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b11. Calculation method for moment of resistance of a rectangular section in bending at the SLS by stress limitation, Mser

Eurocode 2 - Design of concrete sections

Figure b11-1 introduces a reinforced concrete section with notations. The serviceability moment of resistance Mser is unknown.

Assumptions

1. Plane sections remain plane after straining, so that there is a linear distribution of strains across the section.

2. Reinforcing steels have the same deformation as the nearby concrete.

3. The tensile strength of concrete is ignored.

4. A triangular distribution of the compressive stress in the concrete is assumed.

5. The serviceability limit state occurs when the tensile stress in the reinforcement reaches the limit σs,ser = k3fyk (Pivot A) and/or the compressive stress in the concrete reaches the limit σc,ser = k1fck (Pivot B). The parameters k1 and k3 are chosen by National Annex, see § 7.2 (2) and § 7.2 (5) respectively.

Considering the depth of the neutral axis/the effective depth of the cross-section:

 αser = x/d (b11.1)

The linear distribution of strains and the stress diagram gives:

 αser = εc/(εc + εs) = ne σc/(ne σc + σs) (b11.2)

where:

ne
is the effective modular ratio
ne = Es / Ec,eff

with:

Es
the design value of the modulus of elasticity of the reinforcing steel, see § 3.2.7 (4)
Ec,eff
the effective modulus of elasticity for concrete.

Balanced section AB

We consider a balanced section for which the Pivot A and Pivot B are reached at the same time: σs = σs,ser and σc = σc,ser.

For the balanced section, calculating:

 αAB = ne σc,ser/(ne σc,ser + σs,ser) (b11.3)

If αser > αAB ⇔ the Pivot B is reached first. Otherwise, the Pivot A is reached first.

Compression depth ratio αser

The stress diagram ⇒ the compressive stresses in the reinforcement the concrete are respectively calculated as follows:

 σsc = σs (αser - d'/d) /(1 - αser) (b11.4) σc = (σs /ne)⋅αser /(1 - αser) (b11.5)

For equilibrium, the sum of forces acting on the section gives:

 Fs = Fsc + Fc ⇔ As σs = Asc σsc + 0,5 b d αser σc (b11.6)

Substituting (b11.4) and (b11.5) in (b11.6), (b11.6) becomes a quadratic equation of αser:

 b d αser2 + 2 ne (As + Asc) αser - 2 ne (As + Asc d' /d) (b11.7)

Comparing the root αser ∈ (d'/d, 1) of this equation with the balanced value αAB.

αserαAB ⇒ Pivot A

The tensile stress in the reinforcement σs = σs,ser.

The compressive stresses in the reinforcement and the concrete are respectively calculated according to (b11.4) and (b11.5).

αser > αAB ⇒ Pivot B

The compressive stress in the concrete σc = σc,ser.

The stress diagram ⇒ the tensile stress in the reinforcement and the compressive stress in the reinforcement are respectively calculated as follows:

 σs = ne σc,ser (1 - αser) /αser (b11.8) σsc = ne σc,ser (αser - d'/d) /αser (b11.9)

Serviceability moment of resistance Mser

For equilibrium, the sum of moments to the centre of gravity of the tensile reinforcement allows the calculation of the moment of resistance:

 Mser = Asc σsc (d - d') + 0,5 b d2 αser (1 - αser /3) σc (b11.10)