A steel rod of length `6.0 m` and diameter `20 mm` is fixed between two rigid supports. Determine the stress in the rod, when the temperature increases by `80^@`C if
(a) the ends do not yield (b) the ends yield by 1mm
Take `Y=2.0xx10^(6) kg//cm^(2)`and`alpha=12xx10^(-6) per ^@C.`

Explanation

Given , length of the rod `l=6 m=600cm `
Diameter of the rod ` d=20mm=2 cm`
Increase in temperature t =`80^@C`
Young's modulus
`Y=2.0xx10^6 kg//cm^2`
and thermal coefficient odf linear expansion `alpha=12xx10^-6 per ^@C`
(a) When the ends do not yield
Let, `sigma_1`=stress in the rod
Using the relation `sigma=alpha tY`
`:. sigma_1=(12xx10^-6)(80)(2xx10^6)`
`=1920 kg//cm^2`
(b) When the ends yield by 1mm.
Increase in length due to increase in temperature `Delta l=lalphat`
of this `1mm` or `0.1cm`is allowed to expand. Therefore, net compression in then rod
`Delta l_"net"=(lalphat-0.1)`
or compressive strain in the rod,
`epsilon = (Deltal_net)/l=(alphat-0.1/l)`
:. stress sigma_2=`Yepsilon=Y(alphat-0.1/l)`
substituting the values ,
`sigma_2=2xx10^6(12xx10^-6xx80-0.1/600)`
`=1587kg//cm^2`