고체역학 (Beer) — Stress and Strain—Axial Loading: Problem 2.121

*2.121 Narrow bars of aluminum are bonded to the two sides of a thick steel plate as shown. Initially, at \(T_1 = 70^\circ\text{F}\), all stresses are zero. Knowing that the temperature will be slowly raised to \(T_2\) and then reduced to \(T_1\), determine (a) the highest temperature \(T_2\) that does not result in residual stresses, (b) the temperature \(T_2\) that will result in a residual stress in the aluminum equal to 58 ksi. Assume \(\alpha_a = 12.8 \times 10^{-6}/^\circ\text{F}\) for the aluminum and \(\alpha_s = 6.5 \times 10^{-6}/^\circ\text{F}\) for the steel. Further assume that the aluminum is elastoplastic with \(E = 10.9 \times 10^6 \text{ psi}\) and \(\sigma_Y = 58 \text{ ksi}\). (Hint: Neglect the small stresses in the plate.)