Example $3\mathrm{.}2.$ Thermal Stress and Strain in a Beam

A rectangular beam of small thickness $\backslash ($ t $\backslash ),$ depth $\backslash (2$ h $\backslash ),$ and length $\backslash (2$ L $\backslash )$ is subjected to an arbitrary variation of temperature throughout its depth, $\backslash ($ T$=$T$($y$)\backslash ).$ Determine the distribution of stress and strain for the case in which $($a$)$ the beam is entirely free of surface forces $($Fig$.3\mathrm{.}6$a$)$ and $($b$)$ the beam is held by rigid walls that prevent the $\backslash ($ x $\backslash )-$directed displacement only $($Fig$.3\mathrm{.}6$b$).$

Figure $3\mathrm{.}6.$ Example $3\mathrm{.}2.$ Rectangular be am in plane thermal stress: $($a$)$ unsupported; $($b$)$ placed between two rigid walls.