Short Answer:

The slope of the bending moment diagram (BMD) represents the shear force at that particular section of a beam. In mathematical form, it is expressed as

where is the rate of change of bending moment with respect to distance along the beam, and is the shear force.

In simple words, the slope of the BMD shows how quickly the bending moment changes along the beam’s length. A steeper slope means a larger shear force, while a horizontal BMD indicates zero shear force at that section.

Slope of BMD

Detailed Explanation :

The bending moment diagram (BMD) is a graphical representation that shows how bending moments vary along the length of a beam under external loading. The slope of the BMD at any section gives important information about the shear force acting at that point. The relation between bending moment and shear force is fundamental in beam analysis and helps engineers understand the internal stress distribution within a beam.

Definition of Slope of BMD

The slope of the BMD can be defined as:

“The rate of change of bending moment along the length of a beam, which is equal to the shear force at that particular section.”

In mathematical form, this relationship is expressed as:

where,

= bending moment at the section,
= distance along the beam,
= shear force at the section.

This means that the slope of the BMD represents how rapidly the bending moment changes with distance due to the shear force acting within the beam.

Relationship Between Shear Force and Bending Moment

To understand the slope of the BMD, it is necessary to recall the basic relationship between shear force and bending moment in a beam.

If a beam is subjected to transverse loads, internal shear forces and bending moments develop to maintain equilibrium. Consider a small element of the beam of length :

The shear force on the left side of the element is .
The shear force on the right side of the element is .
The bending moment on the left face is .
The bending moment on the right face is .

By taking moments about one face of the element and neglecting higher-order small quantities, we get:

This equation proves that the slope of the BMD equals the shear force at that section.

Physical Meaning of Slope of BMD

The slope of the bending moment diagram indicates how fast or slow the bending moment is changing along the beam.

When the shear force is large, the bending moment changes rapidly, and the BMD has a steep slope.
When the shear force is small, the BMD changes slowly, and the slope is gentle.
When the shear force is zero, the slope of the BMD becomes zero, and the BMD reaches a maximum or minimum point.

Thus, by studying the slope of the BMD, one can easily identify where the maximum bending moments occur, as these correspond to locations where the shear force changes sign or becomes zero.

Nature of Slope for Different Loading Conditions

For No Load:
When no load acts between two points of a beam, the shear force remains constant. Hence,

Therefore, the BMD has a straight-line slope between these points.

For Point Load:
A point load causes a sudden change in shear force at the point of load application, but the bending moment changes linearly between loads. Thus, the slope of the BMD is constant between two loads and changes suddenly where the point load acts.
For Uniformly Distributed Load (UDL):
Under a UDL, the shear force changes linearly along the beam, so the slope of the BMD also changes gradually. The bending moment diagram becomes a parabolic curve since its slope (shear force) varies linearly.
For Uniformly Varying Load (UVL):
When the load varies linearly along the beam, the shear force changes parabolically. Hence, the slope of the BMD varies nonlinearly, and the BMD becomes a cubic curve.
At Points of Zero Shear Force:
The slope of the BMD is zero at points where the shear force is zero. These points correspond to maximum or minimum bending moments, also known as points of contraflexure.

Graphical Interpretation

When you look at a bending moment diagram:

The inclination or slope of the curve at any point indicates the magnitude and sign of shear force.
A positive slope (upward bending moment increase) corresponds to a positive shear force.
A negative slope (downward bending moment decrease) corresponds to a negative shear force.
A horizontal tangent (flat portion) indicates zero shear force, representing a peak or valley on the BMD curve.

Thus, the slope of the BMD directly provides information about the shear force distribution along the beam.

Mathematical Relationship Summary

For any beam under load:

where is the intensity of the distributed load (N/m).

From these equations, it follows that:

Load (w) is related to the rate of change of shear force.
Shear Force (V) is related to the slope of the BMD.

Therefore, knowing one quantity (such as shear or load) allows the determination of the others by integration or differentiation.

Example (Conceptual Explanation)

Consider a simply supported beam carrying a uniformly distributed load (UDL):

The shear force varies linearly from positive at one support to negative at the other.
Therefore, the slope of the BMD (which equals the shear force) changes gradually.
The BMD becomes a smooth parabolic curve, with maximum bending moment occurring at the center (where shear force = 0).

This example demonstrates how the slope of the BMD changes according to the shear force variation along the beam.

Importance of Slope of BMD

Helps locate maximum and minimum bending moments (critical for design).
Provides a direct link between shear force and bending moment distributions.
Assists in understanding beam behavior under various loading types.
Essential for constructing accurate bending moment diagrams from shear force diagrams.
Used to verify structural stability and determine safe load-carrying capacity.

Conclusion

In conclusion, the slope of the bending moment diagram (BMD) represents the shear force at any section of a beam. The relationship shows that when shear force is large, the BMD changes steeply, and when shear force is zero, the BMD becomes horizontal (maximum or minimum moment). Understanding this slope relationship is fundamental in structural analysis because it helps engineers determine where the beam will experience maximum bending and design it accordingly for safety and efficiency.