Rcs1 -chapter6-SLS

Marwan SADEK 1

Reinforced Concrete

Structures 1 - Eurocodes

RCS 1

Professor Marwan SADEKhttps://www.researchgate.net/profile/Marwan_Sadek

https://fr.slideshare.net/marwansadek00

Email : [email protected]

If you detect any mistakes, please let me know at : [email protected]

Marwan SADEK 2

PLAN – RCS1

M. SADEK

Ch 1 : Generalities – Reinforced concrete in practice

Ch 2 : Evolution of the standards – Limit states

Ch 3 : Mechanical Characteristics of materials – Constitutive relations

Ch 4 : Durability and Cover

Ch 5 : Beam under simple bending – Ultimate limit state ULS

Ch 6 : Beam under simple bending – serviceability limit state SLS

Ch 7 : Section subjected to pure tension

Marwan SADEK 3

Selected ReferencesFrench BAEL Code (91, 99)

Règles BAEL 91 modifiées 99, Règles techniques de conception et de calcul des ouvrages et constructions en béton armé, Eyrolles, 2000. J. Perchat (2000), Maîtrise du BAEL 91 et des DTU associés, Eyrolles, 2000. J.P. Mougin (2000), BAEL 91 modifié 99 et DTU associés, Eyrolles, 2000.….

EUROCODES H. Thonier (2013), Le projet de béton armé, 7ème édition, SEBTP, 2013. Jean-Armand Calgaro, Paolo Formichi ( 2013) Calcul des actions sur lesbâtiments selon l'Eurocode 1 , Le moniteur, 2013. J. M. Paillé (2009), Calcul des structures en béton, Eyrolles- AFNOR, 2009. Jean Perchat (2013), Traité de béton armé Selon l'Eurocode 2, Le moniteur,2013 (2ème édition) Manual for the design of concrete building structures to Eurocode 2, TheInstitution of Structural Engineers, BCA, 2006. A. J. Bond (2006), How to Design Concrete Structures using Eurocode 2, Theconcrete centre, BCA, 2006.https://usingeurocodes.com/

M. SADEK

Marwan SADEK 4

In addition to Eurocodes, the references that are mainly used to prepare this course material are : Thonier 2013

Perchat 2013

Paillé 2009

Some figures and formulas are taken from

Cours de S. Multon - BETON ARME Eurocode 2 (available on internet)

Cours béton armé de Christian Albouy

M. SADEK

Marwan SADEK 5

Chapter VI

Beam under simple bending –Serviceability limit state SLS

1. Assumptions

2. Rectangular section

3. T Section

4. Crack Control

Marwan SADEK 6

Introduction / Assumption 2.Rectangular Section 3.T section 4.Crack width Control

INTRODUCTION (SECTION 7 – EC2-1-1)

Stress limitation (Steel & Concrete)

Crack width Control

Deflection control (RCS 2)

Marwan SADEK7

H1) Principle of Navier-Bernoulli : After deformation, plane sections remain planeand normal to the axis of the beam (linear strain diagram)

H2) The tensile strength of the concrete is neglected

H3) bundled bars are treated as a single bar of a diameter derived from the

equivalent total area and placed at the COG of the group

H4) Total bond between concrete and steel (no relative slip)

H5) The design stress-strain diagram for the concrete and the steel are Linear

Elastic: = E.

Equivalence coefficient e

ASSUMPTIONS


Marwan SADEK 8

Effective equivalence coefficient or ratio(also noted n in the previous french standard)

The obtained value is generally greater than that of the previous French

rules (in the BAEL code, n=15)

Equivalence coefficient (EC2 / Professional Rules)

France (professional rules)proposes to reduce e at thecombination SLS Charac.


Marwan SADEK 9

Note 1 :

For only short time action, take a coefficient e= Es/Ecm

Note 2 :

The crack width control is generally conducted under the combinations :

Quasi-permanent for buildings

Frequent for bridges (FNA NF EN 1992-2-NA)

When high precision is not required, we can take e= 15

Note 3 :


Marwan SADEK 10

Stress limitation (Steel & Concrete)

Concrete SteelSLS Caract. c 0,6.fck (pertinent) s 0,8.fck

SLS quasi permanent c 0,45.fck , if not take non linear creep law Crack Control


Marwan SADEK 11

Combination SLS Charac.“it may be appropriate to limit” the compressive stress to a value

c 0.6 fck , for exposure classes XD, XF et XS

Note 1: it’s not an obligation like in BAEL Code

Note 2 : No limitation for the classes X0, XC

Compression Stress in the Concrete c

Combination SLS quasi-permanent (especially for prestress)

if c 0.45 fck linear creep

if not non linear creep


Marwan SADEK 12

Combination SLS Charac.

s 0.8 fyk

Tensile stress in the Steel

Note : s 1fyk if the stress is due to a imposed deformation

In order to prevent inelastic deformation, crack or unacceptable deflections

Introduction / Assumption 2. Rectangular Section 3.T section 4.Crack width Control

Marwan SADEK 13

(e=n )

Rectangular Section – Design at SLSIntroduction / Assumption 2. Rectangular Section 3.T section 4.Crack width Control

Marwan SADEK 14

Equivalent Inertia– Homogenized section

I = b.x3/3 + n As′ (x–d′)² + n As (d–x)²

Note : The moment of inertia of the steel bars relative to their C.O.G, is neglected


Marwan SADEK 15

Stresses under Mser

c = K.x Compression in Concrete

s= n.K. (d – x) Tension Steel As

Compression Steel A’s ’s= n.K. (x – d’)

(K=Mser / I)


Marwan SADEK 16

Internal forces

Fc = c.b.x/2 = K.b.x²/2

Fs=As.s= n.K. As .(d – x)

F’s=A’s.’s= n.K. A’s(x – d’)

Compression in Concrete

Tension Steel As

Compression Steel A’s


Marwan SADEK 17

Force Equilibrium:

Position of Neutral Axis x=?

b x² / 2 + n.(As + A’s ).x – n (As.d + A’s .d’)=0


Marwan SADEK 18

Particular Case – Without Compression Steel

I = b.x3/3 + n As (d–x)²


Marwan SADEK 19

1) Problem 1: As, A’s known Check for Stresses

Position of the Neutral Axis x

Moment of Inertia of Homogenized Section

Determination of Stresses in Concrete and Steel

Comparison with allowable Values

Detailed Calculation:


Marwan SADEK 20

Quick Check using Tables (Thonier 2012)(Without Compression Steel A’s=0)

Limit Value c Corresponding to a maximum stress in the concrete c = 0.6 fck

Limit Value s corresponding to a maximum stress in the steel s = 0.8 fyk

You can also find tables for quick check in Perchat (2013)


Marwan SADEK 21

2) Problem 2 : The compression stress in the concrete is ok c c

The tensile stress in the steel is exceeded s s

Determination of the new value of As = ? (A’s = 0 )

The new section will resist a tensile stress equal s

Unknowns : As, x, c


Marwan SADEK 22

Fc = c.b.x/2 = Fs = As .s

Eq 1 – Equilibrium of the forces

Eq 2 – Equilibrium of the momentsMser = Fc .(d-x/3) = c.b.x/2 .(d-x/3)

Eq 3 – Stress diagram (Linear)

c / x = (s / n ) / (d-x)


Marwan SADEK 23

= Mser / (b.d²) = x / d

3 equations with 3 unknowns

We obtain a third degree equation in

By Substituting

Iterative resolution


Marwan SADEK 24

Using Table (Thonier 2012)Introduction / Assumption 2. Rectangular Section 3.T section 4.Crack width Control

Marwan SADEK 25

3) Problem 3 : The compression stress in the concrete is exceeded c c

2 possible solutions

a) Change the dimensions of the concrete section

or

b) Introduce compression steel


Marwan SADEK 26

3a) Change the dimensions of the concrete section b, h

The Concrete and steel resist at their maximum strengthc et s

The position of the Neutral Axis is fixed

Whenc = 0.6 fck , we obtain :


Marwan SADEK 27

3b) Introducing A’s?? Unknowns : As, A’s, ??

