0.300 0.390 13.365 0.035 1.100 0.501 0.589 219.879 0.262 0.331 12.732 4.735

0.500 0.250 0.015 0.020 0.467 218.089 0.300 0.390 13.963 0.035 1.134 0.543 0.611 247.297 0.252 0.315 13.110 6.113

0.500 0.250 0.025 0.020 0.457 208.849 0.300 0.390 13.664 0.035 1.134 0.531 0.609 236.820 0.252 0.315 12.829 6.113

0.500 0.250 0.035 0.020 0.447 199.809 0.300 0.390 13.365 0.035 1.134 0.520 0.608 226.569 0.252 0.315 12.548 6.113

A.38

Table H-3: Part II (∆c = 0.035 m and 0.5 m * 0.25 m)

IIbeams hRC (m)

b (m) cRC

(m) Ø (m)

dRC (m)

MEd,RC


∆c (m)


dRAC (m) hRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC (cm²/m)

As,RC - As,RAC (%)

C30/37, µ is

0,25, ∆c =0,035

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.221 0.593 0.661 332.746 0.189 0.225 15.341 8.588

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.221 0.580 0.658 318.648 0.189 0.225 15.013 8.588

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.221 0.568 0.656 304.856 0.189 0.225 14.684 8.588

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.100 0.524 0.592 299.994 0.219 0.266 16.052 4.357

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.100 0.513 0.591 287.284 0.219 0.266 15.708 4.357

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.100 0.501 0.589 274.849 0.219 0.266 15.364 4.357

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.134 0.543 0.611 309.122 0.210 0.254 15.838 5.629

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.134 0.531 0.609 296.025 0.210 0.254 15.499 5.629

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.134 0.520 0.608 283.211 0.210 0.254 15.160 5.629

C30/37, µ is

0,25, ∆c =0

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.000 1.000 0.467 0.500 272.611 0.250 0.313 16.783 0.000

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.000 1.000 0.457 0.500 261.061 0.250 0.313 16.423 0.000

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.000 1.000 0.447 0.500 249.761 0.250 0.313 16.064 0.000

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.000 1.000 0.467 0.500 272.611 0.250 0.313 16.783 0.000

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.000 1.000 0.457 0.500 261.061 0.250 0.313 16.423 0.000

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.000 1.000 0.447 0.500 249.761 0.250 0.313 16.064 0.000

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.000 1.000 0.467 0.500 272.611 0.250 0.313 16.783 0.000

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.000 1.000 0.457 0.500 261.061 0.250 0.313 16.423 0.000

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.000 1.000 0.447 0.500 249.761 0.250 0.313 16.064 0.000

A.39

Table H-4: Part III (∆c = 0.035 m and 0.5 m * 0.25 m)

IIIbeams hRC (m)

b (m) cRC

(m) Ø

(m) dRC (m)

MEd,RC


∆c (m)


dRAC (m) hRAC (m) MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC (cm²/m)

As,RC - As,RAC (%)

C30/37, µ is

0,25, ∆c =0,35,

fcd =90%

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.221 0.593 0.661 332.746 0.210 0.254 15.613 6.973

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.221 0.580 0.658 318.648 0.210 0.254 15.278 6.973

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.221 0.568 0.656 304.856 0.210 0.254 14.944 6.973

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.100 0.524 0.592 299.994 0.243 0.302 16.372 2.450

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.100 0.513 0.591 287.284 0.243 0.302 16.021 2.450

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.100 0.501 0.589 274.849 0.243 0.302 15.670 2.450

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.134 0.543 0.611 309.122 0.233 0.287 16.143 3.812

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.134 0.531 0.609 296.025 0.233 0.287 15.797 3.812

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.134 0.520 0.608 283.211 0.233 0.287 15.452 3.812

C30/37, µ is

0,25, ∆c =0,35,

fcd =80%

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.221 0.593 0.661 332.746 0.236 0.292 15.951 4.953

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.221 0.580 0.658 318.648 0.236 0.292 15.610 4.953

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.221 0.568 0.656 304.856 0.236 0.292 15.268 4.953

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.100 0.524 0.592 299.994 0.273 0.348 16.772 0.067

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.100 0.513 0.591 287.284 0.273 0.348 16.412 0.067

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.100 0.501 0.589 274.849 0.273 0.348 16.053 0.067

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.134 0.543 0.611 309.122 0.262 0.331 16.524 1.540

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.134 0.531 0.609 296.025 0.262 0.331 16.170 1.540

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.134 0.520 0.608 283.211 0.262 0.331 15.817 1.540

C30/37, µ is

0,25, ∆c =0,35,

fcd =70%

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.221 0.593 0.661 332.746 0.270 0.343 16.387 2.357

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.221 0.580 0.658 318.648 0.270 0.343 16.036 2.357

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.221 0.568 0.656 304.856 0.270 0.343 15.685 2.357

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.100 0.524 0.592 299.994 0.312 0.410 17.286 -2.998

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.100 0.513 0.591 287.284 0.312 0.410 16.916 -2.998

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.100 0.501 0.589 274.849 0.312 0.410 16.546 -2.998

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.134 0.543 0.611 309.122 0.300 0.389 17.015 -1.381

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.134 0.531 0.609 296.025 0.300 0.389 16.650 -1.381

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.134 0.520 0.608 283.211 0.300 0.389 16.286 -1.381

C30/37, µ is

0,25, ∆c =0,35,

fcd =60%

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.221 0.593 0.661 332.746 0.315 0.415 16.968 -1.104

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.221 0.580 0.658 318.648 0.315 0.415 16.605 -1.104

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.221 0.568 0.656 304.856 0.315 0.415 16.241 -1.104

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.100 0.524 0.592 299.994 0.364 0.497 17.972 -7.083

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.100 0.513 0.591 287.284 0.364 0.497 17.587 -7.083

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.100 0.501 0.589 274.849 0.364 0.497 17.202 -7.083

0.500 0.250 0.015 0.020 0.467 272.611 0.250 0.313 16.783 0.035 1.134 0.543 0.611 309.122 0.349 0.472 17.668 -5.275

0.500 0.250 0.025 0.020 0.457 261.061 0.250 0.313 16.423 0.035 1.134 0.531 0.609 296.025 0.349 0.472 17.290 -5.275

0.500 0.250 0.035 0.020 0.447 249.761 0.250 0.313 16.064 0.035 1.134 0.520 0.608 283.211 0.349 0.472 16.911 -5.275

A.40

H.5 Discussion

Most of the conclusions made for slabs are the same for beams. Only the differences in the trends are

described in this section.

H.5.1 Part I

The tables show that the different covers in RC, cRC, lead to the same differences in design cross-

section of reinforcement. This was not the case for slabs. The same results can be explained by the

fact that they are depend on the ratio between dRAC and dRC. This ratio is for the various possible

covers of a load combination always the same, which is not the case for slabs.

The higher the load combination of the slabs is, the smaller the difference in cross-section of the

reinforcement becomes. This can also be explained by the fact that the ratio between dRAC and dRC

decrease for a higher load combination.

H.5.2 Part II

The same can be concluded like in section A.5.2.

