Beam Load & Deflection Simulator

Why two extra diagrams

Before an engineer works out how far a beam bends, they draw two other things first: the shear force and the bending moment along its length. Shear is how hard the beam is being sliced at each point — imagine trying to cut it there with scissors. Moment is how hard it's being bent. Those two diagrams are where a beam's problems show up: the peak moment tells you where it will break, and it's almost never where you'd guess just by looking.

Notice the shear diagram steps down sharply right where the load sits, and the moment diagram peaks at that same point. Slide the load toward a support and watch the peak moment drop — that's the whole reason a shelf bracket goes near the end of a shelf rather than in the middle of the span.

The math behind it

A beam doesn't just hold a load — it bends slightly under it. How much depends on three things: how heavy the load is, where along the beam it sits, and how stiff the beam itself is. For a load centered on a simply supported beam, that relationship is:

max bending = (load × length³) ÷ (48 × stiffness)

"Stiffness" here combines two real properties engineers call EI: the material's own stiffness (steel resists bending far more than wood does, for the same shape) and the cross-section's shape (a beam standing on its tall edge resists bending much more than the same beam lying flat — which is exactly why I-beams and floor joists are built tall, not wide).

Moving the load also matters: a load near a support barely bends the beam at all, because the support is carrying it almost directly. A load in the middle of an unsupported span has the most beam to bend, so it produces the most deflection — the same reason a shelf sags most in the center, not near the brackets.

This simulator doesn't look the formula above up. It solves the beam the way an analysis tool does: the support reactions come from statics, the shear is the running total of the loads to the left of each point, and moment, slope and deflection are each built by integrating the one before it — then the far support's boundary condition pins the deflected shape down at the end. The formula is still worked out underneath, but only to check the numerical answer against an exact one. That comparison is the error figure printed under the diagrams, and it's there because a result you haven't checked isn't an answer yet.

The bend is drawn at an exaggerated scale, and the diagram says by how much. A real beam built to code bends far less than this relative to its length. Engineers judge that with the span-to-deflection ratio in the readouts — "L over" some number — and codes set limits on it that vary by country and by what the beam is holding up.