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Macaulay's method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams.Use of Macaulay's technique is very convenient for cases of discontinuous and/or discrete loading.
Another important class of problems involves cantilever beams. The bending moments (), shear forces (), and deflections for a cantilever beam subjected to a point load at the free end and a uniformly distributed load are given in the table below. [5]
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 ...
In the Kirchhoff–Love plate theory for plates the governing equations are [1], = and , = In expanded form, + = ; + = and + + = where () is an applied transverse load per unit area, the thickness of the plate is =, the stresses are , and
Shear and Bending moment diagram for a simply supported beam with a concentrated load at mid-span. Shear force and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear forces and bending moments at a given point of a structural element such as a beam.
generation is distributed across a vast geographical area (e.g., a country), and therefore the response of the electrical grid, itself a highly complex system, has to be taken into account: even if the production levels of all units are known, checking whether the load can be sustained and what the losses are requires highly complex power flow ...
To find the effect of a distributed load, the designer can integrate an influence line, found using a point load, over the affected distance of the structure. [5] For example, if a three-foot-long track acts between 5 feet and 8 feet along a beam, the influence line of that beam must be integrated between 5 and 8 feet.
Schematic of the loading on a plane by force P at a point (0, 0) A starting point for solving contact problems is to understand the effect of a "point-load" applied to an isotropic, homogeneous, and linear elastic half-plane, shown in the figure to the right. The problem may be either plane stress or plane strain.