Define bending moment diagram (BMD).

Short Answer:

The bending moment diagram (BMD) is a graphical representation that shows how the bending moment varies along the length of a beam when it is subjected to external loads. It helps identify the points of maximum and minimum bending moments, which are crucial for beam design and safety.

In simple words, a BMD shows how much bending effect (moment) each part of the beam experiences due to the applied loads. Engineers use the bending moment diagram to ensure that the beam’s material and shape can safely resist the bending stresses developed in it.

Bending Moment Diagram (BMD)

Detailed Explanation :

The bending moment diagram (BMD) is one of the most essential tools in the structural analysis of beams. It visually represents the variation of bending moments at different sections of a beam due to applied loads, reactions, or couples.

Every beam under transverse loading develops shear forces and bending moments internally to maintain equilibrium. The bending moment causes the beam to bend or flex, while the shear force causes it to slide within itself. To understand the distribution of bending effects, engineers draw a bending moment diagram, which helps determine the strength requirements of the beam.

Definition of Bending Moment Diagram (BMD)

The bending moment diagram (BMD) can be defined as:

“A diagram which represents the variation of bending moment along the length of a beam under the action of external loads.”

In other words, it is a graphical plot of the bending moment (on the vertical axis) against the beam length (on the horizontal axis).

The bending moment at a section of a beam is the algebraic sum of moments of all forces acting on one side of that section about the section itself. The value of the bending moment depends on the type of load applied and the position of the section along the beam.

Concept of Bending Moment in a Beam

When a beam is subjected to transverse loads, it tends to bend. The internal resistance developed within the beam to counteract this bending is called the bending moment.

At any section of a beam, the bending moment  is given by:

If the beam bends concave upward (tension at the bottom), the bending moment is positive; if it bends concave downward (tension at the top), it is negative.

This internal moment is what causes bending stress in the beam material, which must be kept below the allowable limit for safe design.

Purpose of Bending Moment Diagram

  1. To show how bending moments vary along the beam.
  2. To determine the maximum bending moment and its location, which is critical for beam strength.
  3. To find the points of contraflexure (where bending moment changes sign).
  4. To aid in designing the beam cross-section for safe bending stress limits.
  5. To relate bending moments with shear forces for complete structural analysis.

Hence, the bending moment diagram is an important graphical tool used by engineers in designing and checking the structural safety of beams.

Procedure to Draw a Bending Moment Diagram

The steps to draw a BMD are as follows:

  1. Calculate Reactions at Supports:
    First, use static equilibrium equations ( and ) to find the support reactions.
  2. Find Shear Force at Different Sections:
    Determine the shear force at various sections of the beam by summing vertical loads.
  3. Locate Points Where Shear Force is Zero:
    The bending moment is maximum or minimum where the shear force equals zero.
  4. Calculate Bending Moment at Key Points:
    Compute the bending moment at supports, under loads, and at critical points using .
  5. Plot the BMD:
    Plot the bending moment values along the beam length. The x-axis represents the beam’s length, and the y-axis represents bending moment (positive or negative).

Sign Convention for Bending Moment

To draw the bending moment diagram correctly, a standard sign convention is used:

  • Positive Bending Moment: When the beam bends in a sagging shape (concave upward), i.e., the top fibers are in compression and bottom fibers are in tension.
  • Negative Bending Moment: When the beam bends in a hogging shape (concave downward), i.e., the top fibers are in tension and bottom fibers are in compression.

The positive bending moment is plotted above the baseline, while the negative moment is plotted below it.

Nature of Bending Moment Diagram for Different Types of Loads

  1. For a Point Load:
    The bending moment varies linearly between two loads or reactions. The BMD consists of straight lines connecting the moments at key points.
  2. For a Uniformly Distributed Load (UDL):
    The bending moment varies parabolically along the loaded length. The BMD becomes a curved line (parabola).
  3. For a Uniformly Varying Load (UVL):
    The bending moment varies as a cubic curve, resulting in a smooth, nonlinear diagram.
  4. For No Load Section:
    The bending moment remains constant; hence, the BMD is a horizontal line.

Relation Between Shear Force Diagram (SFD) and Bending Moment Diagram (BMD)

The shear force and bending moment are mathematically related as:

This means:

  • The slope of the bending moment diagram at any point equals the shear force at that point.
  • Where the shear force is zero, the bending moment is maximum or minimum.
  • The area under the shear force diagram between two points gives the change in bending moment between those points.

Thus, both SFD and BMD are drawn together for a complete beam analysis.

Example (Conceptual Understanding)

For a simply supported beam carrying a single central point load:

  • The shear force changes sign at the midpoint.
  • The bending moment increases linearly from the supports to the center and decreases symmetrically to the other end.
  • The maximum bending moment occurs at the midpoint and equals:

where  is the load and  is the span of the beam.

  • The BMD is a triangle (or a straight-line peak) with maximum value at the center.

Importance of Bending Moment Diagram

  1. Helps locate points of maximum stress in beams.
  2. Determines section size and material requirements for safe design.
  3. Identifies points of contraflexure (where bending moment changes sign).
  4. Used to check deflection and structural safety.
  5. Forms the basis for designing reinforcement in beams (for reinforced concrete or steel).
Conclusion

In conclusion, the bending moment diagram (BMD) is a graphical representation that shows the variation of bending moment along the beam’s length. It provides crucial information about the points of maximum bending stress and helps ensure that beams are designed to resist the applied loads safely. The BMD, along with the shear force diagram (SFD), forms the foundation for structural analysis, ensuring the beam’s stability and strength under different loading conditions.