The analysis of simple end-loaded cantilever (Updated 18th September 2002)
Using the general Beam Bending Equation and taking moments about xx as shown in figure 1 gives the second order differential equation of the cantilever.




Figure 1. The deflection curve of the cantilever under an applied end force F[N] is shown in red. Please note y=f(x) and that arbitrary magnitudes have been chosen for the flexural rigidity [N/m^2] and beam length [m]. The end deflection has a calculated magnitude of 0.033[m].

Note that I use (x+ve, y+ve) axes in the conventional sense, and the end force F is a vector quantity denoted by a +ve magnitude in the upward direction. The resultant moment would tend to cause a +ve curvature, in this case, for the entire length of the beam. The maximum bending moment will be at the wall with a magnitude of +FL. Please note that this is not the general case. Most authors adopt a sign convention to suit the needs of their analysis. When a beam is loaded with multiple transverse forces (including distributed loads) then it becomes much more complex because the curvature will change sign at differnent portions along beam.

Integrating wrt x gives the slope equation


where A is a constant of integration yet to be determined.
Again integrating wrt to x to derive the deflection equation where B becomes the second constant of integration.


To solve the constants of integration we examine the boundary conditions. In this case we know that y=0 at x=0 and the slope dy/dx=0 at x=0. If we substitute those conditions into the deflection and slope equations respectively we can see that B=0 and A=0.
We now know the deflection curve, and we can determine the deflected distance anywhere along the beam.


For example, the end deflection of the cantilever is given by

In many mechanical design books the authors derive beam stiffness in the form F=ky where k is the spring stiffness coefficient. Simply transpose the end deflection equation into the spring equation form. Remember that y is only an arbitrary direction and you'll often see F=kx or similar.

Therefore

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