Gere & Goodno — Axially Loaded Members: Problem 2.6-21

2.6-21 Plastic bar \(AB\) of rectangular cross section (\(b = 19\text{ mm}\) and \(h = 38\text{ mm}\)) and length \(L = 0.6\text{ m}\) is fixed at \(A\) and has a spring support (\(k = 3150\text{ kN/m}\)) at \(C\) (see figure). Initially, the bar and spring have no stress. When the temperature of the bar is raised by \(48^\circ\text{C}\), the compressive stress on an inclined plane \(pq\) at \(L_\theta = 0.46\text{ m}\) becomes \(5.3\text{ MPa}\). Assume the spring is massless and is unaffected by the temperature change. Let \(\alpha = 95 \times 10^{-6}/^\circ\text{C}\) and \(E = 2.8\text{ GPa}\). (a) What is the shear stress \(\tau_\theta\) on plane \(pq\)? What is angle \(\theta\)? (b) Draw a stress element oriented to plane \(pq\), and show the stresses acting on all faces of this element. (c) If the allowable normal stress is \(\pm 6.9\text{ MPa}\) and the allowable shear stress is \(\pm 3.9\text{ MPa}\), what is the maximum permissible value of spring constant \(k\) if the allowable stress values in the bar are not to be exceeded? (d) What is the maximum permissible length \(L\) of the bar if the allowable stress values in the bar are not be exceeded? (Assume \(k = 3150\text{ kN/m}\).) (e) What is the maximum permissible temperature increase \((\Delta T)\) in the bar if the allowable stress values in the bar are not to be exceeded? (Assume \(L = 0.6\text{ m}\) and \(k = 3150\text{ kN/m}\).)