H.5.3 Part III

In the case of ∆c = 0.035 m, it is possible to use a RAC that has a loss in compressive strength of almost

40%. This can be seen in Table H-3. The lower load combinations are less conditioning: this is contrary to

the conclusions made for slabs. It can be explained by the fact that lower loads combinations have a bigger

margin because of the bigger ratio between dRAC and dRC (which depends on the loads of the slab in RAC

and RC).

H.5.4 Comparison with other cover increases (∆c = 0.020 m)

The same can be concluded like in section A.5.4: lower differences in cover lead to smaller differences

in cross-section of reinforcement, which means that the margin for losses in compressive strength

reduces. It is possible to go to a loss in compressive strength of almost 30% if the cases with ∆c =

0.020 m are considered.

A.41

Annex I: Tables with results of the compliance of

the deformation serviceability limit state (beams)

The results of the verification formula for the beam with the intermediate dimensions (0.50 m * 0.25 m)

are already presented in section 3.4.2.2.

The tables below show the results for the beams with other dimensions: 0.40 m * 0.20 m and 0.60 m *

0.30 m. Only the results of the verification formula are shown; the complete tables consist of a

combination of these tables with Tables A-4 and H-2. It can be seen that the lower beam obtains the

less conditioning results, but the differences are negligible.

Table I-1: Calculated α2/α6 for beams (0.40 m * 0.20 m) in function of ∆c and load combinations

Load combination

(slabs)

dRC (m)


∆cslabs

0.005 0.005 0.010 0.015 0.020 0.020 0.025 0.030 0.035 0.035 0.040

∆cbeams

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055

min (α2/α6)

1

0.369 1.169 0.850 0.821 0.707 0.613 0.537 0.522 0.460 0.409 0.365 0.356 0.320

0.359 1.169 0.854 0.825 0.713 0.621 0.545 0.530 0.469 0.417 0.373 0.364 0.328

0.349 1.169 0.858 0.829 0.719 0.629 0.554 0.538 0.478 0.426 0.382 0.373 0.336

2

0.369 1.072 0.913 0.880 0.807 0.742 0.685 0.663 0.614 0.569 0.530 0.515 0.480

0.359 1.072 0.914 0.882 0.810 0.747 0.690 0.668 0.619 0.575 0.536 0.521 0.486

0.349 1.072 0.916 0.884 0.814 0.751 0.695 0.673 0.625 0.581 0.542 0.527 0.493

3

0.369 1.109 0.888 0.857 0.767 0.689 0.623 0.604 0.548 0.500 0.457 0.445 0.409

0.359 1.109 0.891 0.860 0.771 0.695 0.629 0.610 0.555 0.507 0.465 0.452 0.416

0.349 1.109 0.894 0.863 0.776 0.701 0.636 0.617 0.563 0.515 0.473 0.460 0.424

Table I-2: Calculated α2/α6 for beams (0.60 m * 0.30 m) in function of ∆c and load combinations

Load combination

(slabs)

dRC (m)


∆cslabs

0.005 0.005 0.010 0.015 0.020 0.020 0.025 0.030 0.035 0.035 0.040

∆cbeams

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055

min (α2/α6)

1

0.565 1.169 0.857 0.837 0.725 0.633 0.556 0.546 0.483 0.430 0.385 0.379 0.341

0.555 1.169 0.860 0.840 0.730 0.638 0.562 0.552 0.489 0.437 0.392 0.385 0.347

0.545 1.169 0.863 0.843 0.734 0.644 0.568 0.558 0.496 0.443 0.398 0.391 0.353

2

0.565 1.072 0.922 0.900 0.833 0.773 0.718 0.703 0.655 0.612 0.573 0.562 0.527

0.555 1.072 0.923 0.902 0.835 0.776 0.722 0.707 0.659 0.617 0.578 0.566 0.532

0.545 1.072 0.925 0.903 0.838 0.779 0.726 0.710 0.663 0.621 0.582 0.571 0.537

3

0.565 1.109 0.897 0.876 0.790 0.715 0.650 0.637 0.582 0.533 0.490 0.481 0.443

0.555 1.109 0.899 0.877 0.793 0.719 0.655 0.642 0.587 0.538 0.495 0.486 0.449

0.545 1.109 0.900 0.879 0.796 0.723 0.660 0.646 0.592 0.544 0.501 0.492 0.454

A.42

Annex J: Tables with results of the compliance of the bending ultimate limit

state (beams)

Tables J-1 and J-2 of beams correspond to Tables B-1 and B-2 of slabs. The load combinations in B-1 and B-2 are used in the parametric study for beams,

but are not presented again in the tables below.

Table J-1: Compliance of the bending ULS for beams (∆c = 0.000 m and 0.015 m)

hRC (m)

b (m) cRC (m)

Ø (m) dRC (m)

MEd,RC

(kNm/m) µRC ωRC

As,RC

(cm²) ∆cbeam

s (m) (pEd,RAC/

pEd,RC)slabs

dRAC (m)

hRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m) As,RC - As,RAC (%) α1

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 1.000 0.467 0.500 227.176 0.250 0.313 13.986 0.000 1.000

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 1.000 0.457 0.500 217.551 0.250 0.313 13.686 0.000 1.000

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 1.000 0.447 0.500 208.134 0.250 0.313 13.387 0.000 1.000

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 1.000 0.467 0.500 227.176 0.250 0.313 13.986 0.000 1.000

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 1.000 0.457 0.500 217.551 0.250 0.313 13.686 0.000 1.000

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 1.000 0.447 0.500 208.134 0.250 0.313 13.387 0.000 1.000

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 1.000 0.467 0.500 227.176 0.250 0.313 13.986 0.000 1.000

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 1.000 0.457 0.500 217.551 0.250 0.313 13.686 0.000 1.000

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 1.000 0.447 0.500 208.134 0.250 0.313 13.387 0.000 1.000

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.015 1.088 0.517 0.565 247.221 0.222 0.271 13.444 3.872 0.828

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.015 1.088 0.506 0.564 236.747 0.222 0.271 13.156 3.872 0.828

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.015 1.088 0.495 0.563 226.499 0.222 0.271 12.868 3.872 0.828

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.015 1.040 0.490 0.538 236.304 0.237 0.293 13.727 1.849 0.915

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.015 1.040 0.479 0.537 226.292 0.237 0.293 13.433 1.849 0.915

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.015 1.040 0.469 0.537 216.497 0.237 0.293 13.139 1.849 0.915

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.015 1.054 0.497 0.545 239.346 0.232 0.286 13.645 2.433 0.889

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.015 1.054 0.487 0.545 229.206 0.232 0.286 13.353 2.433 0.889

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.015 1.054 0.476 0.544 219.284 0.232 0.286 13.061 2.433 0.889

A.43

Table J-2: Compliance of the bending ULS for beams (∆c = 0.020 m, 0.035 m and 0.040 m)

hRC (m)

b (m) cRC (m)

Ø (m) dRC (m)

MEd,RC

(kNm/m) µRC ωRC

As,RC

(cm²) ∆cbeams

(m) (pEd,RAC/

pEd,RC)slabs dRAC (m)

hRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m) As,RC - As,RAC (%) α1

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 1.132 0.542 0.595 257.243 0.210 0.254 13.207 5.571 0.761

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 1.132 0.531 0.594 246.345 0.210 0.254 12.924 5.571 0.761

