Bending Moment - Formula, Diagram

By Aina Parasher|Updated : August 7th, 2022

In the strength of material application of load, force, and stress causes the deflection in the structural member. This deflection is can be positive or negative. Generally, it is called bending, and the effect generated by the applied force is known as the bending moment. The bending moment also depends on the direction of the application of stress.

In this article, we are about to know what a bending moment is and what type of bending moment occurs in the standard case of structural cross-section or loading conditions. These standard cases are based on the structure member support type or direction of loading.

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

Where;

  • 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

S.NO.

Standard Cases

Maximum Bending Moment

1.

If a simply supported beam carrying a central point load as W with span L.

WL/4

2.

If a simply supported beam carrying a uniformly distributed load as W (N/m) with span L.

WL/4xL/2

3.

If a simply supported beam carrying a uniformly distributed load as W (KN) with span L.

WL/8

4.

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.

Wab/L

5.

If a simply supported beam with L span carries two point load L/3 distance away from both supports.

WL/3

6.

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

7.

A simply supported beam carrying U.V.L at zero on left support and W on right support with span L.

WL/6xL/2

8.

A fixed beam that carries a central point load as W.

At center = WL/8

At support = WL/8

9.

A fixed beam that carries a U.D.L N/m at span L.

At center = WL/12xL/2

At support = WL/6xL/2

10.

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

11.

A cantilever beam carrying a point load on the free end with span L.

WL

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.

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Bending Moment FAQs

  • Bending moment is the resistance against the rotation of a member caused by applying load on the structure to the distance from the reference point. The applied moment depends on intensity of the load, nature of the load and distance from the reference point and support type cause the effect on bending moment.

  • Bending moment is classified in two terms as positive and negative. When all loads deflect the structure downward it is known as a positive bending moment or sagging moment. When all loads deflect the structure in upward direction it is known as negative bending moment or hogging bending moment.

  • Shear force is the net force acting on the support due to all applied force. Shear force can be termed as V. But the bending moment is the derivative of the shear force as in terms of M. In other words, the bending moment is the dv/dx.

  • When a load is applied throughout the span of a fixed beam, bending moment changes sign from positive to negative or negative to positive. At that point where the bending moment changes, the sign is called the point of contra flexure.

  • We know as the bending moment is directly proportional to the modulus of elasticity in the bending moment equation (M/I = f/y = E/R). When a material has high deflection due to a high modulus of elasticity, the bending moment is also great for that material. Hence we can say the bending moment depends on the modulus of elasticity.

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