How Is the Deflection Formula for a Simply Supported Beam Derived?

[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


Thanks for your time

Claire

 

[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

 

[oswaler]

Hello Claire, welcome to the forum! Structural design is a fascinating field and I am happy to help with your question about simply supported beams. The formula you are looking for is known as the "Euler-Bernoulli beam equation," which is used to calculate the deflection of a beam under a uniformly distributed load. The steps involved in deriving this formula can be quite complex and involve the use of calculus, so it is understandable that it may be difficult to understand at first.

Essentially, the formula takes into account the material properties of the beam (represented by E, the modulus of elasticity, and I, the area moment of inertia), as well as the length of the beam (L) and the magnitude of the load (W). The 5/384 term is a constant that is derived from the integration process.

To understand the steps involved in deriving this formula, it may be helpful to break it down into smaller parts. Firstly, the deflection of a beam under a uniformly distributed load can be calculated by integrating the load function. This gives us the "moment equation" for the beam. Next, we use the "slope-deflection equation" to relate the moment equation to the deflection equation. Finally, by applying boundary conditions (such as the simply supported ends), we can solve for the unknown constant in the deflection equation and arrive at the final formula.

I hope this explanation helps and I encourage you to continue learning and exploring the fascinating world of structural design. Good luck with your studies!