Question

In a stressed body, an elementary cube of material is taken at a point with its faces perpendicular to X and Y reference axes. Tensile stresses equal to $15kN/c{m}^{2}and9kN/c{m}^2$ are observed on these respective faces. They are also accompanied by shear equal to 4 $kN/c{m}^2$. The magnitude of the principal stresses at the point are

• A. 12 $kN/c{m}^2$ tensile and 3 $kN/c{m}^2$ tensile
• B. 17 $kN/c{m}^2$ tensile and 7 $kN/c{m}^2$ tensile
• C. 9.5 $kN/c{m}^2$ compressive and 6.5$kN/c{m}^2$ tensile compressive
• D. 12 $kN/c{m}^2$ tensile and 3 $kN/c{m}^2$ tensile

Answer 1

The correct option is B 17 $kN/c{m}^2$ tensile and 7 $kN/c{m}^2$ tensile

Principal stresses,
${\sigma }_{1.2}=\frac{{\sigma }_x+{\sigma }_y}{2}\pm \sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\tau }_{xy}^2}$

Given ${\sigma }_x=15kN/c{m}^2$

${\sigma }_y=9kN/c{m}^2$
${\tau }_{xy}=4kN/c{m}^2$

$\therefore {\sigma }_{1.2}=12\pm \sqrt{{\left(3\right)}^2+{\left(4\right)}^2}$

$=12\pm 5kN/c{m}^2$

${\sigma }_1=17kN/c{m}^2$

${\sigma }_2=7kN/c{m}^2$
Both of these are tensile in nature.