What is the relation between load, S.F., and B.M.?

Short Answer:

The relation between load, shear force (S.F.), and bending moment (B.M.) shows how these three quantities are connected in a loaded beam. The load acting on a beam causes a change in shear force, and the shear force in turn causes a change in bending moment. The rate of change of shear force along the beam equals the intensity of the load, and the rate of change of bending moment equals the shear force.

In simple terms, if the load increases, the shear force changes accordingly, and this variation of shear force leads to a change in the bending moment. This relation helps engineers in constructing shear force and bending moment diagrams to analyze and design beams safely.

Detailed Explanation:

Relation between Load, Shear Force, and Bending Moment

In the study of beams and their behavior under loads, the three most important terms are Load (w), Shear Force (F), and Bending Moment (M). These three are interrelated and describe how a beam resists external loads through internal reactions. Understanding their relationship helps in analyzing the internal forces and designing safe and strong beams.

When a beam is subjected to external loads, these loads create internal stresses and reactions inside the beam. These internal reactions are represented by the shear force and bending moment. The relationship between load, shear force, and bending moment can be expressed mathematically and graphically, showing how each affects the other.

  1. Relation between Load and Shear Force

The rate of change of shear force along the length of a beam is equal to the intensity of the distributed load acting on that portion of the beam.
Mathematically,

where,
= Shear Force,
= Distance along the beam,
= Intensity of load per unit length.

The negative sign indicates that the shear force decreases in the direction of increasing load.

Explanation:
If a uniformly distributed load acts on the beam, the shear force will decrease linearly along the length. If there is a concentrated load, the shear force diagram will show a sudden jump or drop at that point. Hence, the load acts as the cause, and the shear force is the effect.

Example:
If a beam is carrying a uniform load of 10 kN/m, then the shear force will decrease at the rate of 10 kN per meter along the length of the beam.

  1. Relation between Shear Force and Bending Moment

The rate of change of bending moment along the length of the beam is equal to the shear force at that section.
Mathematically,

where,
= Bending Moment,
= Shear Force.

This means that if the shear force at a section is known, the bending moment at that section can be found by integrating the shear force over the length of the beam.

Explanation:

  • When the shear force is zero, the bending moment is either maximum or minimum.
  • When the shear force is positive, the bending moment increases in the positive direction.
  • When the shear force is negative, the bending moment decreases.

Thus, shear force acts as the cause for bending moment.

Example:
If the shear force between two points on a beam is constant, the bending moment between those two points will change linearly.

  1. Combined Relationship between Load, Shear Force, and Bending Moment

By combining both relationships, we can write:

If we differentiate the second equation with respect to , we get:

Thus,

This equation shows that the second derivative of bending moment with respect to length gives the load intensity. Hence, load, shear force, and bending moment are connected by differentiation and integration.

Summary of Relationships:

  • Load (w) = Rate of change of Shear Force (F) with respect to distance (x).
  • Shear Force (F) = Rate of change of Bending Moment (M) with respect to distance (x).
  • Bending Moment (M) = Area under the Shear Force Diagram.

These relations can also be expressed in integral form as:

 

Graphical Representation:

Though we are not using diagrams here, you can visualize the relationship as follows:

  • If the loading diagram is given, the slope of the shear force diagram at any point gives the load intensity.
  • The slope of the bending moment diagram at any point gives the shear force at that point.
  • The area under the load curve gives the change in shear force, and the area under the shear force curve gives the change in bending moment.

Hence, these three diagrams—Load, Shear Force Diagram (SFD), and Bending Moment Diagram (BMD)—are directly connected to each other.

Physical Meaning of the Relation

When a load acts on a beam, it tends to bend it downward. This external load is resisted internally by the beam through shear forces and bending moments. The load causes shear deformation, and the shear deformation produces bending. Therefore, the bending moment is a result of the cumulative effect of the load acting through the shear force distribution.

This relation helps engineers in understanding how the beam behaves under different load conditions, and how to design the beam safely against bending and shear failure.

Conclusion:

The relation between load, shear force, and bending moment is very important in beam analysis. The load determines how the shear force changes, and the shear force determines how the bending moment changes. Mathematically, the load is the derivative of shear force, and the shear force is the derivative of bending moment. This relationship is the foundation for constructing shear force and bending moment diagrams, which are essential for designing strong and stable beams in mechanical and structural systems.