SLS analysisIntroductionCircular sectionNotesGeometryEffortsStressesSLS check principleCalculation assumptionsVerification of the reinforced concrete section under SLSGeneration of the interaction diagramCalculation of the resisting forcesResistance force of concreteResistance force of steelDiscrete distribution of steel barsContinuous and homogeneous distribution of the steel sectionCalculation of the mobilisation rate of the bending strengthLimit stresses at SLS required by standard NF P 94-262Rectangular sectionSymbolsGeometryForcesContraintesSLS check principleCalculation assumptionsModes of behavior of the sectionSLS reinforced concrete section checkCalculation of reinforcement sectionPartially and fully compressedSimple tractionCrack opening (
This document details how the SLS analysis is performed for circular sections and for rectangular reinforced concrete sections.
This chapter details the calculation of a circular reinforced concrete section subjected to biaxial bending (SLS).
Symbol | Unit | Description |
---|---|---|
D ou B | m | Diameter of the section |
Symbol | Unit | Description |
---|---|---|
MN | Normal force applied to the centre of gravity of the section | |
MNm | Bending moment applied to the centre of gravity of the section |
Symbol | Unit | Description |
---|---|---|
MPa | Stress in steels | |
MPa | Compressive stress in concrete | |
MPa | Allowable compressive stress of concrete (characteristic value) | |
MPa | Allowable steel stress (design value) | |
MPa | Allowable steel stress (characteristic value) |
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.
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.
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.
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.
Each homogenized stress diagram is described by the following equation:
Where a is the slope and b is the ordinate at the origin.
For each bar of section Ai :
Stress in bar "i" from the stress diagram of the homogenized section:
Where n is the equivalence coefficient:
.
Resulting force on all steel bars :
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.
Scage allows to respect the two limit stresses required by the NF P 94-262 standard:
Symbol | Unit | Description |
---|---|---|
b | m | Width of the section |
h | m | Total height of the section |
m² | Lower steel section | |
m² | Upper steel section | |
m | Distance between the top of the section and the center of gravity of the lower steels | |
m | Distance between the bottom of the section and the center of gravity of the upper steels | |
m | Position of the reinforcements | |
m | Position 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 |
Symbol | Unit | Description |
---|---|---|
MN | Normal force applied to G | |
MNm | Bending moment applied to G | |
MNm | Equivalent bending moment applied to the lower fibre of the section | |
MN | Equivalent normal force applied to the lower fibre of the section | |
MNm | Equivalent bending moment applied to G'' | |
MN | Equivalent normal force applied to G'' | |
MN | Equivalent normal force applied to G'' | |
MN | Force taken up by the compressed reinforcement |
Symbol | Unit | Description |
---|---|---|
MN/m² | Stress in tensioned steels | |
MN/m² | Stress in compressed steels | |
MN/m² | Compressive stress of concrete | |
MN/m² | Compressive stress of concrete (characteristic value) | |
MN/m² | Allowable compression/tension stress of steel (design value) | |
MN/m² | Stress of the steel (characteristic value) |
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.
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.
Based on the stress diagram generated by the external load (
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.
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.
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:
The stresses in steels are considered equal to the allowable stress in the SLS.
Symbol | Unit | Description |
---|---|---|
mm | Opening cracks | |
mm | Maximum crack spacing | |
- | Average reinforcement elongation, under the combination of actions considered, taking into account the contribution of tensioned concrete | |
- | Average concrete elongation between cracks |
Symbol | Unit | Description |
---|---|---|
c | mm | Coating of longitudinal reinforcement |
- | Coefficient function of the adhesion properties of the bars (0.8 for HA bars) | |
- | Coefficient taking into account a distribution of longitudinal bars With | |
- | Coefficient of coating adjustment, equal to 3.4 | |
mm | Equivalent diameter of the bars | |
- | Ratio between the steel section and the effective concrete section ( | |
m² | Concrete area surrounding the reinforcement stretched over a height |
Symbol | Unit | Description |
---|---|---|
MN/m² | SLS stress on tensioned reinforcement, calculated by assuming the cracked section | |
MN/m² | Elastic modulus of steel | |
MN/m² | Average value of the tensile strength of the concrete effective at the time the cracks are expected to occur ( | |
- | Coefficient taking into account the loading time |