To calculate the deflection at points B and C using the Area Moment Method, we will break the problem into steps. Here's how we proceed:

Step 1: Analyze the beam

The beam has:

• A moment of $5 \, \text{kN-m}$ at point B.
• A point load of $8 \, \text{kN}$ at point C.
• Supports at points A and D.

The distances are:

• $AB = 3 \, \text{m}$
• $BC = 3 \, \text{m}$
• $CD = 3 \, \text{m}$

Reactions at supports:

• Reaction at A, $R_A = 2.11 \, \text{kN}$
• Reaction at D, $R_D = 5.89 \, \text{kN}$

Step 2: Bending moment diagram

For this method, the bending moment diagram is essential. The bending moments can be calculated as follows:

Section AB (0 ≤ x ≤ 3):

• The bending moment at point A (x = 0) is $0$ since it's a pinned support.
• The moment at point B (x = 3 m): $M_B = R_A \times 3 - 5 = 2.11 \times 3 - 5 = 6.33 - 5 = 1.33 \, \text{kN-m}$

Section BC (3 ≤ x ≤ 6):

• At point C, just before the 8 kN force, the moment is: $M_C = M_B - 8 \times 3 = 1.33 - 24 = -22.67 \, \text{kN-m}$

Step 3: Calculate the areas under the bending moment diagram

Using the bending moment diagram, divide the diagram into two key areas:

Area between A and B:

• This is a triangle with height $M_B = 1.33 \, \text{kN-m}$ and base $3 \, \text{m}$. $A_1 = \frac{1}{2} \times 1.33 \times 3 = 1.995 \, \text{kN-m}^2$ The centroid of this triangle is $\frac{2}{3} \times 3 = 2 \, \text{m}$ from point A.

Area between B and C:

• This is a trapezoid with heights $M_B = 1.33 \, \text{kN-m}$ and $M_C = -22.67 \, \text{kN-m}$, and a base $3 \, \text{m}$. $A_2 = \frac{1}{2} \times (1.33 + (-22.67)) \times 3 = \frac{1}{2} \times (-21.34) \times 3 = -32.01 \, \text{kN-m}^2$ The centroid of this trapezoid is calculated based on the shape, but generally lies slightly closer to the larger moment (closer to C).

Step 4: Deflection calculations using the Area Moment Method

To calculate the deflection at points B and C:

• Deflection at B: Use the area $A_1$ and the centroidal distance for point B to calculate the deflection.

• Deflection at C: Use areas $A_1$ and $A_2$ along with their respective centroidal distances to calculate the deflection at C.

We also need the modulus of elasticity $E$ and the moment of inertia $I$ of the beam to finish the calculation.

Would you like to provide the values for $E$ and $I$, or should we assume standard values to continue the calculations?

Follow-up Questions:

1. What is the modulus of elasticity $E$ and moment of inertia $I$ for the beam?
2. Would you like to see the detailed bending moment diagram?
3. Do you need assistance with drawing the shear force diagram?
4. Would you like the final deflection values once we have $E$ and $I$?
5. Would you prefer this explanation in another method like the conjugate beam method?

Tip: In structural analysis, always confirm boundary conditions and load distributions before starting calculations.