SLS analysis

Introduction

This document details how the SLS analysis is performed for circular sections and for rectangular reinforced concrete sections.

 

Circular section

This chapter details the calculation of a circular reinforced concrete section subjected to biaxial bending (SLS).

Notes

Geometry

SymbolUnitDescription
D ou BmDiameter of the section

 

Efforts

SymbolUnitDescription
NMNNormal force applied to the centre of gravity of the section
MMNmBending moment applied to the centre of gravity of the section

Stresses

SymbolUnitDescription
σsMPaStress in steels
σcMPaCompressive stress in concrete
fckMPaAllowable compressive stress of concrete (characteristic value)
fydMPaAllowable steel stress (design value)
fykMPaAllowable steel stress (characteristic value)

SLS check principle

The SLS check is performed using a stress based analysis.

This stress based analysis requires a calculation on an homogenized cross section considering the different deformation moduli of steel and concrete.

Calculation assumptions

The SLS check of reinforced concrete sections takes into account the following hypotheses:

The constitutive laws of concrete and steel are provided in the Materials chapter of this manuals.

Verification of the reinforced concrete section under SLS

The verification of the SLS reinforced concrete section is performed by homogenising the section in order to take into account the presence of two materials of different stiffnesses. The equilibrium of the section is based on the equilibrium of forces and moments, which allows to find the stress diagram of the section.

In the case of a fully tensioned section, the concrete does not provide any tensile strength. Only the reinforcement forces balance the external forces.

Generation of the interaction diagram

The principle of interaction diagram generation is based on the analysis of all possible bending modes of the section. In order to optimize the work of the section, this scan is done based on the limit stresses of the steel (pivot A) and the concrete (pivot B) at SLS. The stress diagram is homogenized to take into account the different strain moduli of the steel and concrete.

SLS SectionSection circulaire - SLS - 2 regions

Calculation of the resisting forces

Each homogenized stress diagram is described by the following equation:

(1)σc(y)=ay+b
(2)σc(φ)=aRcosφ+b

Where a is the slope and b is the ordinate at the origin.

Resistance force of concrete
(3)Nc=0xσc(φ)b(φ)dy
(4)Mc=0xσc(φ)b(φ)ydy
Resistance force of steel
Discrete distribution of steel bars
(5)σi=nσhomogénéisée

Where n is the equivalence coefficient: n=Es/Ec.

(6)Fi=σiAi

Resulting force on all steel bars :

(7)Ns=iFi

 

Continuous and homogeneous distribution of the steel section
(8)Ns,rectangular=2φ1φ2fyasrsdφ
(9)Ms,rectangular=2φ1φ2fyasrsrscosφdφ
(10)Ns,triangular=2φ2φ3σs,i(φ)asrsdφ
(11)Ms,triangular=2φ2φ3σs,i(φ)asrsrscos(φ)dφ

Calculation of the mobilisation rate of the bending strength

The mobilisation rate of the bending resistance is the ratio between the resistant moment and the applied one for the same normal force. It is deduced from the strain diagram calculated for the section under examination.

Limit stresses at SLS required by standard NF P 94-262

Scage allows to respect the two limit stresses required by the NF P 94-262 standard:

image-20221103184325409

Rectangular section

Symbols

Geometry

SymbolUnitDescription
bmWidth of the section
hmTotal height of the section
As1Lower steel section
As2Upper steel section
c1mDistance between the top of the section and the center of gravity of the lower steels
c2mDistance between the bottom of the section and the center of gravity of the upper steels
d2mPosition of the reinforcements As2 in relation to the lower fiber of the section
vmPosition of G' with respect to the lower fiber of the section
G-Center of gravity of the rectangular section
G’-Center of gravity of the homogenized section

 

Forces

SymbolUnitDescription
NsMNNormal force applied to G
MsMNmBending moment applied to G
MMNmEquivalent bending moment applied to the lower fibre of the section
NMNEquivalent normal force applied to the lower fibre of the section
MMNmEquivalent bending moment applied to G''
NMNEquivalent normal force applied to G''
Fs1MNEquivalent normal force applied to G''
Fs2MNForce taken up by the compressed reinforcement

Contraintes

SymbolUnitDescription
σs1MN/m²Stress in tensioned steels
σs2MN/m²Stress in compressed steels
σcMN/m²Compressive stress of concrete
fckMN/m²Compressive stress of concrete (characteristic value)
fydMN/m²Allowable compression/tension stress of steel (design value)
fykMN/m²Stress of the steel (characteristic value)

SLS check principle

The SLS check is performed using a stress based analysis.

