Consider the frame shown in the figure under the loading of 100 kN.m couples at the joints B and G. Considering only the effects of flexural deformations, which of the following statements is/are true:
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In moment distribution and stiffness matrix methods, ensure to calculate the moment transfers carefully and check the forces in each member based on the boundary conditions and applied loads.
The magnitude of bending moment developed in the member BC at the end C is more than 50 kN.m
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The Correct Option isB, C
Solution and Explanation
From the given frame, we can analyze the behavior of the members under the applied loads using the moment distribution method and the stiffness matrix approach. Let's examine each statement:
(A) Axial force in the member CD: In this case, the axial force is not zero in member CD because it is part of the frame where moments are transferred. Thus, this statement is false.
(B) Shear force in the member CD: The shear force in member CD is indeed zero. Since there is no vertical load applied directly to the member CD, it does not develop a shear force. Thus, this statement is true.
(C) No rotation at joint C: Joint C does not undergo any rotation because the moment transferred to the joint is balanced, and thus, there is no angular displacement at joint C. Thus, this statement is true.
(D) Magnitude of bending moment developed in member BC at end C: The bending moment at joint C in member BC is \( 30 \) kN.m, which is less than \( 50 \) kN.m. Thus, this statement is false.
Thus, the correct answers are (B) and (C).
Detailed Moment Transfer Calculation:
For member BA:
\[
M_{BA} = DF_{BA} \times M = \frac{2}{5} \times 100 = 40 \text{ kN.m}
\]
For member BC:
\[
M_{BC} = DF_{BC} \times M = \frac{3}{5} \times 100 = 60 \text{ kN.m}
\]
Thus, the bending moment transfer calculations show that the moment in member BC at joint C is \( 30 \) kN.m, which is below \( 50 \) kN.m.
