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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 solid mechanics and structural engineering, section modulus is a geometric property of a given cross-section used in the design of beams or flexural members.Other geometric properties used in design include: area for tension and shear, radius of gyration for compression, and second moment of area and polar second moment of area for stiffness.
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.
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.
The second polar moment of area, also known (incorrectly, colloquially) as "polar moment of inertia" or even "moment of inertia", is a quantity used to describe resistance to torsional deformation (), in objects (or segments of an object) with an invariant cross-section and no significant warping or out-of-plane deformation. [1]
In this case, the equation governing the beam's deflection can be approximated as: = () where the second derivative of its deflected shape with respect to (being the horizontal position along the length of the beam) is interpreted as its curvature, is the Young's modulus, is the area moment of inertia of the cross-section, and is the internal ...
When a joint is released, balancing moment occurs to counterbalance the unbalanced moment. The balancing moment is initially the same as the fixed-end moment. This balancing moment is then carried over to the member's other end. The ratio of the carried-over moment at the other end to the fixed-end moment of the initial end is the carryover factor.
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.