ULS analysis

Introduction

This document details how the ULS 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 (ULS).

Notes

Geometry

SymbolUnitDescription
D or BmDiameter of the section

Forces

SymbolUnitDescription
NMNNormal force applied to the section
MxMNmBending moment applied to the section around the positive semi-axis OX
MyMNmBending moment applied to the section around the positive semi-axis OY
VxMNShear force applied to the section along the positive semi-axis OX
VyMNShear force applied to the section along the positive semi-axis OY

The right hand rule applies:

Section circulaire - Bending moment signs

Areas compressed by positive Mx and My moments

Stresses

SymbolUnitDescription
σs,iMPaStress in steel ("i" bar)
σcMPaCompressive stress in concrete
fcdMPaAllowable compressive stress of concrete (design value)
fckMPaAllowable compressive stress of concrete (characteristic value)
fydMPaAllowable steel stress (design value)
fykMPaAllowable steel stress (characteristic value)

 

Strains

SymbolUnitDescription
εs-Strain in the steel section
εc-Strain of concrete
εe-Maximum elastic strain of the steel

Principle of ULS calculation

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.

Calculation assumptions

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 (ε) of each material, in particular concrete (εc) and steel (εs), are defined by the user and therefore fixed before the interaction diagrams are established.

 

Generation of the interaction diagram

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:

Section circulaire - 5 regions

5 bending regions of the ULS section

Calculation of the resisting forces

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).

Schema section circulaire ELU

 

Resistance force of concrete

The force of the concrete is calculated on the basis of the parabola-rectangle law.

(1){σc=(εc0.25εc2)fc0<εc<2σc=fc2εc3.5
Resistance effort of the steel

The force provided by the steel is accounted for on the basis of the bilinear law.

(6){σs=Esεs0<εs<εyσc=σyεy<εsεud
Discrete distribution of steel bars
(7)σi=σ(εi)
(8)Fi=σiAi
(9)Ns=iFi
Continuous and homogeneous distribution of the steel section

ULS Steel Strains and Stress - Continuous Steel Ring

as represents the linear density of the steel section

(10)Ns,rectangular=2φ1φ2fyasrsdφ
(11)Ms,rectangular=2φ1φ2fyasrsrscosφdφ
(12)Ns,triangular=2φ2φ3σs,i(φ)asrsdφ
(13)Ms,triangular=2φ2φ3σs,i(φ)asrsrscos(φ)dφ

 

Calculation of the bending resistance mobilisation rate

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.

 

Rectangular section

Notes

Geometry

SymbolUnitDescription
bmWidth of the section
hmTotal height of the section
dmEffective section height
c1mDistance between the top of the section and the centre of gravity of the tension steels
c2mDistance between the bottom of the section and the centre of gravity of compressed steels
xmPosition of the neutral axis from the most compressed fibre
As1Tension steel section
As2Compressed steel section
zRmLever arm associated with MR

 

Forces

SymbolUnitDescription
NMNNormal force applied to the reinforced concrete section
MMNmBending moment applied to the reinforced concrete section (at the bottom reinforcement)
McMNmBending moment taken up by concrete alone
NcMNNormal force taken up by the concrete alone
Fs1MNForce taken up by the tensioned reinforcement
Fs2MNForce taken up by the compressed reinforcement
MRMNmLimit moment of the reinforced concrete section beyond which a compressed reinforcement must be provided in order to ensure an optimised behaviour of tension reinforcement
NRMNNormal force taken up by the concrete associated with MR

Stresses

SymbolUnitDescription
σs1MN/m²Stress in tension steels
σs2MN/m²Stress in compressed steels
σcMN/m²Compression stress of concrete
fcdMN/m²Allowable compression stress of concrete (design value)
fckMN/m²Allowable compression stress of concrete (characteristic value)
fydMN/m²Allowable stress of steel (design value)
fykMN/m²Allowable stress of the steel (characteristic value)

 

Strains

SymbolUnitDescription
εs1-Strain of the tension steel section
εs2-Strain of the compressed steel section
εc-Concrete strain
εe-Maximum elastic strain of steels

Principle of ULS calculation

The design of reinforced concrete section at the ULS is performed using a strain based analysis.

 

Calculation assumptions

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.

Modes of behaviour of the section

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:

[1] Cross-section geometry
[2] Strains diagram with the 3 pivots

x is the position of the neutral axis from the most compressed fibre.

If the strain diagram passes through both pivots A and B, then:

(14)x=εcuεcu+εudd=3.53.5+45d=0.072d

The strain diagrams of the section can be found in one of the three domains defined by the pivots:

Partially compressed section

Depending on the position of the neutral axis, 2 cases are possible:

The strain diagram of the section is in domain 2

 

[1] Diagramme de déformations d'une section partiellement comprimée
[2] Diagramme de contraintes d'une section partiellement comprimée

The shortening of the most compressed fibre is εc=3.5, which corresponds to a stress equal to fcd.

The stress diagram has two parts: a straight part over a height of 3/7x and a parabolic part over a height of 4/7x.

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 fcd.
  • Concrete strain below 2‰ follow a parabolic stress distribution ranging from 0 and fcd.

2‰ corresponds to the maximum strain of the elastic domain, i. e. 2/3.5=4/7 of the compressed height.

Normal force and moment taken up by the concrete:

(15)Nc=0.81bxfcd
(16)Mc=0.81bxfcd(h0.416x)

The relationship between Mc and Nc can be deduced from the previous expressions:

(21)(15),(16)Mc=Nc(h0.514Ncbfcd)

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.

Stress diagram of a partially compressed section

In this case: :

  1. For a value of x between 0 and 0.072d, the value of the force lever arm z=Mc/Nc is between:

    z=d for x=0

    z=0.97d for x=0.072d

  2. 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 (15), (16) and (21) as in domain 2.

 

Fully compressed section

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 3/7h and a parabolic part over a total height equal to 4/7h.

Strain diagram (fully compressed section)

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.

Fully tension section

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.

Strain diagram for fully tensioned section

Calculation of the required reinforcement sections

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 εe. If this is not possible, the force in the reinforcement will be minimised.

 

Partially compressed section

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:

(18)(dc2)NM<(0.3370.81c2h)bh2fcd

In this case, no compression reinforcement is required.

The moment is counterbalanced by the concrete alone. The lower reinforcement elongation is greater than εe, this reinforcement is thus efficiently used.

[1] Reinforced concrete section
[2] Force equilibrium

 

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.

Strains diagram and force equilibrium

The equilibrium calculation leads to the necessary steel sections on each side to ensure that the acting forces are correctly taken up.

Fully compressed section

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.

 

Tension only section

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:

(19)As2=Nes1(es1+es2)σs2
(20)As1=Nes2(es1+es2)σs1

The stresses in steel are considered equal to σ=σ(εud).

Bibliography