b3. Design method for ultimate moment of resistance of a rectangular section in bending at the ULS, M_{Rd}
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 | (b3.1) |
The linear distribution of strains gives:
α_{u} = ε_{c}/(ε_{c} + ε_{s}) | (b3.2) |
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}) | (b3.3) |
If α_{u} > α_{AB} ⇔ the Pivot B is reached first. Otherwise, the Pivot A is reached first.
Iterative calculation of the compression depth ratio α_{u}
This application considers d'/d ≤ α_{u} < 1. Beginning by the minimum value α_{u} = d'/d.
• 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 tensile reinforcement
ε_{s} = ε_{cu3}⋅(1 - α_{u})/α_{u}
• the strain in the compression reinforcement
ε_{sc} = ε_{cu3}⋅(α_{u} - d/d')/α_{u}.
The design stress-strain diagram for reinforcing steel
⇒ the stresses in the reinforcements σ_{s}, σ_{sc}.
For equilibrium, the sum of forces acting on the section gives:
α_{u*} = (A_{s} σ_{s} - A_{sc} σ_{sc}) / (b λ η f_{cd} d) | (b3.4) |
This calculated value of α_{u*} must be compared with the initial value α_{u}. If the difference is greater than 5 %, we start over with a greater value of α_{u}. If the difference is lower than or equal to 5 %, we accept the initial value α_{u} as the compression depth ratio.
Ultimate moment of resistance M_{Rd}
For equilibrium, the sum of moments to the centre of gravity of the tensile reinforcement gives:
M_{Rd} = A_{sc} σ_{sc} (d - d') + b λ x η f_{cd} (d - λ x / 2) | (b3.5) |
Flowchart
Figure b3-2 resumes this design method for a rectangular reinforced concrete section in bending at the ULS: