Design Examples Steel Reinforced Concrete...

Design Examples for High Strength Steel Reinforced

Concrete ColumnsA Eurocode 4 Approach

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Design Examples for High Strength Steel Reinforced

Concrete ColumnsA Eurocode 4 Approach

Sing-Ping ChiewYan-Qing Cai

CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

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Library of Congress Cataloging-in-Publication Data

Names: Chiew, Sing-Ping, author. | Cai, Y. Q. (Yan Qing), author.Title: Design of high strength steel reinforced concrete columns : a Eurocode 4 approach / S.P. Chiew and Y.Q. Cai.Description: Boca Raton : CRC Press, [2018] | Includes bibliographicalreferences and index. Identifiers: LCCN 2017057555 (print) | LCCN 2018000768 (ebook) |ISBN 9781351203944 (Adobe PDF) | ISBN 9781351203937 (ePub) |ISBN 9781351203920 (Mobipocket) | ISBN 9780815384601(hardback : acid-free paper) | ISBN 9781351203951 (ebook)Subjects: LCSH: Composite construction--Specifications--Europe. | Building, Iron and steel--Specifications--Europe. | Reinforced concrete construction--Specifications--Europe.Classification: LCC TA664 (ebook) | LCC TA664 .C48 2018 (print) | DDC 624.1/83425--dc23LC record available at https://lccn.loc.gov/2017057555

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v

Contents

List of symbols xi Preface xiii Authors xv

Design examples 1

Steel-reinforced concrete column subjected to axial compression 1 Steel-reinforced concrete column with normal- strength material 2

Design data 2Design strengths and modulus 2Cross-sectional areas 3Second moments of area 3Check the reinforcement ratio 4Check the local buckling 4Check the steel contribution factor 4Long-term effects 5 Elastic modulus of concrete considering long-term effects 6Effective flexural stiffness of cross-section 6Elastic critical normal force 7Relative slenderness ratio 7Buckling reduction factor 7Buckling resistance 8

Steel-reinforced concrete column with high-strength concrete 8Design strengths and modulus 8 Cross-sectional areas and second moments of area 9Check the steel contribution factor 9Long-term effects 10

vi Contents

Elastic modulus of concrete considering long-term effects 11Effective flexural stiffness of cross-section 11Elastic critical normal force 11Relative slenderness ratio 12Buckling reduction factor 12Buckling resistance 13

Steel-reinforced concrete column with high-strength steel 13Design strengths and modulus 13 Cross-sectional areas and second moments of area 16Check the steel contribution factor 16Long-term effects 16 Elastic modulus of concrete considering long-term effects 16Effective flexural stiffness of cross-section 17Elastic critical normal force 17Relative slenderness ratio 17Buckling reduction factor 17Buckling resistance 18

Steel-reinforced concrete column with high-strength concrete and steel 18

Design strengths and modulus 19 Cross-sectional areas and second moments of area 19Check the steel contribution factor 19Long-term effects 19 Elastic modulus of concrete considering long-term effects 20Effective flexural stiffness of cross-section 20Elastic critical normal force 20Relative slenderness ratio 20Buckling reduction factor 20Buckling resistance 21Alternative design 22Design data 22Design strengths and modulus 22 Cross-sectional area and second moments of area 23Check the reinforcement ratio 23Check the local buckling 23Check the steel contribution factor 23Long-term effects 24

Contents vii

Elastic modulus of concrete considering long-term effects 25Effective flexural stiffness of cross-section 25Elastic critical normal force 25Relative slenderness ratio 26Buckling reduction factor 26Buckling resistance 26

Steel-reinforced concrete column subjected to combined compression and bending 27 Steel-reinforced concrete column with normal-strength material 27

Design data 27Design strengths and modulus 28 Cross-sectional areas and second moments of area 29Check the reinforcement ratio 29Check the local buckling 29Check the steel contribution factor 29Long-term effects 30 Elastic modulus of concrete considering long-term effects 31Effective flexural stiffness of cross-section 31Elastic critical normal force 32Relative slenderness ratio 32Buckling reduction factor 32Buckling resistance 33Interaction curve 33 Check the resistance of steel-reinforced concrete column in combined compression and uniaxial bending 35

Steel-reinforced concrete column with high-strength concrete 38Design strengths and modulus 38 Cross-sectional areas and second moments of area 38Check the steel contribution factor 38Long-term effects 39 Elastic modulus of concrete considering long-term effects 40Effective flexural stiffness of cross-section 40Elastic critical normal force 41Relative slenderness ratio 41Buckling reduction factor 41

viii Contents

Buckling resistance 42Interaction curve 42 Check the resistance of steel-reinforced concrete column in combined compression and uniaxial bending 44

Steel-reinforced concrete column with high-strength steel 46Design strengths and modulus 47 Cross-sectional areas and second moments of area 49Check the steel contribution factor 49Long-term effects 49 Elastic modulus of concrete considering long-term effects 50Effective flexural stiffness of cross-section 50Elastic critical normal force 50Relative slenderness ratio 50Buckling reduction factor 51Buckling resistance 51Interaction curve 52 Check the resistance of steel-reinforced concrete column in combined compression and uniaxial bending 54

Steel-reinforced concrete column with high-strength materials 56

Design strengths and modulus 56 Cross-sectional areas and second moments of area 56Check the steel contribution factor 56Long-term effects 57 Elastic modulus of concrete considering long-term effects 57Effective flexural stiffness of cross-section 57Elastic critical normal force 57Relative slenderness ratio 58Buckling reduction factor 58Buckling resistance 58Interaction curve 59 Check the resistance of steel-reinforced concrete column in combined compression and uniaxial bending 61

Steel-reinforced concrete column with different degree of confinement 63Original design 63

Contents ix

High-strength concrete 63 High-strength steel and high-strength concrete 64

Appendix A: Design resistance of shear connectors 69Appendix B: Design chart 73Index 77

xi

List of symbols

Aa Area of the structural steelAc Area of concreteAch Area of highly confined concreteAcp Area of partially confined concreteAcu Area of unconfined concreteAs Area of reinforcementEa Modulus of elasticity of structural steelEc,eff Effective modulus of elasticity of concreteEcm Secant modulus of elasticity of concreteEc(t) Tangent modulus of elasticity of concrete at time tEs Modulus of elasticity of reinforcement(EI)eff Effective flexural stiffnessGa Shear modulus of structural steelI Second moment of area of the composite sectionIa Second moment of area of the structural steelIc Second moment of area of the concreteIs Second moment of area of the reinforcementKe Correction factorL LengthMEd Design bending momentMpl,a,Rd The plastic resistance moment of the structural steelMpl,Rd The plastic resistance moment of the composite sectionNcr Elastic critical force in composite columnsNEd The compressive normal forceNpl,Rd The plastic resistance of the composite sectionNpl,Rk Characteristic value of the plastic resistance of the

composite sectionNpm,Rd The resistance of the concrete to compressive normal forcePRd The resistance of per shear stud

xii List of symbols

VEd The shear forceVpl,a,Rd The shear resistance of the steel sectionWpa The plastic section modulus of the structural steelWpc The plastic section modulus of the concreteWps The plastic section modulus of the reinforcing steelbc Width of the composite sectionbf Width of the steel flangecy, cz Thickness of concrete coverd Diameter of shank of the headed stude Eccentricity of loadingfck The cylinder compressive strength of concretefcd The design strength of concretefc,p The compressive strength of partially confined concretefc,h The compressive strength of highly confined concretefs The yield strength of reinforcementfu Tensile strengthfy The yield strength of structural steelfyd The design strength of structural steelfyh The yield strength of transverse reinforcementha Depth of steel sectionhc Depth of composite sectionhn Distance from centroidal axis to neutral axishsc Overall nominal height of the headed studs Spacing center-to-center of linkstf Thickness of steel flangetw Thickness of the steel webΔσ Stress rangeΨ Coefficientα Coefficient; factorβ Factor; coefficientγ Partial factorδ Steel contribution ratioη Coefficientεc u, Strain of unconfined concreteεc,p Strain of partially confined concreteεc,h Strain of highly confined concreteµ Factors related to bending momentsλ Relative slendernessρs Reinforcement ratioχ Reduction factor of bucklingϕ Creep coefficient

xiii

Preface

This book is the companion volume to Design of High Strength Steel Reinforced Concrete Columns—A Eurocode 4 Approach.

Guidance is much needed on the design of high strength steel reinforced concrete (SRC) columns beyond the remit of Eurocode 4 for composite steel concrete structures. Given the much narrower range of permitted concrete and steel material strengths in comparison to Eurocode 2 for concrete structures and Eurocode 3 for steel structures, and the better ductility and buckling resistance of SRC columns compared to steel or reinforced concrete, there is a clear need for design beyond the current guidelines. The design principles to do so are set out in the companion volume to this book, Design of High Strength Steel Reinforced Concrete Columns—A Eurocode 4 Approach. This book provides a number of design examples for high strength SRC columns using these principles which are based on the Eurocode 4 approach. Special considerations are given to resistance calculations that maximize the full strength of the materials, with concrete cylinder strength up to 90 N/mm2, yield strength of structural steel up to 690 N/mm2 and yield strength of reinforcing steel up to 600 N/mm2 respectively. These design examples will allow the readers to practice and understand the Eurocode 4 methodology easily.

Structural engineers and designers who are familiar with basic Eurocode 4 design should find these design examples particularly helpful, whilst civil engineering students who are studying composite steel concrete design and construction should gain further understanding from working through the design examples which are set out clearly in a step-by-step fashion.

xv

Authors

Sing-Ping Chiew is a professor and the Civil Engineering Programme director at the Singapore Institute of Technology, Singapore, and coauthor of Structural Steelwork Design to Limit State Theory, 4th Edition.

Yan-Qing Cai is a project officer in the School of Civil and Environmental Engineering at Nanyang Technological University, Singapore.

1

Design examples

STEEL-REINFORCED CONCRETE COLUMN SUBJECTED TO AXIAL COMPRESSION

Determine the axial buckling resistance of SRC columns (concrete-encased I-section) subject to pure compression with an effective length of 4 m, as shown in Figure 1.

bc

cy cy

tw

h cc z

c z

l f

h

b

z

y

Figure 1 Cross-section of SRC column.

2 Design Examples for High Strength Steel Reinforced Concrete Columns

Steel-reinforced concrete column with normal-strength material

Design data

Design strengths and modulus

According to Tables 2.12 and 2.13 in the companion book, for the strain compatibility between steel and concrete, the two materials (concrete

Design axial force NEd = 8000 kNPermanent load NG,Ed = 4000 kNColumn length L = 4.0 mEffective length Leff = 4.0 mStructural steel Grade S355, fy = 355 N/mm2

Concrete C30/37, fck = 30 N/mm2

Reinforcement fsk = 500 N/mm2

Properties of cross-section

Concrete depth hc = 500 mmConcrete width bc = 500 mmConcrete cover c = 30 mmCover cy = 95.4 mmCover cz = 89.8 mm

Section properties of steel section305 × 305 UC 137

Depth h = 320.5 mmWidth b = 309.2 mmFlange thickness tf = 21.7 mmWeb thickness tw = 13.8 mmFillet r = 15.2 mmSection area Aa = 174.4 cm2

Second moment of area/y Iay = 32810 cm4

Second moment of area/z Iaz = 10700 cm4

Plastic section modulus/y Wpl,a,y = 2297 cm3

Plastic section modulus/z Wpl,a,z = 1053 cm3

Reinforcement

Longitudinal reinforcement number n = 8, diameter dl,s = 20 mmTransverse reinforcement diameter dt,s = 10 mm, spacing

s = 200 mm

Design examples 3

class C30/37 and steel grade S355) are strain compatible. Therefore, the steel section can reach its full strength when the composite concrete section reaches its ultimate strength, without considering the confinement effect from the lateral hoops and steel section.

