The problem asks us to determine the reactions at the supports of the given structure using Castigliano's theorem. The structure is a continuous beam with supports at A, B, and C. There is a point load of 10 kN at the middle of the span AB, and a uniformly distributed load of 2 kN/m over the span BC. The span lengths are given as AB = 4 + 2 = 6 m, BC = 4 m.

1. Problem Description

The problem asks us to determine the reactions at the supports of the given structure using Castigliano's theorem. The structure is a continuous beam with supports at A, B, and C. There is a point load of 10 kN at the middle of the span AB, and a uniformly distributed load of 2 kN/m over the span BC. The span lengths are given as AB = 4 + 2 = 6 m, BC = 4 m.

2. Solution Steps

Castigliano's second theorem states that the partial derivative of the total strain energy $U$ with respect to a force $F_i$ at a point is equal to the displacement at that point in the direction of the force:

$\frac{\partial U}{\partial F_i} = \delta_i$

For reactions, since the supports are fixed, the displacement is zero. Therefore,

$\frac{\partial U}{\partial R_A} = 0$

$\frac{\partial U}{\partial R_B} = 0$

$\frac{\partial U}{\partial R_C} = 0$

The strain energy due to bending is given by:

$U = \int \frac{M^2}{2EI} dx$

However, since it's a statically indeterminate problem, we will assume one of the reactions as redundant and solve for it. Then we can determine the other reactions using equilibrium equations.

First, determine support reactions based on static equilibrium and release redundant reaction. The picture is not detailed and the support configuration at B and C are not clear to proceed and the assumption about the type of support needs to be clearly provided to solve the problem.

Let's analyze the free body diagram of the beam.

Let R_A, R_B, and R_C be the reactions at supports A, B, and C, respectively. We will use the equations of static equilibrium.

Sum of vertical forces:

$R_A + R_B + R_C - 10 - (2 * 4) = 0$

$R_A + R_B + R_C = 18$

Sum of moments about point A:

$R_B * 6 + R_C * 10 - 10 * 3 - (2 * 4) * (6+4/2) = 0$

$6R_B + 10R_C = 30 + 56$

$6R_B + 10R_C = 86$

These two equations are not sufficient to solve for three unknowns. This problem must be solved by considering the deflections, making use of Castigliano's theorem.

Without more details, it is impossible to proceed further in a meaningful way to provide an exact solution.

3. Final Answer

Due to the incomplete information provided in the image and ambiguity in the support types, a numerical final answer cannot be determined. More information about the support types (hinged, fixed, etc.) is needed to proceed with the calculation using Castigliano's theorem.