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DC EC 10: stress limites 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
Reinforcement required for ULS: A_{s1.ULS}=1189mm^2
A_{s2.ULS}=107mm^2
Creep factor: \phi_{stress}=1.36
Loads M_{Ed.SLS.RC}=144.9kNm
M_{Ed.SLS.RC}=104kNm
M_{Ed.SLS.QP}=83kNm
Standard EN 1992-1-1 [- -]
Note: this model is similar to DC EC 04 and DC EC 08, but we’ve increased the loads so the additional reinforcement due to the stress limits would be clearly present.

Independent reference results

Open handcalculations

Determine the limits for the allowable stresses (EN 1992-1-1)

  • Maximum allowable concrete stress in SLS RC

        \[\sigma_{c.RC.max}=0.6 \cdot f_{ck}= 0.6 \cdot 25 \text{MPa}= 15\text{MPa}\]

  • Maximum allowable concrete stress in SLS QP

        \[\sigma_{c.QP.max}=0.45 \cdot f_{ck}= 0.45 \cdot 25 \text{MPa}= 11.25\text{MPa}\]

  • Maximum allowable steel stress in SLS RC

        \[\sigma_{s.RC.max}=0.8 \cdot f_{yk}= 0.8 \cdot 500 \text{MPa}= 400\text{MPa}\]

Verify the concrete and steel stresses with ULS

Determine cracked moment of inertia (creep, theoretical ULS reinforcement)

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

    \[\alpha_{cr}=\frac{E_s}{\frac{E_c}{1+\phi_{stress}}}=15\]

    \[x_{cr.ULS}=175.7mm\]

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

    \[I_{cr.ULS}=995 \cdot 10^6 mm^4\]

Determine concrete and steel stresses
  • Concrete stress in SLS RC

        \[\sigma_{c.RC}=\frac{M_{Ed.SLS RC} \cdot x_{c.ULS}}{I_{cr.ULS}} = \frac{104 kNm \cdot 175.6mm}{995 \cdot 10^6 mm^4}=18.36\text{MPa}\]

  • Concrete stress in SLS QP

        \[\sigma_{c.QP}=\frac{M_{Ed.SLS QP} \cdot x_{c.ULS}}{I_{cr.ULS}} = \frac{83 kNm \cdot 175.6mm}{995 \cdot 10^6 mm^4}=14.65\text{MPa}\]

  • Steel stress in SLS RC

        \[\sigma_{s.RC}=\alpha_{cr} \frac{M_{Ed.SLS RC} \cdot (d- x_{c.ULS})}{I_{cr.ULS}} = 15 \cdot \frac{104 kNm \cdot (360 - 175.6)mm}{995 \cdot 10^6 mm^4}=289.07 \text{MPa}\]

Compare
  • Concrete stress in SLS RC

        \[\sigma_{c.RC} > \sigma_{c.RC.max}\]

        \[18.36\text{MPa} > 15\text{MPa} \Rightarrow \text{NOT OK}\]

  • Concrete stress in SLS QP

        \[\sigma_{c.QP} > \sigma_{c.QP.max}\]

        \[14.65\text{MPa} >11.25\text{MPa} \Rightarrow \text{NOT OK}\]

  • Steel stress in SLS RC

        \[\sigma_{s.RC} \le \sigma_{s.RC.max}\]

        \[289.07 \text{MPa} \le 400\text{MPa} \Rightarrow \text{OK}\]

The concrete stresses in SLS RC and QP are too high. So Diamonds will add upper and lower reinforcement until the requirements are met. Diamonds finally suggests 592mm^2 upper and 1538mm^2 lower reinforcement. Let’s see if the stress limits are met with these reinforcement amounts.

Verify the concrete and steel stresses with ULS + SLS reinforcement

Determine cracked moment of inertia (creep, theoretical ULS + SLS reinforcement)

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

    \[\alpha_{cr}=\frac{E_s}{\frac{E_c}{1+\phi_{stress}}}=15\]

    \[x_{cr.ULS}=176.2mm\]

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

    \[I_{cr.ULS}=1300 \cdot 10^6 mm^4\]

Determine concrete and steel stresses
  • Concrete stress in SLS RC

        \[\sigma_{c.RC}=\frac{M_{Ed.SLS RC} \cdot x_{c.ULS}}{I_{cr.ULS}} = \frac{104 kNm \cdot 175.6mm}{1300 \cdot 10^6 mm^4}=14.09\text{MPa}\]

  • Concrete stress in SLS QP

        \[\sigma_{c.QP}=\frac{M_{Ed.SLS QP} \cdot x_{c.ULS}}{I_{cr.ULS}} = \frac{83 kNm \cdot 175.6mm}{1300 \cdot 10^6 mm^4}=11.25\text{MPa}\]

  • Steel stress in SLS RC

        \[\sigma_{s.RC}=\alpha_{cr} \frac{M_{Ed.SLS RC} \cdot (d- x_{c.ULS})}{I_{cr.ULS}} = 15 \cdot \frac{104 kNm \cdot (360 - 175.6)mm}{1300 \cdot 10^6 mm^4}=220.49 \text{MPa}\]

Compare
  • Concrete stress in SLS RC

        \[\sigma_{c.RC} \le \sigma_{c.RC.max}\]

        \[14.09\text{MPa} \le 15\text{MPa} \Rightarrow \text{OK}\]

  • Concrete stress in SLS QP

        \[\sigma_{c.QP} \le \sigma_{c.QP.max}\]

        \[11,25\text{MPa} \le11.25\text{MPa} \Rightarrow \text{OK}\]

  • Steel stress in SLS RC

        \[\sigma_{s.RC} \le \sigma_{s.RC.max}\]

        \[220,49 \text{MPa} \le 400\text{MPa} \Rightarrow \text{OK}\]

Diamonds results and comparison

It is impossible to display the concrete and steel stresses in Diamonds. The only stresses displayed by Diamonds, are elastic stresses.

But from the calculate above, we can conclude that Diamonds verifies the stresses in a correct way. If you’d do the same calculation but lower A_{s1} and A_{s2} with 2mm^2, the concrete stresses in SLS QP would exceed again.

Detailed result of the reinforcement calculated by Diamonds (EN 1992-1-1 [- -]).
The thin line is the reinforcement required to meet the ULS requirements, the thick is the reinforcement required to meet the ULS and SLS (stress) requirements.

Results Independent reference Diamonds Difference
SLS reinforcement 592 mm²
1538 mm²		 592 mm²
1538 mm²		 0%

References

  • EN 1992-1-1: 2005 + AC: 2010
  • Tested in Diamonds 2024r01.

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