The design strengths of the steel, concrete, and reinforcement are:

f

fyd

y

M

2N/mm=γ= =

3551 0

355.

f

fcd

ck

C

220 N/mm=γ= =

301 5.

f

fsd

sk

S

2N/mm=γ= =

5001 15

435.

Ecm = 33 Gpa

Ea = 210 Gpa

Cross-sectional areas

The cross-sectional areas of the steel, reinforcement, and concrete are:

Aa2mm=17 440,

As

2mm=× ×

=8 20

42512

2π

A b h A Ac c a s2251 mm= − − = × − − =c , ,500 500 17 440 2 230 048

Second moments of area

Iay4mm= ×328 1 106.

Iaz4mm= ×107 106

I A e

i

n

sy s,i i4mm= = ×

×× = ×

=∑ 2

1

42 66

204

200 73 56 10π

.


I A e

i

n

sz s,i i4mm= = ×

×× = ×

=∑ 2

1

42 66

204

200 73 56 10π

.

Ib h

I Icyc c

ay sy= − −

=×

− × − × = ×

3

36 6

12

500 50012

328 1 10 73 56 10 4805 10. . 66 mm4

Ih b

I Iczc c

az sz

m

= − −

=×

− × − × = ×

3

36 6 6

12

500 50012

107 10 73 56 10 5026 10. mm4

Check the reinforcement ratio

ρss= = =

A

Ac

2512230 048

1 1,

. %

The reinforcement ratio is within the range 0.3%–6%.

Check the local buckling

The concrete cover to the flange of the steel section: c = 89.8 mm > maximum (40 mm; bf/6).

Thus, the effect of local buckling is neglected for the SRC column.

Check the steel contribution factor

The design plastic resistance of the composite cross-section in compression is:

N A f A f A fpl,Rd a yd c cd s sd.

. 2

= + +

= × × × +

0 85

17 440 355 0 85 230 048 20( , ,+ 5512

kN

× ×=

−435 10

11 195

3)

,

Design examples 5

δ = =

× ×=

−A f

Na yd

pl,Rd

17 440 355 1011 195

0 553,

,.

which is within the permitted range, 0.2 ≤ δ ≤ 0.9.

Long-term effects

The age of concrete at loading t0 is assumed to be 14 days. The age of concrete at the moment considered t is taken as infinity. The relative humidity RH is taken as 50%.

The notional size of the cross-section is:

h

A

u0

2 2 230 0484 500

230= =××

=c ,mm

Coefficient:

α1

0 7 0 735 35

30 80 94=

= +

=fcm

. .

.

α2

0 2 0 235 35

380 98=

=

=fcm

. .

.

α3

0 5 0 535 35

380 96=

=

=fcm

. .

.

Factor:

ϕ α αRH

/ /= +

−

= +

−×

1

1 100

0 11

1 50 100

0 1 2300 94

03

1 2 3

RH

h. ..

×0 98 1 73. .=

β( )

. ..f

fcm

cm

= = =16 8 16 8

382 73

β( )

( . ) ( . ).. .t

t0

00 20 0 20

10 1

10 1 14

0 56=+

=+

=


ϕ ϕ β β0 RH= = × × =( ) ( ) . . . .f tcm 0 1 73 2 73 0 56 2 64

Factor:

β αH = + +

= + × × + ×

1 5 1 0 012 250

1 5 1 0 012 50 230 250

180 3

18

. [ ( . ) ]

. [ ( . ) ]

RH h

00 96 58. = 5

β

β( , )

( ) ( )

. .

t tt t

t t0

0

0

0 3 0 314

585 14=

−+ −

=

∞−+∞−

H

==1 0.

The creep coefficient is:

ϕ ϕ βt = = × =0 0 2 64 1 0 2 64c( , ) . . .t t

Elastic modulus of concrete considering long-term effects

Long-term effects due to creep and shrinkage should be considered in determining the effective elastic flexural stiffness. The modulus of elasticity of concrete Ecm is reduced to the value Ec,eff:

E

E

N Nc,eff

cm

G,Ed Ed t/ /kN/mm=

+=+ ×

=1

331 4000 8000 2 64

14 22 2

( ) ( ) ..

ϕ

Effective flexural stiffness of cross-section

The effective elastic flexural stiffness taking account of the long-term effects is:

( ) .

. . .

EI E I E I E Ieff,y a ay c,eff cy s sy= + +

= × × + ×

0 6

210 328 1 10 0 6 14 226 ×× × + × ×

= ×

4805 10 210 73 56

1 25 10

6 6

11 2

.

.

10

kNmm

( ) .

.

EI E I E I E Ieff,z a az c,eff cz s sz

.

= + +

= × × + × ×

0 6

210 107 10 0 6 14 22 56 0026 10 73 56

8 04 10

6 6

10 2

× + × ×

= ×

210 10

kNmm

.

.

Design examples 7

Elastic critical normal force

N

EI

Lcry

eff,y

y

kN= =× ××

=π π2

2

2 11

2 6

1 25 104 10

77 000( ) .

,

N

EI

Lcrz

eff,z kN= =× ××

=π π2

2

2 10

2 6

8 04 104 10

49 600( ) .

,z

The characteristic value of the plastic resistance to the axial load is:

N A f A f A fpl,Rk a y c ck s sk.= + +

= × + × × +

0 85

17 440 355 0 85 230 048 30 25( , . , 112 500 10

13 313

3× ×=

)

, kN

Relative slenderness ratio

λy

pl,Rk

cry

= = =N

N

13 31377 000

0 42,,

.

λz

pl,Rk

crz

= = =N

N

13 31349 600

0 52,,

.

The nondimensional slenderness does not exceed 2.0, so the simplified design method is applicable.

Buckling reduction factor

Buckling curve b is applicable to axis y-y, and buckling curve c is applicable to axis z-z in accordance with EN 1994-1-1. The imperfection factor is taken as 0.34 for curve b and 0.49 for curve c. According to EN 1993-1-1, the factor is:

Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 42 0 2 0 42 0 62

2

2

. .

. [ . ( . . ) . ] .

α λ λ


Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 52 0 2 0 52 0 71

2

2

. .

. [ . ( . . ) . ] .

α λ λ

The reduction factor for column buckling is:

χλ

y

y y y

=+ −

=+ −

=1 1

0 62 0 62 0 420 92

2 2 2 2Φ Φ . . .

.

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 71 0 71 0 520 83

2 2Φ Φ . . .

.

Buckling resistance

The minor axis is the more critical, so

N N

N

b Rd y z

Ed

, min( ; )

. ,

=

= × = > =

χ χ pl,Rd

kN kN0 83 11 195 9292 8000

The buckling resistance of the SRC column is adequate.

Steel-reinforced concrete column with high-strength concrete

Concrete class C90/105 is used in this design example. Other design data are same as in Section “Steel-reinforced concrete column with normal-strength material,” such as loading; column length; steel strength; and the dimensions of the SRC column cross-section, steel section, and reinforcement.


According to Tables 2.12 and 2.13 in the companion book, for the strain compatibility between steel and concrete, the two materials (concrete class C90/105 and steel grade S355) are strain compatible, so the steel can reach its full strength when the composite concrete section reaches its ultimate strength without considering the confinement effect from the lateral hoops and steel section.

For high-strength concrete with fck > 50 N/mm2, the effective compressive strength of concrete should be used in accordance with EC2. The effective strength is:

Design examples 9

fck = = × − − =90 90 1 0 90 50 200 72η ( . ( ) )/ N/mm2

Accordingly, the secant modulus for high-strength concrete C90/105 is

E fcm ck / / GPa= + = + =22 8 10 22 72 8 10 41 10 3 0 3[( ) ] [( ) ] .. .η

Then, the design strengths of the steel, concrete, and reinforcement are:

f

fyd

y

M

2N/mm= = =γ

3551 0

355.

f

fcd

ck

C

2

.5N/mm= = =

γ721

48

f

fsd

sk

S

2N/mm= = =γ

5001 15

435.

Cross-sectional areas and second moments of area

The cross-sectional area and second moment area of the steel, reinforcement, and concrete are the same as the design example in Section “Steel-reinforced concrete column with normal-strength material.”

Aa = 17,440 mm2, As = 2512 mm2, Ac = 230,048 mm2

Iay = 328.1 × 106 mm4, Iaz = 107 × 106 mm4

Isy = 73.56 × 106 mm4, Isz = 73.56 × 106 mm4

Icy = 4805 × 106 mm4, Icz = 5026 × 106 mm4



N A f A f A fpl,Rd a yd c cd s sd.= + +

= × + × × +

0 85

17 440 355 0 85 230 048 48 2( , . , 5512 435 10

16 669

3× ×=

−)

, kN


δ = =

× ×=

−A f

Na yd

pl,Rd

17 440 355 1016 669

0 373,

,.


Long-term effects



h0 = 2Ac/u = 230 mm

Coefficient:

α1

0 7 0 735 35

72 80=

=

=fcm

. .

+.56

α2

0 2 0 235 35

800 85=

=

=fcm

. .

.

α3

0 5 0 535 35

800 66=

=

=fcm

. .

.

Factor:

ϕ α αRH

/ /=

−

=

−×

1

1 100

0 11

1 50 100

0 1 2300 56

03

1 2 3+ +

RH

h. ..

× =0 85 1 24. .

β( )

. .f

fcm

cm

1.88= = =16 8 16 8

80

β( )

( . ) ( . ).. .t

t0

00 20 0 20

10 1

10 1 14

0 56=+

=+

=

ϕ ϕ β β0 RH 3= = × × =( ) ( ) . . . .f tcm 0 1 24 1 88 0 56 1

Design examples 11

Factor:

β αH = + +

= + × × + ×

1 5 1 0 012 250

1 5 1 0 012 50 230 250

180 3

18

. [ ( . ]

. [ ( . ) ]

RH) h

00 66. = 510

β

β( , )

( ) ( )

. .

t tt t

t t0

0

0

0 3 0 314

510 14=

−+ −

=

∞−+∞−

H

==1 0.


ϕ ϕ βt = = × =0 0 1 30 1 0 1 30c( , ) . . .t t



E

EN N

c,effcm

G,Ed Ed t/ /kN/mm=

+=+ ×

=1

41 11 4000 8000 1 3

24 9 2

( ).

( ) ..

ϕ


( ) .

. . .


= × × + × ×

0 6

210 328 1 10 0 6 24 96 44805 10 210 73 56 10

1 56 10

6 6

11 2

× + × ×

= ×

.

. kNmm

( ) .

. .

EI E I E I E Ieff,z a az c,eff cz s sz= + +

= × × + × ×

0 6

210 107 10 0 6 24 9 506 226 10 210 73 56 10

1 13 10

6 6

11 2

× + × ×

= ×

.

. kNmm


N

EI

Lcry

eff,y

y

kN= =× ××

=π2

2

2 11

2 6

1 56 104 10

96 100( ) .

,π


N

EI

Lcrz

eff,z

z

kN= =× ××

=π2

2

2 11

2 6

1 13 104 10

69 500( ) .

,π


N A f A f A fpl,Rk a y c ck s sk.

0.85 25

= + +

= × + × × +

0 85

17 440 355 230 048 72( , , 112

kN

× ×=

500 10

21 526

3)

,


λypl,Rk

cry

= = =N

N

21 52696 100

0 47,,

.

λz

pl,Rk

crz

= = =N

N

21 52669 500

0 56,,

.


