ULS analysisIntroductionCircular sectionNotesGeometryForcesStressesStrainsPrinciple of ULS calculationCalculation assumptionsGeneration of the interaction diagramCalculation of the resisting forcesResistance force of concreteResistance effort of the steelDiscrete distribution of steel barsContinuous and homogeneous distribution of the steel sectionCalculation of the bending resistance mobilisation rateRectangular sectionNotesGeometryForcesStressesStrainsPrinciple of ULS calculationCalculation assumptionsModes of behaviour of the sectionPartially compressed sectionFully compressed sectionFully tension sectionCalculation of the required reinforcement sectionsPartially compressed sectionFully compressed sectionTension only sectionBibliography
This document details how the ULS 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 (ULS).
Symbol | Unit | Description |
---|---|---|
D or B | m | Diameter of the section |
Symbol | Unit | Description |
---|---|---|
MN | Normal force applied to the section | |
MNm | Bending moment applied to the section around the positive semi-axis OX | |
MNm | Bending moment applied to the section around the positive semi-axis OY | |
MN | Shear force applied to the section along the positive semi-axis OX | |
MN | Shear force applied to the section along the positive semi-axis OY |
The right hand rule applies:
Symbol | Unit | Description |
---|---|---|
MPa | Stress in steel ("i" bar) | |
MPa | Compressive stress in concrete | |
MPa | Allowable compressive stress of concrete (design value) | |
MPa | Allowable compressive stress of concrete (characteristic value) | |
MPa | Allowable steel stress (design value) | |
MPa | Allowable steel stress (characteristic value) |
Symbol | Unit | Description |
---|---|---|
- | Strain in the steel section | |
- | Strain of concrete | |
- | Maximum elastic strain of the steel |
The principle of calculation of circular sections (piles) under ULS consists in comparing the resulting torsor applied to the section with the interaction diagram (N, M) of the reinforced concrete section.
This interaction diagram determines the allowable range (inner area of the interaction diagram, including the boundary) in which the torques (N, M) that can be taken up by the reinforced concrete circular section.
The ULS design of reinforced concrete sections takes into account the following assumptions:
The constitutive laws of concrete and steel are provided in the Materials chapter of this manual.
The design strain levels (
The principle of generating interaction diagrams is based on the analysis of all possible bending modes of the section. In order to optimise the work of the section, this scan is done based on the usual pivots A, B and C. The range of possible regions are described as follows:
The resistant force of the section is calculated for each strain diagram. It is composed of a part coming from the steel and another part coming from the concrete (only if it is compressed, the tensile contribution is neglected).
The force of the concrete is calculated on the basis of the parabola-rectangle law.
Resistant forces on the "rectangle" part:
Resistant normal force:
Resistant moment:
Resistant forces on the "parabola" part
Resistant normal force:
Resistant moment:
The force provided by the steel is accounted for on the basis of the bilinear law.
For each bar of section
The bending resistance mobilization rate is the ratio between the resisting moment and the applied moment for the same normal force. It is deduced from the strain diagram calculated for the section under examination.
Symbol | Unit | Description |
---|---|---|
b | m | Width of the section |
h | m | Total height of the section |
d | m | Effective section height |
m | Distance between the top of the section and the centre of gravity of the tension steels | |
m | Distance between the bottom of the section and the centre of gravity of compressed steels | |
x | m | Position of the neutral axis from the most compressed fibre |
m² | Tension steel section | |
m² | Compressed steel section | |
m | Lever arm associated with |
Symbol | Unit | Description |
---|---|---|
MN | Normal force applied to the reinforced concrete section | |
MNm | Bending moment applied to the reinforced concrete section (at the bottom reinforcement) | |
MNm | Bending moment taken up by concrete alone | |
MN | Normal force taken up by the concrete alone | |
MN | Force taken up by the tensioned reinforcement | |
MN | Force taken up by the compressed reinforcement | |
MNm | Limit moment of the reinforced concrete section beyond which a compressed reinforcement must be provided in order to ensure an optimised behaviour of tension reinforcement | |
MN | Normal force taken up by the concrete associated with |
Symbol | Unit | Description |
---|---|---|
MN/m² | Stress in tension steels | |
MN/m² | Stress in compressed steels | |
MN/m² | Compression stress of concrete | |
MN/m² | Allowable compression stress of concrete (design value) | |
MN/m² | Allowable compression stress of concrete (characteristic value) | |
MN/m² | Allowable stress of steel (design value) | |
MN/m² | Allowable stress of the steel (characteristic value) |
Symbol | Unit | Description |
---|---|---|
- | Strain of the tension steel section | |
- | Strain of the compressed steel section | |
- | Concrete strain | |
- | Maximum elastic strain of steels |
The design of reinforced concrete section at the ULS is performed using a strain based analysis.
