Deflection of beams

Strength is not the only requirement for a beam; it must also not sag too much. Excess deflection cracks partitions and finishes, ponds water on roofs, makes floors feel bouncy and jams doors and windows. Codes therefore set limits such as span divided by a number, and designers calculate deflections or use span-to-depth rules.

The elastic curve follows from bending theory: E I d²y/dx² = M(x). Integrate the bending moment once to get slope and twice to get deflection, using the support conditions to fix the constants. Macaulay's method handles several point loads in one expression. The moment-area method reads slope change as the area of the M/EI diagram and deflection as its first moment, convenient for cantilevers. Conjugate beam and unit load methods give the same answers by other routes.

A handful of standard results cover most hand checks. A simply supported beam under a uniform load w deflects 5 w L⁴ / (384 E I) at mid-span; under a central point load P, P L³ / (48 E I). A cantilever with an end load deflects P L³ / (3 E I) and with a uniform load w L⁴ / (8 E I). Deflection rises with the fourth power of span for distributed loads, so doubling span multiplies sag sixteen times. Reinforced concrete adds cracking, creep and shrinkage, which can make long-term deflection two to three times the short-term value.