Buckling curve b is applicable to axis y-y, and buckling curve c is applicable to axis z-z in accordance with EN 1994-1-1. The imperfection factor is taken as 0.34 for curve b and 0.49 for curve c.

Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 47 0 2 0 47 0 66

2

2

. .

. [ . ( . . ) . ] .

α λ λ

Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 56 0 2 0 56 0 74

2

2

. .

. [ . ( . . ) . ] .

α λ λ


χλ

y

y y y

=+ −

=+ −

=1 1

0 66 0 66 0 470 90

2 2 2 2Φ Φ . . .

.

Design examples 13

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 74 0 74 0 560 81

2 2Φ Φ . . .

.

Buckling resistance


N N

N

b Rd

Ed

, min( ; )

. , ,

=

= × = > =

χ χy z pl,Rd

kN kN0 81 16 669 13 502 8000

The buckling resistance of the SRC column is adequate.Compared to the SRC column with concrete class C30/37, the buckling

resistance ratio is:

N

Nb Rd C

b Rd C

, , /

, , /

,.90 105

30 37

13 5029292

1 45= =

The buckling resistance of an SRC column with high-strength concrete C90/105 is increased by 45% compared to the resistance of a column with C30/37 concrete.

Steel-reinforced concrete column with high-strength steel

Steel grade S550 is used in this design example. Other design data are same as in Section “Steel-reinforced concrete column with normal-strength material,” such as loading; column length; concrete strength; and dimensions of the SRC column cross-section, steel section, and reinforcement.


According to Tables 2.12 and 2.13 in the companion book, for the strain compatibility between steel and concrete, the two materials (concrete class C30/37 and steel grade S550) are not strain compatible, so the high-strength concrete reaches its peak strain much earlier than the yield strain of steel. This implies that the concrete will fail earlier than the steel, resulting in a partial utilization of the steel strength. Using the strain-compatibility method, the strength of steel is limited to the stress corresponding to the crushing strain of concrete. The confinement effect from the lateral hoops and steel section is considered as follows.


Longitudinal reinforcement ratio:

ρs

s 1.1= =A

Ac

%

Factor:

k

b b h s b s hin

e

i c c c c/ / /=−∑( ) − −

−=

=1 6 1 2 1 2

10 514

12(( ) ) ( ( ))( ( ))

.ρs

The effective volume ratio of the hoops is:

ρ ρse e s h= = × =k , . . % . %0 514 0 2 0 1

The real stress of the hoops is calculated by the modified confinement model:

κρ ε

= =× ×

=f

Ec,u

se s c

300 001 210 0 0022

65. .

ff

Er,hc,u

sec s=

−

=×

max.( )

; .

max.

0 2510

0 43

0 25 300

ρ κε

.. ( ); . . ,

001 65 100 43 0 0022 210 000 199

−× ×

= N/mm2

The effective lateral confining pressure for PCC from the hoops is:

f fl,p se r,h N/mm= = × =ρ 0 001 199 0 199 2. .

The strain of PCC is:

ε εc,pl,p

c,uc= +

= +1 35 1 35

0 191 2

f

f

.. 9930

0 0022 0 00241 2

× =

.

. .

Design examples 15

Factor:

′ =

−=

−=k

A A

Ae

c,f c,r

c,f

40 927 12 79740 927

0 69, ,

,.

Factor:

k

t

la = =f

2

230 0072.

The effective lateral confining pressure from the steel section is:

f k k fl,s e a r,y N/mm= ′ = 2 5 2.

The effective lateral confining stress for HCC is:

f f fl,h l,p l,s N/mm= + = + =0 199 2 5 2 699 2. . .

The strain of HCC is:

ε εc,pl,h

c,uc= +

= +1 35 1 35

2 691 2

f

f

.. 9930

0 0022 0 0061 2

× =

.

. .

To ensure the yield strain of steel is less than the compressive strain of concrete, the maximum steel strength can be determined accordingly.

The real stress of the steel flange in partially confined concrete is:

f Er,f N/mm= = × =εc p a, . ,0 0024 210 000 504 2

The real stress of the steel web in highly confined concrete is:

f E fr,w y N/mm= = × =min( ; ) min( . , ; ),εc h a 0 006 210 000 550 550 2

The steel strength in partially confined concrete is lower than the yield strength of steel, 550 N/mm2. The confinement pressure is insufficient to ensure the utilization of steel’s full strength. A higher confinement level


is needed. Thus, the conservative value of the steel flange is taken as the steel strength in the following design.

Then, the design strength of steel is:

f

fyd

y

M

2N/mm= = =γ

5041 0

504.



Aa = 17,440 mm2, As = 2512 mm2, Ac = 230,048 mm2

Iay = 328.1 × 106 mm4, Iaz = 107 × 106 mm4

Isy = 73.56 × 106 mm4, Isz = 73.56 × 106 mm4

Icy = 4805 × 106 mm4, Icz = 5026 × 106 mm4




. 2

= + +

= × × ×

0 85

17 440 504 0 85 230 048 20( , ,+ + 5512

kN

× ×=

−435 10

13 792

3)

,

δ = =

× ×=

−A f

Na yd

pl,Rd

17 440 504 1013 792

0 643,

,.


Long-term effects

The creep coefficient is 2.64 (refer to design example 1, Section “Steel-reinforced concrete column with normal-strength material”).


The modulus of elasticity of concrete Ec,eff due to long-term effects is 14.22 GPa (refer to Section “Steel-reinforced concrete column with normal-strength material”).

Design examples 17


The effective elastic flexural stiffness (refer to Section “Steel-reinforced concrete column with normal-strength material”) is:

(EI)eff,y = 1.25 × 1011 kN mm2

(EI)eff,z = 8.04 × 1010 kN mm2


Refer to Section “Steel-reinforced concrete column with normal-strength material”:

Ncry = 77,000 kNNcrz = 49,600 kN



= × + × × +

0 85

17 440 504 0 85 230 048 30 25( , . , 112 500 10

15 912

3× ×=

)

, kN


λypl,Rk

cry

= = =N

N

15 91277 000

0 45,,

.

λz

pl,Rk

crz

= = =N

N

15 91249 600

0 57,,

.


Buckling curve b is applicable to axis y-y, and buckling curve c is applicable to axis z-z in accordance with EN 1994-1-1. The imperfection factor is taken as 0.34 for curve b and 0.49 for curve c. So:

Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 45 0 2 0 45 0 65

2

2

. .

. [ . ( . . ) . ] .

α λ λ


Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 57 0 2 0 57 0 75

2

2

. .

. [ . ( . . ) . ] .

α λ λ


χλ

y

y y y

=+ −

=+ −

=1 1

0 65 0 65 0 450 90

2 2 2 2Φ Φ . . .

.

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 75 0 75 0 570 81

2 2Φ Φ . . .

.

Buckling resistance


N N

N

b Rd

Ed

, min( ; )

. , ,

=

= × = > =

χ χy z pl,Rd

kN kN0 81 13 793 11 172 8000

The buckling resistance of the SRC column is adequate.Compared to the SRC column with steel grade S355, the buckling

resistance ratio is:

N

Nb Rd

b Rd

, ,

, ,

,.S

S

550

355

11 1729292

1 20= =

The buckling resistance of the SRC column with high-strength steel S550 is increased by 20% compared to the resistance of the column with S355 steel.

Steel-reinforced concrete column with high-strength concrete and steel

Steel grade S550 and concrete class C90/105 are used in this design example. Other design data are the same as in Section “Steel-reinforced concrete column with normal-strength material,” such as loading; column length; dimensions of the SRC column cross-section, steel section, and reinforcement; and so on.

Design examples 19



The effective compressive strength and elastic modulus of concrete C90/105 are:

fck = 72 N/mm2; fcd = 48 N/mm2; Ecm = 41.1 GPa;

The design strength of steel is:

fy = 550 N/mm2; fyd = 550 N/mm2;



Aa = 17,440 mm2, As = 2512 mm2, Ac = 230,048 mm2

Iay = 328.1 × 106 mm4, Iaz = 107 × 106 mm4

Isy = 73.56 × 106 mm4, Isz = 73.56 × 106 mm4

Icy = 4805 × 106 mm4, Icz = 5026 × 106 mm4




= × + × × +

0 85

17 440 550 0 85 230 048 48 2( , . , 5512 435 10

20 070

3× ×=

−)

, kN

δ = =

× ×=

−A f

Na yd

pl,Rd

17 440 550 1011 195

0 853,

,.


Long-term effects

The creep coefficient is 1.30 (refer to design example 2, Section “Steel-reinforced concrete column with high-strength concrete”).



The modulus of elasticity of concrete Ec,eff due to long-term effects is 24.9 GPa (refer to Section “Steel-reinforced concrete column with high-strength concrete”).


The effective elastic flexural stiffness (refer to Section “Steel-reinforced concrete column with high-strength concrete”) is:

(EI)eff,y = 1.56 × 1011 kNmm2

(EI)eff,z = 1.13 × 1011 kNmm2


Refer to Section “Steel-reinforced concrete column with high-strength concrete”:

Ncry = 96,100 kNNcrz = 69,500 kN



= × + × × +

0 85

17 440 550 0 85 230 048 72 25( , . , 112 500 10

24 927

3× ×=

)

, kN


λy

pl,Rk

cry

= = =N

N

24 92796 100

0 51,,

.

λz

pl,Rk

crz

= = =N

N

24 92769 500

0 60,,

.


Buckling curve b is applicable to axis y-y, and buckling curve c is applicable to axis z-z in accordance with EN 1994-1-1. The

Design examples 21

imperfection factor is taken as 0.34 for curve b and 0.49 for curve c. So:

Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 51 0 2 0 51 0 68

2

2

. .

. [ . ( . . ) . ] .

α λ λ

Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 60 0 2 0 60 0 78

2

2

. .

. [ . ( . . ) . ] .

α λ λ


χλ

y

y y y

=+ −

=+ −

=1 1

0 68 0 68 0 510 88

2 2 2 2Φ Φ . . .

.

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 78 0 78 0 600 79

2 2Φ Φ . . .

.

Buckling resistance


N N

N

b Rd y

Ed

, min( ; )

. , ,

=

= × = > =

χ χz pl,Rd

kN kN0 79 20 070 15 855 8000

The buckling resistance of the SRC column is adequate.Compared to the SRC column with normal-strength material S355 and

C30/37, the buckling resistance ratio is:

N

Nb Rd H

b Rd N

, ,

, ,

,.= =

15 8559292

1 71

The buckling resistance of the SRC column with high-strength steel S550 and high-strength concrete C90/105 is increased by 71% compared to the resistance of the column with normal-strength steel S355 and normal-strength concrete C30/37.


Alternative design

Alternatively, the column size can be reduced when high-strength steel and concrete materials are used, but the buckling resistance is almost the same as in design example 1 in Section “Steel-reinforced concrete column with normal-strength material.”

Design data

Structural steel Grade S550Concrete C90/105

Properties of cross-section



Depth h = 260.3 mmWidth b = 256.3 mmFlange thickness tf = 27.3 mmWeb thickness tw = 10.3 mmFillet r = 12.7 mmSection area Aa = 113.3 cm2

Second moment of area/y Iay = 14,270 cm4




Other data are the same as in design example 1 in Section “Steel-reinforced concrete column with normal-strength material.”


