고체역학 (Gere) — Axially Loaded Members: Problem 2.5-5

2.5-5 A bar \( AB \) of length \( L \) is held between rigid supports and heated nonuniformly in such a manner that the temperature increase \( \Delta T \) at a distance \( x \) from end \( A \) is given by the expression \( \Delta T = \Delta T_B x^3/L^3 \), where \( \Delta T_B \) is the increase in temperature at end \( B \) of the bar (see figure part a). (a) Derive a formula for the compressive stress \( \sigma_c \) in the bar. (Assume that the material has modulus of elasticity \( E \) and coefficient of thermal expansion \( \alpha \)). (b) Now modify the formula in part (a) if the rigid support at \( A \) is replaced by an elastic support at \( A \) having a spring constant \( k \) (see figure part b). Assume that only bar \( AB \) is subject to the temperature increase.