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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: d_1=d_2=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 [- -]

Results

Handcalculation
  • 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.

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

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

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

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

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 §7.4.3
  • 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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