According to Tables 2.12 and 2.13 in the companion book, for the strain compatibility between steel and concrete, the two materials (concrete class

Design examples 23

C90/105 and steel grade S550) are strain compatible, so the steel can reach its full strength when the composite concrete section reaches its ultimate strength without considering the confinement effect from the lateral hoops and steel section.


fck = 72 N/mm2; fcd = 48 N/mm2; Ecm = 41.1 GPa


fy = 550 N/mm2; fyd = 550 N/mm2

Cross-sectional area and second moments of area

The cross-sectional area and second moment area of the steel, reinforcement, and concrete are:

Aa = 11,330 mm2, As = 2512 mm2, Ac = 146,158 mm2

Iay = 142.7 × 106 mm4, Iaz = 48.57 × 106 mm4

Isy = 42.39 × 106 mm4, Isz = 42.39 × 106 mm4

Icy = 1948 × 106 mm4, Icz = 2042 × 106 mm4


ρs

c

= = =A

As 7

2512146 158

1,

. %









= × + × × +

0 85

11 330 550 0 85 146 158 48 2( , . , 5512 435 10

13 287

3× ×=

−)

, kN

δ = =

× ×= <

−A f

Na yd

pl,Rd

11 330 550 1013 287

0 47 0 93,

,. .

Long-term effects


h0 = 2Ac/u = 183 mm

Coefficient:

α1

0 7 0 735 35

72 80 56=

= +

=fcm

. .

.

α2

0 2 0 235 35

800 85=

=

=fcm

. .

.

α3

0 5 0 535 35

800 66=

=

=fcm

. .

.

Factor:

ϕ α αRH/

= +−

= +−

×

11 100

0 1

11 50 100

0 1 1830 56

03

1 2

3

RH

h.

/

..

× =0 85 1 27. .

β( ). .

.ff

cm = = =16 8 16 8

801 88

cm

β( )

( . ) ( . ).. .t

t0

00 20 0 20

10 1

10 1 14

0 56=+

=+

=

Design examples 25

ϕ ϕ β β0 RH= = × × =( ) ( ) . . . .f tcm 0 1 27 1 88 0 56 1 34

Factor:

β αH = + +

= + × × + ×

1 5 1 0 012 250

1 5 1 0 012 50 183 250

180 3

18

. [ ( . ) ]

. [ ( . ) ]

RH h

00 66 440. =

β

β( , )

( ) ( )

. .

t tt t

t t0

0

0

0 3 0 314

440 14=

−+ −

∞−+∞−

H

= ==1 0.


ϕ ϕ βt = = × =0 0 1 34 1 0 1 34c( , ) . . .t t


E

E

N Nc,eff

cm

G,Ed Ed t/ /kN/mm=

+=+ ×

=1

41 11 4000 8000 1 34

24 6( )

.( ) .

.ϕ

22


( ) . .EI E I E I E Ieff,y a ay c,eff cy s sy kNmm= + + = ×0 6 6 73 1010 2

( ) . .EI E I E I E Ieff,z a az c,eff cz s sz kNmm= + + = ×0 6 4 89 1010 2


NEI

Lcry

eff,y

y

kN= =× ××

=π2

2

2 10

2 6

6 73 104 10

41 500( ) .

,π

NEIL

crzeff,z

z

kN= =× ××

=π2

2

2 10

2 6

4 89 104 10

30 200( ) .

,π



N A f A f A fpl,Rk a y c ck s sk. kN= + + =0 85 16 432,


λy

pl,Rk

cry

= = =N

N

16 43241 500

0 63,,

.

λz

pl,Rk

crz

= = =N

N

16 43230 200

0 74,,

.


Buckling curve b is applicable to axis y-y, and buckling curve c is applicable to axis z-z in accordance with EN 1994-1-1. The imperfection factor is taken as 0.34 for curve b and 0.49 for curve c. So:

Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 63 0 2 0 63 0 77

2

2

. .

. [ . ( . . ) . ] .

α λ λ

Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 74 0 2 0 74 0 90

2

2

. .

. [ . ( . . ) . ] .

α λ λ


χλ

y

y y y

=+ −

=+ −

=1 1

0 77 0 77 0 630 82

2 2 2 2Φ Φ . . .

.

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 90 0 90 0 740 70

2 2Φ Φ . . .

.

Buckling resistance


Design examples 27

N N

N

b Rd y

Ed

, min( ; )

. ,

=

= × = > =

χ χz pl,Rd

kN kN0 70 13 287 9300 8000

The buckling resistance of the SRC column is adequate.Compared to the SRC column with normal-strength materials S355

and C30/37, the buckling resistance ratio is:

N

Nb Rd H

b Rd N

, ,

, ,

.= ≈93009292

1 0

The buckling resistance is almost the same as that of the SRC column with normal-strength materials S355 and C30/37.

The cross-section area ratio of the SRC column is:

A

AH

N

=××

=400 400500 500

0 64.

The cross-section area of the SRC column with high-strength materials S550 and C90/105 is reduced by 36% compared to the SRC column with S355 steel and C30/37 concrete. Similarly, the amount of the steel section is also reduced by 36% compared to the SRC column with normal-strength material.

STEEL-REINFORCED CONCRETE COLUMN SUBJECTED TO COMBINED COMPRESSION AND BENDING

Determine the resistance of SRC columns subjected to compression and bending about the major axis.

Steel-reinforced concrete column with normal-strength material

Design data

Design axial force NEd = 9000 kNPermanent load NG,Ed = 4000 kNDesign moment Mb,y = 300 kNm Mt,y = 200 kNm

Continued


Column length L = 4.0 mEffective length Leff = 4.0 mStructural steel Grade S355, fy = 355 N/mm2

Concrete C50/60, fck = 50 N/mm2

Reinforcement fsk = 500 N/mm2

Properties of cross-section:



Depth h = 314.5 mmWidth b = 307.4 mmFlange thickness tf = 18.7 mmWeb thickness tw = 12 mmFillet r = 15.2 mmSection area Aa = 150.2 cm2

Second moment of area/y Iay = 272,670 cm4




Reinforcement

Longitudinal reinforcement

number n = 8, diameter dl,s = 20 mm

Transverse reinforcement diameter dt,s = 10 mm, spacing s = 200 mm



Design examples 29

Then, the design strengths of the steel, concrete, and reinforcement are:

f

fyd

y

M

2

.N/mm= = =

γ3551 0

355

f

fcd

ck

C

2N/mm= = =γ

501 5

33 3.

.

f

fsd

sk

S

2N/mm= = =γ

5001 15

435.

Ecm = 37 Gpa

Ea = 210 Gpa


Aa = 15,020 mm2, As = 2512 mm2, Ac = 232,468 mm2

Iay = 276.7 × 106 mm4, Iaz = 90.6 × 106 mm4

Isy = 75.36 × 106 mm4, Isz = 75.36 × 106 mm4

Icy = 4856 × 106 mm4, Icz = 5042 × 106 mm4


ρs = = =AA

s

c

2512232 468

1 1,

. %









= × + × ×

0 85

15 020 355 0 85 232 468 33 3( , . , . ++ × ×=

−2512 435 10

13 011

3)

, kN

δ= =

× ×= <

−A f

Na yd

pl,Rd

15 020 355 1013 011

0 41 0 93,

,. .

Long-term effects



h

A

u0

2 2 232 4684 500

232= =××

=c ,mm

Coefficient:

α1

0 7 0 735 35

580 70=

=

=fcm

. .

.

α2

0 2 0 235 35

580 90=

=

=fcm

. .

.

α3

0 5 0 535 35

580 78=

fcm

=

=

. .

.

Factor:

ϕ α αRH = +−

= +

−×

1

1 100

0 11

1 50 100

0 1 2320 70

03

1 2 3

RH

h

/

.

/

..

× =0 90 1 41. .

β( )

. ..f

fcm

cm

= = =16 8 16 8

582 21

Design examples 31

β( )

( . ) ( . ).. .t

t0

00 20 0 20

10 1

10 1 28

0 49=+

=+

=

ϕ ϕ β β0 RH= = × × =( ) ( ) . . . .f tcm 0 1 41 2 21 0 49 1 53

Factor:

β αH = + +

= + × × + ×

1 5 1 0 012 250

1 5 1 0 012 50 232 250

180 3

18

. [ ( . ) ]

. [ ( . ) ]

RH h

00 78. = 543

β

β( , )

( ) ( )

. .

t tt t

t t0

0

0

0 3 0 328

543 28=

−+ −

=

∞−+∞−

H

==1 0.


ϕ ϕ βt = = × =0 0 1 53 1 0 1 53c( , ) . . .t t



E

E

N Nc,eff

cm

G,Ed Ed t/ /kN/mm=

+=+ ×

=1

371 4000 9000 1 53

22 2 2

( ) ( ) ..

ϕ



( ) .

. . .


= × × + × ×

0 6

210 276 7 10 0 6 22 26 44856 10 210 75 36 10

1 38 10

6 6

11 2

× + × ×

= ×

.

. kN mm


( ) .

. . .

EI E I E I E Ieff,z a az c,eff cz s sz= + +

= × × + × ×

0 6

210 90 6 10 0 6 22 2 56 0042 10 210 75 36 10

1 01 10

6 6

11 2

× + × ×

= ×

.

. kN mm


NEI

Lcry

eff,y

y

kN= =× ××

=π2

2

2 11

2 6

1 38 104 10

85 100( ) .

,π

NEIL

crzeff,z

z

kN= =× ××

=π2

2

2 11

2 6

1 01 104 10

62 500( ) .

,π



= × + × × +

0 85

15 020 355 0 85 232 468 50 25( , . , 112 500 10

16 468

3× ×=

)

, kN


λy

pl,Rk

cry

= = =N

N

16 46885 100

0 44,,

.

λz

pl,Rk

crz

= = =N

N

16 46862 500

0 51,,

.



Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 44 0 2 0 44 0 64

2

2

. .

. [ . ( . . ) . ] .

α λ λ

Design examples 33

Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 51 0 2 0 51 0 71

2

2

. .

. [ . ( . . ) . ] .

α λ λ


χλ

y

y y y

=+ −

=+ −

=1 1

0 64 0 64 0 440 91

2 2 2 2Φ Φ . . .

.

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 71 0 71 0 510 84

2 2Φ Φ . . .

.

Buckling resistance


N N

N

b Rd

Ed

, min( ; )

. , ,

=

= × = > =

χ χy z pl,Rd

kN kN0 84 13 011 10 929 9000


Interaction curve

The polygonal interaction diagram for major-axis bending is calculated, using the notation shown in Figure 2.

Point A (0, Npl,Rd)

The full cross-section is under compression without the bending moment.

MA = 0NA = Npl,Rd = 13,011 kN

Point B (Mpl,Rd, 0)

Assuming the neutral axis lies in the web of the steel section (hn ≤ h/2 − tf), the 2 reinforcement bar lies within the region 2hn, Asn = 628 mm2, so,


hA f A f f

b f t f fn

c cd sn sd cd

c cd w yd cd

. .

. .=

− −+ −

0 85 2 0 852 0 85 2 2 0 85

( )( )

==× × − × × − ×

× × × +232 468 0 85 33 3 628 2 435 0 85 33 32 500 0 85 33 3 2

, . ( . . ).

.. ×× × × − ×

=12 2 355 0 85 33 3

136( . ).

mm

Hence,

h

htn fmm mm= − =136

2138 6< .

The assumption for the plastic neutral axis is verified. The neutral axis lies in the web of the steel section.

The plastic section moduli for the steel section, reinforcement, and concrete are:

Wpa3mm= ×1 958 106.

W A eps si i

3.3768 mm= = × = ×∑ [ ]1

661884 200 0 10

Wb h

W Wpcc c

pa ps= − − =×

− × − ×

= ×

2 26 6

4500 500

41 958 10 0 3768 10

28 915 1

. .

. 006 3mm

bc

ez hchchc

h

b

BD

C

BD

C

Figure 2 Plastic neutral axes for encased I-section.

