We are given a simply supported beam AB of length $L$, carrying a point load $W$ at a distance $a$ from one end and $b$ from the other end ($a + b = L$). We need to find the expression for the total strain energy of the beam and the deflection under the load.

1. Problem Description

We are given a simply supported beam AB of length $L$, carrying a point load $W$ at a distance $a$ from one end and $b$ from the other end ($a + b = L$). We need to find the expression for the total strain energy of the beam and the deflection under the load.

2. Solution Steps

First, let's find the reactions at the supports. Let $R_A$ be the reaction at support A and $R_B$ be the reaction at support B.

Taking moments about point B:

$R_A * L = W * b$

$R_A = (W * b) / L$

Taking moments about point A:

$R_B * L = W * a$

$R_B = (W * a) / L$

Now, let's consider the bending moment in the two sections of the beam:

Section 1 (0 <= x <= a):

$M_1(x) = R_A * x = (W * b * x) / L$

Section 2 (0 <= x' <= b):

$M_2(x') = R_B * x' = (W * a * x') / L$

The strain energy $U$ is given by:

$U = \int (M^2 / (2EI)) dx$

Where $E$ is the Young's modulus and $I$ is the moment of inertia.

So, the total strain energy is:

$U = \int_0^a (M_1(x)^2 / (2EI)) dx + \int_0^b (M_2(x')^2 / (2EI)) dx'$

$U = \int_0^a (((W * b * x) / L)^2 / (2EI)) dx + \int_0^b (((W * a * x') / L)^2 / (2EI)) dx'$

$U = (W^2 * b^2 / (2EI * L^2)) \int_0^a x^2 dx + (W^2 * a^2 / (2EI * L^2)) \int_0^b x'^2 dx'$

$U = (W^2 * b^2 / (2EI * L^2)) [x^3 / 3]_0^a + (W^2 * a^2 / (2EI * L^2)) [x'^3 / 3]_0^b$

$U = (W^2 * b^2 * a^3) / (6EI * L^2) + (W^2 * a^2 * b^3) / (6EI * L^2)$

$U = (W^2 * a^2 * b^2 * (a + b)) / (6EI * L^2)$

Since $a + b = L$:

$U = (W^2 * a^2 * b^2) / (6EI * L)$

To find the deflection under the load, we can use Castigliano's theorem:

$\delta = \partial U / \partial W$

$\delta = \partial / \partial W [(W^2 * a^2 * b^2) / (6EI * L)]$

$\delta = (2 * W * a^2 * b^2) / (6EI * L)$

$\delta = (W * a^2 * b^2) / (3EI * L)$

3. Final Answer

The total strain energy of the beam is: $U = (W^2 a^2 b^2) / (6 E I L)$

The deflection under the load is: $\delta = (W a^2 b^2) / (3 E I L)$