Read this to become a structural engineer.

The type of beam we will be studying is the cantilever. One end is firmly fixed to a support so that it can neither rotate or move up or down. The other end of the beam is not supported in any way. A load is distributed along the top surface of the beam. The load may vary and is described by the function w(x), with units of force per length, where x is the spatial coordinate measured along the neutral axis of the beam. (The neutral axis is a reference axis which does not stretch or compress during bending.) A very common load is w=constant. Here is a sketch.

To model the behavior of the beam it is important to understand the force and couple (or torque) that the built in support must supply. As the following figure indicates, the force, which is called the shear at the wall, is the integral of w(x) from 0 to L. You can see this by thinking of w(x) dx as the force due to the load's application over an infinitesimal length in the neighborhood of x. The force supplied must equal the sum of all these infinitesimal load contributions. In the case of w=constant, the shear at the wall is just wL.

The couple or torque that the wall must supply to the beam is intended to offset the total torque produced by the load distributed over the beam. As the figure indicates, this torque, called the bending moment at the wall, is the integral of x w(x) from 0 to L. You can see this by thinking of x w(x) as the torque of the infinitesimal load w(x) dx.

We now turn our attention to answering the question, "How much does the beam bend?" What we really need is a function y(x) which describes the vertical deflection at each point x between 0 and L.We use the neutral axis as a reference as the figure indicates.

We wish to relate deflection to loading. There is a simple way to do this provided that the deflections y and slope y' is small. Here is that bending theory in a nutshell:

E I y'''' = -w(x)

The fourth derivative of the deflection is proportional to the loading at each point x. The constant term E I depends on the material the beam is made from and the shape of the cross section. For the case of the cantilever beam we have the following boundary conditions.

E I y'''(0) = V0

EI y''(0) = M0

y'(0) = 0

y(0) = 0

With this information, the beam bending problem is reduced to a problem in integral calculus. Here we can take you through a solution.