**Problem Statement**

A beam AB of span 11 m is fixed at A and B and carries a point load of 12 kN at a distance of 4 m from the left support A, as shown in the figure. We need to calculate the support moments at A and B using the method of consistent deformation. Assume the elastic modulus (E) to be constant.

**Solution**

To calculate the support moments at A and B using the method of consistent deformation, we follow the steps given below:

1. **Calculate the reactions at A and B**

Since the beam is fixed at both ends, the reactions at A and B can be calculated using the equations of equilibrium. Taking moments about B, we have:

12 kN * (11 m - 4 m) - RA * 11 m = 0

RA = (12 kN * 7 m) / 11 m = 7.636 kN

Similarly, taking moments about A, we have:

RA * 11 m - 12 kN * (11 m - 4 m) - RB * 11 m = 0

RB = (RA * 11 m - 12 kN * 7 m) / 11 m = -3.09 kN

Therefore, the reactions at A and B are RA = 7.636 kN and RB = -3.09 kN, respectively.

2. **Determine the slope at A**

To determine the slope at A, we consider a fictitious beam AB' with a hinge at B'. The slope at A, θA, can be calculated using the equation:

θA = -RB * L^2 / (2EI)

where L = 11 m is the span of the beam and I is the moment of inertia of the beam cross-section.

3. **Determine the displacement at A**

The displacement at A, δA, can be calculated using the equation:

δA = -RB * L^3 / (6EI)

4. **Calculate the support moments**

The support moments at A and B can be calculated using the equations:

MA = -RA * δA - RB * θA
MB = -RA * L - MA

Substituting the values of RA, RB, δA, and θA into these equations, we can calculate the support moments at A and B.

In conclusion, by following the steps mentioned above, we can calculate the support moments at A and B using the method of consistent deformation.