The field of bending theory contains the subfield of the bending equation. According to this hypothesis, a beam will bend when a force is applied to a point along the beam’s longitude axis. Hence, bending or flexural theory pertains to studying axial deformation generated by such forces.
Bending moment: Equation
A bending stress equation, often known as a bending equation, is a mathematical formula used to calculate the force exerted on a beam during a bending operation. Nevertheless, one must consider a set of assumptions imposed by the bending moment equation to get accurate data about flexural stresses.
Bending moment: Key concepts
The greatest bending moment occurs at the point along the beam when the shear force reverses direction. The point of contra flexure is determined by locating the location at which the bending moment changes. It is said that a pure bending span is one in which the shear force value is 0 and the bending moment is constant.
Bending moment: Assumptions
- You can’t have a crooked beam. Moreover, its cross-section must be consistent and free of distortions.
- The beam must be constructed from a single, consistent material. It also has to be longitudinally symmetrical.
- As the bending moment equation is derived from the location of the applied load, it follows that the load must be centred along the member’s longitudinal axis.
- The assumption that buckling and not bending would lead to failure is fundamental to the bending equation.
- The elastic limit, denoted by “E,” is the same under tension as it is under compression.
- When the object bends, the plane cross-section maintains its flat shape.
Bending moment: Formula for bending force
Calculating the bending moment requires adding the applied load to the distance away from the origin using some algebraic formula. The total applied moments from the reference point will also induce bending moments.
The bending moment formula is used in structural analysis, material strength, reinforced cement concrete and steel analysis. The bending moment is proportional to the product of the following in the bending moment equation:
M/I = f/y = E/R
Where:
M = Bending moment, I = Moment of Inertia
f = Bending stress
y = Distance of outer fibre from C.G
E = Modulus of Elasticity
R = Radius of Curvature
Bending moment: Stress-strain graph
The bending moment is determined for a wide variety of support and load configurations, including but not limited to simply supported, cantilever supported, propped cantilever supported, overhanging supported, and continuously supported, and for a variety of load configurations, including point load, uniform load, gradually varied load, and direct moment. The bending moment of a structural part is calculated by multiplying the applied load or force by the span distance from a fixed point.
- In the area of the stress-strain graph where Hooke’s Law holds, the ratio of stress to strain remains proportional to the strain across the whole graph, and this region is known as the proportional limit. Young’s modulus is the name given to this fixed numerical number.
- The stress-strain curve’s “Breaking Point,” also known as the “Fracture or Breaking Point,” indicates where a material fails catastrophically, resulting in a break.
- When a load is removed from a material, the material will return to its original position up to a point on the stress-strain graph known as the elastic limit. Once a certain elastic limit is passed, plastic deformation begins to set in.
- The plastic deformation of a material begins at a point called the yield point on a Stress-strain diagram. When the yield point is exceeded, plastic deformation is irreversible.
- When a load is removed from a material, the material will return to its original position up to a point on the stress-strain graph known as the elastic limit. Once a certain elastic limit is passed, plastic deformation begins to set in.
- A single point on the stress-strain diagram, called the Ultimate Stress Point, represents the stress at which a material fails irreversibly.
Bending moment: Standard cases
| 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 is 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 an 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 centre = WL/8
At support = WL/8 |
| 9. | A fixed beam that carries a U.D.L N/m at span L. | At centre = 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 |
FAQs
What causes the effect of bending moment on structural members?
The bending moment is the resistance to rotation of a member caused by applying load to the structure to the distance from the reference point. The bending moment is affected by the force of the load, the type of the load, the distance from the reference point, and the type of support.
What is a bending moment capacity?
Moment capacity is the maximum amount of bending an element can handle before it breaks.