Rectangular section - Compound bending (ULS)

Rectangular section - Compound bending (ULS)NotesGeometryForcesStressesStrainsULS design principleCalculation assumptionsModes of behaviour of the sectionPartially compressed sectionFully compressed sectionFully tension sectionCalculation of the required reinforcement sectionsPartially compressed sectionFully compressed sectionTension only sectionBibliography

Notes

Geometry

Symbol	Unit	Description
b	m	Width of the section
h	m	Total height of the section
d	m	Effective section height
$c_1$	m	Distance between the top of the section and the centre of gravity of the tension steels
$c_2$	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
$A_{s1}$	m²	Tension steel section
$A_{s2}$	m²	Compressed steel section
$z_R$	m	$M_R$

Forces

Symbol	Unit	Description
$N$	MN	Normal force applied to the reinforced concrete section
$M$	MNm	Bending moment applied to the reinforced concrete section (at the bottom reinforcement)
$M_c$	MNm	Bending moment taken up by concrete alone
$N_c$	MN	Normal force taken up by the concrete alone
$F_{s1}$	MN	Force taken up by the tensioned reinforcement
$F_{s2}$	MN	Force taken up by the compressed reinforcement
$M_R$	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
$N_R$	MN	$M_R$

Stresses

Symbol	Unit	Description
$\sigma_{s1}$	MN/m²	Stress in tension steels
$\sigma_{s2}$	MN/m²	Stress in compressed steels
$\sigma_c$	MN/m²	Compression stress of concrete
$f_{cd}$	MN/m²	Allowable compression stress of concrete (design value)
$f_{ck}$	MN/m²	Allowable compression stress of concrete (characteristic value)
$f_{yd}$	MN/m²	Allowable stress of steel (design value)
$f_{yk}$	MN/m²	Allowable stress of the steel (characteristic value)

Strains

Symbol	Unit	Description
$\varepsilon_{s1}$	-	Strain of the tension steel section
$\varepsilon_{s2}$	-	Strain of the compressed steel section
$\varepsilon_c$	-	Concrete strain
$\varepsilon_e$	-	Maximum elastic strain of steels

ULS design principle

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:

Hypothesis 1: The cross sections remain plane and there is no relative slippage between the reinforcement and the concrete. The deformation diagram is therefore linear.

Hypothesis 2: The tensile strength of concrete is neglected.

Hypothesis 3: The strain diagram of the section corresponds to a limit state if it passes through one of the pivots A, B, C. These pivots are defined hereafter.

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:

Pivot A $\varepsilon_{ud}$ )
Pivot B $\varepsilon_{cu}$ )
Pivot C $\varepsilon_{ce}$ ) on the compressed height of the concrete.

[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:

x=\frac{\varepsilon_{cu}}{\varepsilon_{cu}+\varepsilon_{ud}}d=\frac{3.5‰}{3.5‰+45‰}d=0.072d

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

Domain 1 $0<x<0.072d$ → Partially compressed section
Domain 2 $0.072d\leq x<h$ → Partially compressed section
Domain 3 $h\leq x$ → Fully compressed section

Partially compressed section

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

$0.072d\leq x<h$

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

$\varepsilon_c=3.5‰$ $f_{cd}$ .

$3/7x$ $4/7x$ .

This distribution of heights is due to the layout of the concrete constitutive law. The strain diagram is linear over the compressed height:
$f_{cd}$ .
$f_{cd}$ .
$2‰/3.5‰=4/7$ of the compressed height.

Normal force and moment taken up by the concrete:

N_c=0.81bxf_{cd} \label{Nc}

M_c=0.81bxf_{cd}(h-0.416x) \label{Mc}

$M_c$ $N_c$ can be deduced from the previous expressions:

\eqref{Nc},\eqref{Mc} → M_c=N_c(h-0.514\frac{N_c}{bf_{cd}}) \label{McNcRelation}

$0<x<0.072d$

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: :

$z=M_c/N_c$ is between:
$z=d$ $x=0$
$z=0.97d$ $x=0.072d$
Since the concrete section is overabundant, it is not necessary to add a compressed reinforcement.

$\eqref{Nc}$ $\eqref{Mc}$ $\eqref{McNcRelation}$ as in domain 2.

Fully compressed section

The strain diagram of the section is in domain 3 and passes through pivot C.

$3/7h$ $4/7h$ .

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

$\varepsilon_e$ . 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:

\left(d-c_2\right)N-M<\left(0.337-0.81\frac{c_2}h\right)bh^2f_{cd}

$M\leq M_R$

In this case, no compression reinforcement is required.

$\varepsilon_e$ , 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.

$M_R< M$

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:

$e_{s1}$ distance between the point of application of normal force and the tension reinforcement
$e_{s2}$ distance between the point of application of normal force and the compressed reinforcement

The steel sections are obtained from moment equilibrium:

A_{s2}=\frac{Ne_{s1}}{\left(e_{s1}+e_{s2}\right)\sigma_{s2}}

A_{s1}=\frac{Ne_{s2}}{\left(e_{s1}+e_{s2}\right)\sigma_{s1}}

$\sigma=\sigma(\varepsilon_{ud})$ .

Bibliography

Eurocode 2
Simple or compound bending design at the ultimate limit state of the rectangular reinforced concrete sections (A. CAPRA, M. HAUTCOEUR)
Calculation of reinforced concrete structures (Perchat)