b9. Design method for longitudinal reinforcements of a rectangular section in bending at the SLS, A_{sc} and A_{s}
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  (b9.1) 
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})  (b9.2) 
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})  (b9.3) 
If α_{ser} > α_{AB} ⇔ the Pivot B is reached first. Otherwise, the Pivot A is reached first.
Serviceability limit state at Pivot B
For equilibrium, the ultimate design moment must be balanced by the moment of resistance of the section:
M_{ser} = F_{c} z = 0,5 b σ_{c,ser} x (d  x/3) = 0,5 b d^{2} σ_{c,ser} α_{c,ser} (1  α_{c,ser}/3)  (b9.4) 
Substituting the following in (b9.4):
μ_{c,ser} = M_{ser} /(b d^{2} σ_{c,ser})  (b9.5) 
(b9.4) ⇒ μ_{c,ser} = 0,5 α_{c,ser} ( 1  α_{c,ser}/3)
⇔ α_{c,ser}^{2}  3 α_{c,ser} + 6 μ_{c,ser} = 0  (b9.6) 
If μ_{c,ser} ≤ 0,375, this quadratic equation has a solution:
α_{c,ser} = 1,5 [1  (1  8 μ_{c,ser}/3)^{0.5}]  (b9.7) 
Serviceability limit state at Pivot A
For equilibrium, the ultimate design moment must be balanced by the moment of resistance of the section:
M_{ser} = F_{c} z = 0,5 b σ_{c} x (d  x/3) = 0,5 b d^{2} σ_{c} α_{s,ser} (1  α_{s,ser}/3)  (b9.8) 
The stress σ_{c} is deducted from the stress diagram:
σ_{c} = σ_{s,ser} α_{s,ser} /[n_{e} (1  α_{s,ser})]  (b9.9) 
Calculating:
μ_{s,ser} = M_{ser} /(b d^{2} σ_{s,ser})  (b9.10) 
Substituting (b9.9) and (b9.10) in (b9.8),
(b9.8) ⇒ μ_{s,ser} = 0,5 n_{e} α_{s,ser}^{2} ( 1  α_{s,ser}/3)
⇔ α_{s,ser}^{3}  3 α_{s,ser}^{2}  6 n_{e} μ_{s,ser} α_{s,ser} + 6 n_{e} μ_{s,ser} = 0  (b9.11) 
By solving this cubic equation, we get a root α_{s,ser} ∈ (0,1).
• α_{c,ser} > α_{AB} ⇒ Pivot B
The tensile reinforcement A_{s} may be determined in two ways:
 Based solely on the Pivot B and no compressive reinforcement will be provided;
 Based on the balanced section AB. Thus, the tensile reinforcement A_{s} is optimized because its stress σ_{s} is taken equal to the limit σ_{s,ser}. This optimization requires a calculation of the compressive reinforcement A_{sc}.
In the first case, the tensile stress in the reinforcement is deducted from the stress diagram:
σ_{s} = n_{e} σ_{c,ser} (1  α_{c,ser}) /α_{c,ser}  (b9.12) 
The equilibrium of the section allows the calculation of the tensile reinforcement:
A_{s} = M_{ser} / [σ_{s} d (1  α_{c,ser}/3)]  (b9.13) 
In the second case, we consider that the section is an overlap of two fictive sections (see Figure b92):
 One concrete section with only a tensile reinforcement A_{s1} This section reaches the Pivot A and Pivot B at the same time (balanced section).
 One section with only two reinforcements A_{s2} and A_{sc}.
The bending moment supported by the first section is equal to:
M_{AB} = 0,5 b d^{2} α_{AB} σ_{c,ser} (1  α_{AB}/3)  (b9.14) 
The tensile reinforcement A_{s1} of the first section:
A_{s1} = M_{AB} /[σ_{s,ser} d (1  α_{AB}/3)]  (b9.15) 
The tensile reinforcement A_{s2} of the second section:
A_{s2} = (M_{ser}  M_{AB}) /[σ_{s,ser} (d  d')]  (b9.16) 
The compressive stress in the reinforcement:
σ_{sc} = σ_{s,ser} [α_{AB}  d'/d) /(1  α_{AB})]  (b9.17) 
The compressive reinforcement A_{s2}:
A_{sc} = A_{s2} σ_{s,ser} /σ_{sc}  (b9.18) 
Finally, the total tensile reinforcement is equal to:
A_{s} = A_{s1} + A_{s2}  (b9.19) 
• α_{c,ser} ≤ α_{AB} ⇒ Pivot A
No compressive reinforcement is needed.
The tensile reinforcement is calculated using α_{s,ser} from (b9.11):
A_{s} = M_{ser} /[σ_{s,ser} d (1  α_{s,ser}/3)]  (b9.20) 
Flowchart
Figure b93 resumes this design method for a rectangular reinforced concrete section in bending at the serviceability limit state: