b10. Design method for tensile reinforcement of a rectangular section in bending at the SLS with known compressive reinforcement, A_s

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 = k₃ f_yk (Pivot A) and/or the compressive stress in the concrete reaches the limit σ_c,ser = k₁ f_ck (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

(b10.1)

The linear distribution of strains and the stress diagram gives:

αser = εc/(εc + εs) = ne σc/(ne σc + σs)

(b10.2)

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:

αAB = ne σc,ser/(ne σc,ser + σs,ser)

(b10.3)

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:

Mser = Fc z = 0,5 b σc,ser x (d - x/3) = 0,5 b d2 σc,ser αc,ser (1 - αc,ser/3)

(b10.4)

Substituting the following in (b10.4):

μc,ser = Mser /(b d2 σc,ser)

(b10.5)

(b10.4) ⇒ μ_c,ser = 0,5 α_c,ser ( 1 - α_c,ser/3)

⇔ αc,ser2 - 3 αc,ser + 6 μc,ser = 0

(b10.6)

If μ_c,ser ≤ 0,375, this quadratic equation has a solution:

αc,ser = 1,5 [1 - (1 - 8 μc,ser/3)0.5]

(b10.7)

Serviceability limit state at Pivot A

For equilibrium, the ultimate design moment must be balanced by the moment of resistance of the section:

Mser = Fc z = 0,5 b σc x (d - x/3) = 0,5 b d2 σc αs,ser (1 - αs,ser/3)

(b10.8)

The stress σ_c is deducted from the stress diagram:

σc = σs,ser αs,ser /[ne (1 - αs,ser)]

(b10.9)

Calculating:

μs,ser = Mser /(b d2 σs,ser)

(b10.10)

Substituting (b10.9) and (b10.10) in (b10.8),
(b10.8) ⇒ μ_s,ser = 0,5 n_e α_s,ser² ( 1 - α_s,ser/3)

⇔ αs,ser3 - 3 αs,ser2 - 6 ne μs,ser αs,ser + 6 ne μs,ser = 0

(b10.11)

By solving this cubic equation, we get a root α_s,ser ∈ (0, 1).

Two fictive sections

Considering that the section is an overlap of two fictive sections (see Figure b10-2):

• One concrete section with only a tensile reinforcement A_s1;
• One section with only two reinforcements A_s2 and A_sc.

Flowchart

Figure b10-3 resumes this design method for a rectangular reinforced concrete section in bending at the SLS with known compressive reinforcement: