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# b9. Design method for longitudinal reinforcements of a rectangular section in bending at the SLS, Asc and As

## Eurocode 2 - Design of concrete sections

Figure b9-1 introduces a reinforced concrete section with notations. The reinforcements Asc and As are 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 (b9.1)

The linear distribution of strains and the stress diagram gives:

 αser = εc/(εc + εs) = ne σc/(ne σc + σs) (b9.2)

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) (b9.3)

If αser > αAB ⇔ 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) (b9.4)

Substituting the following in (b9.4):

 μc,ser = Mser /(b d2 σc,ser) (b9.5)

(b9.4) ⇒ μc,ser = 0,5 αc,ser ( 1 - αc,ser/3)

 ⇔ αc,ser2 - 3 αc,ser + 6 μc,ser = 0 (b9.6)

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

 αc,ser = 1,5 [1 - (1 - 8 μc,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:

 Mser = Fc z = 0,5 b σc x (d - x/3) = 0,5 b d2 σc αs,ser (1 - αs,ser/3) (b9.8)

The stress σc is deducted from the stress diagram:

 σc = σs,ser αs,ser /[ne (1 - αs,ser)] (b9.9)

Calculating:

 μs,ser = Mser /(b d2 σs,ser) (b9.10)

Substituting (b9.9) and (b9.10) in (b9.8),
(b9.8) ⇒ μs,ser = 0,5 ne αs,ser2 ( 1 - αs,ser/3)

 ⇔ αs,ser3 - 3 αs,ser2 - 6 ne μs,ser αs,ser + 6 ne μs,ser = 0 (b9.11)

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

αc,ser > αAB ⇒ Pivot B

The tensile reinforcement As 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 As is optimized because its stress σs is taken equal to the limit σs,ser. This optimization requires a calculation of the compressive reinforcement Asc.

In the first case, the tensile stress in the reinforcement is deducted from the stress diagram:

 σs = ne σc,ser (1 - αc,ser) /αc,ser (b9.12)

The equilibrium of the section allows the calculation of the tensile reinforcement:

 As = Mser / [σs d (1 - αc,ser/3)] (b9.13)

In the second case, we consider that the section is an overlap of two fictive sections (see Figure b9-2):

• One concrete section with only a tensile reinforcement As1 This section reaches the Pivot A and Pivot B at the same time (balanced section).
• One section with only two reinforcements As2 and Asc.

The bending moment supported by the first section is equal to:

 MAB = 0,5 b d2 αAB σc,ser (1 - αAB/3) (b9.14)

The tensile reinforcement As1 of the first section:

 As1 = MAB /[σs,ser d (1 - αAB/3)] (b9.15)

The tensile reinforcement As2 of the second section:

 As2 = (Mser - MAB) /[σs,ser (d - d')] (b9.16)

The compressive stress in the reinforcement:

 σsc = σs,ser [αAB - d'/d) /(1 - αAB)] (b9.17)

The compressive reinforcement As2:

 Asc = As2 σs,ser /σsc (b9.18)

Finally, the total tensile reinforcement is equal to:

 As = As1 + As2 (b9.19)

αc,serαAB ⇒ Pivot A

No compressive reinforcement is needed.

The tensile reinforcement is calculated using αs,ser from (b9.11):

 As = Mser /[σs,ser d (1 - αs,ser/3)] (b9.20)

## Flowchart

Figure b9-3 resumes this design method for a rectangular reinforced concrete section in bending at the serviceability limit state: