Short Answer:

The simple bending theory is based on several assumptions that help in analyzing the bending of beams easily. These assumptions simplify the real behavior of materials under bending into an idealized form. The main assumptions include that the beam is initially straight, the material is homogeneous and isotropic, and the bending is within the elastic limit.

It is also assumed that the plane sections before bending remain plane after bending, the radius of curvature is large compared to the depth of the beam, and the stress is directly proportional to strain. These assumptions make it easier to apply bending equations accurately for most practical cases.

Detailed Explanation :

Assumptions in Simple Bending Theory

The simple bending theory, also known as the theory of pure bending, provides a relationship between bending stress, bending moment, and beam curvature. It is used to analyze how beams behave when subjected to bending loads. To make the analysis simple and mathematically workable, several assumptions are made. These assumptions idealize the beam behavior and allow the use of formulas such as the bending equation:

Where,

•  = Bending moment
•  = Moment of inertia
•  = Bending stress
•  = Distance from the neutral axis
•  = Modulus of elasticity
•  = Radius of curvature

Let us now discuss all the assumptions of the simple bending theory in detail.

1. The beam is initially straight and has a uniform cross-section

It is assumed that the beam is perfectly straight before any load is applied and that its cross-section is the same throughout its length. This ensures that bending occurs uniformly along the beam. If the beam were curved or irregular, the bending analysis would become complex because of uneven distribution of stresses and strains.

2. The material of the beam is homogeneous and isotropic

The material is assumed to be homogeneous, meaning it has the same composition and properties throughout, and isotropic, meaning its mechanical properties are the same in all directions. This ensures that the stress-strain relationship is uniform, allowing the use of Hooke’s law.

For example, steel and aluminum are generally treated as homogeneous and isotropic materials in engineering calculations.

3. The beam is subjected to pure bending only

It is assumed that the bending moment acts on the beam without any shear force or axial load. In pure bending, only the bending moment is present between two sections of the beam. This simplifies the analysis since shear stresses are ignored, and the bending stress alone is considered.

4. Plane sections before bending remain plane after bending

This is one of the most important assumptions in bending theory. It means that a cross-section of the beam that was flat and perpendicular to the axis before bending remains flat and perpendicular even after bending.
In other words, there is no warping or distortion of the cross-section.
This assumption ensures that the strain varies linearly across the depth of the beam and makes the stress distribution linear as well.

5. The radius of curvature is large compared to the depth of the beam

It is assumed that the beam bends into a circular arc with a large radius of curvature compared to the beam depth. This means the slope of the deflection curve is small, and the beam experiences small deformations.
This allows small-angle approximations in the derivation of the bending equation and ensures that the geometry remains simple and predictable.

6. The material of the beam obeys Hooke’s law

According to this assumption, the material behaves elastically and follows Hooke’s law, which states that stress is directly proportional to strain within the elastic limit.
This assumption is essential because it ensures a linear relationship between bending stress and curvature, making the bending equation valid only within the elastic range of the material.

If the material goes beyond the elastic limit, plastic deformation starts, and the linear relationship no longer holds true.

7. The stress induced in the beam is within the elastic limit

This assumption ensures that after the removal of load, the beam will return to its original shape without any permanent deformation. The bending theory is only valid as long as the stresses remain within the elastic range. If the load exceeds this limit, yielding will occur, and the simple bending theory cannot be used.

8. Every layer of the beam is free to expand or contract longitudinally

It is assumed that the longitudinal layers in the beam can expand or contract independently when bending occurs. The top fibers shorten (compression), the bottom fibers lengthen (tension), and the neutral axis remains unchanged in length. This assumption allows the formation of a linear strain distribution across the beam depth.

9. The effect of shear is neglected

In simple bending theory, only normal stresses due to bending are considered, while shear stresses are neglected. This assumption is valid when the beam is long and slender, and the shear force is relatively small compared to the bending moment.

Importance of These Assumptions

These assumptions help simplify the real, complex behavior of materials and structures under bending into manageable forms for analysis and design.
By applying these assumptions, engineers can predict the bending stresses, deflections, and moments in beams accurately enough for most practical engineering applications. Although real structures may not perfectly follow all these assumptions, the theory gives results that are close to reality within the elastic limit.

Limitations of Simple Bending Theory

While the assumptions make calculations easy, they also create limitations. The theory is not valid for:

• Beams with large deflections.
• Non-homogeneous or anisotropic materials (like composite beams).
• Cases where shear deformation is significant.
• Plastic bending, where material yields beyond the elastic range.

In such cases, more advanced theories like elastic-plastic bending theory or shear deformation theory are used.

Conclusion

The assumptions in simple bending theory provide the foundation for analyzing beam behavior under bending loads. They simplify complex mechanical behavior into a linear model by assuming the beam is straight, homogeneous, isotropic, and within the elastic limit. These assumptions allow accurate calculation of bending stress and deflection for most engineering structures. However, for large deformations or non-linear materials, these assumptions must be modified to maintain accuracy.