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DC EC 11: longitudonal reinforcement in beam under pure bending (doubly reinforced – ULS)

Description


Geometry Cross-section: b=200mm
h=400mm
Material: C25/30
Concrete cover: c=40mm
Load Self-weight:
Dead loads:
neglected
P=30kN at end of span
Internal forces M_{Ed.ULS}=162\text{kNm}
Standard EN 1992-1-1 [- -]

Independent reference results

Open handcalculations

This model will require both upper and lower reinforcement. If you’re into the formula’s for designing reinforced concrete, you know that there’s only two equilibrium equations [Van Hooymissen.L, §5.3.4.3] while there are 5 unknows (b, d, \xi, A_{s1} and A_{s2}).
In Diamonds, you impose b and d by defining the cross-section and concrete cover. That leaves 3 unknows \xi, A_{s1} and A_{s2} with 2 equations…. Diamonds will solve this system by looking for a solution so that A_{s1} + A_{s2} is as small as possible (= the most economic reinforcement).

However that’s not a condition we can easily impose when we only have the tables from [Minne, P. Gewapend beton: numeri] at our disposal. As a work around we’re going to assume that the compressive reinforcement A_{s2} calculated by Diamonds, is also given. Then we can calculate \xi and A_{s1}.

Required reinforcement:

    \[d=h-c=400mm-40mm=360mm\]

    \[\delta_1=h-c=\frac{d_1}{d}=\frac{40mm}{360mm}=0.11\]

    \[\mu _d=\frac{M_{Ed.ULS}}{b\cdot d^2\cdot f_{cd}}=\frac{162\text{kNm}}{200mm\cdot (360mm)^2\cdot 16.7\text{MPa}}=0.375\]

    \[\omega _2=\frac{A_{s2}\cdot f_{yd}}{b \cdot d \cdot f_{cd}}=0.052\]

Table 5.7b from Gewapend beton: numeri (click here to open the Table) shows \omega _1 and \xi as a function of \mu _d and A_{s2}. We find:

    \[\omega _1=0.487\]

    \[A_{s1}=\omega _1 \cdot b\cdot d\cdot \frac{f_{cd}}{f_{yd}}=0.487 \cdot 200mm \cdot 360mm \cdot \frac{16.7\text{MPa}}{434.8\text{MPa}}=1344mm^2\]

Minimum reinforcement:

(9.1N)   \begin{align*} A_{s.min}&=\max (0.26\cdot \frac{f_{ctm}}{f_{yk}}; 0.0013) \cdot b \cdot h \\ &=\max (0.26 \cdot \frac{2.56\text{MPa}}{500\text{MPa}}; 0.0013) \cdot 200mm \cdot 400mm\\ &=107mm^2 \end{align*}

Diamonds calculates the minimum reinforcement with h instead of d because for more complex cross-section shapes, d is not unambiguously defined. Therefor h is a safe alternative.

Diamonds results and comparison

Longitudonal reinforcement calculated by Diamonds (EN 1992-1-1 [- -])

Results Independent reference Diamonds Difference
Compressive longitudonal reinforcement A_{s2} 144 mm² 144 mm² 0,00%
Tensile longitudonal reinforcement A_{s1} 1344 mm² 1304 mm² -2,98%1
1. Although the difference here is certainly acceptable, it is still larger than expected. But that’s because we’re using tables. If you solve the equilibrium equations analytically, you will obtain higher accuracy that matches the Diamonds results even better.

References

  • Tested in Diamonds 2024r01.

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