This stress based analysis requires a calculation on an homogenized cross section considering the different deformation moduli of steel and concrete.

Calculation assumptions

The SLS check of reinforced concrete sections takes into account the following hypotheses:

The constitutive laws of concrete and steel are provided in the Materials chapter of this manuals.

Modes of behavior of the section

Based on the stress diagram generated by the external load (Ms, Ns), the reinforced concrete section can be in:

Stress diagram of a simple compressed section

Stress diagram of a partially compressed section

Stress diagram of a fully compressed section

Stress diagram of a fully tensioned section

SLS reinforced concrete section check

The verification of the reinforced concrete section with SLS is performed by homogenizing the section in order to take into account the presence of two materials with different stiffnesses. The equilibrium of the section is based on the equilibrium of forces and moments, which makes it possible to obtain the stress diagram of the section.

In the case of a fully tensioned section, concrete does not provide any tensile strength. Only the reinforcement forces balance the external forces.

Calculation of reinforcement section

Partially and fully compressed

The principle of dimensioning in SLS consists in looking for the minimum cross section verifying the equilibrium of the cross section while guaranteeing that the limit stresses of each material are not exceeded.

Simple traction

In simple traction, the tensile strength of concrete is neglected. Only the reinforcement forces counterbalance the forces applied to the section. The most economical solution is to guarantee that the center of gravity of the reinforcement is at the point of application of normal force.

Let's note:

The steel sections are obtained from the moments equilibrium:

(12)As2=Nes1(es1+es2)σs2
(13)As1=Nes2(es1+es2)σs1

The stresses in steels are considered equal to the allowable stress in the SLS.

 

Crack opening (wk)

(14)wk=sr,max(εsmεcm)
SymbolUnitDescription
wkmmOpening cracks
sr,maxmmMaximum crack spacing
εsm-Average reinforcement elongation, under the combination of actions considered, taking into account the contribution of tensioned concrete
εcm-Average concrete elongation between cracks
(15)sr,max=k3c+0.425k1k2ϕeqρp,eff
SymbolUnitDescription
cmmCoating of longitudinal reinforcement
k1-Coefficient function of the adhesion properties of the bars
(0.8 for HA bars)
k2-Coefficient taking into account a distribution of longitudinal bars
k2=0.5 in bending
k2=1 in tension
k2=ε1+ε22ε1 in eccentric tension
With ε1 the largest and ε2 the lowest of the relative elongations of the relative elongations of the fibers and the cross-section considered, evaluated on the basis of a cracked cross-section.
k3-Coefficient of coating adjustment, equal to 3.4
ϕeqmmEquivalent diameter of the bars
ρp,eff-Ratio between the steel section and the effective concrete section (As/Ac,eff)
Ac,effConcrete area surrounding the reinforcement stretched over a height
hc,ef=min{2.5(hd);(hx)/3;h/2}
(16)εsmεcm=max{0.6σsEs;σs+ktfct,effρp,eff(1+αeρp,eff)Es}
SymbolUnitDescription
σsMN/m²SLS stress on tensioned reinforcement, calculated by assuming the cracked section
EsMN/m²Elastic modulus of steel
fct,effMN/m²Average value of the tensile strength of the concrete effective at the time the cracks are expected to occur (fct,eff=fctm)
kt-Coefficient taking into account the loading time
kt=0.6 for short term loading
kt=0.4 for long term loading

Bibliography