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The problem involves determining the vertical displacement of joint \( D \) in a structure consisting of steel members made from A-36 steel with a cross-sectional area of 400 mm².

### Steps to Solve:
1. **Calculate the forces in the truss members**:
   - First, we need to determine the forces in each truss member using the method of joints or method of sections. The forces at joints can be determined by applying equilibrium conditions:
     - \(\Sigma F_x = 0\) (sum of horizontal forces is zero)
     - \(\Sigma F_y = 0\) (sum of vertical forces is zero)

2. **Determine the elongation of each member**:
   - After finding the forces in the truss members, the elongation (or shortening) of each member can be calculated using Hooke's Law:
     \[
     \delta = \frac{PL}{AE}
     \]
     where:
     - \( P \) is the force in the member
     - \( L \) is the length of the member
     - \( A \) is the cross-sectional area of the member (400 mm²)
     - \( E \) is the Young's modulus for A-36 steel (approximately \( 200 \, \text{GPa} = 200 \times 10^9 \, \text{Pa} \))

3. **Calculate the vertical displacement**:
   - The vertical displacement of joint \( D \) is obtained by summing the contributions of the elongations of the members connected to joint \( D \), taking into account their direction and angle.

Let me calculate the forces in the truss members and then proceed to compute the displacement of joint \( D \).The length of member \( CD \) is calculated as 2.5 meters using the Pythagorean theorem.

Next, let's proceed to determine the forces in the truss members using the method of joints. We'll start by analyzing joint \( A \) to calculate the internal forces in members \( AB \) and \( AD \).The internal forces in the truss members connected to joint \( A \) are as follows:
- The force in member \( AD \) is approximately \( -16.67 \, \text{kN} \) (negative sign indicates compression).
- The force in member \( AB \) is approximately \( -10.00 \, \text{kN} \) (also in compression).

Now, I'll calculate the force in member \( CD \) by analyzing joint \( D \), and then proceed to calculate the vertical displacement at joint \( D \).The force in member \( CD \) is approximately \( -16.67 \, \text{kN} \), which also indicates compression.

Now that we have the forces in all members, we can calculate the elongation (or compression) in each member using the formula:
\[
\delta = \frac{PL}{AE}
\]
I'll compute the vertical displacement at joint \( D \) by considering the deformation of members \( AD \) and \( CD \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to discuss the previous steps or ask for further clarification? Here are some related questions to explore:

1. How would you calculate the angle between members in other types of trusses?
2. What role does Young's modulus play in determining structural deformation?
3. How would the displacement change if the material had a different cross-sectional area?
4. What if there were additional horizontal forces acting on the truss?
5. How can you use the method of sections for more complex trusses?

**Tip:** Always start with calculating the internal forces using static equilibrium before determining elongation or deformation in members.

Determine the vertical displacement of joint D. Each A-36 steel member has a cross-sectional area of 400 mm².

Calculate the vertical displacement of joint D in a truss structure made from A-36 steel with a cross-sectional area of 400 mm². This problem involves applying static equilibrium conditions, determining forces in truss members, and calculating deformation using Hooke's Law. Suitable for college-level students studying structural mechanics.

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