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 1.132 0.519 0.592 235.682 0.210 0.254 12.641 5.571 0.761

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 1.060 0.501 0.554 240.867 0.230 0.283 13.605 2.719 0.877

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 1.060 0.490 0.553 230.662 0.230 0.283 13.314 2.719 0.877

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 1.060 0.480 0.553 220.678 0.230 0.283 13.023 2.719 0.877

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 1.080 0.512 0.565 245.431 0.224 0.275 13.489 3.554 0.842

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 1.080 0.501 0.564 235.033 0.224 0.275 13.200 3.554 0.842

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 1.080 0.490 0.563 224.859 0.224 0.275 12.911 3.554 0.842

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.035 1.221 0.593 0.661 277.288 0.189 0.225 12.785 8.588 0.650

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.035 1.221 0.580 0.658 265.540 0.189 0.225 12.511 8.588 0.650

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.035 1.221 0.568 0.656 254.046 0.189 0.225 12.237 8.588 0.650

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.035 1.100 0.524 0.592 249.995 0.219 0.266 13.376 4.357 0.809

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.035 1.100 0.513 0.591 239.403 0.219 0.266 13.090 4.357 0.809

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.035 1.100 0.501 0.589 229.041 0.219 0.266 12.803 4.357 0.809

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.035 1.134 0.543 0.611 257.601 0.210 0.254 13.198 5.629 0.758

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.035 1.134 0.531 0.609 246.687 0.210 0.254 12.916 5.629 0.758

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.035 1.134 0.520 0.608 236.010 0.210 0.254 12.633 5.629 0.758

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.040 1.265 0.619 0.692 287.311 0.180 0.212 12.596 9.935 0.605

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.040 1.265 0.606 0.689 275.138 0.180 0.212 12.326 9.935 0.605

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.040 1.265 0.593 0.686 263.229 0.180 0.212 12.057 9.935 0.605

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.040 1.121 0.535 0.608 254.559 0.213 0.259 13.268 5.130 0.778

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.040 1.121 0.524 0.607 243.774 0.213 0.259 12.984 5.130 0.778

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.040 1.121 0.512 0.605 233.222 0.213 0.259 12.700 5.130 0.778

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.040 1.161 0.558 0.631 263.686 0.203 0.244 13.064 6.593 0.722

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.040 1.161 0.546 0.629 252.515 0.203 0.244 12.784 6.593 0.722

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.040 1.161 0.535 0.628 241.585 0.203 0.244 12.504 6.593 0.722

A.44

Annex K: Tables with results of the compliance of the cracking serviceability

limit state (beams)

Like in Annex C, there are different spread sheets to include the 2 differences in covers (0.020 m and 0.035 m) and the different heights of the beams (0.40 m,

0.50 m and 0.60 m). The composition of the spreadsheets with the various groups of columns and sections of rows is the same as for slabs and is not

repeated in this Annex. Tables K-1, K-2 and K-3 (for intermediate beam) correspond to Tables C-1, C-2 and C-3 and also show only Section A. The other

sections follow the same sequence and consist of the same calculations.

Table K-1: Compliance of the cracking SLS for beams (first 2 groups of columns, section A)

hRC (m) b (m) cRC (m) Ø (m) dRC (m) MEd,RC (kNm/m) µRC ωRC As,RC (cm²) ∆cbeams (m) hRAC (m) dRAC (m) MEd,RAC (kNm/m) µRAC ωRAC As,RAC (cm²/m) As,RC - As,RAC (%)

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 0.500 0.467 227.176 0.250 0.313 13.986 0.000

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 0.500 0.457 217.551 0.250 0.313 13.686 0.000

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 0.500 0.447 208.134 0.250 0.313 13.387 0.000

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 0.500 0.467 227.176 0.250 0.313 13.986 0.000

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 0.500 0.457 217.551 0.250 0.313 13.686 0.000

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 0.500 0.447 208.134 0.250 0.313 13.387 0.000

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 0.500 0.467 227.176 0.250 0.313 13.986 0.000

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 0.500 0.457 217.551 0.250 0.313 13.686 0.000

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 0.500 0.447 208.134 0.250 0.313 13.387 0.000

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 0.616 0.563 257.243 0.195 0.233 12.557 10.217

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 0.614 0.551 246.345 0.195 0.233 12.288 10.217

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 0.612 0.539 235.682 0.195 0.233 12.019 10.217

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 0.561 0.508 240.867 0.224 0.275 13.359 4.480

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 0.560 0.497 230.662 0.224 0.275 13.073 4.480

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 0.559 0.486 220.678 0.224 0.275 12.787 4.480

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 0.582 0.529 245.431 0.211 0.255 12.932 7.537

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 0.580 0.517 235.033 0.211 0.255 12.655 7.537

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 0.579 0.506 224.859 0.211 0.255 12.378 7.537

A.45

Table K-2: Compliance of the cracking SLS for beams (third and fourth group of columns, section A)

(pqp,RAC

/pqp,RC)

slabs

Mqp,RC

(kNm/m) Mqp,RAC

(kNm/m) α2 α6 α2/α6 ∆RC

x1RC (m)

x2RC (m) ∆RAC x1RAC (m)

x2RAC (m)

σs,RC (kN/m²)

σs,RAC (kN/m²)

Ø (m) 2,5*(h-

d)RC (m)

2,5*(h-d)RAC (m)

(h-x)/3

RC (m)

(h-x)/3

RAC (m)

h/2 RC (m)

h/2

RAC (m)


(h-x)/3 ; h/2] RC

(m)


(h-x)/3 ; h/2] RAC

(m)

1.000 132.148 132.148 0.960 1.050 0.914 0.030 0.143 -0.206 0.033 0.148 -0.217 225337.386 226294.629 0.020 0.083 0.083 0.119 0.117 0.250 0.250 0.083 0.083

1.000 126.549 126.549 0.960 1.050 0.914 0.029 0.140 -0.202 0.032 0.145 -0.213 225337.386 226294.629 0.020 0.108 0.108 0.120 0.118 0.250 0.250 0.108 0.108

1.000 121.072 121.072 0.960 1.050 0.914 0.028 0.137 -0.197 0.031 0.142 -0.208 225337.386 226294.629 0.020 0.133 0.133 0.121 0.119 0.250 0.250 0.121 0.119

1.000 140.633 140.633 0.960 1.050 0.914 0.030 0.143 -0.206 0.033 0.148 -0.217 239805.276 240823.979 0.020 0.083 0.083 0.119 0.117 0.250 0.250 0.083 0.083

1.000 134.674 134.674 0.960 1.050 0.914 0.029 0.140 -0.202 0.032 0.145 -0.213 239805.276 240823.979 0.020 0.108 0.108 0.120 0.118 0.250 0.250 0.108 0.108

1.000 128.845 128.845 0.960 1.050 0.914 0.028 0.137 -0.197 0.031 0.142 -0.208 239805.276 240823.979 0.020 0.133 0.133 0.121 0.119 0.250 0.250 0.121 0.119

1.000 124.406 124.406 0.960 1.050 0.914 0.030 0.143 -0.206 0.033 0.148 -0.217 212135.436 213036.597 0.020 0.083 0.083 0.119 0.117 0.250 0.250 0.083 0.083

