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

Description (same model as DC EC 01)

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		$M_{Ed.ULS}=81kNm$
Standard		EN 1992-1-1 [- -] and [BE]

How to model a column in Diamonds that can be compared with handcalculations

According to Eurocode 2, reinforcement is required for the forces in ULS, possibly increased with additional reinforcement to limit the stresses in SLS. It is not known in advance whether additional reinforcement will be required for SLS. Since we are only doing a ULS calculation in this example, we turn off the stress limits in SLS in Diamonds (by making a copy of C25/30 and editing the copy).

Independent reference results

Handcalculation according to EN 1992-1-1 --

Required reinforcement:

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

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

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

$\omega _1=0.210$

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

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.

Handcalculation according to EN 1992-1-1 BE

Required reinforcement:

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

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

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

$\omega _1=0.216$

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

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.

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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