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The image contains a boundary value problem for analyzing beam structures, asking to understand weak forms and solutions through Galerkin's method. Let's break down the key tasks from the image:

1. **Problem Overview**: 
   - A beam with length \( L = 8 \) m is subjected to a distributed load \( q = 2 \, \text{kN/m} \) and a point load \( F = 1 \, \text{kN} \) at the free end. 
   - The boundary conditions are specified as the left end is fixed, and the right end is free. The task is to determine the bending moment function \( M(x) \) using linear approximations and Galerkin's method.

2. **Steps required**:
   a. **Boundary Value Problem**: You are asked to express the equations for internal forces:
      - Use symbols for shear force \( Q \), moment \( M \), the \( x \)-coordinate, the load \( q \), force \( F \), and derivative of the displacement function \( f(x) \).
      - **Equilibrium**: Define force and moment balance equations.

   b. **Weak Form of Boundary Problem**: You are tasked with formulating the weak form:
      - Define the weight function \( w \) and displacement function \( f(x) \), integrating over the domain.

Would you like a more detailed step-by-step breakdown of the weak formulation, the balance equations, or the Galerkin method solution?

Here are five related questions you may want to explore further:
1. How is Galerkin's method applied in boundary value problems for beams?
2. What are the governing differential equations for beam bending?
3. How do boundary conditions influence the solution for beam deflection?
4. Can you derive the weak form for a beam's equation of motion from the strong form?
5. How does one interpret the physical meaning of the bending moment and shear force distributions?

**Tip**: Understanding the weak form requires familiarity with functionals and integration by parts.

Analyze a beam subjected to a distributed load and point load using Galerkin's method to determine the bending moment and shear force.

This problem focuses on analyzing a beam structure subjected to a distributed load and a point load using Galerkin's method. It involves determining the bending moment, shear force, and solving the boundary value problem using weak formulations. Suitable for engineering students familiar with beam deflection theory and approximation methods.

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