A cantilever beam, made of a material with a Young's modulus, E with a rectangular cross section with a moment of inertia I, is under a uniformly distributed load, q, across the length L of the cantilever at a point load P at the free end of the cantilever.

What is the maximum deflection of the beam? Express your answer in terms of q,E,I,L,P.

-L^3*(8*P+3*q*L)/(24*E*I)

To determine the maximum deflection of the cantilever beam, we can use the Euler-Bernoulli beam theory. This theory states that the maximum deflection occurs at the fixed end of a cantilever beam and can be calculated using the formula:

δ_max = (qL^4 / (8EI)) + (PL^3 / (3EI))

where:
δ_max = maximum deflection
q = uniformly distributed load
E = Young's modulus (a measure of the stiffness of the material)
I = moment of inertia of the beam's cross-section
L = length of the cantilever
P = point load at the free end of the cantilever

Now, let's break down the steps to calculate the maximum deflection:

1. Determine the value of the moment of inertia (I):
- The moment of inertia depends on the cross-sectional shape of the beam. For a rectangular cross-section, the moment of inertia (I) can be calculated using the formula I = (b * h^3) / 12, where "b" is the width of the beam and "h" is its height.

2. Plug in the values:
- Substitute the given values of q, E, I, and L into the formula for the maximum deflection.

3. Calculate the maximum deflection (δ_max):
- Simplify the equation with the given values and perform the necessary mathematical operations to find the maximum deflection.

Therefore, by following these steps and using the given values of q, E, I, L, and P, you can calculate the maximum deflection (δ_max) of the cantilever beam.

-L^3*(8*P+3*q*L)/24

P*L^3/(3*E*I)+q*L^4/(6*E*I)