Short Answer:

The point of contraflexure is the point along a beam where the bending moment changes its sign, that is, it changes from positive to negative or vice versa. At this point, the bending moment becomes zero. It usually occurs in a beam that is subjected to both positive and negative bending moments due to different types of loading or support conditions.

In simple terms, the point of contraflexure is the location where the beam changes its curvature—from sagging (concave upward) to hogging (concave downward). Identifying this point is very important for beam design because it helps in determining where the bending stress reverses direction.

Detailed Explanation:

Point of Contraflexure

The point of contraflexure is an important concept in the study of beams and bending moments. It refers to the location along the length of a beam where the bending moment changes its sign, passing through zero. This means that at this point, the bending moment is neither positive nor negative.

When a beam is loaded, it bends under the influence of the applied loads and reactions at the supports. The beam can bend in two ways:

1. Sagging bending moment (Positive bending moment): The beam bends in such a way that it forms a concave-up shape.
2. Hogging bending moment (Negative bending moment): The beam bends in the opposite way, forming a concave-down shape.

The point where the beam changes from sagging to hogging or vice versa is known as the point of contraflexure. This point indicates that the bending stresses on the beam change direction, and hence, it is of great importance in structural analysis and design.

Meaning and Occurrence of Point of Contraflexure

The term contraflexure means “opposite curvature.” When a beam has both positive and negative bending moments along its length, it must have a point where the bending moment is zero. This zero point marks the transition between the two curvatures.

The point of contraflexure generally occurs in beams that are continuous or overhanging, where the loading and support conditions cause moments of opposite nature. It is rarely found in simply supported beams, as such beams usually have only positive (sagging) moments.

For example:

• In a fixed beam, one part near the support experiences a negative bending moment (hogging), while the middle part experiences a positive bending moment (sagging). The point where these two regions meet (where bending moment = 0) is the point of contraflexure.

Mathematical Expression for Point of Contraflexure

The bending moment equation for a beam is derived based on the applied loads and support conditions. The point of contraflexure is determined by setting the bending moment equation equal to zero:

Solving this equation gives the location of the point of contraflexure along the beam.

Example:
Suppose the bending moment at a point along the beam is given by

To find the point of contraflexure, set :

Hence, the point of contraflexure occurs at .

This shows that the bending moment changes its sign at 5 m from one end of the beam.

Significance of Point of Contraflexure

The point of contraflexure has great importance in beam and structural design:

1. Change in curvature:
   It represents the location where the curvature of the beam changes direction — from sagging to hogging or vice versa.
2. Zero bending moment:
   At this point, the bending moment is zero, which means the internal bending stress in the beam is also zero at that exact location.
3. Design importance:
   The point of contraflexure helps engineers in designing beams efficiently. Reinforcement or additional support may be provided near this point to handle the reversal of bending stresses.
4. Safety and strength:
   Knowing the position of this point allows for accurate prediction of stress reversal, which is crucial for preventing beam failure.
5. Practical applications:
   It is important in designing bridges, frames, and other mechanical or civil structures where both positive and negative moments exist.

Graphical Representation in Bending Moment Diagram

In a bending moment diagram (BMD), the point of contraflexure is identified as the point where the curve crosses the zero line.

• When the diagram moves from above the zero line (positive) to below it (negative), it indicates a change in moment direction.
• This intersection point is called the point of contraflexure.

The graphical representation makes it easy to locate and visualize the transition between the two bending regions.

Examples of Beams Having a Point of Contraflexure

1. Fixed Beam:
   In a fixed beam carrying a central load, one part near the support is hogging, and the middle portion is sagging. The point where the bending moment curve crosses zero is the point of contraflexure.
2. Overhanging Beam:
   In an overhanging beam, the overhang portion produces a negative bending moment while the main span has a positive moment. The point separating these two regions is the point of contraflexure.
3. Continuous Beam:
   Continuous beams supported at more than two points also experience both positive and negative bending moments, leading to multiple points of contraflexure.

Conclusion:

The point of contraflexure is the location in a beam where the bending moment changes its sign, passing through zero. It indicates the point of zero bending stress and a change in curvature of the beam. This concept is essential in analyzing and designing beams because it helps determine where tension and compression regions shift within the structure. Recognizing this point ensures safe and efficient design of mechanical and structural components under various loading conditions.