b2. Design method for tensile reinforcement of a rectangular section in bending at the ULS with known compressive reinforcement, 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 rectangular distribution of the compressive stress in the concrete is assumed, see 3.1.7 (3).
5. The ultimate limit state occurs when the strain in the reinforcing steel reaches the limit ε_{ud} (Pivot A) and/or the strain in the concrete reaches the limit ε_{cu3} (Pivot B).
Considering the depth of the neutral axis/the effective depth of the cross-section:
α_{u} = x/d | (b2.1) |
The linear distribution of strains gives:
α_{u} = ε_{c}/(ε_{c} + ε_{s}) | (b2.2) |
Balanced section AB
We consider a balanced section for which the Pivot A and Pivot B are reached at the same time: ε_{c} = ε_{cu3} and ε_{s} = ε_{ud}.
For the balanced section, calculating:
α_{AB} = ε_{cu3}/(ε_{cu3} + ε_{ud}) | (b2.3) |
If α_{u} > α_{AB} ⇔ the Pivot B is reached first. Otherwise, the Pivot A is reached first.
Two fictive sections
Considering that the section is an overlap of two fictive sections (see Figure b2-2):
- One concrete section with only a tensile reinforcement A_{s1};
- One section with only two reinforcements A_{s2} and A_{sc}.
Iterative calculation of the stress σ_{sc}
Assumimg the stress in the compressive reinforcement σ_{sc} = f_{yd}.
The bending moment M_{u} supported by the section without compression reinforcement:
M_{u} = M_{Ed} - A_{sc} σ_{sc} (d - d') | (b2.4) |
For equilibrium, this bending moment must be balanced by the moment of resistance of the section:
M_{u} = F_{c} z = b η f_{cd} λ x (d - λ x/2) | (b2.5) |
M_{u} = F_{s} z = A_{s1} σ_{s} (1 - λ α_{u} /2) d ⇔ A_{s1} = M_{u} /[(1 - λ α_{u} /2) d σ_{s}] | (b2.6) |
Substituting the following in (b2.5):
μ_{u} = M_{u} /(b d^{2} η f_{cd}) | (b2.7) |
(b2.5) ⇒ 2 μ_{u} = 2 λ (x/d) - λ^{2} (x/d)^{2}
⇔ 2 μ_{u} = 2 λ (α_{u}) - λ^{2} (α_{u})^{2} |
(b2.8) |
If μ_{u} ≤ 0,5, this quadratic equation has a solution:
α_{u} = [1 - (1 - 2μ_{u})^{0.5}] /λ | (b2.9) |
• α_{u} ≤ α_{AB} ⇒ Pivot A
The strain in the tensile reinforcement ε_{s1} = ε_{ud}.
The strain diagram ⇒ the strain in the compression reinforcement ε_{sc} = ε_{ud}⋅(α_{u} - d/d')/(1 - α_{u}).
• α_{u} > α_{AB} ⇒ Pivot B
The strain in the concrete ε_{c} = ε_{cu3}.
The strain diagram ⇒ the strain in the compression reinforcement ε_{sc} = ε_{cu3}⋅(α_{u} - d/d')/α_{u}.
By reading the design stress-strain diagram for reinforcing steel ⇒ the stress in the compressive reinforcement σ_{sc*}.
This calculated value of σ_{sc*} must be compared with the initial value σ_{sc}. If the difference is greater than 5 %, we start over with a lower value of σ_{sc}. If the difference is lower than or equal to 5 %, we accept the initial value σ_{sc} as the stress in the compressive reinforcement.
Reinforcement A_{s1}
For the first fictive section without compression steel, the strain in the tensile reinforcement ε_{s1} is equal to ε_{ud} in case of Pivot A, is equal to ε_{cu3}⋅(1 - α_{u})/α_{u} in case of Pivot B.
From the design stress-strain diagram for reinforcing steel ⇒ the stress in the tensile reinforcement σ_{s}.
(b2.6) ⇒ the tensile reinforcement A_{s1} = M_{u} /[(1 - λ α_{u} /2) d σ_{s}].
Reinforcement A_{s2}
The second fictive section at equilibrium gives:
A_{s2} = A_{sc} σ_{sc} /σ_{s2} | (b2.10) |
Finally, the total tensile reinforcement is:
A_{s} = A_{s1} + A_{s2} | (b2.11) |
Flowchart
Figure b2-3 resumes this design method for a rectangular reinforced concrete section in bending at the ULS with known compressive reinforcement: