Example 1.2 Using the slope-deflection method, determine the end moments and the reactions at the supports of the beam shown in Figure 1.6 and draw the shearing force and the bending moment diagrams. The Modulus of elasticity and moment of inertia of the beam are 29000 ksi and $$ 8000 \mathrm{In}^{4} $$ respectively.

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To determine the end moments and reactions at the supports of the beam using the **slope-deflection method**, we'll follow a systematic approach. Although the specific geometry and loading details from **Figure 1.6** aren't provided, I will outline the general steps involved in the slope-deflection method. You can apply these steps to your specific beam configuration.

### **Given:**
- **Modulus of Elasticity (E)** = 29,000 ksi
- **Moment of Inertia (I)** = 8,000 in⁴

### **Steps Using the Slope-Deflection Method:**

1. **Identify Beam Elements and Supports:**
   - Divide the beam into spans between supports.
   - Identify the type of supports (fixed, pinned, etc.).
   - Determine the length of each span (L).

2. **Assume End Rotations:**
   - Assign rotation variables (θ) at each support (both fixed and free).
   - For example, if the beam has two spans with three supports (A, B, C), assign θ_A, θ_B, and θ_C.

3. **Apply Slope-Deflection Equations:**
   - For each beam element, write the slope-deflection equations at both ends.
   - The general slope-deflection equation for a beam element between supports $$ i $$ and $$ j $$ is:

$$
     M_{ij} = \frac{2EI}{L}\theta_i + \frac{EI}{L}\theta_j - \frac{6EI}{L^2}\delta_i
     $$
     $$
     M_{ji} = \frac{EI}{L}\theta_i + \frac{2EI}{L}\theta_j - \frac{6EI}{L^2}\delta_j
     $$
     
     Where:
     - $$ M_{ij} $$ = Moment at end $$ i $$ due to end $$ j $$
     - $$ \theta_i, \theta_j $$ = Rotations at ends $$ i $$ and $$ j $$
     - $$ \delta_i, \delta_j $$ = Displacements (typically zero for fixed supports)

4. **Apply Boundary Conditions:**
   - For fixed supports, rotations are typically zero ($$ \theta = 0 $$).
   - For other support types (pinned, roller), apply appropriate conditions based on continuity and equilibrium.

5. **Set Up Equilibrium Equations:**
   - Ensure that at each internal support, the sum of moments equals zero.
   - This will create a system of equations based on the slope-deflection equations.

6. **Solve for Rotations ($$ \theta $$):**
   - Solve the system of equations derived from the slope-deflection equations to find the unknown rotations at each support.

7. **Calculate End Moments:**
   - Once the rotations are known, substitute them back into the slope-deflection equations to find the end moments at each support.

8. **Determine Reactions at Supports:**
   - Use equilibrium equations (sum of forces and moments) to find the reactions at the supports based on the calculated moments and applied loads.

9. **Draw Shear Force and Bending Moment Diagrams:**
   - **Shear Force Diagram (SFD):**
     - Start from the left end of the beam.
     - Plot the shear force values, considering the loads and reactions.
     - Ensure that the areas under the shear force diagram correspond to the bending moments.
   
   - **Bending Moment Diagram (BMD):**
     - Integrate the shear force diagram to plot the bending moments.
     - Reflect the end moments and ensure continuity where the moments match the support moments.

### **Example Outline:**

Assuming a simple continuous beam with two spans (A-B and B-C), fixed at both ends (A and C) and support at B:

1. **Assign Rotations:**
   - $$ \theta_A = 0 $$ (fixed support)
   - $$ \theta_C = 0 $$ (fixed support)
   - $$ \theta_B $$ = Unknown rotation at internal support

2. **Apply Slope-Deflection Equations to Both Spans:**
   - **Span A-B:**
     $$
     M_{AB} = \frac{2EI}{L}(0) + \frac{EI}{L}\theta_B - \frac{6EI}{L^2}(0) = \frac{EI}{L}\theta_B
     $$
     $$
     M_{BA} = \frac{EI}{L}(0) + \frac{2EI}{L}\theta_B - \frac{6EI}{L^2}\theta_A = \frac{2EI}{L}\theta_B
     $$
   
   - **Span B-C:**
     $$
     M_{BC} = \frac{2EI}{L}\theta_B + \frac{EI}{L}(0) - \frac{6EI}{L^2}(0) = \frac{2EI}{L}\theta_B
     $$
     $$
     M_{CB} = \frac{EI}{L}\theta_B + \frac{2EI}{L}(0) - \frac{6EI}{L^2}\theta_C = \frac{EI}{L}\theta_B
     $$

3. **Apply Moment Equilibrium at Support B:**
   - Sum of moments at B: $$ M_{BA} + M_{BC} = 0 $$
   - Substitute the expressions:
     $$
     \frac{2EI}{L}\theta_B + \frac{2EI}{L}\theta_B = 0
     $$
     $$
     \frac{4EI}{L}\theta_B = 0 \Rightarrow \theta_B = 0
     $$

4. **Calculate End Moments:**
   - Substitute $$ \theta_B = 0 $$ back into the moment equations:
     $$
     M_{AB} = \frac{EI}{L}(0) = 0
     $$
     $$
     M_{BA} = \frac{2EI}{L}(0) = 0
     $$
     $$
     M_{BC} = \frac{2EI}{L}(0) = 0
     $$
     $$
     M_{CB} = \frac{EI}{L}(0) = 0
     $$
   
   (In this simplified example, all end moments are zero, which would imply no fixed moments at the supports. This result depends heavily on the assumed loading and support conditions.)

5. **Determine Reactions:**
   - Using equilibrium (sum of vertical forces = 0), calculate the reactions based on applied loads and end moments.

6. **Draw SFD and BMD:**
   - Illustrate how shear forces and bending moments change along the length of the beam based on the reactions and moments.

### **Important Notes:**

- **Units Consistency:** Ensure all units are consistent. Here, E is in ksi, I in in⁴, so lengths (L) should be in inches.
- **Loading Conditions:** The nature and position of applied loads significantly affect the moments and reactions. Ensure all loads are accounted for in the equilibrium equations.
- **Boundary Conditions:** Accurately apply the boundary conditions based on the type of supports.

### **Conclusion:**

The slope-deflection method is a powerful technique for analyzing indeterminate beams. By systematically applying the slope-deflection equations, considering boundary conditions, and solving for unknown rotations and moments, you can determine the end moments and support reactions. Drawing the shear force and bending moment diagrams will then provide a complete understanding of the internal forces within the beam.

If you can provide the specific details or a schematic of **Figure 1.6**, I can offer a more tailored solution with numerical results.
To determine the end moments and reactions of the beam using the slope-deflection method, follow these steps: 1. **Identify Beam Elements and Supports:** Divide the beam into spans and note the support types. 2. **Assume End Rotations:** Assign rotation variables at each support. 3. **Apply Slope-Deflection Equations:** Use the equations to relate moments and rotations. 4. **Apply Boundary Conditions:** Set rotations to zero at fixed supports. 5. **Solve for Rotations:** Solve the resulting equations to find unknown rotations. 6. **Calculate End Moments:** Substitute rotations back into the slope-deflection equations. 7. **Determine Reactions:** Use equilibrium equations to find support reactions. 8. **Draw Shear Force and Bending Moment Diagrams:** Plot the diagrams based on the calculated moments and reactions. Ensure all units are consistent and consider the specific loading and support conditions of your beam.

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