Short Answer:

The bending equation relates bending moment, material stiffness, geometry and stress in a beam:

It means the bending moment per unit section inertia equals the normal stress at a fiber divided by its distance from the neutral axis, and equals Young’s modulus divided by the radius of curvature. This simple formula lets engineers compute bending stress from a known moment, or moment capacity from a section.

In plain words, under pure bending the beam curves with radius ; fibers at distance  from the neutral axis stretch or compress by , producing stress . Integrating those stresses across the section gives , and rearranging yields the common form .

Derive bending equation

Detailed Explanation :

Assumptions used (simple bending theory)

1. Material is linear elastic and homogeneous (Hooke’s law applies).
2. Cross-sections are plane and remain plane after bending (no warping).
3. The beam is subjected to pure bending (no transverse shear in the derived element).
4. The radius of curvature  is large compared with beam depth (small strains).

These assumptions let us relate geometry (curvature) to strain and then to stress.

Geometry and strain
Consider a small segment of a beam bent into an arc of radius . Choose the neutral axis (NA) as the layer of fibers that experiences no longitudinal strain. A fiber at a distance  from the NA changes length when the beam bends: its original length  becomes  while the neutral layer length is , where  is the subtended angle. The longitudinal strain in that fiber is

Sign: fibers on one side are in tension (), the other side in compression (). Important point: strain varies linearly with distance from NA:

Stress from strain (Hooke’s law)
Using linear elasticity, stress is proportional to strain:

So bending produces a linear stress distribution across the section: zero at the neutral axis and maximum at the outermost fiber :

Internal moment from stress distribution
The internal bending moment  at the section is the resultant moment of internal stresses about the neutral axis:

Substitute :

The integral  is the second moment of area (moment of inertia) about the neutral axis, denoted . Thus

Rearrange:

Relating stress directly to moment
From  and , eliminate :

Or written as the familiar flexural formula:

Also we can write the compact relation that appears in many texts:

Interpretation and use

•  is the internal bending moment at the section (N·m).
•  is the area moment of inertia about the neutral axis (m⁴).
•  is the normal stress at distance  from NA (N/m²).
•  is the perpendicular distance from the NA to the fiber under consideration (m).
•  is Young’s modulus (N/m²).
•  is radius of curvature of the neutral axis (m).

The equation  shows stress is proportional to : outer fibers (largest ) carry the highest stress. Sign indicates tension or compression depending on bending sense.

Limits and remarks

• The derivation assumes elastic behaviour. If material yields, linear relation fails.
• Plane-sections assumption breaks down for deep beams or when shear deformation is significant. In such cases Timoshenko beam theory or more complex models are needed.
• For composite sections or non-homogeneous materials, replace  by an equivalent transformed-section approach before applying the formula.
• The neutral axis location depends on section geometry and material; for symmetric homogeneous sections it passes through centroid.

Practical example (short)
For a rectangular beam of depth  and width , . For a bending moment , maximum stress at outer fiber  is

Engineers use this to choose section sizes so  stays below allowable stress.

Conclusion

The bending equation  follows directly from geometry of bending and Hooke’s law under simple bending assumptions. It compactly links internal moment, section geometry, material stiffness and stress distribution. The flexural formula  is a core tool in beam design: it lets you compute stresses from moments and guide safe, economical section selection as long as elastic, small-strain conditions apply.