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Typically partial uniformly distributed loads (u.d.l.) and uniformly varying loads (u.v.l.) over the span and a number of concentrated loads are conveniently handled using this technique. The first English language description of the method was by Macaulay . [ 1 ]
For a uniformly-distributed load, we have q ( x , y ) = q 0 {\displaystyle q(x,y)=q_{0}} The deflection of a simply-supported plate with centre ( a 2 , 0 ) {\displaystyle \left({\frac {a}{2}},0\right)} with uniformly-distributed load is given by
Boundary conditions are, however, often used to model loads depending on context; this practice being especially common in vibration analysis. By nature, the distributed load is very often represented in a piecewise manner, since in practice a load isn't typically a continuous function. Point loads can be modeled with help of the Dirac delta ...
Simply supported beam with a uniform distributed load The elastic deflection (at the midpoint C) on a beam supported by two simple supports, under a uniform load (as pictured) is given by: [ 1 ] δ C = 5 q L 4 384 E I {\displaystyle \delta _{C}={\frac {5qL^{4}}{384EI}}} where
A statically determinate beam, bending (sagging) under a uniformly distributed load. A beam is a structural element that primarily resists loads applied laterally across the beam's axis (an element designed to carry a load pushing parallel to its axis would be a strut or column).
Moments are calculated by multiplying the external vector forces (loads or reactions) by the vector distance at which they are applied. When analysing an entire element, it is sensible to calculate moments at both ends of the element, at the beginning, centre and end of any uniformly distributed loads, and directly underneath any point loads.
When an arch carries a uniformly distributed vertical load, the correct shape is a parabola. When an arch carries only its own weight, the best shape is a catenary. [3] A catenary, in blue, graphed against a parabola, in red
The two cases with distributed loads can be derived from the case with concentrated load by integration. For example, when a uniformly distributed load of intensity q {\displaystyle q} is acting on a beam, then an infinitely small part d x {\displaystyle dx} distance x {\displaystyle x} apart from the left end of this beam can be seen as being ...