The problem asks us to determine the vertical displacement at a point I (assumed to be a point within the given frame structure, though its precise location is not clear from the image) due to an external load. The structure is a rectangular frame with supports at points A and B along the bottom horizontal member. The horizontal distance between A and B is $L$. The top horizontal member is subjected to a uniformly distributed load of $q = 9$ kN/m.

1. Problem Description

The problem asks us to determine the vertical displacement at a point I (assumed to be a point within the given frame structure, though its precise location is not clear from the image) due to an external load. The structure is a rectangular frame with supports at points A and B along the bottom horizontal member. The horizontal distance between A and B is $L$. The top horizontal member is subjected to a uniformly distributed load of $q = 9$ kN/m.

2. Solution Steps

To solve this problem, we would typically use the principle of virtual work or Castigliano's second theorem. However, without knowing the location of point I and without specifying the material properties (Young's modulus $E$ and moment of inertia $I$) and the dimensions of the frame (specifically the height), we cannot provide a numerical answer. We can, however, outline the general steps involved using the method of virtual work.

a. Define Coordinates: Establish a coordinate system to define the geometry of the frame.

b. Determine Support Reactions: Calculate the support reactions at A and B due to the applied distributed load. Since the load is uniformly distributed on the top member, and the supports are likely simple supports (pinned or roller), the vertical reactions at A and B are equal.

$R_A = R_B = \frac{qL}{2}$

c. Calculate Internal Moments (Real System): Determine the bending moment distribution throughout the frame due to the external load $q$. This will involve analyzing the horizontal and vertical members. Let $M(x)$ represent the bending moment as a function of position $x$ along each member.

d. Apply Virtual Unit Load: Apply a virtual unit vertical load at point I. The location of point I needs to be well-defined. Assuming point I is on the bottom beam at a distance $x$ from $A$. Then apply a vertical downward load of $1$.

e. Calculate Internal Moments (Virtual System): Determine the bending moment distribution throughout the frame due to the virtual unit load. Let $m(x)$ represent the bending moment due to the virtual load.

f. Apply the Principle of Virtual Work: The vertical displacement $\Delta$ at point I is given by:

$\Delta = \int \frac{M(x)m(x)}{EI} dx$

The integral is taken over the entire length of the frame, summing up the contributions from each member. This includes all integrals from horizontal and vertical beams.

Since specific dimensions and material properties are not given, it is impossible to proceed further to obtain a numerical value for the displacement.

3. Final Answer

Without specific dimensions, material properties, and location of point I, we can only provide the general formula for the vertical displacement at point I using the principle of virtual work. The vertical displacement is given by:

$\Delta = \int \frac{M(x)m(x)}{EI} dx$

where:

$M(x)$ is the bending moment distribution due to the external load,

$m(x)$ is the bending moment distribution due to a virtual unit vertical load at point I,

$E$ is the Young's modulus of the frame material, and

$I$ is the moment of inertia of the frame members.

The integral is taken over the entire length of the frame.