Determine the vertical displacement at the point $I$ of the given structure, due to the effect of the external load of $9 \, kN/m$. The frame has supports at points $A$ and $B$ and the horizontal distance between these supports is $L$.

1. Problem Description

Determine the vertical displacement at the point $I$ of the given structure, due to the effect of the external load of $9 \, kN/m$. The frame has supports at points $A$ and $B$ and the horizontal distance between these supports is $L$.

2. Solution Steps

The problem asks to determine the vertical displacement, but the information required to solve this problem is missing.

Specifically, the location of point $I$, the dimensions of the structure (e.g., the height), and the material properties (Young's Modulus $E$ and the moment of inertia $I$) of the structure are required to calculate displacement.

Assuming the question asks to find the vertical displacement at the center of the top beam, and given the distributed load of $q = 9 \, kN/m$ along the length $L$, we can assume this is a fixed-end beam. However, we still require additional information to provide a numerical solution.

Assuming we can use the principle of virtual work, we would remove the real load ($9 \, kN/m$ along the top beam) and apply a virtual unit load at the point $I$ where we wish to calculate the deflection.

$\delta = \int \frac{M \, m}{EI} \, dx$

Where:

$M$ is the bending moment due to the real load.

$m$ is the bending moment due to the virtual unit load.

$E$ is the Young's modulus of the material.

$I$ is the moment of inertia of the cross-section.

However, without the detailed structure and cross-sectional information, we can't proceed further.

3. Final Answer

It is impossible to determine a numerical value for the vertical displacement without additional information about the structure's dimensions, material properties (E and I), and location of point I. Therefore, I am unable to provide a final numerical answer.