Bending Stress – Definition, Formula

What is Bending Stress?

Bending stress is the normal stress that an object withstands when it is subjected to external load at any cross-section.

Bending Stress Definition

The bending stress is also defined as the ratio between the bending moment and the section modulus of the section.

When a beam having an arbitrary cross-section is subjected to transverse loads, the beam will bend. In addition to bending, other effects such as twisting and buckling may occur, and investigating a problem that includes all the combined effects of bending, twisting, and buckling could become complicated. Thus we are interested in investigating the bending effects alone.

The magnitude of bending stress is calculated by

My/I = M/Z

Where Z: Section modulus of the section: I/y

Bending Stress Formula

To calculate the bending stress in a beam we use the bending stress formula. Using this formula we can calculate the bending moment along with bending stress.

Bending Stress Assumptions

This formula is derived for the beam which has the following assumptions made:

• The beam is initially straight and has a constant cross-section.
• The beam is composed of homogeneous material and has a longitudinal plane of symmetry.
• The resultant force of the applied load lies in the plane of symmetry.
• The geometry of the entire component is such that buckling rather than bending is the primary cause of failure.
• The elastic limit is never exceeded and ‘E’ is the same in tension and compression.
• A flat cross-section remains flat before and after bending.

The bending stress formula is:

M/I = σ/y = E/R

where;

• M: Bending moment of the section passing through a point
• I: Moment of Inertia of the section
• σ: Bending stress at a point
• y: Distance from NA
• E: Modulus of elasticity of the material
• R: Radius of Curvature

NOTE:

• The radius of curvature is represented by EI/M
• The curvature of the section is represented by M/EI
• The radius of curvature is represented by EI

Bending Stress in a Beam

A simply supported beam subjected to load tends to make the beam bend as below.

The bending action of the load will be resisted by the material of the horizontal beam. Therefore, the material of the beam will provide internal resistance against load. Bending stress is nothing but this internal resistance per unit area

Bending is of the following types.

• Unsymmetric Bending: When a bending couple does not take place in the plane of symmetry of a member, it is called Unsymmetric Bending
• Symmetric Bending: When loading acts in the plane of symmetry of the object, the type is called Symmetric bending.
• Non-uniform Bending: Bending in presence of Shear force is termed Non-uniform bending.
• Pure Bending: Bending of a beam under a constant bending moment is termed Pure bending.

Bending Stress Variation in a Beam

The beam is initially straight and has a constant cross-section. The beam is made of homogeneous material and the beam has a longitudinal plane of symmetry. Elastic limit is nowhere exceeded and ‘E’ is the same in tension and compression. For such a case, the bending stress variation can be found by the following formula:

σ = My/I

Hence, with this, we can conclude that

σ ∝y

In simple bending of a beam; variation of bending stress is linear. Variation of bending stress is linear from zero at NA to a maximum at the outer surface.

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