To solve this problem using the slope-deflection method, we'll go through the following steps:

1. Identify the spans and fixed supports:

   • Span AB = 5m
   • Span BC = 3m
   • Span CD = 5m
   • Supports at A and D are fixed, while B and C are roller supports.

2. Calculate fixed-end moments (FEM):

   • Fixed-end moments occur due to the distributed load on span BC.

3. Write the slope-deflection equations:

   • The slope-deflection equations for each span will consider the rotation at each node and the fixed-end moments.

4. Apply equilibrium conditions at the roller supports (B and C):

   • This will give us the equations needed to solve for the rotations at B and C.

5. Calculate moments at A, B, C, and D:

   • Use the slope-deflection equations and the calculated rotations to find the moments.

6. Draw the moment diagram:

   • Based on the calculated moments, sketch the moment diagram for the beam.

Step 1: Fixed-End Moments (FEM)

For a uniformly distributed load $w = 20 \, \text{kN/m}$ on span BC (length $L = 3 \, \text{m}$):

$\text{FEM}_{BC} = \text{FEM}_{CB} = \frac{wL^2}{12} = \frac{20 \times 3^2}{12} = 15 \, \text{kNm}$

Step 2: Slope-Deflection Equations

For spans AB, BC, and CD, the slope-deflection equations are as follows:

$M_{AB} = \frac{2EI}{L_{AB}} \theta_A + \frac{EI}{L_{AB}} \theta_B$ $M_{BA} = \frac{2EI}{L_{AB}} \theta_B + \frac{EI}{L_{AB}} \theta_A$

$M_{BC} = \frac{2EI}{L_{BC}} \theta_B + \frac{EI}{L_{BC}} \theta_C + \text{FEM}_{BC}$ $M_{CB} = \frac{2EI}{L_{BC}} \theta_C + \frac{EI}{L_{BC}} \theta_B + \text{FEM}_{CB}$

$M_{CD} = \frac{2EI}{L_{CD}} \theta_C + \frac{EI}{L_{CD}} \theta_D$ $M_{DC} = \frac{2EI}{L_{CD}} \theta_D + \frac{EI}{L_{CD}} \theta_C$

Step 3: Equilibrium Equations

At points B and C (roller supports), the sum of moments must be zero:

$M_{BA} + M_{BC} = 0$ $M_{CB} + M_{CD} = 0$

Step 4: Solving the Equations

Substitute the FEM and the given lengths into the slope-deflection equations and solve the system of linear equations for $\theta_A$, $\theta_B$, $\theta_C$, and $\theta_D$.

Step 5: Moment Diagram

After finding the moments at A, B, C, and D, plot these moments to create the moment diagram.

Calculation Execution

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