A beam ABC of length (l + a) has one support at the left end and the other support at a distance 1 from the left end. The beam carries a point load W at the right end. We need to find the slopes over each support and at the right end, the deflection at the right end, and the maximum deflection between the supports.


To solve this problem, we will make the following assumptions:
1. The beam is homogeneous and isotropic, meaning it has the same properties throughout its length.
2. The beam is initially straight and the deflections are small.
3. The beam is subjected to a point load at the right end only.


To find the slopes and deflections of the beam, we will use the principles of statics and the Euler-Bernoulli beam theory. The Euler-Bernoulli beam theory assumes that the beam is slender and that the deflections are small compared to the dimensions of the beam.


Since the left end of the beam is fixed, the slope at the left support (A) will be zero. This is because a fixed support prevents any rotation.


To find the slope at the right support (C), we can use the equation of the deflection curve. The deflection curve is given by the equation:

y(x) = (Wx^2 / (6EI)) * (l^2 - x^2)

where:
- y(x) is the deflection at a given point x along the beam
- W is the point load at the right end
- E is the Young's modulus of the beam material
- I is the moment of inertia of the beam cross-section
- l is the length of the beam
- x is the distance from the left end of the beam

To find the slope at the right support (C), we can differentiate the deflection curve equation with respect to x and then substitute x = l + a:

dy/dx = (Wx / (3EI)) * (l^2 - x^2)

Slope at C = dy/dx (at x = l + a)


Since the right end of the beam is free, there is no external moment acting on it. Therefore, the slope at the right end (B) will be zero.


To find the deflection at the right end (B), we can substitute x = l + a into the deflection curve equation:

Deflection at B = y(l + a) = (W(l + a)^2 / (6EI)) * (l^2 - (l + a)^2)


To find the maximum deflection between the supports (A and C), we need to find the point along the beam where the deflection is maximum. We can do this by finding the point where the slope is zero. Since we already know the slope at the left support (A) is zero, we can solve the equation dy/dx = 0 to find the maximum deflection point. Once we find this