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AJ54 · Jun 14, 2008, 07:02 PM

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

JanvdW · Sep 10, 2008, 05:02 AM

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

ebaines · Sep 10, 2008, 10:08 AM

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.