Design examples 35

The plastic section moduli for the region of depth 2hn are:

Wpsn3mm= 0

W t hpan w n2 3.222 mm= = × = ×12 136 0 102 6

W b h W Wpcn c n pan psn

3

0.222 10

mm

= − − = × − × −

= ×

2 2 6

6

500 136 0

9 026 10.

The bending resistance at point B is determined from:

M W W f W W f W W fpl,Rd pa pa,n yd ps,n sd pc pc,n c cd= − + − + −

=

( ) ( ) . ( )

(

ps 0 5 α

11 958 0 222 355 0 3768 0 435

0 5 28 915 9 026 0 85 3

. . ) ( . )

. ( . . ) .

− × + − ×+ × − × × 33 3 1062. = kNm

Point C (Mpl,Rd, Npm,Rd)

The axial force is equal to the full cross-section compression resistance of concrete. The value is determined from:

N A fpm,Rd c cd. . kN= = × × × =−0 85 0 85 232 468 33 3 10 68503, .

Point D (Mmax,Rd, 0.5Npm,Rd)

The maximum moment resistance is determined from:

M f W f W f Wmax,Rd yd pa cd pc sd ps.

. .

= + × +

= × + × ×

0 5 0 85

355 1 958 0 5 0 85 33

.

. .. .3 28 915 435 0

1268

× + ×=

.3768

kNm

The relative information for plotting the interaction curve is shown in Table 1. Then, the interaction curve is plotted as shown in Figure 3.

Check the resistance of steel-reinforced concrete column in combined compression and uniaxial bending

The effective flexural stiffness considering second-order effects is determined from:


( ) ( )

( .

EI K E I K E I E Ieff,II,y o a ay e,II c,eff cy s sy

.

= + +

= × ×0 9 210 276 7×× + ×

× × × × = ×

10 0 22 2

4856 10 75 36 1 15 10

6

6 6 11 2

.5

+210 10 kNmm

.

. ) .

Hence, the elastic critical force is:

NEI

Lcr,y,eff

eff,II,y

y

kN= =× ××

=π2

2

2 11

2 6

1 15 104 10

70 600( ) .

,π

The result is less than 10NEd for major axis, so the second-order effects must be considered for the moment from first-order analysis and the moment from imperfection.

The member imperfection for the major axis according to EN 1994-1-1 is:

e L0,y / mm= =200 20

Table 1 The resistance for interaction curve

Point Resistance to bending (kNm) Resistance to compression (kN)

A 0 13,011B 1062 0C 1062 6850D 1268 3425

15,000

1500

12,000

1200

9000

900

A

C

D

B

6000

600

Axi

al lo

ad (k

N)

Moment (kNm)

3000

3000

0

Figure 3 Interaction curve for major axis.

Design examples 37

For the major axis, the midlength bending moments due to NEd and imperfection are calculated by:

N eEd 0,y . kNm = × =9000 0 02 180

According to EN 1994-1-1, the factor β is equal to 1.0 for the bending moment from member imperfection. Then, the amplification factor is:

k

N Nimp,y

Ed cr,y,eff/./

5=−

=−

=β

11 0

1 9000 70 6001 1

,.

For the first-order bending moment, My,top = 200 kNm, My,bot = 300 kNm, so the ratio of the end moment is:

r = 200/300 = 0.667

Then, the factor β is:

β = max (0.66 + 0.44 r; 0.44) = 0.95

thus, the amplification factor is:

k

N Ny = =

β1

01 9000 70600

1 09− −

=Ed cr,y,eff/

.95/

.

Hence, the design moment considering second-order effects is:

M k M k N ey,Ed y y,Ed,top imp,y Ed ,z .15

kNm

= = × ×

=

+ +0 1 09 300 1 200

557

.

For NEd > Npm,Rd, the factor is determined from:

µd

pl,Rd Ed

pl,Rd pm,Rd

=−−

=−−

=N N

N N

13 011 900013 011 6850

0 65,,

.

Thus,

M

M

M

My,Ed

pl,N,y,Rd

y,Ed

pl,y,Rd

= =×

= <µd

5570 65 1062

0 81 0 9.

. .

So, the resistance of the SRC column to compression and uniaxial bending is satisfied.


Steel-reinforced concrete column with high-strength concrete

The concrete class C90/105 is used in this design example. Other design data are same as in Section “Steel-reinforced concrete column with normal-strength material,” such as loading; column length; steel strength; and dimensions of the SRC column cross-section, steel section, and reinforcement.



For high-strength concrete with fck > 50 N/mm2, the effective compressive strength of concrete should be used in accordance with EC2. The effective strength is:

fck = = × − − =90 90 1 0 90 50 200 72η ( . ( ) )/ N/mm2

Accordingly, the secant modulus for high-strength concrete C90/105 is:

E fcm ck / / GPa= + +22 8 10 22 72 8 10 41 10 3 0 3[( ) ] [( ) ] .. .η = =

The design strength of concrete is:

f

fcd

ck

C

248N/mm= = =γ

721 5.


Aa = 15,020 mm2, As = 2512 mm2, Ac = 232,468 mm2

Iay = 276.7 × 106 mm4, Iaz = 90.6 × 106 mm4

Isy = 75.36 × 106 mm4, Isz = 75.36 × 106 mm4

Icy = 4856 × 106 mm4, Icz = 5042 × 106 mm4



Design examples 39


. 2

= + +

= × × ×

0 85

15 020 355 0 85 232 468 48( , ,+ + 5512

kN

× ×=

−435 10

15 909

3)

,

δ= =

× ×= <

−A f

Na yd

pl,Rd

15 020 355 1015 909

0 34 0 93,

,. .

Long-term effects



h

A

u0

2 2 232 4684 500

232= =××

=c ,mm

Coefficient:

α1

0 7 0 735 35

800 56=

=

=fcm

. .

.

α2

0 2 0 235 35

800 85=

=

=fcm

. .

.

α3

0 5 0 535 35

800 66=

=

=fcm

. .

.

Factor:

ϕ α αRH

/ /=

−

= +

−×

1

1 100

0 11

1 50 100

0 1 2320 56

03

1 2 3+

RH

h. ..

× =0 85 1 24. .

β( )

. ..f

fcm

cm

= = =16 8 16 8

801 88


β( ). ( . )

.. .t

t0

00 20 0 20

1

0 1

10 1 28

0 49=+( )

=+

=

ϕ ϕ β β0 RH cm= = × × =( ) ( ) . . . .f t0 1 24 1 88 0 49 1 14

Factor:

β αH = + +

= × × + ×

1 5 1 0 012 250

1 5 1 0 012 50 232 250

180 3

18

. [ ( . ) ]

. [ ( . ) ]

RH h

+ 00 66. = 513

β

β( , )

( ) ( )

. .

t tt t

t t0

0

0

0 3 0 328

513 28=

−+ −

=

∞−+∞−

H

==1 0.


ϕ ϕ βt = = × =0 0 1 14 1 0 1 14c( , ) . . .t t



E

E

N Nc,eff

cm

G,Ed Ed t/ /kN/mm=

+=+ ×

=1

41 11 4000 9000 1 14

27 2( )

.( ) .

.ϕ

22


The effective elastic flexural stiffness taking account of the long-term effects is: ( ) .

. .

EI E I E I E Ieff,y a ay c,eff cy s sy

.

= + +

= × × × ×

0 6

210 276 7 10 0 6 27 26+ 44856 10 75 36

1 53 10

6 6

11 2

× × ×

= ×

+210 10

kNmm

.

.

( ) .

. .

EI E I E I E Ieff,z a az c,eff cz s sz

.

= + +

= × × + × ×

0 6

210 90 6 10 0 6 27 2 56 0042 10 75 36

1 17 10

6 6

11 2

× × ×

= ×

+210 10

kNmm

.

.

Design examples 41


N

EI

Lcry

eff,y

y

kN= =× ××

=π π2

2

2 11

2 6

1 53 104 10

94 200( ) .

,

N

EIL

crzeff,z

z

kN= =× ××

=π π2

2

2 11

2 6

1 17 104 10

71 900( ) .

,



0.85 25

= + +

= × × ×

0 85

15 020 355 232 468 72( , ,+ + 112

kN

× ×=

500 10

20 815

3)

,


λy

pl,Rk

cry

= = =N

N

20 81594 200

0 47,,

.

λz

pl,Rk

crz

= = =N

N

20 81571 900

0 54,,

.



Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 47 0 2 0 47 0 66

2

2

. .

. [ . ( . . ) . ] .

α λ λ

Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 54 0 2 0 54 0 73

2

2

. .

. [ . ( . . ) . ] .

α λ λ



χλ

y

y y y

=+ −

=+ −

=1 1

0 66 0 66 0 470 90

2 2 2 2Φ Φ . . .

.

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 73 0 73 0 540 82

2 2Φ Φ . . .

.

Buckling resistance


N N

N

b Rd

Ed

, min( ; )

. , ,

=

= × = > =

χ χy z pl,Rd

kN kN0 82 15 909 13 045 9000


Interaction curve

Point A (0, Npl,Rd)


MA = 0NA = Npl,Rd = 15,909 kN

Point B (Mpl,Rd, 0)

Assuming the neutral axis lies in the flange of the steel section (h/2 − tf < hn ≤ h/2), the 2 reinforcement bar lies within the region 2hn, Asn = 628 mm2, so,

Neutral axis in the flange, h/2 − tf < hn < h/2

hA f A f f b t h t f

b fn

c c cd sn sd c cd w f yd c cd

c c

f=

− − + − − −α α αα

( ) ( )( )( )2 2 2

2 ccd yd c cd+ −

=

× × − × × − ×+

2 2

232 468 0 85 48 628 2 435 0 85 48

30

b f f( )

, . ( . )

(

α

77 4 12 314 5 2 18 7 2 355 0 85 482 500 0 85 48 2 307 4

. )( . . )( . ). .− − × × − ×

× × × + × ×× × − ×=

( . )2 355 0 85 48

141mm

Design examples 43

Hence,

h

hn mm mm= =141

2157<

The assumption for the plastic neutral axis is verified. The neutral axis lies in the flange of the steel section.


Wpa3mm= ×1 958 106.

W A eps si i3.3768 mm= = × ×∑ [ ]

1

661884 200 0 10=

Wb h

W Wpcc c

ps .3768= − − =×

− × − ×

= ×

2 26 6

4500 500

41 958 10 0 10

28 915 1

pa .

. 006 mm3


Wpsn mm= 0 3

W bhb t h t

pa,n n2 w f

= −− −( )

= × −− − ×

( )

.( . )( .

2

4

307 4 141307 4 12 314 5 2 1

2

2 88 74

0 441 102

6. ).= ×


3

10

mm

= − − = × − × −

= ×

2 2 6

6

500 141 0 441 0

9 5 10

.

.



=

( ) ( ) . ( )

(

ps 0 5 α

11 958 0 441 355 0 3768 0 435

0 5 28 915 9 5 0 85 48

. . ) ( . )

. ( . . ) .

− × + − ×+ × − × × =11099kNm




N A fpm,Rd c cd. . kN= = × × × =−0 85 0 85 232 468 48 10 94853,




. .

= + × +

= × + × ×

0 5 0 85

355 1 958 0 5 0 85 48

.

. ×× ×=

28 915 435 0

1499

. + .3768

kNm




( ) ( )

( .

EI K E I K E I E Ieff,II,y o a ay e,II c,eff cy s sy

.

= + +

= × ×0 9 210 276 7×× ×

× × × ×

= ×

10 0 27 2

4856 10 75 36

1 26 10

6

6 6

11 2

+

+

.5

210 10

kNmm

.

. )

.



A 0 15,909B 1099 0C 1099 9485D 1499 4742

Design examples 45


N

EI

Lcr,y,eff

eff,II,y

y

kN= =× ××

=π π2

2

2 11

2 6

1 26 104 10

77 400( ) .