1.000 119.135 119.135 0.960 1.050 0.914 0.029 0.140 -0.202 0.032 0.145 -0.213 212135.436 213036.597 0.020 0.108 0.108 0.120 0.118 0.250 0.250 0.108 0.108

1.000 113.978 113.978 0.960 1.050 0.914 0.028 0.137 -0.197 0.031 0.142 -0.208 212135.436 213036.597 0.020 0.133 0.133 0.121 0.119 0.250 0.250 0.121 0.119

1.169 132.148 154.420 0.960 1.050 0.914 0.030 0.143 -0.206 0.040 0.166 -0.235 225337.386 217455.307 0.020 0.083 0.133 0.119 0.150 0.250 0.308 0.083 0.133

1.169 126.549 147.878 0.960 1.050 0.914 0.029 0.140 -0.202 0.038 0.162 -0.230 225337.386 217455.307 0.020 0.108 0.158 0.120 0.151 0.250 0.307 0.108 0.151

1.169 121.072 141.477 0.960 1.050 0.914 0.028 0.137 -0.197 0.037 0.159 -0.225 225337.386 217455.307 0.020 0.133 0.183 0.121 0.151 0.250 0.306 0.121 0.151

1.072 140.633 150.775 0.960 1.050 0.914 0.030 0.143 -0.206 0.036 0.156 -0.225 239805.276 236555.452 0.020 0.083 0.133 0.119 0.135 0.250 0.280 0.083 0.133

1.072 134.674 144.387 0.960 1.050 0.914 0.029 0.140 -0.202 0.035 0.153 -0.220 239805.276 236555.452 0.020 0.108 0.158 0.120 0.136 0.250 0.280 0.108 0.136

1.072 128.845 138.137 0.960 1.050 0.914 0.028 0.137 -0.197 0.033 0.149 -0.215 239805.276 236555.452 0.020 0.133 0.183 0.121 0.137 0.250 0.279 0.121 0.137

1.109 124.406 137.928 0.960 1.050 0.914 0.030 0.143 -0.206 0.038 0.160 -0.229 212135.436 207473.654 0.020 0.083 0.133 0.119 0.141 0.250 0.291 0.083 0.133

1.109 119.135 132.085 0.960 1.050 0.914 0.029 0.140 -0.202 0.036 0.156 -0.224 212135.436 207473.654 0.020 0.108 0.158 0.120 0.141 0.250 0.290 0.108 0.141

1.109 113.978 126.367 0.960 1.050 0.914 0.028 0.137 -0.197 0.035 0.153 -0.219 212135.436 207473.654 0.020 0.133 0.183 0.121 0.142 0.250 0.289 0.121 0.142

A.46

Table K-3: Compliance of the cracking SLS for beams (last 2 groups of columns, section A)

srmax (m) Numerator Denominator α5 εcm-εsm

(respective α5) 0,6*σsRAC/Es

Control wk (mm)

εcm-εsm ( α5=1)


0.101 -513.189 31.225 -16.435 0.003 0.001 0.300 0.001 0.103

0.152 -231.372 37.516 -6.167 0.002 0.001 0.300 0.001 0.151

0.195 -109.453 40.382 -2.710 0.002 0.001 0.300 0.001 0.191

0.101 -492.869 31.225 -15.784 0.003 0.001 0.300 0.001 0.110

0.152 -211.487 37.516 -5.637 0.002 0.001 0.300 0.001 0.162

0.195 -90.003 40.382 -2.229 0.002 0.001 0.300 0.001 0.205

0.101 -531.731 31.225 -17.029 0.003 0.001 0.300 0.001 0.096

0.152 -249.517 37.516 -6.651 0.002 0.001 0.300 0.001 0.141

0.195 -127.201 40.382 -3.150 0.002 0.001 0.300 0.001 0.178

0.200 -116.435 44.225 -2.633 0.002 0.001 0.300 0.001 0.185

0.247 -35.476 48.721 -0.728 0.001 0.001 0.300 0.001 0.224

0.283 7.224 48.641 0.149 0.001 0.001 0.300 0.001 0.256

0.200 -89.722 44.225 -2.029 0.002 0.001 0.300 0.001 0.204

0.237 -22.274 44.862 -0.496 0.001 0.001 0.300 0.001 0.242

0.274 23.233 44.867 0.518 0.001 0.001 0.300 0.001 0.278

0.200 -130.395 44.225 -2.948 0.002 0.001 0.300 0.001 0.175

0.241 -57.094 46.314 -1.233 0.001 0.001 0.300 0.001 0.209

0.277 -12.029 46.286 -0.260 0.001 0.001 0.300 0.001 0.240

A.47

Annex L: Results of the equivalent functional unit

in RAC, concerning durability (beams)

The figures showing the results of the equivalent functional unit in RAC are shown for the intermediate

slabs in Chapter 4. The figures below show the results for the other slabs in function of α3 or α4. Only

structural class S4 is considered as this one is the most relevant for standard framed buildings.

Figure L-1: hRAC/hRC in function of α3 for S4 and smallest slab (beams)

Figure L-2: hRAC/hRC in function of α3 for S4 and thickest slab (beams)

0.800

0.900

1.000

1.100

1.200

1.300

1.400

1.500

1.600

1.700

0.8

00

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=40cm)

XC1 (hRC=40cm)

XC2/XC3 (hRC=40cm)

XC4 (hRC=40cm)

X0 (hRC=50cm)

XC1 (hRC=50cm)

XC2/XC3 (hRC=50cm)

XC4 (hRC=50cm)

X0 (hRC=60cm)

XC1 (hRC=60cm)

XC2/XC3 (hRC=60cm)

XC4 (hRC=60cm)

0.900

0.950

1.000

1.050

1.100

1.150

1.200

1.250

1.300

1.350

0.8

00

0.9

00

1.0

00

1.1

00

1.2

00

1.3

00

1.4

00

1.5

00

1.6

00

1.7

00

1.8

00

1.9

00

2.0

00

2.1

00

2.2

00

2.3

00

2.4

00

2.5

00

hRAC/hRC

α3

X0 (hRC=40cm)

XC1 (hRC=40cm)

XC2/XC3 (hRC=40cm)

XC4 (hRC=40cm)

X0 (hRC=50cm)

XC1 (hRC=50cm)

XC2/XC3 (hRC=50cm)

XC4 (hRC=50cm)

X0 (hRC=60cm)

XC1 (hRC=60cm)

XC2/XC3 (hRC=60cm)

XC4 (hRC=60cm)

A.48

Figure L-3: hRAC/hRC in function of α4 for S4 and smallest slab (beams)

Figure L-4: hRAC/hRC in function of α4 for S4 and thickest slab (beams)

1.000

1.050

1.100

1.150

1.200

1.250

1.0

00

1.0

50

1.1

00

1.1

50

1.2

00

1.2

50

1.3

00

1.3

50

hRAC/hRC

α4

XD1/XS1 (hRC=40cm)

XD2/XS2 (hRC=40cm)

XD3/XS3 (hRC=40cm)

XD1/XS1 (hRC=50cm)

XD2/XS2 (hRC=50cm)

XD3/XS3 (hRC=50cm)

XD1/XS1 (hRC=60cm)