The ULS design of reinforced concrete sections takes into account the following assumptions:
The constitutive laws of concrete and steel are provided in the Materials chapter of this manual.
The optimised design of a reinforced concrete section requires a strain diagram that contains a limit strain, either in concrete or steel. These limit strains are characterised by pivots
Three pivots are used to define the behaviour mode of the section:
If the strain diagram passes through both pivots A and B, then:
The strain diagrams of the section can be found in one of the three domains defined by the pivots:
Depending on the position of the neutral axis, 2 cases are possible:
The strain diagram of the section is in domain 2
The shortening of the most compressed fibre is
The stress diagram has two parts: a straight part over a height of
This distribution of heights is due to the layout of the concrete constitutive law. The strain diagram is linear over the compressed height:
- Concrete strain greater than 2‰ generate a stress equal to
. - Concrete strain below 2‰ follow a parabolic stress distribution ranging from 0 and
. 2‰ corresponds to the maximum strain of the elastic domain, i. e.
of the compressed height.
Normal force and moment taken up by the concrete:
The relationship between
The strain diagram of the section is in domain 1.
The shortening of the most compressed fibre is less than 3.5 ‰ and the diagram can have one of the two layouts shown below.
In this case: :
For a value of x between 0 and 0.072d, the value of the force lever arm
Since the concrete section is overabundant, it is not necessary to add a compressed reinforcement.
We find that the relative variation of the lever arm is very small and that it is not necessary to evaluate x with great precision, which allows the use of the equations
The strain diagram of the section is in domain 3 and passes through pivot C.
The stress diagram is composed of a straight part over a total height equal to
The equilibrium state of the section is obtained from the balance of forces and moments, allowing the deformation diagram associated with the applied torsor to be obtained.
The strain diagram does not cause any compression on the section, the tensile strength of the concrete is neglected. The forces in the reinforcement are the only ones that counterbalance the external forces.
The formulation described below consists of the proposition of the most economical solution. In principle, we avoid to propose a reinforcement whose relative strain is smaller than the elastic strain
A section is partially compressed if the forces N and M counterbalanced by the section and calculated at the lower reinforcement verify the following equation:
In this case, no compression reinforcement is required.
The moment is counterbalanced by the concrete alone. The lower reinforcement elongation is greater than
The equilibrium calculation allows to find the necessary steel section in the lower part.
In this case, a compressed reinforcement should be added to ensure that the tension reinforcement is used properly and that the concrete stress does not exceed the permissible stress.
The equilibrium calculation leads to the necessary steel sections on each side to ensure that the acting forces are correctly taken up.
A section is fully compressed if the forces N and M (calculated at the level of the lower reinforcement) result in a purely compressive deformation diagram (no tensile zone). In this case, the deformation diagram passes through point C.
If there are two reinforcements in the section, the lower reinforcement is the less well used of the two since it has the lowest relative shortening. Therefore, a balance of forces should be considered without establishing a bottom reinforcement.
The equilibrium calculation leads to the steel sections necessary to ensure that the acting forces are correctly taken up by the steel and concrete.
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 centre of gravity of the reinforcement is at the point of application of the normal force.
Let’s note:
The steel sections are obtained from moment equilibrium:
The stresses in steel are considered equal to