What is the point of contraflexure?

Short Answer:

The point of contraflexure is a point along a beam where the bending moment changes its sign from positive to negative or vice versa. At this point, the bending moment value is zero. It indicates a change in the curvature of the beam — from sagging (concave upward) to hogging (concave downward) or the reverse.

In simple words, it is the location in a bending beam where the nature of bending changes. The beam curves in one direction on one side of the point and in the opposite direction on the other side. The point of contraflexure is important in design because it represents a region of reduced bending stress.

Detailed Explanation:

Point of Contraflexure

In the study of bending of beams, engineers analyze how a beam bends under different types of loads such as point loads, uniformly distributed loads (UDL), and varying loads. When a beam bends, it develops internal stresses and bending moments that cause it to curve either upward or downward.

The point of contraflexure is defined as the point on the beam where the bending moment changes its sign — from positive (sagging) to negative (hogging) or vice versa. The term contraflexure literally means “opposite bending.”

At this point, the bending moment is zero, and the beam changes the direction of its curvature. This phenomenon usually occurs in beams with overhangs, fixed beams, or continuous beams, where both positive and negative bending moments exist.

Meaning of Positive and Negative Bending Moments

Before understanding the point of contraflexure completely, it is important to know what positive and negative bending moments mean:

  • Positive Bending Moment (Sagging):
    When the beam bends in a concave upward shape (like a smile), the top fibers are in compression and the bottom fibers are in tension. This is called sagging.
  • Negative Bending Moment (Hogging):
    When the beam bends in a concave downward shape (like a frown), the top fibers are in tension and the bottom fibers are in compression. This is called hogging.

The point of contraflexure is the point between these two regions where the bending moment changes from positive to negative or the other way around.

Definition

The point of contraflexure can be defined as:

“The point on the bending moment diagram where the bending moment curve crosses the zero line, indicating a change in the nature of bending.”

At this point:

Where M is the bending moment.

This point separates the portions of the beam experiencing sagging and hogging.

Location of Point of Contraflexure

The location of the point of contraflexure can be found using the bending moment equation. It is the value of x (distance along the beam) at which the bending moment M = 0.

For example:
If the bending moment equation for a beam is given as:

then to find the point of contraflexure:

Thus, the point of contraflexure occurs at .

This means that at 4 meters from the origin, the bending moment is zero, and the beam changes its curvature.

Bending Moment Diagram and Point of Contraflexure

When the bending moment diagram (BMD) is drawn for a beam, the point of contraflexure can easily be identified where the BMD curve crosses the zero line.

  • The portion above the zero line represents positive bending moment (sagging).
  • The portion below the zero line represents negative bending moment (hogging).
  • The point of intersection of the BMD with the zero line indicates the point of contraflexure.

This graphical representation helps engineers to visualize how the beam bends and where the bending nature changes.

Importance of Point of Contraflexure

  1. Design Optimization:
    The point of contraflexure divides the beam into two parts — one in sagging and the other in hogging. This helps engineers decide where reinforcement is needed in beams made of materials like reinforced concrete (RCC).
  2. Reduction in Bending Stresses:
    At this point, bending moment is zero, hence bending stresses are minimal. Designers often take advantage of this to reduce material usage without compromising strength.
  3. Structural Safety:
    Identifying this point ensures that the beam sections are strong enough to handle both positive and negative bending moments safely.
  4. Used in Continuous and Fixed Beams:
    In continuous or overhanging beams, multiple points of contraflexure may exist. Knowing their positions helps in accurate structural analysis.

Occurrence of Point of Contraflexure

The point of contraflexure does not occur in every beam. It is found only in beams where both positive and negative bending moments occur together. Common cases include:

  1. Fixed Beams: Due to the development of negative bending moments at the supports and positive moments at the mid-span.
  2. Continuous Beams: When the beam extends over multiple supports, alternate regions of sagging and hogging appear.
  3. Overhanging Beams: Where the overhanging part produces negative bending moments near supports and positive moments on the main span.

In simply supported beams under uniform loads or point loads at the center, the bending moment does not change sign — hence no point of contraflexure exists.

Example

Consider a fixed beam of span  carrying a uniformly distributed load of .

The bending moment equation for such a beam is:

To find the point of contraflexure, we set :

Solving for x gives the distance of the point of contraflexure from the left end.

This calculation helps in identifying the section of the beam where curvature changes.

Practical Applications

  1. Reinforced Concrete Design (RCC):
    In reinforced beams, the top and bottom reinforcements are provided differently based on the bending moment’s nature. Knowing the point of contraflexure helps in positioning these reinforcements correctly.
  2. Bridge and Roof Beams:
    Engineers identify points of contraflexure to strengthen critical regions and reduce unnecessary material usage.
  3. Machine Components:
    Shafts and levers under varying loads also exhibit similar points where bending direction changes, useful in mechanical design.
Conclusion

The point of contraflexure is a location on a beam where the bending moment changes sign, indicating a shift in curvature from sagging to hogging or vice versa. At this point, the bending moment equals zero, and the beam experiences minimal bending stress. It is important in the design and analysis of continuous, fixed, and overhanging beams to ensure structural strength, safety, and material economy. Identifying the point of contraflexure helps engineers optimize reinforcement and improve the performance of beams under varying loads.