DC EC [G1] 11: longitudinal 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 1992-1-1:2005 [- -]

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

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

Results	Independent reference	Diamonds	Difference
Compressive longitudinal reinforcement $A_{s2}$	144 mm²	144 mm²	0,00%
Tensile longitudinal 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

EN 1992-1-1: 2005 + AC: 2010
Van Hooymissen, L., Spegelaere, M., Van Gysel, A., & De Vylder, W. (2002). Gewapend beton. Academia Press
Keep in mind that the calculations in this book are made done using the NBN B15-002/ ENV 1992-1. Both old standards. However, this book still remains a good reference if you want to understand how reinforcement calculations work.
Gruyaert, E., & Minne, P. (2019). Gewapend beton: numeri.
This reference is a summary of Gewapend beton (2002) but with updated formula and principles according to EN 1992-1-1. This document contains multiple graphs and tables helping the design of reinforced concrete.

Tested in Diamonds 2026.

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