XD2/XS2 (hRC=60cm)

XD3/XS3 (hRC=60cm)

1.000

1.020

1.040

1.060

1.080

1.100

1.120

1.0

00

1.0

50

1.1

00

1.1

50

1.2

00

1.2

50

1.3

00

1.3

50

hRAC/hRC

α4

XD1/XS1 (hRC=40cm)

XD2/XS2 (hRC=40cm)

XD3/XS3 (hRC=40cm)

XD1/XS1 (hRC=50cm)

XD2/XS2 (hRC=50cm)

XD3/XS3 (hRC=50cm)

XD1/XS1 (hRC=60cm)

XD2/XS2 (hRC=60cm)

XD3/XS3 (hRC=60cm)

A.49

Annex M: Results of the equivalent functional unit

in RAC, concerning deformation (beams)

Tables M-1 and M-2 correspond to Tables E-1 and E-2 of the slabs; the ratio pqp,RAC/pqp,RC of the slabs is

used to determine the results of the beams. The difference in cover is not increased to the corresponding

value of ∆cslabs = 0.050 m as this is not feasible in practice. ∆cbeams ranges from 0.000 m to 0.040 m. Only

the tables concerning the beam with dimensions of 0.50 m * 0.25 m are presented because the

calculations for the other beams follow the same sequence.

Table M-1: Equivalent unit in RAC in function of α6/α2 (∆c = 0.000 m, 0.010 m, 0.015 m) (beams)

hRC (m) b (m) cRC (m) Ø (m) dRC (m) ∆cbeams

(m) (pqp,RAC/

pqp,RAC)slabs dRAC (m) hRAC (m) α6/α2 hRAC/hRC

0.500 0.250 0.015 0.020 0.467 0.000 1.000 0.467 0.500 1.000 1.000

0.500 0.250 0.025 0.020 0.457 0.000 1.000 0.457 0.500 1.000 1.000

0.500 0.250 0.035 0.020 0.447 0.000 1.000 0.447 0.500 1.000 1.000

0.500 0.250 0.015 0.020 0.467 0.000 1.000 0.467 0.500 1.000 1.000

0.500 0.250 0.025 0.020 0.457 0.000 1.000 0.457 0.500 1.000 1.000

0.500 0.250 0.035 0.020 0.447 0.000 1.000 0.447 0.500 1.000 1.000

0.500 0.250 0.015 0.020 0.467 0.000 1.000 0.467 0.500 1.000 1.000

0.500 0.250 0.025 0.020 0.457 0.000 1.000 0.457 0.500 1.000 1.000

0.500 0.250 0.035 0.020 0.447 0.000 1.000 0.447 0.500 1.000 1.000

0.500 0.250 0.015 0.020 0.467 0.010 1.056 0.499 0.542 1.204 1.083

0.500 0.250 0.025 0.020 0.457 0.010 1.056 0.488 0.541 1.199 1.082

0.500 0.250 0.035 0.020 0.447 0.010 1.056 0.477 0.540 1.195 1.081

0.500 0.250 0.015 0.020 0.467 0.010 1.024 0.481 0.524 1.121 1.047

0.500 0.250 0.025 0.020 0.457 0.010 1.024 0.470 0.523 1.119 1.046

0.500 0.250 0.035 0.020 0.447 0.010 1.024 0.460 0.523 1.117 1.046

0.500 0.250 0.015 0.020 0.467 0.010 1.036 0.487 0.530 1.152 1.061

0.500 0.250 0.025 0.020 0.457 0.010 1.036 0.477 0.530 1.149 1.060

0.500 0.250 0.035 0.020 0.447 0.010 1.036 0.467 0.530 1.146 1.059

0.500 0.250 0.015 0.020 0.467 0.015 1.112 0.531 0.579 1.393 1.157

0.500 0.250 0.025 0.020 0.457 0.015 1.112 0.519 0.577 1.384 1.155

0.500 0.250 0.035 0.020 0.447 0.015 1.112 0.508 0.576 1.374 1.152

0.500 0.250 0.015 0.020 0.467 0.015 1.048 0.494 0.542 1.216 1.084

0.500 0.250 0.025 0.020 0.457 0.015 1.048 0.483 0.541 1.212 1.083

0.500 0.250 0.035 0.020 0.447 0.015 1.048 0.473 0.541 1.208 1.082

0.500 0.250 0.015 0.020 0.467 0.015 1.072 0.508 0.556 1.281 1.112

0.500 0.250 0.025 0.020 0.457 0.015 1.072 0.497 0.555 1.275 1.110

0.500 0.250 0.035 0.020 0.447 0.015 1.072 0.486 0.554 1.269 1.108

A.50

Table M-2: Equivalent unit in RAC in function of α6/α2 (∆c = 0.020 m, 0.025 m, 0.035 m, 0.040 m) (beams)

hRC

(m) b (m)

cRC

(m) Ø

(m) dRC (m)

∆cbeams (m)

(pqp,RAC/ pqp,RAC)slabs

dRAC (m) hRAC (m)

α6/α2 hRAC/hRC

0.500 0.250 0.015 0.020 0.467 0.020 1.169 0.563 0.616 1.600 1.232

0.500 0.250 0.025 0.020 0.457 0.020 1.169 0.551 0.614 1.584 1.228

0.500 0.250 0.035 0.020 0.447 0.020 1.169 0.539 0.612 1.568 1.224

0.500 0.250 0.015 0.020 0.467 0.020 1.072 0.508 0.561 1.315 1.121

0.500 0.250 0.025 0.020 0.457 0.020 1.072 0.497 0.560 1.309 1.120

0.500 0.250 0.035 0.020 0.447 0.020 1.072 0.486 0.559 1.303 1.118

0.500 0.250 0.015 0.020 0.467 0.020 1.109 0.529 0.582 1.419 1.163

0.500 0.250 0.025 0.020 0.457 0.020 1.109 0.517 0.580 1.410 1.160

0.500 0.250 0.035 0.020 0.447 0.020 1.109 0.506 0.579 1.400 1.158

0.500 0.250 0.015 0.020 0.467 0.025 1.225 0.596 0.654 1.824 1.307

0.500 0.250 0.025 0.020 0.457 0.025 1.225 0.583 0.651 1.801 1.302

0.500 0.250 0.035 0.020 0.447 0.025 1.225 0.570 0.648 1.778 1.296

0.500 0.250 0.015 0.020 0.467 0.025 1.096 0.521 0.579 1.419 1.159

0.500 0.250 0.025 0.020 0.457 0.025 1.096 0.510 0.578 1.411 1.156

0.500 0.250 0.035 0.020 0.447 0.025 1.096 0.499 0.577 1.402 1.154

0.500 0.250 0.015 0.020 0.467 0.025 1.145 0.549 0.607 1.565 1.215

0.500 0.250 0.025 0.020 0.457 0.025 1.145 0.538 0.606 1.552 1.211

0.500 0.250 0.035 0.020 0.447 0.025 1.145 0.526 0.604 1.538 1.208

0.500 0.250 0.015 0.020 0.467 0.035 1.281 0.629 0.697 2.111 1.393

0.500 0.250 0.025 0.020 0.457 0.035 1.281 0.615 0.693 2.079 1.386

0.500 0.250 0.035 0.020 0.447 0.035 1.281 0.602 0.690 2.048 1.379

0.500 0.250 0.015 0.020 0.467 0.035 1.120 0.535 0.603 1.567 1.206

0.500 0.250 0.025 0.020 0.457 0.035 1.120 0.524 0.602 1.556 1.203

0.500 0.250 0.035 0.020 0.447 0.035 1.120 0.512 0.600 1.544 1.200

0.500 0.250 0.015 0.020 0.467 0.035 1.181 0.570 0.638 1.761 1.277

0.500 0.250 0.025 0.020 0.457 0.035 1.181 0.558 0.636 1.743 1.272

0.500 0.250 0.035 0.020 0.447 0.035 1.181 0.546 0.634 1.725 1.268

0.500 0.250 0.015 0.020 0.467 0.040 1.337 0.662 0.735 2.373 1.470

0.500 0.250 0.025 0.020 0.457 0.040 1.337 0.648 0.731 2.333 1.461

0.500 0.250 0.035 0.020 0.447 0.040 1.337 0.633 0.726 2.294 1.453

0.500 0.250 0.015 0.020 0.467 0.040 1.144 0.549 0.622 1.682 1.244

0.500 0.250 0.025 0.020 0.457 0.040 1.144 0.537 0.620 1.668 1.240

0.500 0.250 0.035 0.020 0.447 0.040 1.144 0.525 0.618 1.654 1.237

0.500 0.250 0.015 0.020 0.467 0.040 1.217 0.591 0.664 1.927 1.329

0.500 0.250 0.025 0.020 0.457 0.040 1.217 0.579 0.662 1.904 1.323

0.500 0.250 0.035 0.020 0.447 0.040 1.217 0.566 0.659 1.881 1.318

A.51

Annex N: Results of the equivalent functional unit in RAC, concerning bending

(beams)

Tables N-1 and N-2 correspond to Tables F-3 and F-4 of slabs; the ratio pqp,RAC/pqp,RC of the slabs is used to determine the results of the beams.


hRC

(m) b (m)

cRC (m)

Ø (m)

dRC (m)

MEd,RC

(kNm/m) µRC ωRC

As,RC (cm²)

∆c (m)


dRAC (m)

hRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m) As,RC -

As,RAC (%) α1 hRAC/hRC

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 1.000 0.467 0.500 227.176 0.250 0.313 13.986 0.000 1.000 1.000 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 1.000 0.457 0.500 217.551 0.250 0.313 13.686 0.000 1.000 1.000 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 1.000 0.447 0.500 208.134 0.250 0.313 13.387 0.000 1.000 1.000 0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 1.000 0.467 0.500 227.176 0.250 0.313 13.986 0.000 1.000 1.000 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 1.000 0.457 0.500 217.551 0.250 0.313 13.686 0.000 1.000 1.000 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 1.000 0.447 0.500 208.134 0.250 0.313 13.387 0.000 1.000 1.000 0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.000 1.000 0.467 0.500 227.176 0.250 0.313 13.986 0.000 1.000 1.000 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.000 1.000 0.457 0.500 217.551 0.250 0.313 13.686 0.000 1.000 1.000 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.000 1.000 0.447 0.500 208.134 0.250 0.313 13.387 0.000 1.000 1.000

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.010 1.044 0.492 0.535 237.199 0.235 0.291 13.703 2.023 0.907 1.070 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.010 1.044 0.481 0.534 227.149 0.235 0.291 13.409 2.023 0.907 1.069 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.010 1.044 0.471 0.534 217.317 0.235 0.291 13.116 2.023 0.907 1.068 0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.010 1.020 0.478 0.521 231.740 0.243 0.302 13.854 0.944 0.956 1.043 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.010 1.020 0.468 0.521 221.921 0.243 0.302 13.557 0.944 0.956 1.042 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.010 1.020 0.458 0.521 212.316 0.243 0.302 13.260 0.944 0.956 1.042 0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.010 1.027 0.482 0.525 233.261 0.241 0.299 13.811 1.250 0.942 1.050 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.010 1.027 0.472 0.525 223.378 0.241 0.299 13.515 1.250 0.942 1.049 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.010 1.027 0.461 0.524 213.709 0.241 0.299 13.219 1.250 0.942 1.049

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.015 1.088 0.517 0.565 247.221 0.222 0.271 13.444 3.872 0.828 1.130 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.015 1.088 0.506 0.564 236.747 0.222 0.271 13.156 3.872 0.828 1.128 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.015 1.088 0.495 0.563 226.499 0.222 0.271 12.868 3.872 0.828 1.125 0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.015 1.040 0.490 0.538 236.304 0.237 0.293 13.727 1.849 0.915 1.075 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.015 1.040 0.479 0.537 226.292 0.237 0.293 13.433 1.849 0.915 1.074 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.015 1.040 0.469 0.537 216.497 0.237 0.293 13.139 1.849 0.915 1.073 0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.015 1.054 0.497 0.545 239.346 0.232 0.286 13.645 2.433 0.889 1.090 0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.015 1.054 0.487 0.545 229.206 0.232 0.286 13.353 2.433 0.889 1.089 0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.015 1.054 0.476 0.544 219.284 0.232 0.286 13.061 2.433 0.889 1.088

A.52


hRC

(m) b (m)

cRC (m)

Ø (m)

dRC (m)

MEd,RC

(kNm/m) µRC ωRC

As,RC (cm²)

∆c (m)


dRAC (m)

hRAC (m)

MEd,RAC

(kNm/m) µRAC ωRAC

As,RAC

(cm²/m) As,RC - As,RAC

(%) α1 hRAC/hRC

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 1.132 0.542 0.595 257.243 0.210 0.254 13.207 5.571 0.761 1.190

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 1.132 0.531 0.594 246.345 0.210 0.254 12.924 5.571 0.761 1.187

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 1.132 0.519 0.592 235.682 0.210 0.254 12.641 5.571 0.761 1.184

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 1.060 0.501 0.554 240.867 0.230 0.283 13.605 2.719 0.877 1.108

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 1.060 0.490 0.553 230.662 0.230 0.283 13.314 2.719 0.877 1.106

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 1.060 0.480 0.553 220.678 0.230 0.283 13.023 2.719 0.877 1.105

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.020 1.080 0.512 0.565 245.431 0.224 0.275 13.489 3.554 0.842 1.131

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.020 1.080 0.501 0.564 235.033 0.224 0.275 13.200 3.554 0.842 1.129

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.020 1.080 0.490 0.563 224.859 0.224 0.275 12.911 3.554 0.842 1.127

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.025 1.176 0.568 0.626 267.266 0.199 0.239 12.987 7.138 0.702 1.251

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.025 1.176 0.555 0.623 255.942 0.199 0.239 12.709 7.138 0.702 1.247

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.025 1.176 0.543 0.621 244.864 0.199 0.239 12.431 7.138 0.702 1.243

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.025 1.080 0.512 0.570 245.431 0.224 0.275 13.489 3.554 0.842 1.141

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.025 1.080 0.501 0.569 235.033 0.224 0.275 13.200 3.554 0.842 1.139

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.025 1.080 0.490 0.568 224.859 0.224 0.275 12.911 3.554 0.842 1.137

