Chapter $4$ : Tutorial Questions

$4.$ A stress state at a material point in a solid is given by$,[]=\frac{1}{4}{}_o\left[\begin{array}{ccc}3& 1& 0\\ 1& 3& 0\\ 0& 0& 4\end{array}\right]$ where ${}_o>0$

$($a$)$ Determine the principal stresses and their directions.

$($b$)$ Determine the maximum shear stress.

Ans:

$($a$)$

${}_1={}_2={}_0,{}_3=\frac{1}{2}{}_0$

$l_1=m_1,n_1$ is not unique

$l_2=m_2,n_2$ is not unique

vec$(i_3)=+-\frac{1}{\sqrt[\phantom{2}]{2}}(vec(i)-$vec$(j))$

$($b$){}_{\text{m}\text{a}\text{x}}=\frac{{}_0}{4}$