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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 .
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.
One first-order effect is the initial deflection of the structure in reaction to the lateral load. The magnitude of the P-delta effect depends on the magnitude of this initial deflection. P-delta is a moment found by multiplying the force due to the weight of the structure and applied axial load, P, by the first-order deflection, Δ or δ.
An influence line for a given function, such as a reaction, axial force, shear force, or bending moment, is a graph that shows the variation of that function at any given point on a structure due to the application of a unit load at any point on the structure. An influence line for a function differs from a shear, axial, or bending moment diagram.
Although the moment () and displacement generally result from external loads and may vary along the length of the beam or rod, the flexural rigidity (defined as ) is a property of the beam itself and is generally constant for prismatic members. However, in cases of non-prismatic members, such as the case of the tapered beams or columns or ...
In the beam equation, the variable I represents the second moment of area or moment of inertia: it is the sum, along the axis, of dA·r 2, where r is the distance from the neutral axis and dA is a small patch of area. It measures not only the total area of the beam section, but the square of each patch's distance from the axis.
The bending stiffness is the resistance of a member against bending deflection/deformation. It is a function of the Young's modulus E {\displaystyle E} , the second moment of area I {\displaystyle I} of the beam cross-section about the axis of interest, length of the beam and beam boundary condition.
The bending moment applied to the beam also has to be specified. The rotation φ {\displaystyle \varphi } and the transverse shear force Q x {\displaystyle Q_{x}} are not specified. Clamped beams : The displacement w {\displaystyle w} and the rotation φ {\displaystyle \varphi } are specified to be zero at the clamped end.