Simply supported beam—second-order ODE
May 19, 2026 at 7:52 AM by Dr. Drang
Here’s the first of the derivations for the center deflection of a simply supported beam with a uniform load.
We start with the differential relationship between the bending moment, M, and the deflection, y:
The second derivative of y is the curvature of the beam (for small deflections, which is one of the fundamental assumptions of beam theory), and the negative sign is there to account for the usual sign conventions for moment and displacement.
M is a parabola that passes through 0 at each end of the beam and peaks at at the center. Its formula is
Therefore,
where I’ve started using primes for differentiation.
Integrating once gives
Symmetry tells us the slope at the center of the beam is zero, so
Which means
Plugging this result in and integrating again gives us
Because the deflection is zero at ,
and the deflection at the center of the beam is
which is the answer we were expecting.