What a beam deflection curve represents
A deflection curve, also called the elastic curve, shows how much a loaded beam bends away from its original straight axis at each point along its length. Deflection is governed by the flexural rigidity EI, the product of the material's modulus of elasticity (E) and the cross section's moment of inertia (I). For a given load, a stiffer material or a deeper section reduces deflection sharply, since I grows with the cube of section depth.
Engineers care about maximum deflection because serviceability limits, not just strength, often govern design. Excessive bending can crack finishes, cause vibration, or violate code limits such as span over 360 for floors, even when the beam is nowhere near failing structurally.
Standard cases and their formulas
Common configurations have closed-form maximum deflections. A simply supported beam with a central point load P deflects PL cubed over 48EI at midspan. The same beam under a uniform load w deflects 5wL to the fourth over 384EI. A cantilever with an end point load deflects PL cubed over 3EI at the free tip, the most flexible standard case.
Deflection scales with the cube or fourth power of span length, which is why doubling a span increases bending dramatically. Superposition lets engineers add the deflections of individual loads on the same beam to find the combined elastic curve.