$(1)$ A rectangular bar is subjected to uniaxial stretching with a stress ${}_0.$ Draw the Mohr's circle for such a

loading condition.

$($m$)$ What relationship must be satisfied between the local shear force and local bending moment generated in a

beam due to lateral loading?

$($n$)$ A square plate in the $x-y$ plane with Youngs modulus E and Poisson ratio $v$ experiences the following state

of stress $($in any prescribed units$).$ What will be the strains generated? Schematically draw the shape of the

plate after deformation.

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

$($o$)$ Write down the equation that relates the meridional and the hoop stresses in a thin vessel whose shape has

an axial symmetry $($a surface of revolution$),$ a thickness of $t$ and the inside pressure of $p.$

$(10$ points$)$ Calculate the traction vector on a plane that intersects the $x,y,$ and $z$ axes respectively at distances

of $2,3,$ and $4$ units, if the state of stress is the same as in Problem $1($a$).$

$(10$ points$)$ A conical vesssel of wall thickness $1cm,=60,$ and base radius $R=0\mathrm{.}5m$ is made of a material

that can not sustain a normal stress higher than $10$ MPa and will undergo permanent deformation. If this vessel

is subjected to an internal pressure of $5{10}^{5}Pa,$ which part of the vessel will undergo permanent deformation?

$(15$ points$)$ A rectangular plate of side lengths $4m3m$ is made of a material with given $E={10}^{7}Pa$ and $v=$

$0\mathrm{.}3.$ This plate is to be bent into a spherical surface with a radius of curvature $R=10m,$ by applying bending

moments along its sides. How much bending moments will need to be applied along each side? $($Ignore the end

effects.$)$

$(15$ points$)$ Find out the value of Poisson ratio for an incompressible deformation. $($Hint: Use the fact that

change in volume is zero in an incompressible deformation for any stress applied.$)$

points$)$ A cylindrical vessel is to be made of IS:$2002-1962$ grade $2$B steel having allowable

design stress of $118$ MPa $.$ The weld joint efficiency factor is $J=0\mathrm{.}85,$ and Poisson ratio $v=0\mathrm{.}3.$

$($a$)$ Find out the required thicknesses of the cylindrical vessel with an outer diameter of $d_0=1\mathrm{.}2m,$ if the

maximum operating inside pressure is $500$ kPa $.$

$($b$)$ If the closure is a flat plate type welded to the cylindrical vessel at its periphery, what will be its minimum

thickness for safe operation. You can assume that a plate welded at its periphery behaves like a circular

plate with clamped ends.

$($c$)$ Find out the maximum deflection in the head? Where does the maximum deflection appear?

$($d$)$ Find out the values of $M_1$ and $M_2,$ the bending moments along the two orthogonal directions of principal

radii of curvature, at a distance of $\frac{d_0}{4}$ from the center.

The deflection in a circular plate is given by the following formula: $w(x)=\frac{q}{64D}(a^2-x^2{)}^2$ where each

symbol carry its usual meaning. Young's modulus of the steel is given to be $E=200$GPa.