25.3 : 荷重分布からの弾性曲線

When a beam carries a distributed load, the shear force and bending moment at any point on the beam can be expressed in a differential form.

A third-order linear differential equation is created by further differentiating the expression and assuming the flexural rigidity to be constant.

Differentiating once again, results in a fourth-order linear differential equation. This equation essentially governs the shape or elastic curve the beam will take when supporting a distributed load.

By multiplying the fourth-order linear differential equation by the flexural rigidity constant and integrating it four times, an expression for the curve can be obtained.

The constants in the equation defining the beam's curve are dictated by end conditions, such as tilt or deflection from supports and whether the shear force and bending moment are null at a cantilever beam's free end or only the bending moment is zero at both ends of a supported beam.

This technique accurately calculates curves for cantilevered or supported beams with distributed loads. However, overhanging beams' support reactions cause shear irregularities, requiring distinct functions for the full-length curve definition.