The Concrete and steel resist at their maximum strengthc et s


Marwan SADEK 28

3b) Introducing A’s2 imaginary sections

The moment that could be resisted by the concrete in compression (c = 0.6 fck) :


Marwan SADEK 29

3b) Introducing A’s

The moment resisted by A’s (or A2) :


Marwan SADEK 30

3b) Introducing A’s


Marwan SADEK 31

T Section – Design at SLS

Introduction / Assumption 2. Rectangular Section 3. T section 4. Crack width Control

Marwan SADEK 32

Neutral Axis

Homogenized moment of inertia

c = K.xs= n.K. (d – x) K= Mser / I


Marwan SADEK 33

Note :

In principle, when we find x<hf , the neutral Axis is located in the flange.

The design is carried out as a rectangular section.

In general, we neglect the resistance of concrete between 0.8 x and x,

that’s why, we can consider that the neutral axis is still located in the

flange until x 1.25 hf (Thonier 2012)


Marwan SADEK 34

T Section – Steel Reinforcement If the stress in the Steel is exceeded, we should recalculate the new

section of steel reinforcement As. If the neutral Axis is located in the web,

the calculation is relatively complex. However we can approximate As using

a simplified formula for the lever arm :

In this formula, the concrete strength is assumed sufficient and can resist

the resultant compression force, which is case in general (A’s = 0)


Marwan SADEK 35

Crack Control


Marwan SADEK 36

Cracking

(Perchat 2013)


Marwan SADEK 37

Recommended values of wmax (mm) - EC2 -7.3 (SLS quasi-permanent)

Modifications by the French National Annex (FNA)


Marwan SADEK 38

Control of cracking without direct calculation (SLS quasi-permanent)For cracks caused mainly by loading, either the provisions of Table 7.2N or the provisions ofTable 7.3N are complied with. The steel stress should be calculated on the basis of a crackedsection under the relevant combination of actions. (Table 7.2N when cracking is causeddominantly by restraint)

or

The conditions given by these

tables are very conservative for the

design. Instead, It is recommended

to proceed to the calculation of the

crack width.


Marwan SADEK 39

The maximum bar diameter should be modified as follows:

a) Bending (at least part of section in compression)

b) Tension (uniform axial tension)

kc = 0.4 (simple bending)

hcr: is the depth of the tensile zone immediately prior to cracking, It

could be approximated by hcr = 0.5 h for simple bending

Control of cracking without direct calculation (SLS quasi-permanent)


Marwan SADEK 40

For reinforced or prestressed slabs in buildings subjected to bending without

significant axial tension, specific measures to control cracking are not necessary

where the overall depth does not exceed 200 mm and the provisions of 9.3 have

been applied (detailing requirement for solid slabs)

Control of cracking without direct calculation


Marwan SADEK 41

Control of cracking without direct calculation (Bridge FNA NF-EN 1992-2-NA)

France proposes in its National Annex for bridge the following method :

1. longitudinal Bar Spacing is less than 5 (c + /2) ;

2. The steel stress σs should not exceed 1000.wk under the SLS frequent

combination

Bending

(Method verified by SETRA, Paillé 2009)

σs ≤ 1000.wk (σs expressed in MPa and wk in mm)


Marwan SADEK 42

The second condition (2.) becomes :

σs ≤ 600.wk (instead of 1000.wk)

Tension

Control of cracking without direct calculation (Bridge FNA NF-EN 1992-2-NA)


Marwan SADEK 43

Minimum Reinforcement (Cracking )

Act : is the area of concrete within tensile zone. The tensile zone is that part of the

section which is calculated to be in tension just before formation of the first crack

Act = 0,5.b.h (for a rectangular section bh)

fct,eff = fctm

k = 1 for webs with h 300 mm or flanges with widths less than 300 mm

k = 0.65 for webs with h 800 mm or flanges with widths greater than 800 mm

(intermediate values may be interpolated)


Marwan SADEK 44

For rectangular sections and webs of box sections and T sections:

k1 = 1 simple bending

h* = Min[h,1m]

c : is the mean stress of the concrete acting on the part of the section under consideration

c = Ned/Ac (=0 simple bending)

kc= 0.4 (simple bending)

Note : For a rectangular section subjected to bending, The ratio of minimum steel reinforcement

required for cracking control is lower than that for preventing the brittleness of concrete. In practice,

it is not necessary to calculate it.

