Question

The principal stresses at a point inside a solid object are ${\sigma }_1$ = 100 MPa, ${\sigma }_2$ = 100 MPa and ${\sigma }_3$ = 0 MPa. The yield strength of the material is 200 MPa. The factor of safety calculated using Tresca (maximum shear stress) theory is $n_T$ and the factor of safety calculated using Von Mises (maximum distortional energy) theory is $n_v$, Which one of the following relations is TRUE?

• A. $n_T=\left(\frac{\sqrt{3}}{2}\right)n_V$
• B. $n_T=\left(\sqrt{3}\right)n_V$
• C. $n_T=n_V$
• D. $n_V=\left(\sqrt{3}\right)n_T$

Answer 1

The correct option is C $n_T=n_V$

${\sigma }_1=100MPa,{\sigma }_2=100MPa,$
${\sigma }_3=0;{\sigma }_y=200MPa$
According to Tresca theory

$\tau$${}_{max}\le \frac{{\sigma }_y}{2FOS}=\frac{{\sigma }_y}{2n_T}$
now $\tau$${}_{ma{x}_{abs}}=\frac{{\sigma }_1-{\sigma }_3}{2}=50MPa$
$\therefore 50=\frac{200}{2n_T}$
$\therefore n_T=2$
According to Von - Mises theory
${\sigma }_1^2+{\sigma }_2^2-{\sigma }_1{\sigma }_2\le {\left(\frac{{\sigma }_y}{n_V}\right)}^2$
${100}^2+{100}^2-100\times 100\le {\left(\frac{200}{n_V}\right)}^2$
$\therefore 100=\frac{200}{n_V}$
or $n_V=2$
$\therefore n_T=n_V$