Question

Consider the linearly elastic plane frame shown in the figure. Members HF, FK, and FG are welded together at joint F. Joints K, G, and H are fixed supports. A counter-clockwise moment \( M \) is applied at joint F. Consider flexural rigidity \( EI = 10^5 \, \text{kN-m}^2 \) for each member and neglect axial deformations. \includegraphics[width=0.35\linewidth]{image51.png} If the magnitude (absolute value) of the support moment at H is 10 kN-m, the magnitude (absolute value) of the applied moment \( M \) (in kN-m) to maintain static equilibrium is \(\underline{\hspace{1cm}}\) (round off to the nearest integer).

Hint:

To maintain static equilibrium in frame structures, apply the moment-curvature relationships considering the flexural rigidity and the length of the members.

Answer 1

Correct Answer: 57

We need to maintain static equilibrium for the given structure. To do so, we can use the principle of superposition for the moments at the joints.
Given that the support moment at \( H \) is \( 10 \, \text{kN-m} \), the applied moment \( M \) at joint \( F \) will cause deformations at the structure that need to counterbalance the applied forces. Using moment-curvature relationships and considering the flexural rigidity \( EI \), we can apply the following formula to calculate the required moment: \[ M = \frac{EI}{L} \times (\text{support moment at H}) \] Using the given values:
- \( EI = 10^5 \, \text{kN-m}^2 \)
- Length of each member (\( L \)) is 4 m for the vertical member and 3 m for the horizontal member
Substituting these values into the equation, we get the magnitude of \( M \) required to balance the system: \[ M = \boxed{60 \, \text{kN-m}}. \] Thus, the magnitude of the applied moment \( M \) is 60 kN-m.