points$)$ Please answer the following questions in not more than three sentences:

$($a$)$ For the state of stress shown below, what is the value of hydrostatic pressure? Also write the deviatoric part

of the stress.

$=\left[\begin{array}{ccc}-1& 2& 3\\ 3& -4& -1\\ 2& -2& -4\end{array}\right]$

$($b$)$ A block is being compressed along $x$ direction with a stress of $10$ kPa and stretched along $y$ direction with a

stress of $20$ kPa $.$ What is maximum shear stress this block is subjected to$?$

$($c$)$ Schematically draw a toropsherical head denoting all important geometrical parameters.

$($d$)$ What will be the bending rigidity of a beam having a rectangular cross section with width $b=5cm$ and

height $h=3cm?$ The shear modulus of the material is given to be $100$ kPa and the Poisson ratio is $0\mathrm{.}3.$

$($e$)$ What will be the values of axial and hoop stresses in a closed thin cylinder having a radius of $1$ m and a

thickness of $2$ cm and pressurized to $10$ atm $.$

$($f$)$ Will the cylinder in part $($e$)$ fail if the maximum shear stress that the material can sustain is ${10}^{7}Pa.$ Explain

your answer.

$($g$)$ At any point in a beam with an axis along $x$ axis and subjected to bending moment along $z$ axis, which

stress components can be assumed zero? Justify.

$($h$)$ A given mass of a material is formed into a hollow pipe in one case and $$ solid rod in another case. Which

of these will be easier to bend? Why?

$($i$)$ What will be the bending rigidity of a plate of thickness $2$ cm $,$ with material's Young's modulus $E={10}^{6}Pa$

and Poisson ratio $v=0\mathrm{.}3.$ Will it be easier or harder to bend if Poisson's ratio is increased?

$($j$)$ A beam is clamped at one end and freely resting on a contact at the other end as shown in the figure below.

Write down the boundary conditions that must be used to solve for the acquired shape of this beam under

an applied bending moment.

$($k$)$ What kind of difficulty may arise in using an ellipsoidal closure if $\frac{a}{b}$ ratio exceeds certain limit$?$ What is

the value of that limit$?1$