Gere & Goodno — Axially Loaded Members: Problem 2.2-20

2.2-20 A framework \(ABC\) consists of two rigid bars \(AB\) and \(BC\), each having a length \(b\) (see the first part of the figure part a). The bars have pin connections at \(A\), \(B\), and \(C\) and are joined by a spring of stiffness \(k\). The spring is attached at the midpoints of the bars. The framework has a pin support at \(A\) and a roller support at \(C\), and the bars are at an angle \(\alpha\) to the horizontal. When a vertical load \(P\) is applied at joint \(B\) (see the second part of the figure part a) the roller support \(C\) moves to the right, the spring is stretched, and the angle of the bars decreases from \(\alpha\) to the angle \(\theta\). (a) Determine the angle \(\theta\) and the increase \(\delta\) in the distance between points \(A\) and \(C\). Also find reactions at \(A\) and \(C\). (Use the following data: \(b = 200 \text{ mm}\), \(k = 3.2 \text{ kN/m}\), \(\alpha = 45^\circ\), and \(P = 50 \text{ N}\).) (b) Repeat part (a) if a translational spring \(k_1 = k/2\) is added at \(C\) and a rotational spring \(k_r = kb^2/2\) is added at \(A\) (see figure part b).