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### **Problem 3: Structural Analysis - Reactions, Shear Force Diagram (SFD), and Bending Moment Diagram (BMD)**

#### **Given:**
1. Frame with a pin connection at A and C, hinge at B, and a rigid connection at E.
2. Distributed load of \(2 \, \text{kips/ft}\) from D to E over a length of 4 feet.
3. Point load of \(20 \, \text{kips}\) at E.
4. Point load of \(10 \, \text{kips}\) at B.
5. Lengths: AB = 3 ft, BC = 10 ft, BD = 3 ft, DE = 4 ft.

#### **Steps to Solve:**
1. **Determine Reactions at Supports:**
   - Apply equilibrium equations: \(\sum F_x = 0\), \(\sum F_y = 0\), and \(\sum M = 0\) to find reactions at points A, C, and E.
2. **Construct the Shear Force Diagram (SFD):**
   - Compute shear force at critical points (A, B, D, E) considering the point loads and distributed load.
3. **Construct the Bending Moment Diagram (BMD):**
   - Calculate bending moments at critical points and plot the bending moment distribution along the beam.

### **Problem 4: Truss Analysis - Horizontal Displacement of Joint B**

#### **Given:**
1. Modulus of Elasticity, \(E = 30,000 \, \text{kips/in}^2\).
2. Area of bars AD and BC = 5 in², all other bars = 4 in².
3. Loading: External loads and bar forces are specified at points A, B, C, and D.

#### **Steps to Solve:**
1. **Use the Method of Virtual Work for Trusses:**
   - To find the horizontal displacement \(\delta_B\) at joint B, use the principle of virtual work.
2. **Calculate the Internal Forces in Each Member:**
   - Determine the internal forces in each member under the given loads.
3. **Apply Virtual Displacement:**
   - Apply a virtual unit load at joint B in the horizontal direction.
   - Compute the internal virtual forces due to this virtual load.
4. **Compute the Displacement:**
   - Use the formula for deflection in trusses:  
   \[
   \delta_B = \frac{1}{AE} \sum \left( \frac{F_i \cdot f_i \cdot L_i}{A_i} \right)
   \]
   where:
   - \(F_i\) = Internal force in member \(i\) due to actual load.
   - \(f_i\) = Internal force in member \(i\) due to virtual unit load.
   - \(L_i\) = Length of member \(i\).
   - \(A_i\) = Cross-sectional area of member \(i\).

### **Would you like me to proceed with the calculations for either or both problems?**

### **Related Questions:**
1. What are the step-by-step calculations for finding the reactions in Problem 3?
2. How do you construct a Shear Force Diagram (SFD) for Problem 3?
3. What is the detailed process of determining internal member forces in Problem 4?
4. How do you apply the method of virtual work to calculate deflections in trusses?
5. What are the common mistakes to avoid when analyzing complex frames or trusses?

### **Tip:**
When solving structural problems, always start by sketching free-body diagrams and clearly identifying all forces, moments, and constraints. This will simplify setting up equilibrium equations and applying methods like virtual work.

Analyze structure below including SFD and BMD. Find all reactions at A and C. Assume the frame is pin connected at A and C. Hinge at B and Rigid connection at E.

Explore detailed structural analysis including reactions, Shear Force Diagram (SFD), Bending Moment Diagram (BMD), and the horizontal displacement of a truss joint. This advanced guide is suitable for engineering students at the university level.

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