Short Answer:

The relationship between load, shear force, and bending moment describes how external loads acting on a beam create internal reactions within it. The rate of change of shear force along the beam equals the intensity of the distributed load, and the rate of change of bending moment equals the shear force at that point.

In simple words, the load produces shear force, and the shear force produces bending moment. Hence, these three quantities are interconnected. By understanding their relationship, engineers can easily draw shear force and bending moment diagrams to analyze and design beams safely and effectively.

Detailed Explanation :

Relationship between Load, Shear Force, and Bending Moment

When a beam is subjected to external loads, internal forces and moments develop within the beam to maintain equilibrium. These internal quantities are called shear force and bending moment, and they are directly related to the type and magnitude of the applied load. The relationship between load (w), shear force (V), and bending moment (M) helps engineers analyze how forces are distributed along the length of the beam.

This relationship is fundamental to the study of strength of materials and forms the basis for drawing shear force diagrams (SFD) and bending moment diagrams (BMD). It allows engineers to locate critical points where maximum stress and bending occur, ensuring that the beam is designed for both safety and efficiency.

Relation between Load and Shear Force

Let the load acting on the beam be represented by , which is the intensity of distributed load per unit length (measured in N/m).

If we consider a small element of the beam between two sections separated by a small distance , the total load acting on this segment is .

The change in shear force (dV) over this small distance is equal to the negative of the load acting on the segment.

Mathematically,

This equation shows that the rate of change of shear force with respect to the beam length is equal to the negative intensity of the load.

Explanation:

When the load is uniformly distributed, the shear force changes linearly along the beam.
When the load is uniformly varying, the shear force changes parabolically.
When there is no load, the shear force remains constant along that length.

Example:
For a simply supported beam with a uniformly distributed load (UDL), the shear force decreases linearly from the left support to the right support because of the constant load intensity.

Relation between Shear Force and Bending Moment

Now consider the same small element of the beam having a shear force and bending moment at one end, and and at the other end.

The bending moment at any section of a beam is caused by the algebraic sum of moments of all forces acting to one side of that section.

For a small beam element of length , the relationship between bending moment and shear force can be derived as:

This means that the rate of change of bending moment along the beam is equal to the shear force at that section.

Explanation:

If the shear force is positive, the bending moment increases in the positive direction.
If the shear force is negative, the bending moment decreases in the positive direction.
When shear force is zero, the bending moment is maximum or minimum at that point.

Example:
In a simply supported beam with a central point load, the shear force changes sign at the midspan, and the bending moment is maximum at the same location.

Relation between Load and Bending Moment

The relationship between the load intensity and bending moment can be obtained by combining the two earlier equations.

From

By differentiating the second equation with respect to , we get:

Substituting :

Hence,

This means that the second derivative of bending moment with respect to the beam length is equal to the negative intensity of load.

Summary of Relationships:

→ Relation between load and shear force.
→ Relation between shear force and bending moment.
→ Relation between load and bending moment.

Graphical Interpretation

The relationship between load, shear force, and bending moment can be represented graphically using three diagrams:

Load Diagram: Represents the type of external load (point load, UDL, or varying load).
Shear Force Diagram (SFD): Represents the variation of shear force along the length of the beam.
Bending Moment Diagram (BMD): Represents the variation of bending moment along the length of the beam.

Important Points:

The slope of the shear force diagram equals the negative load intensity.
The slope of the bending moment diagram equals the shear force.
The area under the load diagram gives the change in shear force.
The area under the shear force diagram gives the change in bending moment.

Example:
If a beam is loaded with a uniformly distributed load:

The load diagram is a rectangle.
The shear force diagram is a straight line.
The bending moment diagram is a parabola.

Significance of the Relationship

Understanding the relationship between load, shear force, and bending moment is essential because it allows engineers to:

Determine the maximum bending moment and shear force in a beam.
Locate points of zero shear, where the bending moment is maximum or minimum.
Design safe and economical beams that can resist bending and shearing stresses.
Draw accurate diagrams (SFD and BMD) for analysis and design.
Predict the behavior of beams under different types of loads.

This relationship is a fundamental concept in strength of materials and structural analysis.

Conclusion

In conclusion, the relationship between load, shear force, and bending moment can be summarized as follows:

The rate of change of shear force is equal to the negative load intensity,

The rate of change of bending moment is equal to the shear force,

These equations show that load produces shear force, and shear force produces bending moment. This relationship helps engineers analyze beams, locate critical sections, and design structures that are safe, strong, and efficient under various loading conditions.