Ask Me Help Desk - How was the defelction formula for beam deflection derived?

How was the defelction formula for beam deflection derived?

Does anyone know how the deflection formula for beam delfection was derived? Was it through integration? Any ideas would really help! :)

Most of these formulae are derived using the Unit-Load method, which also use integration.

The basic equation in beam theory is:

$ \frac {d^2} {dx^2} ( EI \frac {d^2 u} {dx^2} ) = w(x) $

where u is displacement, E is Young's modulus ( a measure of how stiff the beam material is), I is the moment of inertia of the beam cross-section, and w(x) is the load applied to the beam. So yes, integration is definitely involved. For a simple point load P at the end of a cantilevered beam of length L (like a person standing on the end of a diving board), this collapses to

$ EI ( \frac {d^2 u} {dx^2} ) = PL $

And the deflection out at the end of the diving board is:

$ u(L) = \frac {PL^3} {2 EI} $

See Euler-Bernoulli beam equation - Wikipedia, the free encyclopedia for a primer on beam theory.