What is bending stress formula?

Short Answer:

The bending stress formula is used to calculate the internal stress developed in a beam when it is subjected to a bending moment. It relates the bending moment, moment of inertia, and the distance from the neutral axis to the bending stress at any point in the beam.

In simple words, the bending stress in a beam is directly proportional to the bending moment and the distance from the neutral axis, and inversely proportional to the beam’s moment of inertia. The general bending stress formula is given by:

where  is the bending stress,  is the bending moment,  is the distance from the neutral axis, and  is the moment of inertia.

Detailed Explanation :

Bending Stress Formula

When a beam is subjected to external loads perpendicular to its length, it bends. This bending produces internal stresses within the material to resist deformation. These stresses are known as bending stresses.

The bending stress formula helps engineers determine the magnitude of these stresses so that beams can be designed safely and economically. It is derived from the bending equation, which expresses the relationship between the bending moment, the geometry of the beam section, and the material’s response to bending.

Concept of Bending in Beams

When a beam bends under the action of external loads, one side of the beam’s longitudinal fibers shortens (compression), while the opposite side elongates (tension). Between these two regions lies a surface known as the neutral surface or neutral axis, where the fibers experience no change in length.

Thus,

  • Above the neutral axis → fibers are in compression,
  • Below the neutral axis → fibers are in tension.

This variation of stress from compression to tension across the beam depth forms a bending stress distribution.

Assumptions in Deriving the Bending Stress Formula

To derive the bending stress formula, the following assumptions are made:

  1. The material of the beam is homogeneous and isotropic (same properties in all directions).
  2. The beam follows Hooke’s law (stress is proportional to strain).
  3. The beam is initially straight and has a constant cross-section.
  4. The cross-sections of the beam remain plane before and after bending (no distortion).
  5. The radius of curvature is large compared to the beam depth.
  6. The loads act perpendicular to the beam axis.

These assumptions are essential for linear elastic bending analysis.

Derivation of Bending Stress Formula

Consider a beam subjected to a bending moment (M), causing it to bend into an arc of a circle with radius .

Let,

  •  = Distance from the neutral axis,
  •  = Bending stress at distance ,
  •  = Modulus of elasticity of the beam material,
  •  = Moment of inertia of the cross-section about the neutral axis.

From the geometry of bending, the strain in a fiber at distance  from the neutral axis is given by:

From Hooke’s law,

Substitute :

Now, consider the internal resisting moment produced by these stresses to balance the external bending moment :

Substitute :

Since  and  are constant for a section,

The term  represents the moment of inertia (I) of the cross-section about the neutral axis.

Therefore,

Rearranging,

From the earlier relation ,

This is the bending equation, and from it, we can write the bending stress formula as:

Explanation of Terms

  •  = Bending stress at a fiber (N/m² or MPa)
  •  = Bending moment at the section (N·m)
  •  = Distance from the neutral axis to the fiber (m)
  •  = Moment of inertia of the section about the neutral axis (m⁴)

Nature of Bending Stress

  • The bending stress varies linearly with the distance from the neutral axis.
  • It is zero at the neutral axis.
  • It is maximum at the outermost fiber of the beam, where .

Hence, the maximum bending stress is given by:

Sign Convention

  1. Positive Bending Moment (Sagging):
    • The beam bends concave upward.
    • Top fibers → Compression, Bottom fibers → Tension.
  2. Negative Bending Moment (Hogging):
    • The beam bends concave downward.
    • Top fibers → Tension, Bottom fibers → Compression.

Application of the Bending Stress Formula

  1. Design of Beams:
    The formula helps in determining the maximum stress in a beam for a given load, ensuring it remains within the safe limit of the material.
  2. Calculation of Safe Load:
    For a known beam size and material, the formula is used to find the maximum allowable bending moment and thus the load the beam can safely carry.
  3. Selection of Beam Section:
    The bending stress equation guides engineers in choosing suitable cross-sectional shapes (rectangular, circular, I-section) with required strength.
  4. Failure Analysis:
    Helps in locating points of maximum stress where cracks or fractures may initiate.

Example

A simply supported rectangular beam of width , depth , and subjected to a bending moment :

 

Thus, the maximum bending stress = 7.5 MPa.

Conclusion

The bending stress formula () is a fundamental relationship in the study of strength of materials. It explains how internal stresses are distributed across the cross-section of a beam when subjected to bending. The stress is directly proportional to the bending moment and the distance from the neutral axis, and inversely proportional to the moment of inertia. This formula is essential for the safe design of beams, bridges, and mechanical structures subjected to bending loads.