In the frame shown in the figure (not to scale), all four members (AB, BC, CD, and AD) have the same length and same constant flexural rigidity. All the joints A, B, C, and D are rigid joints. The midpoints of AB, BC, CD, and AD, are denoted by E, F, G, and H, respectively. The frame is in unstable equilibrium under the shown forces of magnitude \( P \) acting at E and G. Which of the following statements is/are TRUE?
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When analyzing structural frames under unstable equilibrium, check for symmetry and force distribution to determine displacements and shear forces at key joints.
Given that all the joints (A, B, C, and D) are rigid, and the midpoints (E, F, G, H) are connected to the respective members, the following statements hold true:
1. Shear forces at H and F are zero:
Since the frame is in unstable equilibrium under forces \( P \), the shear forces at the joints H and F are zero, as the members are aligned symmetrically. The structure does not transfer vertical shear forces to the joints at the midpoints. Hence, statement (A) is true.
2. Horizontal displacements at H and F are zero:
The system is symmetric, and the forces are applied in a way that causes no horizontal displacement at the midpoints (H and F). The forces applied are balanced at the midpoints, resulting in zero horizontal displacements. Thus, statement (B) is true.
3. Slopes at E, F, G, and H are zero:
The structure is rigid at the joints, and there is no bending moment induced at points E, F, G, and H under the given load. This results in zero angular displacement (slopes) at these points. Hence, statement (D) is true.
Thus, the correct answer is (A), (B), and (D).
