Bending and Shear Stresses in Beams: Pure Bending, Flexure Formula, and Section Modulus

Overview

In this video a beam under load is analyzed to show how bending and internal stresses arise. The presenter explains the vertical shear force and bending moment that accompany deformation, defines bending strains via pure bending, and introduces the neutral axis where fibers remain unchanged in length. The discussion then moves to how bending stresses are calculated using Hooke's law and the flexure formula, and how the area moment of inertia and section modulus govern stress magnitudes. The distribution of bending stresses across common cross sections like I-beams and T-sections is illustrated, followed by a look at shear stresses and their non uniform distribution across the cross section. The video ends with practical notes on real beams where shear and bending coexist and how engineers use these concepts in design.

Introduction to Bending and Shear in Beams

This video explains the mechanical response of a beam when loaded. A load induces bending, which generates internal stresses represented by a vertical shear force and a bending moment. The shear stresses act parallel to the cross section and the bending stresses act normal to it. The presenter emphasizes the importance of calculating these stresses in any beam design or analysis.

Pure Bending and the Neutral Axis

To simplify the analysis, the video first considers pure bending where the shear force along the beam is zero and the bending moment is constant along its length. Under pure bending, fibers near the top shorten (compression) and those near the bottom lengthen (tension). Between these extremes lies the neutral surface, which passes through the cross section centroid and is referred to as the neutral axis in two dimensions. The beam is modeled as fibers bending into a circular arc with center at O, allowing the calculation of strain from the geometric deformation.

Strain, Hooke's Law, and Bending Stress

Strain is defined as the change in length over the original length. For a distance y from the neutral axis, the bending strain is derived from the arc geometry. Assuming elastic behavior, Hooke's law for uniaxial stress gives bending stress as a function of the radius of curvature r. This leads to the key relation between bending stress and bending moment through an integration that yields the area moment of inertia I and introduces the section modulus S = I / y max. The neutral axis passes through the cross-section centroid, and the bending stress increases linearly with both the bending moment and the distance from the neutral axis, peaking at the outer fibers.

The Flexure Formula and Section Modulus

Combining the bending stress expression with the internal bending moment, the flexure formula is obtained: sigma = M y / I. This shows bending stress grows with y and M while being reduced by larger I. The ratio I / y max is the section modulus S, a geometry-dependent quantity often tabulated for common cross sections. The video notes that I-beams are favored in design because they maximize I for a given area, thereby reducing stresses.

Cross-Section Distributions and Examples

Bending stress distributions are described for I-beams and T-sections. In an I-beam, the neutral axis is centered, and bending stress is zero at the neutral axis but reaches a maximum at the outer surfaces of the flanges. In a T-section, shifting of the neutral axis upward changes the stress distribution accordingly. These distributions illustrate why cross-sectional geometry strongly influences stress levels under bending loads.

Shear Stresses and the Tau Equation

The video then moves to shear stresses, denoting them as tau. The shear force V is the resultant of vertical shear stresses that act parallel to the cross section. A standard approach is used to compute the shear stress at a point, assuming tau is constant across the width but varying with y and position along the beam. The equation involves the cross-sectional geometry (width B, height y), the area moment of inertia I, the shear force V, and the first moment of area q. The resulting tau(y) varies parabolically with distance from the neutral axis, attaining a maximum at the neutral axis, which contrasts with the zero bending stress there. A familiar result for rectangular sections is that the maximum shear stress equals 1.5 times the average shear stress. The derivation includes caveats about nonuniform width and alignment of shear stresses with the y axis, as well as adjustments for circular cross sections (constant 4/3 factor) and thin-walled sections like I-beams where the web carries most of the shear force and the flanges carry bending moment.

Practical Insights for Beams

The video concludes with practical notes: in real structures, shear does not drastically alter bending stresses, so the flexure formula derived for pure bending remains a good approximation for many cases. It also explains why wooden beams may fail due to horizontal shear near the neutral axis and how the web and flanges in I-beams work together to carry shear and bending moments. The final invitation encourages viewers to support the channel for more content on bending and shear in beams.

Takeaways

Key takeaways include the separation of bending and shear effects, the neutral axis as the locus of zero strain, the linear increase of bending stress with distance from the neutral axis, and the importance of the area moment of inertia and section modulus in controlling stress magnitudes. The cross-section geometry determines where maximum stresses occur and which parts of a beam carry bending versus shear, guiding practical design choices in structural engineering.