PROBLEM $1.$ The figure shows equal and opposite temperature changes at the upper and lower surfaces of the flexural $($beam$)$ element. At the upper surface the temperature change varies linearly from zero at node $1$ to DeltaT$\_(2)$ at node $2$ ; and at he lower surface the variation is from zero at node $1$ to $-$DeltaT$\_(2)$ at node $2.$ Determine the equivalent load vector $\{$f$\_($T$)\}$ due to these temperature changes, assuming a linear temperature gradient through the depth d $.$