Question

At a point in the web of a girder, the bending and the shearing stresses are 90 $N/m{m}^2$ (tensile) and 45 $N/m{m}^2$ respectively. The principal stresses are

• A. 108.64 $N/m{m}^2$ (tensile) and 18.64 $N/m{m}^2$ (compressive)
• B. 107.60 $N/m{m}^2$ (compressive) and 18.64 $N/m{m}^2$ (tensile)
• C. 108.64 $N/m{m}^2$ (compressive) and 18.64 $N/m{m}^2$ (tensile)
• D. 0.64 $N/m{m}^2$ (tensile) and 0.78 $N/m{m}^2$ (compressive)

Answer 1

The correct option is A 108.64 $N/m{m}^2$ (tensile) and 18.64 $N/m{m}^2$ (compressive)

${\sigma }_x=90N/m{m}^2$

${\sigma }_y=0$

${\tau }_{xy}=45N/m{m}^2$

$\Rightarrow {\sigma }_{1/2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)+{\tau }_{xy}^2}$

$=\frac{90+0}{2}\pm \sqrt{{\left(\frac{90-0}{2}\right)}^2+{45}^2}$

${\sigma }_1=108.63N/m{m}^2$

${\sigma }_2=-18.63N/m{m}^2$