Short Answer:

Pure bending is a condition in which a beam is subjected to a constant bending moment along its length without any shear force. This means the beam bends due to moment only and not because of any transverse or axial force. It occurs when equal and opposite moments act at the two ends of a beam segment, causing it to bend uniformly.

In pure bending, the beam’s cross-section remains plane before and after bending, and the stress distribution across the section is linear. This condition is often used to study the basic bending behavior of materials and to derive the bending equation in mechanics of materials.

Detailed Explanation :

Pure Bending

Definition:
Pure bending refers to the condition of bending of a beam where a constant bending moment acts on the beam, and no shear force is present along the section. This happens when a beam is loaded in such a way that bending moments are produced without any variation along a certain length. In other words, the bending moment remains the same from one section to another, and there is no transverse load in that region.

For example, in a simply supported beam with a concentrated load at the center, the portion between the load and the support experiences both bending moment and shear force. However, if a beam is subjected to equal and opposite moments at its ends, the entire beam is under pure bending, as the shear force becomes zero throughout the length.

Explanation of Pure Bending

When a beam bends under the action of external moments, one part of the beam is subjected to compression and the other to tension. The layer that separates these two regions and experiences no change in length is called the neutral axis.
In pure bending, the stresses vary linearly from the top to the bottom fibers, and the maximum stresses occur at the extreme fibers. The bending is uniform along the entire length, and the curvature of the beam remains constant.

To understand pure bending, let us consider a beam subjected to two equal and opposite moments  at its ends. Since the moments are equal and opposite, the beam bends into an arc of a circle. The upper fibers are shortened due to compression, while the lower fibers are stretched due to tension. The neutral axis remains unchanged in length, separating these two regions.

Mathematical Condition for Pure Bending

For pure bending to exist, the following condition must be satisfied:

This means the bending moment  is constant along the length of the beam, and hence the shear force .

In regions where transverse loads act on a beam, the bending moment changes along the length, and such a condition is called simple bending or ordinary bending, not pure bending.

Stress Distribution in Pure Bending

When a beam is subjected to pure bending, the stress at any point in the cross-section is given by the bending equation:

Where,

•  = Bending moment,
•  = Moment of inertia of the beam cross-section,
•  = Bending stress at a distance  from the neutral axis,
•  = Modulus of elasticity,
•  = Radius of curvature of the neutral axis.

This equation shows that the bending stress varies linearly with the distance from the neutral axis. The top fibers experience compressive stress, and the bottom fibers experience tensile stress of equal magnitude if the material is uniform.

Physical Behavior of Beam Under Pure Bending

When a beam is under pure bending, its shape changes from a straight line to a curved arc. The radius of curvature is constant throughout the beam, meaning that the bending is smooth and uniform.

• The topmost layer shortens due to compressive stress.
• The bottommost layer elongates due to tensile stress.
• The neutral axis remains at the same length, neither stretching nor compressing.

Because the stresses are symmetrical and distributed linearly, the beam does not twist or distort. This condition helps engineers study the pure behavior of materials under bending, free from complications caused by shear or other forces.

Examples of Pure Bending

1. Cantilever Beam with End Moment:
   A cantilever beam subjected to a moment at the free end produces a constant bending moment along its entire length.
2. Simply Supported Beam with Equal End Moments:
   When equal and opposite moments act at both supports, the entire beam is under pure bending.
3. Portion of a Loaded Beam:
   In a simply supported beam with distributed load, the middle region where the shear force is zero may approximately experience pure bending.

Difference Between Pure Bending and Simple Bending

• Pure Bending: Occurs when only a bending moment acts on the beam without shear force.
• Simple Bending: Occurs when both bending moment and shear force are present together due to external loads.

Pure bending is a special case of simple bending, often used to derive theoretical relationships and understand stress distribution in beams.

Importance of Studying Pure Bending

Studying pure bending helps in:

• Deriving the bending equation, which forms the basis of bending stress analysis.
• Understanding how materials behave when subjected only to moments.
• Designing safe and efficient structural members that can resist bending without failure.
• Developing simplified models for complex bending conditions found in engineering structures.

Conclusion

Pure bending is a fundamental concept in mechanics of materials that describes the bending of a beam under a constant bending moment without any shear force. It provides a clear understanding of how internal stresses develop across a beam’s cross-section. The study of pure bending leads to the formulation of the bending equation and helps engineers design structural components that can safely withstand bending moments. In practice, pure bending is an idealized condition, but it forms the foundation for understanding real-world bending behavior.