Beam Bending Stresses

When a structural beam supports transverse loads, internal bending moments develop along its length. These bending moments cause the beam to deform—specifically, to curve. Because of this curvature, the material on one side of the beam is stretched in tension, while the material on the opposite side is compressed. This article covers the calculation of these normal stresses, known as bending stresses, in the elastic region.

The Neutral Axis and Linear Stress Distribution

Within the cross-section of a bending beam, there is a specific axis where the material is neither stretched nor compressed. This is known as the neutral axis. For a beam made of a single uniform material that obeys Hooke's Law, the neutral axis always passes through the geometric centroid of the cross-section.

The fundamental assumption of elastic beam theory is that plane sections remain plane after bending. As a result, the longitudinal strain varies linearly from the neutral axis. Since stress is proportional to strain in the elastic region, the bending stress also varies linearly, reaching its maximum magnitude at the extreme outer surfaces of the beam.

The Elastic Flexure Formula

The normal stress developed by a bending moment in a straight elastic beam is given by the flexure formula:

$\sigma = \frac{My}{I}$

where:

• $\sigma$ = bending stress [Pa or MPa]
• $M$ = internal bending moment at the section [N·m]
• $y$ = perpendicular distance from the neutral axis to the point where stress is being evaluated [m]
• $I$ = second moment of area (area moment of inertia) of the cross-section about the neutral axis [$m^4$]

By definition, at the neutral axis, $y = 0$, thus the bending stress $\sigma = 0$. The maximum bending stress occurs at the maximum distance $y$, often denoted as $c$ (the distance to the extreme fiber).

Sign Convention and Interpretation

Care must be taken to interpret whether the stress is tensile or compressive based on the direction of the applied moment:

• A positive bending moment (often visualized as making the beam "smile") causes compression in the upper portion of the beam and tension in the lower portion.
• A negative bending moment causes tension in the upper portion and compression in the lower portion.

Sometimes the formula is written with a negative sign ($\sigma = -\frac{My}{I}$) to adhere to a strict Cartesian coordinate convention where tensile stresses are positive and compressive stresses are negative. In engineering practice, it is common to calculate the absolute magnitude using $\sigma = \frac{My}{I}$ and apply the tensile/compressive designation through physical inspection.

Worked Example

A detailed interactive calculation example for beam bending stresses can be found at Worked Example: Beam Bending Stresses.

Engineering Check

This method applies strictly to linear elastic behavior. If the calculated stress exceeds the material's yield strength, the linear relationship breaks down, and the elastic flexure formula is no longer valid. In such cases, plastic analysis methods must be employed.