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.025 1.107 0.528 0.586 251.516 0.217 0.264 13.340 4.618 0.798 1.171

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.025 1.107 0.516 0.584 240.860 0.217 0.264 13.054 4.618 0.798 1.169

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.025 1.107 0.505 0.583 230.434 0.217 0.264 12.769 4.618 0.798 1.166

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.035 1.221 0.593 0.661 277.288 0.189 0.225 12.785 8.588 0.650 1.322

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.035 1.221 0.580 0.658 265.540 0.189 0.225 12.511 8.588 0.650 1.317

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.035 1.221 0.568 0.656 254.046 0.189 0.225 12.237 8.588 0.650 1.312

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.035 1.100 0.524 0.592 249.995 0.219 0.266 13.376 4.357 0.809 1.184

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.035 1.100 0.513 0.591 239.403 0.219 0.266 13.090 4.357 0.809 1.181

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.035 1.100 0.501 0.589 229.041 0.219 0.266 12.803 4.357 0.809 1.179

0.500 0.250 0.015 0.020 0.467 227.176 0.250 0.313 13.986 0.035 1.134 0.543 0.611 257.601 0.210 0.254 13.198 5.629 0.758 1.222

0.500 0.250 0.025 0.020 0.457 217.551 0.250 0.313 13.686 0.035 1.134 0.531 0.609 246.687 0.210 0.254 12.916 5.629 0.758 1.219

0.500 0.250 0.035 0.020 0.447 208.134 0.250 0.313 13.387 0.035 1.134 0.520 0.608 236.010 0.210 0.254 12.633 5.629 0.758 1.216

A.53

Annex O: Tables with design results (beams)

Only one example of simply supported beams and one example of continuous beams is showed

because including all the examples would make this part too extensive. The two examples represent

cases in which the fundamental parameters are not available and correspond to Tables G-4, G-5 and

G-6 . If the parameters are available, similar results are obtained. Tables O-1, O-2 and O-3 show the

results of a simply supported RC beam. Tables O-4, O-5 and O-6 demonstrate the same but for a

continuous RC beam.

Table O-1: Design of simply supported beams (fundamental parameters and data)

Simply supported beam, L = 5 m, Amorim et al. (2012)

Mixture RC RAC (20%) RAC (50%) RAC (100%)


α1(available) 1 0.994 0.977 0.955

α2 1 0.869 0.865 0.859

α3 (available) 1 1.007 1.163 1.248

α4 (available) 1 1.033 0.974 0.962

α5 1 0.996 0.984 0.970

α6 1 1.009 1.036 1.069

Data

fcm, cylinder (MPa) 41.280 48.762 48.047 47.172

fck (MPa) research 33.280 40.762 40.047 39.172



fck,cylinder and fck,cube (MPa) EC2 30 and 37 / / /

Ecm (GPa) 33.000 28.693 28.541 28.352


Ec,eff (GPa)= Ecm/ ((creep coefficient+1)*α6)

9.429 8.125 7.871 7.577

Es (GPa) 200.000 200.000 200.000 200.000

fctm (MPa) 2.900 2.889 2.855 2.813


XD1/XS1 XC2/XC3 and

XD1/XS1 XC2/XC3 and

XD1/XS1 XC2/XC3 and

XD1/XS1

Structural class for cmin S4 S4 S4 S4

Minimum cover cmin (m) in function of XC2/XC3

0.025 0.025 0.025 0.025


0.035 0.035 0.035 0.035

hRAC/hRC because of α1 1.000 1.003 1.014 1.027



hRAC/hRC because of α6/α2 1.000 1.064 1.079 1.098

K-value 1.000 1.064 1.079 1.098

A.54

Table O-2: Design of simply supported beams (bending ULS)

Bending ULS

fcd (MPa) 27.333 27.174 26.698 26.115

fyk (MPa) 500.000 500.000 500.000 500.000

fyd (MPa) 434.783 434.783 434.783 434.783

h (m) 0.450 0.479 0.486 0.494

real h (m) 0.450 0.480 0.490 0.490

b (m) 0.220 0.220 0.220 0.220

L (m) 5.000 5.000 5.000 5.000

gbeam = ((hbeam-hslab)*b*25) (kN/m) 1.705 1.815 1.870 1.870

pslab,Ed = (spanslab*1/2)*2*pEd (kN/m) 62.550 64.575 64.575 64.575

pslab,qp = (spanslab*1/2)*2*pqp (kN/m) 36.600 38.100 38.100 38.100

pEd (kN/m) 64.852 67.025 67.100 67.100

pqp (kN/m) 38.305 39.915 39.970 39.970

MEd,support = 0 (kNm) 0.000 0.000 0.000 0.000

MEd,midspan = pEdl²/8 (kNm) 202.662 209.454 209.686 209.686

Mqp,support = 0 (kNm) 0.000 0.000 0.000 0.000

Mqp,midspan =pqp*l²/8 (kNm) 119.703 124.734 124.906 124.906


0.035 0.036 0.034 0.034

Cover c (m) 0.040 0.041 0.039 0.039

Ø (m) 0.020 0.020 0.020 0.020

d = h - c - Ø/2 - Østirb (m) 0.392 0.421 0.433 0.433

µd = MEd,midspan/(fcd*b*d²) 0.219 0.198 0.190 0.194

ω = µd * (1+µd) 0.267 0.237 0.227 0.232

As = ω*fcd*b*d/fyd (cm²/m) 14.499 13.712 13.262 13.293

REAL As 5Ø20 5Ø20 5Ø20 5Ø20

As (cm²/m) 15.710 15.710 15.710 15.710

A.55

Table O-3: Design of simply supported beams (deformation SLS and cracking SLS)

Deformation SLS

y (neutral axis) (m) 0.266 0.287 0.294 0.296

II (m4) 0.002 0.003 0.003 0.003

II (cm4) 234113.267 292542.516 315586.299 318970.728

As*(d-x)*Es/(Ec,eff)-b*x2/2=0 0.000 0.000 0.000 0.000

x (m) 0.225 0.247 0.254 0.257

III (m4) 0.002 0.002 0.002 0.003

III (cm4) 176469.214 227380.982 247971.168 253408.644

w=II/(h-y) (m³) 0.013 0.015 0.016 0.016

Mcr = fctm * w (kNm) 36.809 43.791 46.040 46.223

β 0.500 0.500 0.500 0.500

Mqp,midspan (kNm) 119.703 124.734 124.906 124.906

ξ = 1-β*(Mcr/M)² 0.953 0.938 0.932 0.932

I = ξ*III+(1-ξ)*II (m4) 0.002 0.002 0.003 0.003

I (cm4) 179194.620 231396.600 252564.410 257897.875

δ (m) 0.018 0.017 0.016 0.017

L/250 (m) 0.020 0.020 0.020 0.020

L/250 (mm) 20.000 20.000 20.000 20.000

δ (mm) 18.450 17.278 16.363 16.645


Cracking SLS

φ (m) 0.020 0.020 0.020 0.020

As*(d-x)*Es/(Ec,eff)-b*x2/2=0 0.000 0.000 0.000 0.000

x (m) 0.253 0.247 0.254 0.257

Ac,eff (m²) 0.014 0.017 0.017 0.017

ρp.eff = As/Ac,eff 0.109 0.092 0.091 0.092

k1 0.800 0.800 0.800 0.800

k2 0.500 0.500 0.500 0.500

k3 3.400 3.400 3.400 3.400

k4 0.425 0.425 0.425 0.425

Mqp,midspan (kNm) 119.703 124.734 124.906 124.906

σs (kN/m²) 247541.270 234580.951 228404.097 228794.058

kt 0.400 0.400 0.400 0.400

αe 6.061 6.970 7.008 7.054

fct,eff = fctm (MPa) 2.900 2.889 2.855 2.813

wmax (m) 0.000 0.000 0.000 0.000

wmax (mm) 0.300 0.300 0.300 0.300

sr,max=k3*c+k1*k2*k4*φ/ρp.eff (m) 0.167 0.177 0.170 0.168

εsm-εcm 0.001 0.001 0.001 0.001

wk=sr,max*(εsm-εcm) (m) 0.000 0.000 0.000 0.000

wk (mm) 0.192 0.189 0.177 0.176


A.56

Table O-4: Design of continuous beams (fundamental parameters and data)

