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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=31475\text{MPa}
E_s=200000\text{MPa}
Concrete cover: c=40mm
Practical reinforcement: A_{s1}=798mm^2 divided over 2 bars
A_{s2}=107mm^2 divided over 2 bars
Creep factor: \phi=2
Loads q_{Ed.SLS.RC}=37\text{kN/m}
M_{Ed.SLS.RC}=74\text{kNm}
M_{Ed.SLS.QP}=60\text{kNm}
Standard EN 1992-1-1 [- -]

Independent 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\)\]

        \[x_0=207mm\]

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

        \[I_0=1187 \cdot 10^6 mm^4\]

  • Cracking moment M_r:

        \[M_r=\frac{f_{ctm.fl}\cdot I_0}{h-x_0}=\frac{3.08N/mm^2 \cdot 1187 \cdot 10^6 mm^4}{400mm-207mm}=18.92\text{kNm}\]

  • Uncracked moment of inertia I_{1} (with creep, pract. reinforcement):

        \[0.5\cdot b\cdot x_{1}^{2}+\left ( \alpha _{cr}-1 \right )\cdot A_{s2}\cdot \left (x_1-d_2 \right )=0.5\cdot b\cdot \left ( h-x_1 \right )^2+\left ( \alpha _{cr}-1 \right )\cdot A_{s1}\cdot \left ( d-x_1 \right )\]

        \[\alpha_{cr}=\frac{E_s}{E_{cr.\infty}}=\frac{E_s}{\frac{E_c}{1+\phi}}=\frac{20 000 \text{MPa}}{\frac{31475\text{MPa}}{1+2}}=19.06\]

        \[x_1=220.7mm\]

        \[I_1=\frac{b\cdot x_{1}^3}{3}+\left ( \alpha _{cr}-1 \right )\cdot A_{s2}\cdot \left (x_1-d_2 \right )^2+\frac{b\left ( h-x_1 \right )^3}{3}+\left ( \alpha _{cr}-1 \right ) \cdot A_{s1} \cdot \left ( d-x_1 \right )^2\]

        \[I_1=1444 \cdot 10^6 mm^4\]

  • Cracked moment of inertia I_{2} (with creep, pract. reinforcement):

        \[0.5\cdot b\cdot x_{cr}^{2}+\left ( \alpha _{cr}-1 \right )\cdot A_{s2}\cdot \left (x_{cr}-d_2 \right )=\alpha_{cr} \cdot A_{s1}\cdot \left ( d-x_{cr} \right )\]

        \[\alpha_{cr}=\frac{E_s}{E_{cr.\infty}}=\frac{E_s}{\frac{E_c}{1+\phi}}=\frac{20 000 \text{MPa}}{\frac{31475\text{MPa}}{1+2}}=19.06\]

        \[x_{2}=165.0mm\]

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

        \[I_{2}=908 \cdot 10^6 mm^4\]

  • Further:

        \[\beta = 0.5 \text{ for sustained loads}\]

        \[\zeta = 1-\beta \cdot \left ( \frac{M_r}{M_{Ed.SLS RC}} \right )^2 =1-0.5 \cdot \left ( \frac{18.92\text{kNm}}{74\text{kNm}} \right )^2=0.9673\]

        \[\delta_1 = \frac{5\cdot q\cdot L^4}{384\cdot E_c\cdot I_1}=\frac{5\cdot 30 kN/m \cdot \left (4m \right )^4}{384\cdot 31475\text{MPa} \cdot 1444 \cdot 10^6 mm^4}=2.201mm\]

        \[\delta_2 = \frac{5\cdot q\cdot L^4}{384\cdot E_{c.\infty } \cdot I_2}}=\frac{5\cdot 30 kN/m \left (4m \right )^4}{384\cdot 10491\text{MPa} \cdot 908 \cdot 10^6mm^4}=10.5mm\]

        \[\delta_y=\left ( 1-\zeta \right )\cdot \delta_1+\zeta\cdot \delta_2=10.22mm\]

Diamonds results and comparison

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

Results Independent reference Diamonds Difference
Cracked deformation SLS RC 10.22 mm 10.22 mm 0%

References

  • Tested in Diamonds 2024r01.

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