Short Answer:
Pure bending is a condition in which a beam is subjected only to a constant bending moment along its length and no shear force acts on it. This situation occurs when equal and opposite couples (moments) are applied at two ends of a beam section.
In simple words, pure bending means the beam bends into a circular arc due to bending moments alone, without any influence of shear forces. This type of bending helps in studying the basic behavior of beams and in deriving the bending equation .
Detailed Explanation :
Pure Bending
When a beam or any structural member is subjected to bending, it experiences both bending moments and shear forces due to external loads. However, in certain situations, the beam experiences only bending moment without any shear force — this condition is called pure bending.
In pure bending, the bending moment remains constant along the length of a portion of the beam, and no transverse load acts on that portion. The beam bends into a smooth circular arc, and the stress and strain distribution can be studied easily.
Definition of Pure Bending
Pure bending can be defined as:
“The bending of a beam under the action of constant bending moment and zero shear force.”
It occurs when a beam is subjected to equal and opposite couples at its ends, causing it to bend uniformly along its length.
Mathematically, if and , the beam is said to be under pure bending.
Examples of Pure Bending
- Simply Supported Beam:
The portion of a simply supported beam between the points of maximum bending moment and zero shear force is under pure bending. - Cantilever Beam with End Couple:
When a cantilever beam has an applied couple (moment) at its free end and no transverse load, the entire beam is under pure bending. - Beam under Equal Opposite Couples:
A beam subjected to equal and opposite moments at both ends (without any transverse load) experiences pure bending throughout its length.
These conditions are idealized but form the basis for understanding the behavior of beams under bending.
Bending in Beams
When a beam is loaded, the fibers (layers) along its depth experience different types of stresses:
- Top fibers get compressed and tend to shorten.
- Bottom fibers get stretched and tend to elongate.
- Between these two, there is a layer of fibers that remains unchanged in length — this is called the neutral axis.
Under pure bending, this stress distribution is linear along the depth of the beam, and the beam curves into a circular shape.
Conditions for Pure Bending
For pure bending to occur in a beam, the following conditions must be satisfied:
- The beam must be subjected to constant bending moment only.
This means there should be no variation in bending moment along the beam length. - No transverse load should act on the beam section.
Transverse loads cause variation in bending moment and introduce shear forces, leading to “simple bending,” not pure bending. - Shear force at every section must be zero.
The condition must hold true throughout the length where pure bending exists. - Material must remain within elastic limit.
The stress must not exceed the elastic limit of the material to maintain linear behavior as per bending theory.
Theory of Pure Bending
The theory of pure bending helps in deriving the fundamental bending equation:
This theory is based on the following assumptions:
- The material is homogeneous and isotropic.
- The beam has a constant cross-section and is initially straight.
- Plane sections before bending remain plane after bending.
- The radius of curvature is large compared to beam depth.
- Stresses are within elastic limits and follow Hooke’s law.
Derivation of Relation Between Bending Stress and Bending Moment
Consider a beam under pure bending forming an arc of radius .
Let:
- = distance of a fiber from the neutral axis,
- = bending stress at that fiber,
- = modulus of elasticity,
- = moment of inertia of the beam’s cross-section about the neutral axis,
- = bending moment applied.
- Strain in a fiber:
During bending, the strain in a fiber at distance from the neutral axis is given by:
- Stress in that fiber:
According to Hooke’s law,
- Moment of resistance:
The total moment of these stresses about the neutral axis must equal the applied bending moment :
Substitute :
The term is the moment of inertia (I) of the cross-section about the neutral axis.
Therefore,
or,
and since , we get,
This is the bending equation, derived under the condition of pure bending.
Difference Between Pure Bending and Simple Bending
| Pure Bending | Simple Bending |
| Occurs due to constant bending moment only. | Occurs due to variable bending moment and shear force. |
| No transverse load acts on the beam. | Beam carries transverse loads such as UDL or point loads. |
| Shear force is zero. | Shear force is not zero. |
| Moment is constant along the length. | Moment varies along the beam. |
| Easier to analyze for stress distribution. | More complex due to combined bending and shear. |
Importance of Pure Bending
- Foundation of Bending Theory:
Pure bending provides the simplest condition to derive the bending equation used in beam analysis. - Determination of Stress Distribution:
It helps to understand how bending stresses vary linearly along the beam’s depth. - Design of Structural Members:
Used for designing beams, levers, axles, and shafts to ensure they can resist bending safely. - Material Testing:
Experimental studies of bending strength are often done under pure bending conditions for accuracy. - Simplified Analysis:
The assumption of pure bending helps engineers predict beam behavior without involving complex shear effects.
Conclusion
In conclusion, pure bending is a special case of beam bending where the beam is subjected to a constant bending moment with no shear force acting on it. It occurs when equal and opposite moments are applied at the beam ends, causing uniform curvature. This condition simplifies analysis and helps in deriving the bending equation . The concept of pure bending forms the basis of understanding beam behavior and is essential for safe and efficient structural design.