Problem 2.11. Spiral stirrups

The failure strength of the column according to EC2 is

f_c,c [N/mm²]=

The failure strength of the column using the von Mises criterion is

f_c,c [N/mm²]=

Step 1. Determine the contact pressure in the concrete at failure. 

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The axially loaded column is subjected to the compression of the surrounded spiral stirrups. At failure the stress in the stirrups reaches the yield strength, the tensile force in the stirrups is

$N_t=f_y{A}_{sw}$

From the tension of the stirrups radial compression arise in the concrete. The concrete pressure is given by the pressure vessel formula:

See Example 2.8.

$P=\frac{N_t}{R}=\frac{f_y{A}_{sw}}{R}$

We distribute the above pressure uniformly along the height of the column:

$p=\frac{P}{s}=\frac{f_y{A}_{sw}}{sD/2}=\frac{435\times 78.5}{100\times 280/2}=2.439\frac{\mathrm{N}}{{\mathrm{mm}}^2}$

Step 2. Determine failure strength according to EC2.

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Step 3. Determine the principal stresses of the concrete. Write the von Mises criterion in 3D stress stage. 

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Eq.(2-92)

${\left({\sigma }_1–{\sigma }_2\right)}^2+{\left({\sigma }_1–{\sigma }_3\right)}^2+{\left({\sigma }_2–{\sigma }_3\right)}^2\le 2f_{}^2$

here

${\mathrm{\sigma }}_1=f_{\mathrm{c},\mathrm{c}}, {\mathrm{\sigma }}_2={\mathrm{\sigma }}_3=p$

thus the criterion has the following form

${\left(f_{\mathrm{c},\mathrm{c}}–p\right)}^2+{\left(f_{\mathrm{c},\mathrm{c}}–p\right)}^2+{\left(p\mathit{–}p\right)}^2\le 2f_{\mathrm{c}}^2$

Step 4. Express failure strength, f_c,c from the von Mises criterion.

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The axially loaded column is in 3D stress stage. Beneath the compression of the axial load, it is also subjected to the compression of the surrounded spiral stirrups.

At failure the stress in the stirrups reaches the yield strength, the tensile force in the stirrups is

$N_t=f_y{A}_{sw}$

From the tension of the stirrups radial compression arise in the concrete. The concrete pressure is given by the pressure vessel formula:

See Example 2.8.

We distribute the above pressure uniformly along the height of the column:

$p=\frac{P}{s}=\frac{f_y{A}_{sw}}{sD/2}=\frac{435\times 78.5}{100\times 280/2}=2.439\frac{\mathrm{N}}{{\mathrm{mm}}^2}$

According to EC 2 the failure strength can be calculated as

$f_{c,c}=\min \left\{\begin{array}{c}\begin{array}{c}{f}_{c\mathit{,}c}+\frac{10}{3}p=25+8.13=33.13\frac{\mathrm{N}}{{\mathrm{mm}}^2}\\ 1.125f_{c\mathit{,}c}+\frac{5}{3}p=28.125+4.065=32.19\frac{\mathrm{N}}{{\mathrm{mm}}^2}\end{array}\end{array}\right\}$

The von Mises failure criterion in 3D stress stage is

Eq.(2-92)

here

${\mathrm{\sigma }}_1=f_{\mathrm{c},\mathrm{c}}, {\mathrm{\sigma }}_2={\mathrm{\sigma }}_3=p$

thus the criterion has the following form

${\left(f_{\mathrm{c},\mathrm{c}}–p\right)}^2+{\left(f_{\mathrm{c},\mathrm{c}}–p\right)}^2+{\left(p\mathit{–}p\right)}^2\le 2f_{\mathrm{c}}^2$

The failure strength, f_c,c can be expressed from the above criterion.

$2f_{c,c}^2–4p{f}_{c,c}^{}+2p^2–2f_c^2=0 \to {\mathit{f}}_{\mathbf{c}\mathbf{,}\mathbf{c}}=p+f_c=2.439+25\mathbf{=}\mathbf{27}\mathbf{.}\mathbf{439}\frac{\mathbf{N}}{{\mathbf{mm}}^{\mathbf{2}}}$