DC EC 04: cracked deformation in beam under pure bending (SLS design)

Description

Geometry	Cross-section:	$b=200mm$ $h=400mm$
	Material:	C25/30 $E_c=31475MPa$ $E_s=200000MPa$
	Concrete cover:	$c=40mm$
	Practical reinforcement:	$A_1=942mm^2$ $A_2=226mm^2$
	Creep factor:	$\phi=2$
Loads		$q_{Ed.SLS.RC}=37kN/m$
Standard		EN 1992-1-1 [- -]

Indepent reference results

Open handcalculations

Uncracked moment of inertia $I_{0}$ (no creep, pract. reinforcement):
$0.5\cdot b\cdot x_{0}^{2}+\left ( \alpha _{0}-1 \right )\cdot A_{s2}\cdot \left (x_0-d_2 \right )=0.5\cdot b\cdot \left ( h-x_0 \right )^2+\left ( \alpha _{0}-1 \right )\cdot A_{s1}\cdot \left ( d-x_0 \right )$

$\alpha_{0}=\frac{E_s}{E_c}=6.35\)$

$x_0=207mm$

$I_0=\frac{b\cdot x_{0}^3}{3}+\left ( \alpha _{0}-1 \right )\cdot A_{s2}\cdot \left (x_0-d_2 \right )^2+\frac{b\left ( h-x_0 \right )^3}{3}+\left ( \alpha _{0}-1 \right ) \cdot A_{s1} \cdot \left ( d-x_0 \right )^2$

$I_0=1222 \cdot 10^6 mm^4$
Cracking moment $M_r$ :
$M_r=\frac{f_{ctm.fl}\cdot I_0}{h-x_0}=\frac{3.08N/mm^2 \cdot 1222 485 304mm^4}{400mm-207mm}=19.51kNm$
Cracked moment of inertia $I_{cr}$ (with creep, pract. reinforcement):
$0,5\cdot b\cdot x_{cr}^{2}+\left ( \alpha _{cr}-1 \right )\cdot A_{s2}\cdot \left (x_{cr}-d_2 \right )=\alpha_{cr} \cdot A_{s1}\cdot \left ( d-x_{cr} \right )$

$\alpha_{cr}=\frac{E_s}{E_{cr.\infty}}=19.06$

$x_0=169mm$

$I_0=\frac{b\cdot x_{cr}^3}{3}+\left ( \alpha_{cr}-1 \right )\cdot A_{s2} \cdot \left (x_{cr}-d_2 \right )^2+ \alpha_{cr} \cdot A_{s1} \cdot \left ( d-x_{cr} \right )^2$

$I_0=1045 \cdot 10^6 mm^4$
Further:
For a sustained load $\beta = 0.5$ .

$\zeta = 1-\beta \cdot \left ( \frac{M_r}{M_{Ed.SLS RC}} \right )^2 =1-0.5 \cdot \left ( \frac{19.51kNm}{74kNm} \right )^2=0.9653$

$\delta_1 = \frac{5\cdot q\cdot L^4}{384\cdot E_c\cdot I_0}=\frac{5\cdot 37 kN/m \cdot \left (4m \right )^4}{384\cdot 31475MPa \cdot I_0}=3.205mm$

$\delta_2 = \frac{5\cdot q\cdot L^4}{384\cdot E_{c.\infty } \cdot I_{cr}}=\frac{5\cdot 37 kN/m \left (4m \right )^4}{384\cdot 10491MPa \cdot 1045\cdot 10^6mm^4}=11.25mm$

$\delta_y=\left ( 1-\zeta \right )\cdot \delta_1+\zeta\cdot \delta_2=10.97mm$

Diamonds results and comparison

Cracked deformation calculated by Diamonds (EN 1992-1-1 [- -])

Results	Independent reference	Diamonds	Difference
Cracked deformation SLS RC	10,97 mm²	11,0 mm²	0,29%

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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Description

Indepent reference results

Diamonds results and comparison

References

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