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To determine the displacement of each node and the stress at each element for the given two-member truss, we need to follow these steps:

### Step-by-Step Solution

1. **Identify the Geometry and Properties of Each Member:**
   - **Member 1 (Horizontal):**
     - Length, \( L_1 = 10 \) inches
     - Cross-sectional area, \( A_1 = 0.5 \text{ in}^2 \)
     - Young's Modulus, \( E_1 = 30 \times 10^6 \text{ psi} \)
   - **Member 2 (Diagonal):**
     - Length, \( L_2 = \sqrt{10^2 + 10^2} = 10\sqrt{2} \approx 14.14 \) inches
     - Cross-sectional area, \( A_2 = 0.4 \text{ in}^2 \)
     - Young's Modulus, \( E_2 = 30 \times 10^6 \text{ psi} \)

2. **Determine the Global Stiffness Matrix:**
   Each member has a stiffness matrix, which in global coordinates can be expressed as:

   \[
   k = \frac{AE}{L} 
   \begin{bmatrix}
   c^2 & cs & -c^2 & -cs \\
   cs & s^2 & -cs & -s^2 \\
   -c^2 & -cs & c^2 & cs \\
   -cs & -s^2 & cs & s^2 
   \end{bmatrix}
   \]

   where \( c = \cos \theta \) and \( s = \sin \theta \).

   For member 1, \( \theta = 0^\circ \) and for member 2, \( \theta = 45^\circ \).

3. **Calculate the Element Stiffness Matrices:**
   - **For Member 1 (Horizontal):**
     \[
     k_1 = \frac{(0.5)(30 \times 10^6)}{10} 
     \begin{bmatrix}
     1 & 0 & -1 & 0 \\
     0 & 0 & 0 & 0 \\
     -1 & 0 & 1 & 0 \\
     0 & 0 & 0 & 0 
     \end{bmatrix}
     \]
     \[
     k_1 = 1.5 \times 10^6 
     \begin{bmatrix}
     1 & 0 & -1 & 0 \\
     0 & 0 & 0 & 0 \\
     -1 & 0 & 1 & 0 \\
     0 & 0 & 0 & 0 
     \end{bmatrix}
     \]

   - **For Member 2 (Diagonal, \( \theta = 45^\circ \)):**
     \[
     k_2 = \frac{(0.4)(30 \times 10^6)}{14.14} 
     \begin{bmatrix}
     0.5 & 0.5 & -0.5 & -0.5 \\
     0.5 & 0.5 & -0.5 & -0.5 \\
     -0.5 & -0.5 & 0.5 & 0.5 \\
     -0.5 & -0.5 & 0.5 & 0.5 
     \end{bmatrix}
     \]
     \[
     k_2 = 0.8485 \times 10^6 
     \begin{bmatrix}
     0.5 & 0.5 & -0.5 & -0.5 \\
     0.5 & 0.5 & -0.5 & -0.5 \\
     -0.5 & -0.5 & 0.5 & 0.5 \\
     -0.5 & -0.5 & 0.5 & 0.5 
     \end{bmatrix}
     \]

4. **Assemble the Global Stiffness Matrix:**
   Combine \( k_1 \) and \( k_2 \) into the global stiffness matrix, considering the degrees of freedom at each node.

5. **Apply Boundary Conditions:**
   - Node 1 and Node 3 are fixed, meaning their displacements are zero.
   - This will reduce the global stiffness matrix, leaving us with only the equations for Node 2.

6. **Solve for Displacements:**
   Use the reduced global stiffness matrix and the applied load vector to solve for the displacements at Node 2.

7. **Calculate Stresses in Each Member:**
   The stress in each member is given by:

   \[
   \sigma = \frac{E \cdot \Delta L}{L}
   \]

   where \( \Delta L \) is the change in length of each member.

I can perform the numerical calculations if you want, or provide further details on any step. Let me know how you'd like to proceed!

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**Relative Questions:**

1. How is the global stiffness matrix constructed in a truss analysis?
2. What are the boundary conditions for this truss problem?
3. How do you calculate the displacement at a node in a truss structure?
4. How is stress in a truss member related to its strain and Young's modulus?
5. Why is it important to know the cross-sectional area and material properties of truss members?

**Tip:** Always ensure you consider boundary conditions and constraints when setting up your global stiffness matrix in structural analysis problems.

For the two-member truss shown below, determine the displacement of each node and the stress at each element. Demonstrate all the steps.

Learn how to determine the displacement of each node and the stress at each element for a two-member truss system. This problem involves finite element analysis, stiffness matrices, and applying boundary conditions to find the displacement and stress. Ideal for engineering students.

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