,

The result is less than 10NEd for the major axis, so the second-order effects must be considered for the moment from the first-order analysis and the moment from imperfection.

The member imperfection for the major axis according to EN1994-1-1 is:

e L0,y / mm= =200 20

For the major axis, the midlength bending moments due to NEd and imperfection are calculated by:

N eEd 0,y . kNm = × =9000 0 02 180

According to EN 1994-1-1, the factor β is equal to 1.0 for the bending moment from the member imperfection. Then, the amplification factor is:

k

N Nimp,y

Ed cr,y,eff/./

.13=−

=−

=β

11 0

1 9000 77 4001

,

20,000

15,000

20001500

10,000

A

C

D

B

1000

Axi

al lo

ad (k

N)

Moment (kNm)

5000

5000

0




r = 200/300 = 0.667

The factor β is:

β = max (0.66 + 0.44 r; 0.44) = 0.95

Thus, the amplification factor is:

k

N Ny = −

=−

=β

10

1 9000 77 4001 08

Ed cr,y,eff/.95/ ,

.



kNm

= = × ×

=

+ +0 1 08 300 1 200

550

.

For 0.5Npm,Rd < NEd ≤ Npm,Rd, the factor is determined from:

µdpm,Rd Ed

pm,Rd

max,Rd

pl,Rd

= +−

−

= +×

12

1

12

( )

(

N N

N

M

M

99485 90009485

14991099

1 1 04−

−

=

).

Thus,

M

M

M

My,Ed

pl,N,y,Rd

y,Ed

d pl,y,Rd

= =×

= <µ

5501 04 1099

0 48 0 9.

. .

The resistance of the SRC column to compression and uniaxial bending is satisfied.

Steel-reinforced concrete column with high-strength steel

The steel grade S550 is used in this design example. Other design data are same as in Section “Steel-reinforced concrete column with normal-strength material,” such as loading; column length; concrete strength; and dimensions of the SRC column cross-section, steel section, and reinforcement.

Design examples 47


According to Tables 2.12 and 2.13 in the companion book, for the strain compatibility between steel and concrete, the two materials (concrete class C50/60 and steel grade S550) are not strain compatible. Therefore, high-strength concrete reaches its peak strain much earlier than the yield strain of steel. This implies that concrete will fail earlier than steel, resulting in a partial utilization of steel strength. Using the strain-compatibility method, the strength of steel is limited to the stress corresponding to the crushing strain of concrete. The confinement effect from the lateral hoops and steel section is considered as follows.

Longitudinal reinforcement ratio:

ρs

s= =A

Ac

1 1. %

The confinement effective coefficient:

k

b b h s b s hi

n

e

i c c c c/ / /=−

− −

−=∑1 6 1 2 1 2

1

2

1(( ) ) ( ( ))( ( ))

ρss

= 0 5135.

The effective volume ratio of the hoops is:

ρ ρse e s h= = × =k , . . % . %0 5135 0 2 0 1

The real stress of the hoops is:

κρ ε

= =× ×

=f

Ec,u

se s c

500 001 210 0 0025

100. .

ff

Er,hc,u

sec s=

−

=×

max.( )

; .

max.

0 2510

0 43

0 25 500

ρ κε

.. ( ); . . ,

001 100 100 43 0 0025 210 000 223

−× ×

= N/mm2


The effective lateral confining pressure for PCC from the hoops is:

f fl,p se r,h N/mm= = × =ρ 0 001 223 0 223 2. .

The strain of PCC is:

ε εc,pl,p

c,uc= +

= +1 35 1 35

0 221 2

f

f

.. 3350

0 0025 0 00261 2

× =

.

. .

Factor:

′ =

−=

−=k

A A

Ae

c,f c,r

c,f

40 927 12 79740 927

0 69, ,

,.

Factor:

k

tl

af= =2

230 0053.

The effective lateral confining pressure from the steel section is:

f k k fl,s e a r,y N/mm= ′ =1 9 2.

The effective lateral confining stress for HCC is:

f f fl,h l,p l,s N/mm= + = + =0 223 1 9 2 123 2. . .

The strain of HCC is:

ε εc,pl,h

c,uc= +

= +1 35 1 35

2 121 2

f

f

.. 3350

0 0025 0 0041 2

× =

.

. .

To ensure the yield strain of steel is less than the compressive strain of concrete, the maximum steel strength can be determined accordingly.

The real stress of the steel flange in partially confined concrete is:

f Er,f N/mm= = × =εc p a, . ,0 0026 210 000 546 2

Design examples 49

The real stress of the steel web in highly confined concrete is:

f E fr,w N/mm= = × =min( ; ) min( . , ; ),εc h a y 0 004 210 000 550 550 2

The steel strength in partially confined concrete is lower than the yield strength of steel, 550 N/mm2. The confinement pressure is insufficient to ensure the utilization of steel’s full strength. A higher confinement level is needed. Thus, the conservative value of the steel flange is taken as the steel strength in the following design.

Then, the design strengths of steel is:

f

fyd

y

M

2546 N/mm= = =γ

5461 0.


Aa = 15,020 mm2, As = 2512 mm2, Ac = 232,468 mm2

Iay = 276.7 × 106 mm4, Iaz = 90.6 × 106 mm4

Isy = 75.36 × 106 mm4, Isz = 75.36 × 106 mm4

Icy = 4856 × 106 mm4, Icz = 5042 × 106 mm4




.

= + +

= × × ×

0 85

15 020 546 0 85 232 468 33 3( , , .+ ++2512

kN

× ×=

−435 10

15 880

3)

,

δ = =

× ×= <

−A f

Na yd

pl,Rd

15 020 546 1015 880

0 52 0 93,

,. .

Long-term effects


ϕt .53=1



The modulus of elasticity of concrete Ecm is reduced to the value Ec,eff:

E

E

N Nc,eff

cm

G,Ed Ed t/kN/mm=

+=

122 2 2

( ).

ϕ



( ) . .EI E I E I E Ieff,y a ay c,eff cy s sy kN mm= + + = ×0 6 1 38 1011 2

( ) . .EI E I E I E Ieff,z a az c,eff cz s sz kN mm= + + = ×0 6 1 01 1011 2


N

EI

Lcry

eff,y

y

kN= =π2

2 85 100( )

,

N

EI

Lcrz

eff,z

z

kN= =π2

2 62 500( )

,



0.85 25

= + +

= × × ×

0 85

15 020 546 232 468 50( , ,+ + 112

kN

× ×=

500 10

19 337

3)

,


λy

pl,Rk

cry

= = =N

N

19 33785 100

0 48,,

.

Design examples 51

λz

pl,Rk

crz

= = =N

N

19 33762 500

0 56,,

.



Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 48 0 2 0 48 0 66

2

2

. .

. [ . ( . . ) . ] .

α λ λ

Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 56 0 2 0 56 0 74

2

2

. .

. [ . ( . . ) . ] .

α λ λ


χλ

y

y y y

=+ −

=+ −

=1 1

0 66 0 66 0 480 89

2 2 2 2Φ Φ . . .

.

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 74 0 74 0 560 81

2 2Φ Φ . . .

.

Buckling resistance


N N

N

b Rd y

Ed

, min( ; )

. , ,

=

= × = > =

χ χz pl,Rd

kN kN0 81 15 880 12 863 9000



Interaction curve

Point A (0, Npl,Rd)


MA = 0NA = Npl,Rd = 15,880 kN

Point B (Mpl,Rd, 0)


hA f A f f

b f t f fn

c cd sn sd cd

c cd w yd cd

. .

. .=

− −+ −

0 85 2 0 852 0 85 2 2 0 85

( )( )

==× × − × × − ×

× × × +232 468 0 85 33 3 628 2 435 0 85 33 32 500 0 85 33 3 2

, . . ( . . ). . ×× × × − ×

=12 2 546 0 85 33 3

113( . . )

mm

Hence,

h


2138 6< .



Wpa3mm= ×1 958 106.

W A eps si i

3.3768 mm= = × = ×∑ [ ]1

661884 200 0 10

Wb h

W Wpcc c

ps .3768= − − =×

− × − ×

= ×

2 26 6

4500 500

41 958 10 0 10

28 915 1

pa .

. 006 mm3


Design examples 53

Wpsn3mm= 0

W t hpan w n2 3.153 mm= = × = ×12 113 0 102 6


3

0.153 10

mm

= − − = × − × −

= ×

2 2 6

6

500 113 0

6 23 10.



=

( ) ( ) . ( )

(

ps 0 5 α

11 958 0 153 546 0 3768 0 435

0 5 28 915 6 23 0 85 33

. . ) ( . )

. ( . . ) .

− × + − ×+ × − × × ..3 1470= kNm



N A fpm,Rd c cd. . kN= = × × × =−0 85 0 85 232 468 33 3 10 68503, .




. .

= + × +

= × × ×

0 5 0 85

546 1 958 0 5 0 85 33

.

. + .. .3 28 915 435 0

1642

× ×=

+ .3768

kNm




A 0 15,880B 1470 0C 1470 6850D 1642 3425




( ) ( ) .EI K E I K E I E Ieff,II,y o a ay e,II c,eff cy s sy kNmm= + + = ×1 15 1011 2


N

EI

Lcr,y,eff

y

eff,II,y kN= =× ××

=π π2

2

2 11

2 6

1 15 104 10

70 600( ) .

,

The result is less than 10NEd for the major axis, so the second-order effects must be considered for the moment from the first-order analysis and the moment from imperfection.


e L0,y / mm= =200 20

For the major axis, the midlength bending moments due to NEd and the imperfection are calculated by:

20,000

15,000

20001500

10,000

A

C

D

B

1000

Axi

al lo

ad (k

N)

Moment (kNm)

5000

5000

0


Design examples 55

N eEd 0,y . kNm = × =9000 0 02 180


k

N Nimp,y

Ed cr,y,eff/./

.15=−

=−

=β

11 0

1 9000 70 6001

,


r = 200/300 = 0.667


β = max (0.66 + 0.44 r; 0.44) = 0.95

thus, the amplification factor is:

k

N Ny

Ed cr,y,eff/ /=−

=−

=β

10 95

1 9000 70 6001 09

.,

.

Hence, the design moment considering the second-order effect is:


kNm

= = × ×

=

+ +0 1 09 300 1 200

557

.

For NEd > Npm,Rd, the factor is determined from:

µd

pl,Rd Ed

pl,Rd pm,Rd

=−−

=−−

=N N

N N

15 880 900015 880 6850

0 76,,

.

Thus,

M

M

M

My,Ed

pl,N,y,Rd

y,Ed

d pl,y,Rd

= =×

= <µ

5570 76 1470

0 5 0 9.

. .



Steel-reinforced concrete column with high-strength materials

Steel grade S550 and concrete class C90/105 are used in this design example. Other design data are same as in Section “Steel-reinforced concrete column with normal-strength material,” such as loading; column length; dimensions of the SRC column cross-section, steel section, and reinforcement; and so on.




fck = 72 N/mm2; fcd = 48 N/mm2; Ecm = 41.1 GPa;


fy = 550 N/mm2; fyd = 550 N/mm2;


Aa = 15,020 mm2, As = 2512 mm2, Ac = 232,468 mm2

Iay = 276.7 × 106 mm4, Iaz = 90.6 × 106 mm4

Isy = 75.36 × 106 mm4, Isz = 75.36 × 106 mm4

Icy = 4856 × 106 mm4, Icz = 5042 × 106 mm4




= × + × × +

0 85

15 020 550 0 85 232 468 48 2( , . , 5512 435 10

18 838

3× ×=

−)

, kN

Design examples 57

δ = =

× ×= <

−A f

Na yd

pl,Rd

15 020 550 1018 838

0 44 0 93,

,. .

