Beam Bending: Point of Inflexion Explained

This question concerns the behavior of a horizontal beam subjected to bending forces. Specifically, it asks about the defining characteristic of a 'point of inflexion' or 'contra flexure' along the beam.

Understanding Point of Inflexion

The point of inflexion, also known as the point of contraflexure, is a critical point on a beam where the internal bending moment ($M$) changes its curvature. In simpler terms, it's the location where the beam transitions from bending concave upwards to concave downwards, or vice versa. This transition is directly related to how the bending moment behaves at that specific location.

Bending Moment and Shear Force Relationship

In beam theory, the relationship between the shear force ($V$) and the bending moment ($M$) along the beam's length ($x$) is fundamental. The shear force at any point is the rate of change of the bending moment with respect to the position along the beam. Mathematically, this is expressed as:

$$ V = \frac{dM}{dx} $$

This equation tells us that the shear force is the derivative of the bending moment. Conversely, the bending moment is the integral of the shear force.

Analyzing the Options

Let's analyze the given options in the context of the point of inflexion:

• Option 1: Shear force is zero
  When the shear force ($V$) is zero ($V = \frac{dM}{dx} = 0$), it indicates that the bending moment ($M$) is at a local maximum or minimum value. This point is typically where the beam experiences its maximum or minimum bending effect, but it does not necessarily signify a change in the sign of the moment itself.
• Option 2: Bending moment changes its sign
  A point of inflexion occurs precisely where the bending moment ($M$) crosses zero and changes from a positive value to a negative value, or from negative to positive. This sign change implies a reversal in the beam's curvature. Mathematically, if $M=0$ at a point and $V = \frac{dM}{dx} \neq 0$, it confirms a point of inflexion. This aligns directly with the definition of contraflexure.
• Option 3: Shear force is maximum
  Maximum shear force usually occurs at points of concentrated loads or supports. While the shear force might be non-zero at the point of inflexion, its maximum value does not define the point of inflexion.
• Option 4: None of the above
  Since Option 2 correctly identifies the condition at the point of inflexion, this option is incorrect.

Conclusion

Based on the fundamental principles of beam bending and the relationship between shear force and bending moment, the point of inflexion (or contra flexure) is defined as the location on the beam where the bending moment ($M$) changes its sign.