DC EC 02: longitudonal reinforcement in beam under pure bending (ULS FC design)

Description

Geometry	Cross-section:	$b=200mm$ $h=400mm$
	Material:	C25/30
	Concrete cover:	$c=40mm$
Load	Self-weight: Dead loads:	neglected $P=15kN$ at end of span
Internal forces		$M_{Ed.ULS}=81\text{kNm}$
Standard		EN 1992-1-1 [- -] and [BE]

Independent reference results

Handcalculation according to EN 1992-1-1 --

Required reinforcement:

$d=h-c=400mm-40mm=360mm$

$\mu _d=\frac{M_{Ed.ULS}}{b\cdot d^2\cdot f_{cd}}=\frac{81\text{kNm}}{200mm\cdot (360mm)^2\cdot 16.7\text{MPa}}=0.190$

Table 3 from Gewapend beton: numeri (click here to open the Table) shows $\omega _1$ as a function of $\mu _d$ , and we find:

$\omega _1=0.210$

$A_{s1}=\omega _1 \cdot b\cdot d\cdot \frac{f_{cd}}{f_{yd}}=0.210 \cdot 200mm \cdot 360mm \cdot \frac{16.7\text{MPa}}{434.8\text{MPa}}=581mm^2$

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.

Handcalculation according to EN 1992-1-1 BE

Required reinforcement:

$d=h-c=400mm-40mm=360mm$

$\mu _d=\frac{M_{Ed.ULS}}{b\cdot d^2\cdot \alpha_{cc} \cdot f_{cd}}=\frac{81\text{kNm}}{200mm\cdot (360mm)^2\cdot 0.85 \cdot 16.7\text{MPa}}=0.216$

Table 3 from Gewapend beton: numeri (click here to open the Table) shows $\omega _1$ as a function of $\mu _d$ .

$\omega _1=0.216$

$A_{s1}=\omega _1 \cdot b\cdot d\cdot \frac{\alpha_{cc} \cdot f_{cd}}{f_{yd}}=0.216 \cdot 200mm \cdot 360mm \cdot \frac{0.85 \cdot16.7\text{MPa}}{434.8\text{MPa}}=595mm^2$

Minimum reinforcement:

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

According to EN 1992-1-1 [- -]

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

Results	Independent reference	Diamonds	Difference
Maximum longitudonal reinforcement	765 mm²	768 mm²	-0,45%
Minimum longitudonal reinforcement	107 mm²	107 mm²	0,00%

According to EN 1992-1-1 [BE]

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

Results	Independent reference	Diamonds	Difference
Maximum longitudonal reinforcement	798 mm²	798 mm²	0,00%
Minimum longitudonal reinforcement	107 mm²	107 mm²	0,00%

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 2024r01.

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