Minimum Reinforcement (Cracking )


Marwan SADEK 45

Skin reinforcement (Crack control) Beams h > 1 m

Beams with a total depth larger than 1 m, should be provided with additionalskin reinforcement to control cracking on the side faces of the beam. Thisreinforcement should be distributed between the level of the tension steel andthe neutral axis and should be located within the links.

skin = As / Act = 0.2 fctm / fyk Bending

skin = As / Act = 0.5 fctm / fyk pure tension

kc=0.4 simple bending, 1 pure tension; k=0.5 ; s = fyk

The spacing and size of suitable bars may be obtained from tables 7.2N and

7.3N, with a steel stress of half the value assessed for the main tension

reinforcement


Marwan SADEK 46

Note 1 : Case of large diameter ( 32 mm)

EC2 (Annex J): Surface reinforcement to resist spalling should beused where the main reinforcement is made up. The surfacereinforcement should consist of wire mesh or small diameter bars,and be placed outside the links. This reinforcement is subjected tothe validation of National Annex and has been rejected by theFrance.

Note 2 : Case of BridgeEC2 (bridge) don’t give any specification. In the French NationalAnnexe, it is required for high depth beams, a skin reinforcementparallel to longitudinal axe, at least 3 cm²/ m of the surfaceperpendicular to the reinforcement , and should not be less than0,10 % of the concrete cross section.For beams located in the exposure classes XD and WS, this valuebecomes 5 cm² /m.

Skin reinforcement (Crack control)


Marwan SADEK 47

Crack width

Homogenized state :

‘RC Tie’, with a cross section Ac, or a RC zonesurrounding the reinforcement steel of a bendedbeam and similar to a tie.


Marwan SADEK 48

Theoretical Aspect

The width opening wm of a crack could be determined by calculating the mean

strain of a steel bar relatively to that of the concrete along the length comprised

between 2 cracks :

Note : For more details on the theory of crack opening , see Paillé 2009, Perchat

2013.

Crack width


Marwan SADEK 49

The crack width, wk, may be calculated as follow :

In Eurocode2

a) In situations where bonded reinforcement is fixed at reasonably close centres within the tension zone (spacing< 5(c+/2)) [case of beam in general ]

k1 = 0,8 for high bond bars, 1,6 bars with an effectively plain surface

k2 = 0,5 for bending , 1 for pure tension

k3 = 3,4 This value has been reviewed by France for high values of cover

take 3,4*(25/c)2/3 if c > 25 mm (FNA, M. Cortade)

Crack width


Marwan SADEK 50

hc,eff =Min[2,5.(h – d) ; h/2 ;(h – x)/3]

x : Distance from the top fibre to the Neutral Axis.

: is the bar diameter (where a mixture of bar diameters is used in a section, an equivalent diameter eq should be used)


Marwan SADEK 51

s : is the stress in the tension reinforcement assuming a cracked section (quasi-

permanent combination)

kt = 0,4 for long term loading, 0,6 for short term loading


Marwan SADEK 52

b) Where the spacing of the bonded reinforcement exceeds 5(c+/2) [case

of slabs in general]

Max of 2 values


53M. SADEK

Exercices Rectangular section without compression steel :

Determination of As at ULS

Check at SLS

Rectangular section with compression steel : :

Determination of As at ULS

Check at SLS

Crack Control

Using tables

method applied for bridges (FNA)

Evaluation of crack width

Rcs1 -chapter6-SLS

Education

Transcript of Rcs1 -chapter6-SLS