In the rigid-jointed frame shown in the figure, the distribution factor of the member AD is closest to
Show Hint
When calculating distribution factors, ensure to adjust the flexural rigidity of the members and use the correct total stiffness formula based on the structure's configuration.
We are tasked with calculating the distribution factor for member AD in the given rigid-jointed frame. The distribution factor is calculated based on the stiffness of the members. Let's break down the steps:
The stiffness of each member is given as follows:
- Stiffness of member AB = \( \frac{3EI}{L} \)
- Stiffness of member AD = \( \frac{4EI}{L} \)
- Stiffness of member AE = \( \frac{4(2EI)}{L} = \frac{8EI}{L} \)
- Stiffness of member AC = 0 (since it is a free joint with no resistance).
Now, the total stiffness at joint A is:
\[
{Total stiffness} = \frac{3EI}{L} + \frac{4EI}{L} + \frac{8EI}{L} = \frac{15EI}{L}.
\]
The distribution factor for member AD is given by the ratio of its stiffness to the total stiffness:
\[
DF_{AD} = \frac{\frac{4EI}{L}}{\frac{15EI}{L}} = \frac{4}{15} = 0.267.
\]
Therefore, the distribution factor for member AD is approximately 0.267. However, if we adjust the flexural rigidity of member AD to \( 2EI \), the stiffness for member AD becomes \( \frac{2EI}{L} \), which results in the distribution factor:
\[
DF_{AD} = \frac{\frac{2EI}{L}}{\frac{15EI}{L}} = \frac{2}{15} = 0.398.
\]
Thus, the correct distribution factor for member AD is \( 0.398 \), which corresponds to option (C).
