Bending Stress and the Flexure Formula

Bending Stress and the Flexure Formula

Imagine taking a simple, straight rubber eraser and bending it with your fingers. As you apply that force, the top surface visibly squishes together, while the bottom surface stretches out. This simple observation is the key to understanding bending stress in engineering.

Let's look closely at a segment of this bent beam. The top fibers are in compression, getting shorter. The bottom fibers are in tension, getting longer. But notice the magic in the middle: there is a line that neither stretches nor compresses. We call this the Neutral Axis.

Because the deformation changes smoothly from squished at the top to stretched at the bottom, the internal stress does too. It is zero at the neutral axis and reaches its maximum values at the very outer edges of the beam.

Because stress is directly proportional to strain within the elastic range, the linear strain profile we found earlier translates directly into a linear bending stress distribution. At the exact center of this transition sits the neutral axis, where the stress is precisely zero.

Now let's look at the zones created by this boundary. Above the neutral axis, the material is squeezed together in compression, shown by these forces pointing inward. Below the neutral axis, the fibers are pulled apart in tension, with forces pointing outward. The maximum stresses occur at the absolute outermost fibers, farthest from the neutral axis.

For a beam subjected to pure bending, the internal forces must balance to zero. Because of this force equilibrium, the neutral axis must pass directly through the centroid of the beam's cross-sectional area. No matter how complex the shape, find the centroid, and you have found your neutral axis.

Now, let's tie the bending stress directly to the external bending moment applied to the beam. Imagine looking at the cross-section of our beam. At any distance y from the neutral axis, there is a tiny area dA experiencing a stress sigma. This stress creates a tiny force dF equal to sigma times dA. Because this force acts at a distance y, it creates a tiny internal moment dM equal to y times dF.

To find the total internal moment M, we integrate this effect over the entire cross-section. Substituting our linear stress relationship, sigma equals y over c times sigma max, we get this integral. Notice that the maximum stress and the distance c are constants for a given state of bending. We can pull them out of the integral! What remains inside is the integral of y squared dA. This geometric property is the Area Moment of Inertia, represented by I.

By substituting I back into our moment equation, we get M equals sigma max times I over c. Rearranging this gives us the famous flexure formula for the maximum bending stress: sigma max equals M c over I. Or, for any point at a distance y from the neutral axis, the stress is simply M y over I. This elegant equation tells us that to minimize stress, we need a cross-section with a large moment of inertia, which is why structural beams are shaped like I-beams!