Bending Moment Formula
The bending moment is the algebraic sum of the applied load to the given distance from the reference point. Bending moments will also be caused due to the sum of applied moments from the reference point. Generally, a moment that will cause the bending in a structural member is known as a bending moment.
In the strength of the material, structural analysis, reinforced cement concrete, and steel analysis all use the bending moment equation. In the bending moment equation bending moment has a relationship with:
M/I = f/y = E/R
- M = Bending moment, I = Moment of Inertia
- f = Bending Stress
- y = Distance of outer fiber from C.G
- E = Modulus of Elasticity
- R = Radius of Curvature
Bending Moment Diagram
The bending moment calculation is done for different types of support conditions as simply supported, cantilever support, propped cantilever support, overhanging support or continuous support with different load combinations as point load, uniformly load, gradually varied load, or direct moment. The basic principle of determination of bending moment is the applied load or force is multiplied by the distance from the reference point on the span of the structural member.
Bending Moment Examples
A simply supported beam has a point load at mid-span. The bending moment can be determined as a central point load of P is divided by the support as 0.5P; this value is known as the reaction of support. We know the length of the span is L. We know that the value of bending moment is zero as pinned support or simply support. We have got the maximum bending moment at the center of the span. Hence this way, the reaction of support is multiplied by the half length of the span. In this way, I got the bending moment at the center as 0.25PL. As per the diagram, we can see this-
Standard Cases of Bending Moment
We know the bending moment can be derived from applied load and given distance. Generally, some cases are derived for standardization, these standard cases are created as reference cases and we can design the structural member by just rearranging the load condition and span dimensions. Some of these are
Maximum Bending Moment
If a simply supported beam carrying a central point load as W with span L.
If a simply supported beam carrying a uniformly distributed load as W (N/m) with span L.
If a simply supported beam carrying a uniformly distributed load as W (KN) with span L.
If an eccentric point load at a beam at “a” distance from left support and “b” distance from right support when the total length of the span is L.
If a simply supported beam with L span carries two point load L/3 distance away from both supports.
If a simply supported beam carrying a moment M at “a” distance from left-hand support or “b” distance away from right-hand support with span L of member
Positive maximum moment = +Mb/L
Negative maximum moment = -Mb/L
A simply supported beam carrying U.V.L at zero on left support and W on right support with span L.
A fixed beam that carries a central point load as W.
At center = WL/8
At support = WL/8
A fixed beam that carries a U.D.L N/m at span L.
At center = WL/12xL/2
At support = WL/6xL/2
A fixed beam that carries U.V.L with intensity zero to W
At zero load support = WL/15xL/2
At W load support = WL/10xL/2
A cantilever beam carrying a point load on the free end with span L.
Some Important Definitions
- At a point on a beam where shear force changes its sign, the point is known as the maximum bending moment position.
- At a point where the bending moment change sign is known as the point of contra flexure.
- At a span where the shear force value is completely zero and the bending moment constant is called a pure bending span.