Long-term effects


ϕt =1 14.


The modulus of elasticity of concrete Ecm is reduced to the value Ec,eff:

E

E

N Nc,eff

cm

G,Ed Ed t/kN/mm=

+=

127 2 2

( ).

ϕ



( ) . .EI E I E I E Ieff,y a ay c,eff cy s sy kN mm= + + = ×0 6 1 53 1011 2

( ) . .EI E I E I E Ieff,z a az c,eff cz s sz kN mm= + + = ×0 6 1 17 1011 2


N

EI

Lcry

eff,y

y

kN= =π2

2 94 200( )

,

N

EI

Lcrz

eff,z

z

kN= =π2

2 71 900( )

,



0.85 25

= + +

= × × ×

0 85

15 020 550 232 468 72( , ,+ + 112

kN

× ×=

500 10

23 744

3)

,



λy

pl,Rk

cry

= = =N

N

23 74494 200

0 50,,

.

λz

pl,Rk

crz

= = =N

N

23 74471 900

0 57,,

.



Φy y y= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 34 0 50 0 2 0 50 0 68

2

2

. .

. [ . ( . . ) . ] .

α λ λ

Φz z z= + −( )+( )= × + × − + =

0 5 1 0 2

0 5 1 0 49 0 57 0 2 0 57 0 76

2

2

. .

. [ . ( . . ) . ] .

α λ λ


χλ

y

y y y

=+ −

=+ −

=1 1

0 68 0 68 0 500 88

2 2 2 2Φ Φ . . .

.

χλ

z

z z2

z2

=+ −

=+ −

=1 1

0 76 0 76 0 570 80

2 2Φ Φ . . .

.

Buckling resistance

The minor axis is the more critical, so,

N N

N

b Rd y

Ed

, min( ; )

. , ,

=

= × = > =

χ χz pl,Rd

kN kN0 80 18 838 15 070 9000

Design examples 59


Interaction curve

Point A (0, Npl,Rd)


MA = 0NA = Npl,Rd = 18,838 kN

Point B (Mpl,Rd, 0)


hA f A f f

b f t f fn

c cd sn sd cd

c cd w yd cd

. .

. .=

− −+ −

0 85 2 0 852 0 85 2 2 0 85

( )( )

==× × − × × − ×

× × × + × ×23 2468 0 85 48 628 2 435 0 85 482 500 0 85 48 2 12 2

, ( . )(

.. ×× − ×

=550 0 85 48

136.

mm)

Hence,

h


2138 6< .



Wpa3mm= ×1 958 106.

W A eps si i

33768 mm= = × = ×∑ [ ] .1

661884 200 0 10

Wb h

W Wpcc c

ps .3768= − − =×

− × − ×

= ×

2 26 6

4500 500

41 958 10 0 10

28 915 1

pa .

. 006 mm3



Wpsn3mm= 0

W t hpan w n2 3.222 mm= = × = ×12 136 0 102 6


3

0.222 10

mm

= − − = × − × −

= ×

2 2 6

6

500 136 0

9 026 10.



=

( ) ( ) . ( )

(

ps 0 5 α

11 958 0 222 550 0 3768 0 435

0 5 28 915 9 026 0 85 4

. . ) ( . )

. ( . . ) .

− × + − ×+ × − × × 88 1524= kNm



N A fpm,Rd c cd. . kN= = × × × =−0 85 0 85 232 468 48 10 94853,




. .

= + × +

= × × ×

0 5 0 85

550 1 958 0 5 0 85 48

.

. + ×× ×=

28 915 435 0 68

1723

. + .37

kNm




A 0 18,838B 1524 0C 1524 9485D 1723 4742

Design examples 61



( ) ( ) .EI K E I K E I E Ieff,II,y o a ay e,II c,eff cy s sy kNmm= + + = ×1 26 1011 2


N

EI

Lcr,y,eff

eff,II,y

y

kN= =× ××

=π π2

2

2 11

2 6

1 26 104 10

77 400( ) .

,

The result is less than 10NEd for the major axis, so the second-order effects must be considered for the moment from the first-order analysis and the moment from the imperfection.


e L0,y / mm= =200 20

25,000

20,000

15,000

20001500

10,000

A

C

D

B

1000

Axi

al lo

ad (k

N)

Moment (kNm)

5000

5000

0



For the major axis, the midlength bending moments due to NEd and the imperfection are calculated by:

N eEd 0,y . kN m = × =9000 0 02 180


k

N Nimp,y

Ed cr,y,eff/./

.13=−

=−

=β

11 0

1 9000 77 4001

,


r = 200/300 = 0.667


β = max (0.66 + 0.44 r; 0.44) = 0.95

Thus, the amplification factor is:

k

N Ny = −

=−

=β

10 95

1 9000 77 4001 08

Ed cr,y,eff/ /.

,.


M k M k N ey,Ed y y,Ed,top imp,y Ed ,z

kNm

= + = × + ×

=0 1 08 300 1 13 200

550

. .

For 0.5Npm,Rd < NEd ≤ Npm,Rd, the factor is determined from:

µdpm,Rd Ed

pm,Rd

max,Rd

pl,Rd

= +−( )

−

= +×

12

1

12

N N

N

M

M

(99485 90009485

17231524

1 1 01−

−

=

).

Thus,

M

M

M

My,Ed

pl,N,y,Rd

y,Ed

d pl,y,Rd

= =×

= <µ

5501 01 1524

0 36 0 9.

. .

Design examples 63


STEEL-REINFORCED CONCRETE COLUMN WITH DIFFERENT DEGREE OF CONFINEMENT

A comparison between the original and alternative designs of the SRC column was conducted to study the effect of confinement on the strength of the column.

Original design

Figure 7 shows the original design of SRC columns from a practical project. In the original design, the concrete cylinder strength was 50 MPa and the yield strengths of the steel and rebar were 355 and 500 MPa, respectively. The diameter and spacing of the transverse reinforcement (hoops) were 10 and 200 mm, respectively. The diameter of the longitudinal reinforcement was 25 mm.

High-strength concrete

High-strength concrete (90 MPa) was proposed as an alternative design. Thus, the concrete and reinforcement bar amounts are reduced and the arrangement is revised. As a result, the dimension of the column is reduced, as shown in Figure 8.

1200

mm

900

mm

600

mm

900 mm1200 mm

(a)

(b)

(c)

600 mm

UC 356 × 406 × 634Φ10@200

UC 356 × 406 × 467Φ10@200

UC 254 × 254 × 107Φ10@200

Figure 7 Original design (S355 steel and C50/60 concrete). (a) N1, (b) N2, (c) N3.


High-strength steel and high-strength concrete

High-strength concrete (90 MPa) with high-strength steel (690 MPa) was proposed as an alternative design. Thus, the concrete and steel amounts were reduced and their arrangement was revised, as shown in Figure 9. To develop the full strength of steel, the spacing of the transverse reinforcement was made smaller than the original design

480 mm

480

mm

700 mm

700

mm

900 mm

900

mm

(a)

(b)

(c)

UC 356 × 406 × 634Φ10@200

UC 356 × 406 × 467Φ10@200

UC 254 × 254 × 107Φ10@200

Figure 8 Alternative design with S355 steel and C90/105 concrete. (a) normal strength steel and High strength concrete (NH)1, (b) NH2, (c) NH3.

900

mm

900 mm 700 mm

700

mm

480 mm

480

mm

(a)

(b)

(c)

UC 356 × 406 × 340Φ10@75

UC 356 × 406 × 235Φ10@100

UC 203 × 203 × 52Φ10@65

Figure 9 Alternative design with S690 steel and C90/105 concrete. (a) HH1, (b) HH2, (c) HH3.

Design examples 65

according to the equations from the modified confinement model. The smaller spacing of the hoops can lead to congestion, which is a practical concern that needs to be addressed.

Figure 10 shows the comparison of the concrete, steel, and rebar amounts of different cross-sections. There is a reduction of 47% (1.5 tons per meter) in the amount of concrete by adopting high-strength concrete C90/105. There is a reduction of 46% (0.3 ton per meter) in the amount of structural steel by adopting high-strength steel S690. However, the transverse reinforcement amount increases, as the spacing of the hoops is smaller than the original design. The transverse reinforcement increases by 200% (0.03 ton per meter) by adopting an S690 steel section, as shown in Figure 10a. Similarly, for original cross-sections N2 and N3, similar results can also be obtained. The concrete and steel amounts are reduced significantly by adopting high-strength material, but the amount of transverse reinforcement is increased by adopting an S690 steel section. Nevertheless, the total amount of concrete and steel (structural and reinforcing steel) is reduced significantly.

Figure 11 shows the effect of the spacing and diameter of links on the steel stress of the three types of SRC columns (high strength steel

5.0(a)

(c)

(b)

4.0

3.0

2.0

1.0

0.0N1 NH1 HH1 N2

ConcreteSteelRebar

ConcreteSteelRebar

ConcreteSteelRebar

NH2 HH2

N3 NH3 HH3

3.0

2.4

1.8

1.2

0.6

0.0

0.3

0.6

0.9

1.2

1.5

0.0

Figure 10 Comparison of material amounts. (a) N1,NH1,HH1, (b) N2,NH2,HH2, (c) N3,NH3,HH3.


700(a)

(c)

(b)C90C70C50650

600

Stee

l str

ess (

MPa

)

500

550

Spacing (mm)0 100 200 300 400 500

C90C70C50

C90C70C50

700

650

600

Stee

l str

ess (

MPa

)

500

550

Spacing (mm)0 100 200 300 400 500

700

650

600

Stee

l str

ess (

MPa

)

500

550

Spacing (mm)0 100 200 300 400 500

Figure 12 Effect of spacing of links and concrete strength. (a) HH1, (b) HH2, (c) HH3.

700(a)

(c)

(b)Φ = 12 mmΦ = 10 mmΦ = 8 mm

650

600Stre

ss (M

Pa)

550

Spacing (mm)0 100 200 300 400 500

700 Φ = 12 mmΦ = 10 mmΦ = 8 mm

650

600Stre

ss (M

Pa)

550

Spacing (mm)0 100 200 300 400 500

700Φ = 12 mmΦ = 10 mmΦ = 8 mm650

600Stre

ss (M

Pa)

550

Spacing (mm)0 100 200 300 400 500

Figure 11 Effect of spacing and diameter of links. (a) HH1, (b) HH2, (c) HH3.

Design examples 67

and high strength concrete [HH]1, HH2, and HH3). It indicates that the real stress of steel increases with the increase in the diameter of the transverse reinforcement when the spacing of links takes a certain value. The steel stress increases with the decreasing of the spacing of links due to insufficient lateral confinement, which has been verified in parametric study. For the three types of SRC columns, if the spacing of the hoops is same as the original design (200 mm); the real stress of steel grade S690 is about 600–650 MPa depending on the diameter of the hoops. For a large spacing of 400 mm, the real stress of steel grade S690 is 590–610 MPa depending on the diameter of the links and the dimensions of the column.

Concrete strength also has an effect on the stress in steel according to the strain-compatibility method because the strains of different concrete classes are different. Figure 12 shows the effect of the spacing of links and concrete cylinder strength on the real stress in steel. It is indicated that the real stress of steel increases with the increasing of the concrete cylinder strength for these three types of SRC columns. The stress in the steel section increases according to the stress–strain relationship of steel even though the lateral confinement pressure is the same. Therefore, for the same arrangement of the reinforcement bar, the stress in steel can reach a higher value when a higher-strength concrete is used in the SRC columns.

