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It looks like you've uploaded an image that contains a problem related to weak form solutions and boundary value problems, possibly in the context of elasticity or thermal expansion of a bar (sauma). The problem involves determining the normal force \( N(z) \) as a function of the coordinate \( z \), with a varying elastic modulus \( EA(x) \) and some thermal effects included.

### Problem Summary:
- The bar has an elastic modulus that varies as \( EA(x) = 20000 \left( 1 + \frac{x}{5\, \text{m}} \right) \) kN.
- The thermal expansion coefficient is given as \( \alpha = 20 - 10 \times 10^{-6} \, ^\circ C^{-1} \).
- The temperature at the bar is \( T = 10^\circ C \), and the reference temperature \( T_0 = 24^\circ C \).
- You are asked to solve for the normal force \( N(z) \) using the weak form and Galerkin method.

### Steps Already Completed:
1. **Differential Equation for Force Balance**:
   The force balance is given as \( \frac{dN}{dz} = 0 \), indicating that the normal force \( N \) is constant along the bar.
   
2. **Boundary Conditions**:
   You have applied the boundary conditions for displacement at:
   - \( u(0) = 0 \)
   - \( u(L) = 0 \), where \( L = 5 \, \text{m} \) is the length of the bar.

### What to Do Next:
- You can now proceed with the weak form solution by considering the displacement \( u(z) \) and the varying modulus of elasticity.
- Use the Galerkin method by choosing appropriate test functions to find the approximate solution for \( N(z) \).

Let me know if you need any further details or clarifications on specific steps, and I can guide you through solving the boundary value problem in detail.

Would you like a detailed explanation of the Galerkin method for this problem? Here are 5 related questions to expand on this topic:

1. How does the varying elasticity modulus \( EA(x) \) affect the force distribution in the bar?
2. What is the weak form of a differential equation, and how is it used in boundary value problems?
3. Can you explain the Galerkin method and its applications in solving differential equations?
4. How do thermal expansion coefficients like \( \alpha \) influence the solution of a problem involving temperature changes?
5. Why is it important to enforce boundary conditions when solving for displacements and forces in structural problems?

### Tip:
In problems involving weak form and the Galerkin method, the choice of test functions is crucial to obtaining an accurate solution. Choose functions that satisfy the boundary conditions for better convergence.

Determine the normal force N(z) for a bar with varying elasticity modulus EA(x) and thermal expansion using the weak form and Galerkin method. The elastic modulus is given as EA(x) = 20000 (1 + x/5m) kN, and the thermal expansion coefficient α = 20 - 10 × 10^-6 °C^-1. The temperature at the bar is T = 10°C, and the reference temperature is T₀ = 24°C.

Solve a boundary value problem involving a bar with varying elasticity modulus and thermal expansion effects using the weak form and Galerkin method. The elastic modulus is EA(x) = 20000 (1 + x/5m) kN, and the thermal expansion coefficient is α = 20 - 10 × 10^-6 °C^-1. Suitable for university-level students.

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