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:

   \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): \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}$$
   • For Member 2 (Diagonal, $\theta = 45^\circ$): \begin{bmatrix} 0.5 & 0.5 & -0.5 & -0.5 \\ 0.5 & 0.5 & -0.5 & -0.5 \\ -0.5 & -0.5 & 0.5 & 0.5 \\ -0.5 & -0.5 & 0.5 & 0.5 \end{bmatrix}$$ $$k_2 = 0.8485 \times 10^6 \begin{bmatrix} 0.5 & 0.5 & -0.5 & -0.5 \\ 0.5 & 0.5 & -0.5 & -0.5 \\ -0.5 & -0.5 & 0.5 & 0.5 \\ -0.5 & -0.5 & 0.5 & 0.5 \end{bmatrix}$$

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!

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.