Short Answer:

Shear force is the internal force developed along a beam when external loads act perpendicular to its length. It represents the tendency of one section of the beam to slide or shear over the adjacent section. Shear force changes the shape of the beam but not its length.

Bending moment is the internal moment (turning effect) developed in a beam due to external loads, which causes the beam to bend or curve. It represents the rotational effect of the loads about a section of the beam. Bending moment changes the curvature of the beam and induces tensile and compressive stresses.

Detailed Explanation :

Shear Force and Bending Moment

When a beam is subjected to external loads, internal forces and moments are developed within the beam to maintain equilibrium. The two most important internal quantities developed in a loaded beam are shear force and bending moment. These quantities are used to analyze the strength, safety, and stability of the beam and its ability to carry loads without failure.

Shear Force

Definition:
Shear force at any cross-section of a beam is defined as the algebraic sum of all vertical forces acting on either side of that section. It is the internal resisting force that prevents one part of the beam from sliding or shearing over the other.

If the left-hand portion of the beam is considered, then:

Shear force is denoted by F or V and is expressed in Newtons (N) or kiloNewtons (kN).

Explanation:
When an external load acts on a beam, it creates a tendency for the beam to slide internally along its cross-section. The internal resisting force developed in response to this sliding tendency is the shear force.

For example, if a beam is supported at two ends and a downward load acts at its center, the beam tries to shear at the point of loading. The internal resistance developed in the material of the beam to counter this is called shear force.

Mathematical Representation:
If is the load per unit length and is the shear force, then:

This means that the rate of change of shear force along the beam is equal to the intensity of the distributed load.

Sign Convention for Shear Force:

If the left portion of the beam tends to move upward, the shear force is positive.
If the left portion tends to move downward, the shear force is negative.

Importance of Shear Force:

Helps determine maximum shear stress in the beam.
Assists in designing beam cross-sections that can resist shearing action.
Used to draw Shear Force Diagrams (SFD) for analysis.

Bending Moment

Definition:
Bending moment at a section of a beam is the algebraic sum of moments of all the forces acting on either side of the section about that section. It represents the internal moment that resists the tendency of the beam to bend under external loading.

Mathematically,

where,

= bending moment at the section,
= load acting on the beam,
= perpendicular distance between the load and the section.

Explanation:
When a beam is subjected to loads, it bends, and the internal fibers experience tension and compression. The bending moment is the internal turning effect developed in the beam to resist this bending action.

For example, in a simply supported beam with a central load, the beam bends downward in the middle. The upper fibers are in compression, and the lower fibers are in tension due to the bending moment.

Mathematical Relation:
The rate of change of bending moment along the beam is equal to the shear force at that section:

Sign Convention for Bending Moment:

When the beam bends concave upward (sagging), the bending moment is positive.
When the beam bends concave downward (hogging), the bending moment is negative.

Importance of Bending Moment:

Determines the maximum bending stress in the beam.
Helps select suitable material and cross-section for safety and performance.
Used to draw Bending Moment Diagrams (BMD) to locate points of maximum bending.

Relationship between Shear Force and Bending Moment

The shear force and bending moment in a beam are closely related to each other. The relationships are given by the following equations:

The slope of the bending moment diagram at any point equals the shear force at that point:

The slope of the shear force diagram at any point equals the negative of the load intensity at that point:

The area under the shear force diagram between two points gives the change in bending moment between those points.

These relationships are fundamental for analyzing how beams carry and distribute loads along their length.

Shear Force and Bending Moment in Different Loading Conditions

For a Simply Supported Beam with a Point Load (W):
- Maximum shear force = at supports.
- Maximum bending moment = at midspan.
For a Beam with Uniformly Distributed Load (UDL):
- Maximum shear force = .
- Maximum bending moment = at center.
For a Cantilever Beam with Point Load at Free End:
- Shear force = constant (equal to W) along the length.
- Maximum bending moment = at fixed end.

These examples show how shear and bending vary depending on the type of beam and load.

Importance in Beam Design

Safety: The beam must be designed so that the maximum shear stress and bending stress do not exceed the material’s allowable limits.
Deflection Control: Bending moments affect deflection; hence, they are minimized for better structural performance.
Failure Prevention: Sudden shear failure or excessive bending can be avoided through proper design.
Economy: Helps select optimal beam size, material, and cross-section to carry loads efficiently.

Conclusion

In conclusion, shear force is the internal resistance developed against sliding of beam sections, while bending moment is the internal moment developed due to external loads causing bending. Both are fundamental for understanding how beams behave under loads. The shear force determines vertical equilibrium, and the bending moment controls bending stresses and deflection. Proper analysis of shear force and bending moment ensures safe, strong, and economical beam design in mechanical and structural engineering applications.