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The problem you provided asks for the **vertical displacement of point C** using **Castigliano's Theorem**. Here's the step-by-step process to approach the problem:

### Given Data:
- Load at point B: \(100 \, \text{N}\) (horizontal force)
- Load at point C: \(200 \, \text{N}\) (vertical force)
- Beam dimensions:
  - \(AB = 4 \, \text{m}\)
  - \(BC = 3 \, \text{m}\)
- Modulus of Elasticity: \(E = 29,000 \, \text{ksi} = 29,000 \times 10^3 \, \text{psi} = 29,000 \times 6.895 \times 10^6 \, \text{Pa}\)
- Moment of Inertia: \(I = 500 \, \text{in}^4 = 500 \times 41.62 \times 10^{-8} \, \text{m}^4\)

### Steps:

1. **Castigliano’s Theorem:**
   Castigliano’s second theorem states that the displacement in a structure subjected to loads can be derived from the strain energy in the system. The vertical displacement of point \(C\), \(\delta_C\), due to the vertical force \(P = 200 \, \text{N}\) can be given by:

   \[
   \delta_C = \frac{\partial U}{\partial P}
   \]

   Where \(U\) is the total strain energy of the system.

2. **Strain Energy in Beams:**
   The strain energy for bending in a beam is:

   \[
   U = \int \frac{M(x)^2}{2EI} \, dx
   \]

   Here, \(M(x)\) is the moment at a point \(x\) along the beam, \(E\) is the modulus of elasticity, and \(I\) is the moment of inertia of the beam's cross-section.

3. **Determine Moment Expressions:**
   The beam is composed of two parts: **vertical beam \(AB\)** and **horizontal beam \(BC\)**.

   - For segment \(AB\), calculate the moment due to the horizontal load at \(B\) and vertical load at \(C\).
   - For segment \(BC\), calculate the moment due to the vertical load at \(C\) and the reaction forces at \(B\).

4. **Apply Boundary Conditions:**
   Use the fact that point \(A\) is fixed to determine the reactions and the moment distribution along the beam.

Once the moment distribution is calculated, integrate the expressions for strain energy for each beam segment and use Castigliano's theorem to determine the vertical displacement at point \(C\).

I'll now compute the detailed solution with these steps.

Let me calculate the necessary components.We have established the basic parameters for the problem. Now, we will proceed step by step:

1. **Reactions at A:**
   Since the beam is fixed at point A, we need to calculate the reaction forces at A using equilibrium equations (sum of forces and moments).

2. **Moments in each segment:**
   For both segments \(AB\) and \(BC\), we'll compute the bending moment expressions caused by the forces applied at points B and C.

3. **Strain Energy:**
   Using the moment expressions, we can integrate them over the lengths of the beams to get the total strain energy.

4. **Displacement at Point C:**
   Apply Castigliano's Theorem to determine the displacement.

Let me calculate these components next.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the manual steps or explain the concepts more clearly if needed.

Would you like a detailed explanation of the intermediate steps? Feel free to ask!

Here are 5 related questions you might find helpful:
1. What are the assumptions made in Castigliano’s Theorem?
2. How do you determine the moments in a beam with multiple loads?
3. What is the physical significance of the strain energy in a beam?
4. Can Castigliano’s Theorem be used for deflections in frames of non-uniform sections?
5. How would adding an additional load affect the displacement at point C?

**Tip:** When solving problems with Castigliano's Theorem, always ensure that the moments in all beam segments are correctly expressed before applying the theorem.

Determine using Castigliano's method the vertical displacement of point C in the frame shown in the figure. Consider E = 29,000 ksi and I = 500 in^4.

Calculate the vertical displacement at point C in a structural frame subjected to horizontal and vertical loads using Castigliano's method. The problem involves strain energy calculations and equilibrium conditions, considering E = 29,000 ksi and I = 500 in^4.

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