Stresses in Beams

Stresses in Beams

$Learning Objectives$

Locate the neutral axis of homogeneous beam sections used in elastic bending.
Apply the flexure formula and section modulus to bending-stress calculations.
Interpret the linear through-depth distribution of elastic flexural stress.
Apply the beam shear formula using the correct first moment of area and local width.
Recognize where common beam-stress formulas are valid and where they are not.

$Neutral Axis$

The neutral axis is the line in a beam cross-section where longitudinal bending stress is zero under the assumptions of ordinary elastic flexure.

Text-free cutaway of a steel I-section with a softly distinguished central web region at its exposed end. — Central Section Reference
The intact cut face makes the section's vertical material arrangement visible; the soft central region is only a visual guide, not a drawn neutral-axis line or a stress result.

$Interpreting the Central Region$

Use the central web region to orient the section before applying the flexure model. The raster intentionally omits a neutral-axis line, zero-stress mark, and dimensions; locate the neutral axis from the section properties and the stated bending assumptions.

Elastic Flexure Formula

Normal bending stress at a distance from the neutral axis.

\sigma_x=-\frac{My}{I}

Section Modulus

Geometric quantity used to express extreme-fiber flexural stress.

S=\frac{I}{c}, \qquad |\sigma_{\max}|=\frac{|M|}{S}

Text-free side-by-side view of two intact steel beam sections, one deep and one shallow, with exposed end faces. — Section Depth Context
The paired end faces make depth visible as a geometric section property; use the section-modulus relation for the quantitative flexural comparison.

$Interpreting Section Depth$

Compare the physical distance between the flanges in the two intact sections, while keeping the raster qualitative. It contains no dimensions or section-modulus ranking; use the actual geometry and $S=I/c$ for quantitative flexural work.

$Flexural Stress Distribution$

For a homogeneous prismatic beam in linear-elastic bending, normal stress varies linearly with $y$ . One side of the neutral axis is in tension and the other in compression. The extreme fibers have the greatest magnitude because $|y|$ is largest there.

$Interactive Exploration$

Vary bending moment and rectangular-section dimensions. Observe the neutral axis, the linear stress distribution, and the strong effect of section depth on $I$ and the resulting extreme-fiber stress.

Controls

15 kN·m

100 mm

200 mm

66.67 ×10⁶ mm⁴

I = bh³/12

Section modulus S

666.7 ×10³ mm³

S = I/c

Extreme-fiber stress

22.5 MPa

Magnitude at y = ±c for the selected positive-moment convention.

$First Moment of Area$

The first moment of area $Q$ is the area on one side of the level of interest multiplied by the distance from that area's centroid to the neutral axis.

Text-free side-by-side comparison of a solid rectangular steel beam and a connected steel I-section. — Shear Section Geometry
The two intact end sections show different material layouts for a beam-shear discussion; exact stress distributions and web contributions are supplied by the formula, not by the raster.

$Interpreting Shear Section Layouts$

Compare the filled rectangular section with the I-section's connected flanges and web. The image supplies no stress curve, dimensions, or exact share of shear; evaluate $VQ/(Ib)$ with the actual section geometry and the local width $b$ .

Beam Shear Formula

Transverse shear stress in a beam section under elementary beam theory.

\tau=\frac{VQ}{Ib}

$Transverse Shear Distribution$

The local width $b$ is measured at the level where the shear stress is being evaluated. For a solid rectangle, the distribution is parabolic and the maximum occurs at the neutral axis. In thin-webbed sections, much of the shear is carried by the web.

Text-free cutaway of a connected steel I-section beam with continuous flanges, web, and a restrained floor edge. — Beam Section and Web Context
The cutaway highlights the connected flanges and web that give a beam its section geometry; exact neutral-axis location and stress distributions remain in the formulas and simulator.

$Reading the Beam Section$

The flanges are far from the centroidal region and the web connects them through the depth, giving a physical context for flexural stiffness and web-dominated shear in thin-webbed sections. The generated section is not dimensioned or to scale; use $\sigma=-My/I$ and $\tau=VQ/(Ib)$ with the actual section properties for quantitative work.

$Beam Stress Evaluation$

STEP-BY-STEP

Solve the beam reactions and obtain the governing $V$ and $M$ at the section.
Determine the neutral axis and section properties $I$ , $S$ , and the required $Q$ .
Evaluate bending stress at the fibers of interest with $\sigma=-My/I$ .
Evaluate transverse shear at the required level with $\tau=VQ/(Ib)$ .
Check signs, units, and whether the assumptions of elementary beam theory are satisfied.

$Flexure Formula Limitations$

The elementary flexure equation assumes small deformation, linear-elastic material behavior, plane sections remaining plane, and a beam geometry compatible with ordinary bending theory. Curved beams, deep-beam effects, plastic bending, severe stress concentrations, and anisotropic behavior require other models.

Text-free cutaway of a steel I-beam with a rounded web opening and a connected bearing plate at a concrete support. — Beam Discontinuity and Support Detail
A clean web opening and local bearing region show where a prismatic beam idealization needs closer scrutiny; no failure or code limit is depicted.

$Interpreting Discontinuities and Supports$

The opening interrupts uniform web geometry while the support introduces a local bearing and connection region. Use this context to question a simple prismatic-beam model, not to infer a stress concentration or a design limit; the raster is not a design detail.

Key Takeaways

Flexural stress is zero at the neutral axis and linear through the depth under ordinary elastic bending.
Section modulus converts moment directly to extreme-fiber stress.
Transverse shear depends on $V$ , $Q$ , $I$ , and the local width $b$ .
Maximum bending and maximum shear generally occur at different locations.
Beam-stress equations are only as valid as their underlying assumptions.