# Structural Design (Beams)

1. Apr 6, 2009

### clairepearl

Hi,
I am only new to this forum so any feedback would be greatly appreciated.
I am wondering if anyone could help me derive the formula for a simply supported beam with a uniformly distributed load. I understand that it is done using integration but I fail to understand the steps involved. The end formula required is that of

Actual Deflection = 5 x WL3(where the L is cubed)
...........................................(384)EI

Claire

2. Apr 15, 2009

### AppleBite

Hi Claire,

sure you have the correct formula for deflection? My notes asy L^4 not L^3....
Anyways, the expression of moment at any part of the beam is

M = (w*L*x)/2 - (w*x^2)/2 as moment is the integration of the shear force (look at the bending moment diagram compared to the shear force diagram) and x is the variable distance from one of the supports

Now, integrate the rotation of the beam as R = integration(M/EI) dx
and integrate this once more to find the deflection, as you would know from definition.

To work out the integration constants that you get from each integration, consider the boundary conditions for the beam, ie where both the deflection v and variable distance x is 0, but also where x=L (L= full length of the beam)

This will give you the full expression for the deflection at any point on the beam.

Now to find the maximum deflection just set x=L/2 , ie half-way along the beam.

Hope this helps!

You really just have to go through the integration following the steps I've provided in order to fully understand what is going on.

A