The problem asks us to determine the reactions at the supports of a given structure using Castigliano's theorem. The structure is a continuous beam with three supports A, B, and C. There is a point load of $10kN$ at a distance of $4m$ from support A. There is also a uniformly distributed load (UDL) of $2kN/m$ acting on the span BC, which is $4m$ long. The distance between A and B is $2m$.

1. Problem Description

The problem asks us to determine the reactions at the supports of a given structure using Castigliano's theorem. The structure is a continuous beam with three supports A, B, and C. There is a point load of $10kN$ at a distance of $4m$ from support A. There is also a uniformly distributed load (UDL) of $2kN/m$ acting on the span BC, which is $4m$ long. The distance between A and B is $2m$.

2. Solution Steps

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

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

Since the supports A, B, and C are fixed, the vertical displacements at these points are zero. Therefore, $\delta_A = \delta_B = \delta_C = 0$. We can apply Castigliano's theorem to find the reactions at these supports.

Let $R_A, R_B,$ and $R_C$ be the vertical reactions at supports A, B, and C, respectively.

We have a continuous beam and can use Castigliano's theorem to determine the support reactions. However, since the problem does not specify the flexural rigidity (EI), a direct numerical answer cannot be derived. The general approach would involve the following steps:

(a) Express the bending moment $M(x)$ as a function of the applied loads and support reactions. For this case, divide the beam into two segments: AB and BC.

(b) Calculate the strain energy $U$ using the bending moment:

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

(c) Apply Castigliano's theorem:

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

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

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

(d) Solve the resulting system of equations to determine $R_A, R_B,$ and $R_C$.

Without specific numerical values for EI, it is impossible to obtain numeric values for reactions.

3. Final Answer

Without the flexural rigidity (EI), a numerical answer for the reactions cannot be determined using Castigliano's theorem.

Therefore, the reactions $R_A$, $R_B$, and $R_C$ are expressed as functions of $EI$ based on the equations in Step 2(c), which cannot be simplified to numerical values without more information.

Final Answer: The reactions at the supports cannot be numerically determined without knowing the value of EI.