69

Appendix A: Design resistance of shear connectors

70 Appendix A

Tabl

e A.

1 D

esig

n re

sist

ance

of s

hear

con

nect

ors

Dim

ensio

n of

co

nnec

tors

Des

ign

resis

tanc

e of

she

ar s

tuds

PR

d (k

N)

d (m

m)

h sc (

mm

)C2

0/25

C25/

30C3

0/37

C35/

45C4

0/50

C45/

55C5

0/60

C55/

67C6

0/75

C70/

85C8

0/95

C90/

105

1650

37.3

43.1

48.8

53.5

57.9

57.9

57.9

57.9

57.9

57.9

57.9

57.9

7545

.252

.357

.957

.957

.957

.957

.957

.957

.957

.957

.957

.910

045

.252

.357

.957

.957

.957

.957

.957

.957

.957

.957

.957

.912

545

.252

.357

.957

.957

.957

.957

.957

.957

.957

.957

.957

.915

045

.252

.357

.957

.957

.957

.957

.957

.957

.957

.957

.957

.917

545

.252

.357

.957

.957

.957

.957

.957

.957

.957

.957

.957

.920

045

.252

.357

.957

.957

.957

.957

.957

.957

.957

.957

.957

.922

545

.252

.357

.957

.957

.957

.957

.957

.957

.957

.957

.957

.925

045

.252

.357

.957

.957

.957

.957

.957

.957

.957

.957

.957

.919

5046

.353

.660

.566

.472

.077

.481

.781

.781

.781

.781

.781

.775

63.1

73.0

81.2

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

100

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

125

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

150

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

175

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

200

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

225

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

250

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

(Con

tinue

d)

Appendix A 71

Tabl

e A.

1 (C

ontin

ued)

Des

ign

resi

stan

ce o

f she

ar c

onne

ctor

s

Dim

ensio

n of

co

nnec

tors

Des

ign

resis

tanc

e of

she

ar s

tuds

PR

d (k

N)

d (m

m)

h sc (

mm

)C2

0/25

C25/

30C3

0/37

C35/

45C4

0/50

C45/

55C5

0/60

C55/

67C6

0/75

C70/

85C8

0/95

C90/

105

275

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

300

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

325

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

350

63.8

73.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

81.7

2250

56.0

64.7

73.1

80.2

87.0

93.5

100.

010

6.3

109.

510

9.5

109.

510

9.5

7575

.487

.298

.510

8.0

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

100

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

125

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

150

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

175

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

200

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

225

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

250

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

275

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

300

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

325

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

350

85.5

98.9

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

109.

510

9.5

(Con

tinue

d)

72 Appendix A

Tabl

e A.

1 (C

ontin

ued)

Des

ign

resi

stan

ce o

f she

ar c

onne

ctor

s

Dim

ensio

n of

co

nnec

tors

Des

ign

resis

tanc

e of

she

ar s

tuds

PR

d (k

N)

d (m

m)

h sc (

mm

)C2

0/25

C25/

30C3

0/37

C35/

45C4

0/50

C45/

55C5

0/60

C55/

67C6

0/75

C70/

85C8

0/95

C90/

105

2550

66.3

76.6

86.6

94.9

102.

911

0.7

118.

312

5.8

133.

114

1.4

141.

475

88.3

102.

111

5.4

126.

513

7.3

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

410

011

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

412

511

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

415

011

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

417

511

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

420

011

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

422

511

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

425

011

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

427

511

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

430

011

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

432

511

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

435

011

0.4

127.

614

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

414

1.4

141.

4

Not

e:

f u =

450

N/m

m2 .

73

Appendix B: Design chart

The design process for SRC column subjected to axial compression is summarized in the chart given in Figure B.1 whereas the design process for SRC column subjected to axial compression and bending is summarized in the chart given in Figure B.2.

74 Appendix B

Find NEd and MEd at both ends of the column

Determine the material properties of steel and concrete, Is high strength steel orhigh strength concrete adopted?

Are the steel and concrete compatible?

Calculate Npl,Rd and Ncr

Calculate λ, and then X

Is NEd ≤ χNpl,Rd? Column not strong enoughNo

Is the column in axial compression only?

Yes

Column verifiedYes

See B2

No

No

Yes

Yes

No

Determine the realstrength of highstrength steel

Figure B.1 Design for SRC column subjected to axial compression.

Appendix B 75

Find Vpl,a,Rd. Is VEd > 0.5Vpl,a,Rd?

Determine Mpl,a,Rd and Mpl,Rd, and hence Va,Ed and Vc,Ed. Is Va,Ed >0.5Vpl,a,Rd?

Determine the interaction curve forthe cross-section.

Calculate ρ and hence reduced fyd. Find member imperfection, e0.

No

Can first-order member analysis be used?

Yes

Yes

From NEd and the interaction diagrams, find µdy and µdz. Check that the cross-section can resist My,Ed,max and Mz,Ed,max.

No

Determine MEd, the maximum first-orderbending moment withinthe column length.If MEd,1 = MEd,2 it isMEd,max = MEd,1 + NEde0

CalculateNcr,eff = π2(EI)eff,II/L2

find β for end momentsMEd,top and MEd,bot andhence k (=k1); find k2 forβ = 1;find the design moment forthe column,MEd,max = k1MEd + k2NEde0

Find MEd,max bysecond-order analysisof the pin-ended column length with force NEd and endmoments MEd,1 andMEd,2.

Yes No

Figure B.2 Design for SRC column subjected to combined compression and bending.

77

Index

Amplification factor, 37, 45, 46, 55, 62Axial compression

design for SRC column subjected to, 74

steel-reinforced concrete column subjected to, 13

Bending resistance, 53, 60Buckling reduction factor

high-strength concrete, 12–13, 20–21, 26, 41–42

high-strength materials, 58high-strength steel, 17–18, 20–21,

26, 51normal-strength material, 7–8,

32–33Buckling resistance

high-strength concrete, 13, 21, 26–27, 42

high-strength materials, 58–59high-strength steel, 18, 21, 26–27,

51normal-strength material, 8, 33

C50/60 concrete, 63, 64C90/105 concrete, 64Combined compression and bending

design for SRC column subjected to, 75

steel-reinforced concrete column subjected to, 27

Combined compression and uniaxial bending, 35–37, 44–46, 54–55, 61–63

Concrete class C50/60, 28Concrete class C90/105, 8, 18, 38, 56Concrete strength, 67Confinement effective coefficient, 47Creep coefficient, 49, 57

Degree of confinement, steel-reinforced concrete column with, 63

high-strength concrete, 63–64high-strength steel and concrete,

64–67original design, 63

Design chart, 73design for SRC column subjected

to axial compression, 74design for SRC column subjected

to combined compression and bending, 75

Design resistance of shear connectors, 69–72

Effective compressive strength and elastic modulus of concrete C90/105, 56

Effective flexural stiffnessconsidering second-order effects,

54, 61high-strength concrete, 11, 20, 25,

40, 57high-strength steel, 17, 20, 25, 50normal-strength material, 6, 31–32

Effective volume ratio of hoops, 47Elastic critical force, 45, 61

78 Index

Elastic critical normal forcehigh-strength concrete, 11–12, 20,

25–26, 41high-strength materials, 57high-strength steel, 17, 20,

25–26, 50normal-strength material, 7, 32


high-strength concrete, 11, 20, 25, 40

high-strength materials, 57high-strength steel, 16, 20, 25, 50normal-strength material, 6, 31

EN 1993–1-1, 41, 51EN 1994–1-1, 36, 37, 45, 54, 55, 58,

61, 62

High-strength concrete and steel, steel-reinforced concrete column with

alternative design, 22buckling reduction factor, 20–21, 26buckling resistance, 21, 26–27cross-sectional areas and second

moments of area, 19, 23design data, 22design strengths and modulus, 19,

22–23effective flexural stiffness of

cross-section, 20, 25elastic critical normal force, 20,

25–26elastic modulus of concrete

considering long-term effects, 20, 25

local buckling checking, 23long-term effects, 19, 24–25reinforcement ratio checking, 23relative slenderness ratio, 20, 26steel contribution factor checking,

19, 23–24High-strength concrete, steel-

reinforced concrete column with, 8, 38

buckling reduction factor, 12–13, 41–42

buckling resistance, 13, 42cross-sectional areas, 9, 38design strengths and modulus,

8–9, 38with different degree of

confinement, 63–67effective flexural stiffness of

cross-section, 11, 40elastic critical normal force,

11–12, 41elastic modulus of concrete


interaction curve, 42–44long-term effects, 10–11, 39–40relative slenderness ratio, 12, 41resistance checking, 44–46second moments of area, 9, 38steel contribution factor checking,

9–10, 38–39High-strength materials, steel-

reinforced concrete column with, 56

buckling reduction factor, 58buckling resistance, 58–59cross-sectional areas and second

moments of area, 56design strengths and modulus, 56effective flexural stiffness of

cross-section, 57elastic critical normal force, 57elastic modulus of concrete

considering long-term effects, 57

interaction curve, 59–61long-term effects, 57relative slenderness ratio, 58resistance checking of steel-

reinforced concrete column, 61–63

steel contribution factor checking, 56–57

High-strength steel, steel-reinforced concrete column with, 13, 46, 64–67

buckling reduction factor, 17–18, 51buckling resistance, 18, 51cross-sectional areas, 16, 49

Index 79

design strengths and modulus, 13–16, 47–49

effective flexural stiffness of cross-section, 17, 50

elastic critical normal force, 17, 50elastic modulus of concrete


interaction curve, 52–54long-term effects, 16, 49relative slenderness ratio, 17,

50–51resistance checking, 54–55second moments of area, 16, 49steel contribution factor checking,

16, 49

Interaction curvehigh-strength concrete, 42–44high-strength materials, 59–61high-strength steel, 52–54normal-strength material, 33–35

Local buckling checkinghigh-strength concrete and steel, 23normal-strength material, 4, 29

Longitudinal reinforcement ratio, 14, 47

Normal-strength material, steel-reinforced concrete column with, 2, 27

buckling reduction factor, 7–8, 32–33

buckling resistance, 8, 33cross-sectional areas, 3, 29design data, 27–28design strengths and modulus,

2–3, 28–29effective flexural stiffness of

cross-section, 6, 31–32elastic critical normal force, 7, 32elastic modulus of concrete


interaction curve, 33–35local buckling checking, 4, 29long-term effects, 5–6, 30–31

reinforcement ratio checking, 4, 29relative slenderness ratio, 7, 32resistance of steel-reinforced

concrete column checking, 35–37

second moments of area, 3–4, 29steel contribution factor checking,

4–5, 29–30

Plastic section moduli, 34, 35, 52–53, 59, 60

Real stress of hoops, 47Reinforcement ratio checking

high-strength concrete and steel, 23

normal-strength material, 4, 29Relative slenderness ratio

high-strength concrete, 12, 20, 26, 41

high-strength materials, 58high-strength steel, 17, 20, 26,

50–51normal-strength material, 7, 32

Resistance checking of steel-reinforced concrete column

high-strength concrete, 44–46high-strength materials, 61–63high-strength steel, 54–55normal-strength material, 35–37

S355 steel grade, 8, 13, 28, 63, 64S550 steel grade, 18, 46, 56S690 steel, 64Shear connectors, design resistance

of, 69–72SRC columns, 1

design for, 63, 73, 74Steel contribution factor checking

high-strength concrete, 9–10, 19, 23–24, 38–39

high-strength materials, 56–57high-strength steel, 16, 19,

23–24, 49normal-strength material, 4–5,

29–30Strain of HCC, 48Strain of PCC, 48

Design Examples Steel Reinforced Concrete...

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