$[2].$ In the figure below, a rigid plate C exists between two columns A and B which are

made of two different materials. The diameters and lengths of columns are given in the

sketch below. Column $A$ is rigidly supported from the top and bottom column $B$ is rigidly

supported from the bottom. The top surface of plate C is bonded rigidly to the bottom

surface of column A$.$ Similarly, the bottom surface of plate C is bonded rigidly to the top

surface of column B$.$ The columns themselves are elastically deformable. A load of $200$

kN is first applied to the rigid plate C as shown in the sketch, and then the two columns are

subjected to a drop in temperature of $30C.$ Determine the resulting axial stresses in

columns A and B$.$ The following elastic modulii $(E)$ and coefficient of thermal expansion

data $()$ is provided; $E_A=100{10}^3\frac{N}{m}{m}^2,E_B=206{10}^3\frac{N}{m}{m}^2,{}_A=12\mathrm{.}5$

$\frac{{10}^{-6}}{}C,{}_B=20\frac{{10}^{-6}}{}C.$ Assume that, $($i$)$ no column buckling occurs, $($ii$)$ only

axial loading and displacements are considered $(1-$D problem$),$ and $($iii$)$ final strain in

column B is tensile due to mechanical and thermal loading. $[3$ marks$]$

Note: This is a statically indeterminate problem. Therefore, static $($force equilibrium in

terms of stress times area terms$)$ alone will not yield the answer requested. One must also

consider all displacements along the axial direction. While calculating the displacement

values both thermal and mechanical contributions of both column members must be

considered.