b11. Calculation method for moment of resistance of a rectangular section in bending at the SLS by stress limitation, M_{ser}
Eurocode 2  Design of concrete sections
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 σ_{s,ser} = k_{3} f_{yk} (Pivot A) and/or the compressive stress in the concrete reaches the limit σ_{c,ser} = k_{1} f_{ck} (Pivot B). The parameters k_{1} and k_{3} 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 crosssection:
α_{ser} = x/d  (b11.1) 
The linear distribution of strains and the stress diagram gives:
α_{ser} = ε_{c}/(ε_{c} + ε_{s}) = n_{e} σ_{c}/(n_{e} σ_{c} + σ_{s})  (b11.2) 
where:
 n_{e}

is the effective modular ratio
n_{e} = E_{s} / E_{c,eff}with:
 E_{s}
 the design value of the modulus of elasticity of the reinforcing steel, see § 3.2.7 (4)
 E_{c,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: σ_{s} = σ_{s,ser} and σ_{c} = σ_{c,ser}.
For the balanced section, calculating:
α_{AB} = n_{e} σ_{c,ser}/(n_{e} σ_{c,ser} + σ_{s,ser})  (b11.3) 
If α_{ser} > α_{AB} ⇔ the Pivot B is reached first. Otherwise, the Pivot A is reached first.
Compression depth ratio α_{ser}
The stress diagram ⇒ the compressive stresses in the reinforcement the concrete are respectively calculated as follows:
σ_{sc} = σ_{s} (α_{ser}  d'/d) /(1  α_{ser})  (b11.4) 
σ_{c} = (σ_{s} /n_{e})⋅α_{ser} /(1  α_{ser})  (b11.5) 
For equilibrium, the sum of forces acting on the section gives:
F_{s} = F_{sc} + F_{c}  
⇔ A_{s} σ_{s} = A_{sc} σ_{sc} + 0,5 b d α_{ser} σ_{c}  (b11.6) 
Substituting (b11.4) and (b11.5) in (b11.6), (b11.6) becomes a quadratic equation of α_{ser}:
b d α_{ser}^{2} + 2 n_{e} (A_{s} + A_{sc}) α_{ser}  2 n_{e} (A_{s} + A_{sc} d' /d)  (b11.7) 
Comparing the root α_{ser} ∈ (d'/d, 1) of this equation with the balanced value α_{AB}.
• α_{ser} ≤ α_{AB} ⇒ Pivot A
The tensile stress in the reinforcement σ_{s} = σ_{s,ser}.
The compressive stresses in the reinforcement and the concrete are respectively calculated according to (b11.4) and (b11.5).
• α_{ser} > α_{AB} ⇒ Pivot B
The compressive stress in the concrete σ_{c} = σ_{c,ser}.
The stress diagram ⇒ the tensile stress in the reinforcement and the compressive stress in the reinforcement are respectively calculated as follows:
σ_{s} = n_{e} σ_{c,ser} (1  α_{ser}) /α_{ser}  (b11.8) 
σ_{sc} = n_{e} σ_{c,ser} (α_{ser}  d'/d) /α_{ser}  (b11.9) 
Serviceability moment of resistance M_{ser}
For equilibrium, the sum of moments to the centre of gravity of the tensile reinforcement allows the calculation of the moment of resistance:
M_{ser} = A_{sc} σ_{sc} (d  d') + 0,5 b d^{2} α_{ser} (1  α_{ser} /3) σ_{c}  (b11.10) 