The problem asks to determine the stiffness component and find the internal stresses of a given frame using the stiffness method. The frame consists of a vertical column of length 4.0 m fixed at the base, a horizontal beam of length 8.0 m (two sections of 4.0 m each) subjected to a uniformly distributed load of 4 kN/m, and another vertical column fixed at the top. Also, there are some numbers like 40/60, 40/40, and 40/0 on the frame. It appears these refer to moment values.

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 consists of a vertical column of length 4.0 m fixed at the base, a horizontal beam of length 8.0 m (two sections of 4.0 m each) subjected to a uniformly distributed load of 4 kN/m, and another vertical column fixed at the top. Also, there are some numbers like 40/60, 40/40, and 40/0 on the frame. It appears these refer to moment values.

2. Solution Steps

Due to the complexity of the stiffness method and the incomplete information provided in the image (e.g., EI values for members, specific points where internal stresses are needed), a complete numerical solution cannot be provided. Instead, I will outline the general steps involved in solving this problem using the stiffness method:

Step 1: Determine the Degrees of Freedom (DOF)

Identify the unconstrained degrees of freedom at the joints of the frame. These typically include rotations and translations (horizontal and vertical displacements). A fixed support has no DOF.

Step 2: Establish Coordinate System

Establish a global coordinate system for the entire frame.

Step 3: Calculate the Stiffness Matrix for Each Member

For each member, calculate the stiffness matrix in the local coordinate system. The stiffness matrix relates forces and moments to displacements and rotations at the member ends.

For a beam element, the local stiffness matrix is a 6x6 matrix related to axial deformation, bending and shear. Assuming no axial deformation:

$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}$

where $E$ is the modulus of elasticity, $I$ is the moment of inertia, and $L$ is the length of the member. This is assuming that only flexural stiffness is relevant.

Step 4: Transform Member Stiffness Matrices to Global Coordinates

Transform each member stiffness matrix from the local coordinate system to the global coordinate system using a transformation matrix.

Step 5: Assemble the Global Stiffness Matrix

Assemble the global stiffness matrix by superimposing the transformed member stiffness matrices.

Step 6: Calculate Fixed-End Moments and Forces due to Loads

Calculate the fixed-end moments and forces for each member due to the applied loads. For the uniformly distributed load $w$ on the beam, the fixed-end moments are:

$M_{fixed} = \pm \frac{wL^2}{12}$

$F_{fixed} = \frac{wL}{2}$

Step 7: Apply Boundary Conditions

Apply the boundary conditions by eliminating the rows and columns corresponding to the constrained degrees of freedom in the global stiffness matrix.

Step 8: Solve for Displacements

Solve the system of equations $KU = F$, where $K$ is the reduced global stiffness matrix, $U$ is the vector of unknown displacements, and $F$ is the vector of equivalent nodal forces (obtained from the fixed-end forces and moments).

Step 9: Calculate Member End Forces and Moments

Calculate the member end forces and moments using the calculated displacements and the member stiffness matrices.

Step 10: Calculate Internal Stresses

Calculate the internal stresses at the desired locations within each member using the calculated member end forces and moments. This usually involves calculating bending moments and shear forces along the member length.

The bending moment is given by $M(x)$. Then the stress is given by $\sigma= My/I$, where $y$ is the distance from the neutral axis to the point where the stress is being calculated.

3. Final Answer

A specific numerical answer cannot be determined with the given information and the complexity of the frame. The outlined steps provide a general solution procedure using the stiffness method. To arrive at a numerical solution, one would need to define the geometry precisely, specify the EI values of all the members and perform the calculations as outlined above. Finally, knowing the stresses relies on knowing the geometry of the cross-section of the members as well. The values such as 40/60 may represent intermediate steps for the solution of a simpler, related problem.