The image you've provided seems to involve a mechanics or structural engineering problem, likely focusing on thermal stress in a rod or beam caused by temperature differences. Here's a summary of the problem and solution steps based on the visible content:

Problem Description:

• A bar of length $L = 5 \, \text{m}$ is shown, subjected to thermal conditions.
• The temperature of the bar changes from $T_0 = 24^\circ C$ (reference temperature) to $T = 10^\circ C$, which causes normal stress due to the thermal expansion or contraction of the material.
• The stiffness of the bar is given by: $EA(x) = 20000 \left( 1 + \frac{x}{5 \, \text{m}} \right) \, \text{kN}$
• The coefficient of thermal expansion $\alpha = 20 \times 10^{-6} \, \text{C}^{-1}$.
• The goal is to determine the normal force $N(x)$ in the bar due to the temperature change, using weak form analysis (Galerkin method).

Steps Already Completed (Visible):

1. Differential Equation Setup:

   • The force balance is expressed as: $\frac{dN}{dx} = 0$ indicating that the normal force $N(x)$ is constant along the length.

2. Constitutive Relation (Material Law):

   • The relation between normal force and deformation is given by: $N = EA$

3. Boundary Conditions:

   • At $x = 0$ and $x = L$, the displacement conditions $u(0) = 0$ and $u(L) = 0$ are imposed, ensuring the beam is fixed at both ends.

Next Steps:

Based on the provided equations and the solution approach, the next steps would likely involve:

• Thermal Strain Calculation: Since the bar undergoes thermal contraction, the thermal strain $\epsilon_T$ would be: $\epsilon_T = \alpha \Delta T = \alpha (T - T_0)$
• Normal Force due to Thermal Expansion: Integrating the effect of thermal strain along the bar should yield the normal force distribution, considering the varying stiffness $EA(x)$.

Do you need further assistance with the next part of the problem, such as solving for the thermal stresses or further expanding the weak form approach?

5 Follow-Up Questions:

1. How is the thermal strain distributed along the beam when the stiffness $EA(x)$ is not constant?
2. What method would you use to compute the exact value of the normal force $N(x)$ given the boundary conditions?
3. How does the varying stiffness $EA(x)$ affect the overall stress distribution compared to a uniform stiffness?
4. Can we assume the bar remains in static equilibrium with the temperature change, or do we need to account for dynamic effects?
5. How would you modify the problem if the temperature distribution along the bar was non-uniform?

Tip:

When dealing with temperature-induced stresses, always ensure to correctly account for boundary conditions, as they significantly impact the displacement and force calculations.