$4\mathrm{.}92$ A $250-mm$ length of steel tubing $($with properties of $E=207{10}^{9}Pa$ and

$=12{10}^{-6}$ per degree Celsius$)$ having a cross$-$sectional area of $625m{m}^2$ is

installed with "fixed" ends so that it is stress$-$free at $26C.$ In operation, the tube is

heated throughout to a uniform $249C.$ Careful measurements indicate that the fixed

ends separate by $0\mathrm{.}20$ mm $.$ What loads are exerted on the ends of the tube, and what

are the resultant stresses?

$4\mathrm{.}46$ The shaft shown in Figure P$4\mathrm{.}46$ is $200$ mm long between self$-$aligning bearings A and

B$.$ Belt forces are applied to a sheave in the center, as shown. The left end of the shaft is

connected to a clutch by means of a flexible coupling. Nothing is attached to the right

end.

$($a$)$ Determine and make a sketch showing the stresses acting on the top and side ele$-$

ments, $T$ and $S,$ located adjacent to the sheave. $($Neglect stress concentration.$)$

$($b$)$ Represent the states of stress at $T$ and $S$ with three$-$dimensional Mohr circles.

$($c$)$ At location $S,$ show the orientation and stresses acting on a principal element,

and on a maximum shear element.