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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.
The starting point is the relation from Euler-Bernoulli beam theory = Where is the deflection and is the bending moment. This equation [7] is simpler than the fourth-order beam equation and can be integrated twice to find if the value of as a function of is known.
The bending moment diagram and the influence line for bending moment at the centre of the left-hand span, B, are shown. In engineering, an influence line graphs the variation of a function (such as the shear, moment etc. felt in a structural member) at a specific point on a beam or truss caused by a unit load placed at any point along the ...
In the absence of a qualifier, the term bending is ambiguous because bending can occur locally in all objects. Therefore, to make the usage of the term more precise, engineers refer to a specific object such as; the bending of rods, [2] the bending of beams, [1] the bending of plates, [3] the bending of shells [2] and so on.
where is the flexural modulus (in Pa), is the second moment of area (in m 4), is the transverse displacement of the beam at x, and () is the bending moment at x. The flexural rigidity (stiffness) of the beam is therefore related to both E {\displaystyle E} , a material property, and I {\displaystyle I} , the physical geometry of the beam.
Strength depends upon material properties. The strength of a material depends on its capacity to withstand axial stress, shear stress, bending, and torsion.The strength of a material is measured in force per unit area (newtons per square millimetre or N/mm², or the equivalent megapascals or MPa in the SI system and often pounds per square inch psi in the United States Customary Units system).
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.