State the relationship between load, shear force, and bending moment.

Short Answer:

The relationship between load, shear force, and bending moment explains how external loads acting on a beam create internal reactions within it. The rate of change of shear force at a point on a beam is equal to the intensity of the load at that point, and the rate of change of bending moment is equal to the shear force at that point.

In simple terms, loads cause shear forces, and shear forces cause bending moments in beams. These three quantities are interrelated and are essential for analyzing and designing safe and efficient structural members like beams, bridges, and frames.

Detailed Explanation:

Relationship between Load, Shear Force, and Bending Moment

In structural and mechanical engineering, when a beam or structural member is subjected to external transverse loads, it experiences internal reactions in the form of shear forces and bending moments. These internal forces help the beam maintain equilibrium and resist deformation or failure. The relationship between load (w), shear force (F), and bending moment (M) forms the foundation for analyzing beam behavior under different loading conditions.

Understanding this relationship helps engineers determine how loads are distributed along a beam, where maximum bending occurs, and how much strength or stiffness is required in design.

  1. Basic Concept of Relationship

When a load acts on a beam:

  • The load (w) causes a change in shear force (F) along the beam.
  • The shear force (F) causes a change in bending moment (M) along the beam.

This means that:

  • The slope of the shear force diagram (SFD) represents the load intensity (w).
  • The slope of the bending moment diagram (BMD) represents the shear force (F).

Thus, the three quantities are mathematically related by differential equations as follows:

 

These two simple relationships help in determining the shear force and bending moment for any type of load acting on a beam.

  1. Explanation of Each Relationship

(a) Relationship between Load and Shear Force

The rate of change of shear force with respect to distance along the beam is equal to the negative intensity of the load acting on it.

Mathematically,

Where:

  • F = Shear force (N or kN)
  • x = Distance along the beam (m)
  • w = Load intensity (N/m or kN/m)

Explanation:
If the load intensity w is positive (acting downward), then the shear force decreases in the direction of the beam. If the load is negative (acting upward), the shear force increases.

Example:
For a uniformly distributed load (UDL) of intensity w, the shear force decreases linearly along the beam because the load is constant.
For a point load (P), the shear force changes suddenly by an amount equal to the magnitude of that load.

Hence, the shear force diagram (SFD) can be easily drawn by integrating or differentiating the load distribution.

(b) Relationship 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:

  • M = Bending Moment (N·m or kN·m)
  • x = Distance along the beam (m)
  • F = Shear Force (N or kN)

Explanation:
If the shear force at a section is positive, the bending moment increases in the direction of the beam.
If the shear force is negative, the bending moment decreases.

In practical terms:

  • A zero shear force at any point indicates the maximum or minimum bending moment at that point.
  • The shape of the bending moment diagram (BMD) depends on the variation of shear force along the beam.

For example:

  • When the shear force is constant, the bending moment varies linearly.
  • When the shear force varies linearly (as in a UDL), the bending moment varies parabolically.
  1. Combined Relationship among Load, Shear Force, and Bending Moment

By combining the two differential relationships, we can derive a direct connection among the three quantities:

  1. Load and Shear Force:
  1. Shear Force and Bending Moment:
  1. Load and Bending Moment (Second Derivative):
    By differentiating the second equation,

Therefore,

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

These three equations are the fundamental relationships used for analyzing and plotting Shear Force Diagrams (SFD) and Bending Moment Diagrams (BMD) in structural analysis.

  1. Graphical Interpretation
  • Load (w): The load acts along the beam and can vary (uniformly or non-uniformly).
  • Shear Force (F): The slope of the shear force diagram represents the rate of change of the load.
  • Bending Moment (M): The slope of the bending moment diagram represents the shear force.

In a graphical sense:

  • A horizontal line in the load diagram indicates constant load (UDL).
  • A sloping line in the shear force diagram indicates uniform load.
  • A parabolic curve in the bending moment diagram indicates uniformly distributed load.

Thus, these diagrams provide a complete understanding of how loads affect the beam internally.

  1. Importance of the Relationship
  1. Design and Safety:
    Engineers use these relationships to calculate maximum shear and bending stresses, ensuring that beams do not fail under load.
  2. Structural Analysis:
    Helps in constructing shear force and bending moment diagrams for beams under various loading conditions.
  3. Determining Critical Sections:
    The points of maximum bending moment and zero shear force indicate critical regions that must be strengthened.
  4. Simplifies Calculations:
    The differential relationships make it easy to analyze any complex loading condition systematically.
  5. Prediction of Deflection:
    Knowing the bending moment distribution helps in determining the deflection and slope of beams.

Example

For a simply supported beam carrying a uniformly distributed load (UDL) of intensity w (kN/m) over span L (m):

  • The shear force at a distance x from the left end is:
  • The bending moment at that section is:

At the center (x = L/2):

This example clearly shows the relationship between load, shear force, and bending moment in a practical situation.

Conclusion

The relationship between load, shear force, and bending moment is fundamental in beam analysis. The load determines the variation of shear force, and shear force determines the variation of bending moment. Mathematically,

This means that the bending moment curve is the integral of the shear force curve, and the shear force curve is the integral of the load curve. Understanding this relationship allows engineers to design strong, stable, and efficient beams that can safely resist applied loads.