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In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. [ 1 ] [ 2 ] The most common or simplest structural element subjected to bending moments is the beam .
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.
This results in a constant bending moment between the two supports. Consequently, a shear-free zone is created, where the specimen is subjected only to bending. This has the advantage that no additional shear force acts on the specimen, unlike in the 3-point bending test. [6] The bending modulus for a flat specimen is calculated as follows:
The deformation of the beam is described by a polynomial of third degree over a half beam (the other half being symmetrical). The bending moments (), shear forces (), and deflections for a beam subjected to a central point load and an asymmetric point load are given in the table below. [5]
where , are the coordinates of a point on the cross section at which the stress is to be determined as shown to the right, and are the bending moments about the y and z centroid axes, and are the second moments of area (distinct from moments of inertia) about the y and z axes, and is the product of moments of area. Using this equation it is ...
As a consequence the three traction components that vary from point to point in a cross-section can be replaced with a set of resultant forces and resultant moments. These are the stress resultants (also called membrane forces, shear forces, and bending moment) that may be used to determine the detailed stress state in the structural element. A ...
where a 1 is the area on the bending moment diagram due to vertical loads on AB, a 2 is the area due to loads on BC, x 1 is the distance from A to the centroid of the bending moment diagram of beam AB, x 2 is the distance from C to the centroid of the area of the bending moment diagram of beam BC.
Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam. Positive directions for forces acting on an element. For a beam with an applied weight w ( x ) {\displaystyle w(x)} , taking downward to be positive, the internal shear force is given by taking the negative ...