DC EC 02: longitudonal reinforcement in beam under pure bending

Description

Geometry	Cross-section:	$b=200mm$ and $h=400mm$
	Material:	C25/30
	Concrete cover:	$d_1=d_2=40mm$
Load	Design bending moment:	$M_{Ed.ULS}=102.9kN$
Standard		EN 1992-1-1 [- -] and [BE]

Results

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

Handcalculation

Required reinforcement:

$\mu _d=\frac{M_{Ed.ULS}}{b\cdot d^2\cdot f_{cd}}=\frac{102.9kNm}{200mm\cdot (360mm)^2\cdot 16.7MPa}=0.238$

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

$\omega _1=0.280$

$A_{s1}=\omega _1 \cdot b\cdot d\cdot \frac{f_{cd}}{f_{yd}}=0.280 \cdot 200mm \cdot 360mm \cdot \frac{16.7MPa}{434.8MPa}=765mm^2$

Minimum reinforcement:

$A_{s.min}=\max (0.26\cdot \frac{f_{ctm}}{f_{yk}}; 0.0013) \cdot b \cdot h=\max (0.26 \cdot \frac{2.56MPa}{500MPa}; 0.0013) \cdot 200mm \cdot 400mm=107mm^2$

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.

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]

Handcalculation

Required reinforcement:

$\mu _d=\frac{M_{Ed.ULS}}{b\cdot d^2\cdot \alfa_{cc} \cdot f_{cd}}=\frac{102.9kNm}{200mm\cdot (360mm)^2\cdot 0.85 \cdot 16.7MPa}=0.280$

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

$\omega _1=0.340$

$A_{s1}=\omega _1 \cdot b\cdot d\cdot \frac{\alfa_{cc} \cdot f_{cd}}{f_{yd}}=0.340 \cdot 200mm \cdot 360mm \cdot \frac{0.85 \cdot16.7MPa}{434.8MPa}=798mm^2$

Minimum reinforcement:

$A_{s.min}=\max (0.26\cdot \frac{f_{ctm}}{f_{yk}}; 0.0013) \cdot b\cdot h$

$A_{s.min}=\max (0.26\cdot \frac{2.56MPa}{500MPa};0.0013)\cdot 200mm\cdot 400mm=107mm^2$

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.

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 §9.2.1 (for the minimum reinforcement)
Van Hooymissen, L., Spegelaere, M., Van Gysel, A., & De Vylder, W. (2002). Gewapend beton. Academia Press.
This reference does not contain this exact example, but it’s a good reference if you want to understand how reinforcement calculations work. However, keep in mind the calculations in this book are made done using the NBN B15-002 which is comparable to ENV 1992-1. The principle is similar to EN 1992-1-1 but some parameters differ slightly.
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 2023r01.

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