The problem asks to determine the stiffness component and find the internal stresses of a given frame using the stiffness method. The frame has fixed supports on the left and right sides. There is a uniformly distributed load of $4 kN/m$ acting on a portion of the horizontal beam. The dimensions are given as $4.0 m$ horizontally for two spans and $4.0 m$ vertically for the column on the left and $4.0 m$ for the column on the right.

1. Problem Description

The problem asks to determine the stiffness component and find the internal stresses of a given frame using the stiffness method. The frame has fixed supports on the left and right sides. There is a uniformly distributed load of $4 kN/m$ acting on a portion of the horizontal beam. The dimensions are given as $4.0 m$ horizontally for two spans and $4.0 m$ vertically for the column on the left and $4.0 m$ for the column on the right.

2. Solution Steps

Due to the complexity of the stiffness method and the lack of information provided (such as the moment of inertia, $I$, and modulus of elasticity, $E$), a complete numerical solution cannot be provided. However, the general steps of the stiffness method are as follows:

Step 1: Define the Coordinates (Degrees of Freedom)

Identify the unconstrained degrees of freedom. Number each coordinate system. In this case, the degrees of freedom are rotations at joints and translations if joints can move.

Step 2: Develop Element Stiffness Matrices

For each element in the frame, develop the element stiffness matrix in local coordinates. The element stiffness matrix relates element end forces to element end displacements. This involves calculating properties like $EI/L$. For example, for a beam element:

$k = \frac{EI}{L^3} \begin{bmatrix}

12 & 6L & -12 & 6L \\

6L & 4L^2 & -6L & 2L^2 \\

-12 & -6L & 12 & -6L \\

6L & 2L^2 & -6L & 4L^2

\end{bmatrix}$

Step 3: Assemble the Structure Stiffness Matrix

Transform each element stiffness matrix from local to global coordinates using transformation matrices. Then, assemble the structure stiffness matrix $K$ by superimposing the element stiffness matrices.

Step 4: Formulate the Global Equilibrium Equations

Express the relationship between applied external loads $P$, the structure stiffness matrix $K$, and the unknown displacements $U$:

$P = KU$

Step 5: Apply Boundary Conditions

Incorporate the support conditions (fixed supports, etc.) by partitioning the equilibrium equations and eliminating the rows and columns corresponding to the constrained degrees of freedom.

Step 6: Solve for Unknown Displacements

Solve the reduced set of equilibrium equations to find the unknown displacements $U$.

$U = K^{-1}P$

Step 7: Calculate Element End Forces

Substitute the calculated displacements $U$ back into the element stiffness equations (in global coordinates) to find the element end forces.

Step 8: Superimpose Fixed-End Moments (FEMs) and Forces

The distributed load will have fixed-end moments and reactions. For a uniformly distributed load $w$ on a fixed-fixed beam of length $L$:

$FEM = \frac{wL^2}{12}$

The reactions are $wL/2$ on each end. Add the fixed-end forces and moments to the element end forces calculated from the displacements to obtain the final internal forces and moments.

Step 9: Calculate Internal Stresses

Once the member end forces and moments are known, internal stresses at any point along each member can be determined using basic stress equations. For example, bending stress $\sigma = My/I$.

Without specific values for $E$ and $I$, and without knowing the exact location and configuration of the supports (there is some ambiguity in the picture), it is impossible to provide a numerical solution.

3. Final Answer

A complete numerical solution cannot be provided without additional information. The steps outlined above provide the process for determining stiffness components and internal stresses using the stiffness method.