Continuous beam, L = 5 m, Amorim et al. (2012)

Mixture RC RAC (20%) RAC (50%) RAC (100%)


α1 (available) 1 0.994 0.977 0.955

α2 1 0.869 0.865 0.859

α3 (available) 1 1.007 1.163 1.248

α4 (available) 1 1.033 0.974 0.962

α5 1 0.996 0.984 0.970

α6 1 1.009 1.036 1.069

Data

fcm, cylinder (MPa) 41.280 48.762 48.047 47.172

fck (MPa) research 33.280 40.762 40.047 39.172



fck,cylinder and fck,cube (MPa) EC2 30 and 37 / / /

Ecm (GPa) 33.000 28.693 28.541 28.352


Ec,eff (GPa) = Ecm/((creep coefficient+1)*α6)

9.429 8.125 7.871 7.577

Es (GPa) 200.000 200.000 200.000 200.000

fctm (MPa) 2.900 2.889 2.855 2.813


XD1/XS1 XC2/XC3 and

XD1/XS1 XC2/XC3 and

XD1/XS1 XC2/XC3 and

XD1/XS1

Structural class for cmin S4 S4 S4 S4

Minimum cover cmin (m) in function of XC2/XC3

0.025 0.025 0.025 0.025


0.035 0.035 0.035 0.035




hRAC/hRC because of α6/α2 1.000 1.064 1.079 1.098

K-value 1.000 1.064 1.079 1.098

A.57

Table O-5: Design of continuous beams (bending ULS)

Bending ULS

fcd (MPa) 27.333 27.174 26.698 26.115

fyk (MPa) 500.000 500.000 500.000 500.000

fyd (MPa) 434.783 434.783 434.783 434.783

h (m) 0.400 0.426 0.432 0.439

real h (m) 0.400 0.430 0.430 0.440

b (m) 0.200 0.200 0.200 0.200

L (m) 5.000 5.000 5.000 5.000

gbeam = (hbeam-hslab)*b*25 (kN/m) 1.300 1.400 1.400 1.450

pslab,Ed = (spanslab*1/2)*2*pEd (kN/m) 62.550 64.575 64.575 64.575

pslab,qp = (spanslab*1/2)*2*pqp (kN/m) 36.600 38.100 38.100 38.100

pEd (kN/m) 64.305 66.465 66.465 66.533

pqp (kN/m) 37.900 39.500 39.500 39.550

MEd,support = pEd*l²/12(kNm) 133.969 138.469 138.469 138.609

MEd,midspan = pEd*l²/24 (kNm) 66.984 69.234 69.234 69.305

Mqp,support = pqp*l²/12(kNm) 78.958 82.292 82.292 82.396

Mqp,midspan = pqp*l²/24(kNm) 39.479 41.146 41.146 41.198


0.035 0.036 0.034 0.034

Cover c (m) 0.040 0.041 0.039 0.039

Ø (m) 0.016 0.016 0.016 0.016

d = h - c - Ø/2 - Østirb (m) 0.344 0.373 0.375 0.385

µd = MEd,support/(fcd*b*d²) 0.207 0.183 0.184 0.179

ω = µd * (1+µd) 0.250 0.217 0.219 0.211

As = ω*fcd*b*d/fyd (cm²/m) 10.812 10.107 10.061 9.752

REAL As 6Ø16 5Ø16 5Ø16 5Ø16

As (cm²/m) 12.060 10.050 10.050 10.050

A.58

Table O-6: Design of continuous beams (deformation SLS and cracking SLS)

Deformation SLS

y (neutral axis) (m) 0.234 0.249 0.250 0.257

II (m4) 0.001 0.002 0.002 0.002

II (cm4) 145407.975 178855.027 181330.759 196052.045

As*(d-x)*Es/(Ec,eff)-b*x2/2=0 0.000 0.000 0.000 0.000

x (m) 0.195 0.204 0.207 0.213

III (m4) 0.001 0.001 0.001 0.001

III (cm4) 106226.564 127124.772 131145.310 143202.087

w=II/(h-y) (m³) 0.009 0.010 0.010 0.011

Mcr = fctm * w (kNm) 25.346 28.567 28.839 30.166

β 0.500 0.500 0.500 0.500

Mqp,support (kNm) 78.958 82.292 82.292 82.396

ξ = 1-β*(Mcr/M)² 0.948 0.940 0.939 0.933

I = ξ*III+(1-ξ)*II (m4) 0.001 0.001 0.001 0.001

I (cm4) 108245.276 130241.788 134227.063 146744.039

δ (m) 0.006 0.006 0.006 0.006

L/250 (m) 0.020 0.020 0.020 0.020

L/250 (mm) 20.000 20.000 20.000 20.000

δ (mm) 6.044 6.076 6.085 5.789


Cracking SLS

φ (m) 0.016 0.016 0.016 0.016

As*(d-x)*Es/(Ec,eff)-b*x2/2=0 0.000 0.000 0.000 0.000

x (m) 0.195 0.204 0.207 0.213

Ac,eff (m²) 0.014 0.015 0.015 0.015

ρp.eff = As/Ac,eff 0.088 0.067 0.068 0.067

k1 0.800 0.800 0.800 0.800

k2 0.500 0.500 0.500 0.500

k3 3.400 3.400 3.400 3.400

k4 0.425 0.425 0.425 0.425

Mqp,support (kNm) 78.958 82.292 82.292 82.396

σs (kN/m²) 234703.894 268673.402 267668.693 260964.504

kt 0.400 0.400 0.400 0.400

αe 6.061 6.970 7.008 7.054

fct,eff = fctm (MPa) 2.900 2.889 2.855 2.813

wmax (m) 0.000 0.000 0.000 0.000

wmax (mm) 0.300 0.300 0.300 0.300

sr,max=k3*c+k1*k2*k4*φ/ρp.eff (m) 0.167 0.181 0.173 0.172

εsm-εcm 0.001 0.001 0.001 0.001

wk=sr,max*(εsm-εcm) (m) 0.000 0.000 0.000 0.000

wk (mm) 0.179 0.220 0.210 0.203


Definition of an equivalent functional unit for structural … · 2015. 12. 5. · Giles Dobbelaere...

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