In the structure shown below, the temperature of element 2 is \(50^{\circ} \mathrm{C}\) above the reference temperature. An external force of \(20,000 \mathrm{~N}\) is applied in the \(x\) direction (horizontal direction) at node 2. Assume \(E=10^{11} \mathrm{~Pa}, A=10^{-4} \mathrm{~m}^{2}\), and \(\alpha=2 \times 10^{-6} /{ }^{\circ} \mathrm{C}\).

a. Write down the stiffness matrices and thermal force vectors for each element.

b. Write down the global matrix equations.

c. Solve the global equations to determine the displacement at node 2 .

d. Determine the forces in each element. State whether it is tension or compression.

e. Show that force equilibrium is satisfied at node 2.

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(1) 1 20,000 N 2 2 1 m 1 m 3