What is the flexure formula (σ = Mc/I)?

The flexure formula calculates the normal bending stress at any point in a beam cross-section. σ is the bending stress in Pascals (Pa), M is the applied bending moment in Newton-meters (N·m), c is the distance from the neutral axis to the outermost fiber in meters, and I is the second moment of area (moment of inertia) of the cross-section in m⁴. Maximum bending stress occurs at the outermost fibers where c is largest.

How is maximum shear stress calculated?

Maximum shear stress is calculated using τ = VQ/(Ib), where V is the shear force (N), Q is the first moment of area of the portion of the cross-section above the point of interest (m³), I is the moment of inertia (m⁴), and b is the width at the point where shear stress is evaluated (m). For rectangular sections the maximum occurs at the neutral axis and equals τ_max = 1.5 × V/A. For circular sections τ_max = 4V/(3A).

What is the moment of inertia (second moment of area)?

The moment of inertia I (also called second moment of area) quantifies a cross-section's resistance to bending. For a rectangle: I = bh³/12. For a solid circle: I = πd⁴/64. For an I-beam: I = (b_f × h³)/12 − 2 × ((b_f − t_w)/2 × h_w³)/12, where b_f is flange width, h is total height, t_w is web thickness, and h_w is web height.

What is the section modulus?

The elastic section modulus S = I/c is a property that combines moment of inertia and the distance to the extreme fiber. It simplifies the flexure formula to σ_max = M/S, making beam selection straightforward — you need only find a section with S ≥ M/σ_allowable.

What units does this calculator use?

All inputs use consistent SI units: cross-section dimensions in millimeters (mm), bending moment in Newton-meters (N·m), and shear force in Newtons (N). Results are given in Megapascals (MPa) for stress, mm⁴ for moment of inertia, and mm³ for section modulus — the conventional units used in structural engineering practice.

Does this account for combined loading?

This calculator evaluates bending stress and transverse shear stress separately. For combined loading (e.g., bending plus torsion, or biaxial bending), you must apply a combined stress theory such as von Mises or the maximum shear stress criterion using the individual stress components as inputs.

Beam Bending Stress Calculator

Name: Beam Bending Stress Calculator
Availability: InStock
Author: Nham Vu

Enter beam loading conditions and cross-section dimensions to compute maximum bending stress (σ = Mc/I) and shear stress (τ = VQ/Ib) for rectangular, circular, or I-beam sections.

Cross-Section Type

Cross-Section Dimensions

Width, b (mm)

Horizontal dimension of the rectangular section.

Height, h (mm)

Vertical dimension (in the plane of bending).

Applied Loads

Bending Moment, M (N·m)

Maximum applied bending moment at the critical section.

Shear Force, V (N)

Transverse shear force at the same cross-section.

Max Bending Stress

—

σ_max = Mc/I (MPa)

Max Shear Stress

—

τ_max = VQ/(Ib) (MPa)

Select a cross-section, enter dimensions and loads, then click Calculate.

Formulas Used

σ_max = M × c / I

τ_max = V × Q / (I × b)

M = moment (N·m), c = dist. to outer fiber (m), I = moment of inertia (m⁴)

Moment of Inertia Formulas by Cross-Section

Rectangular

I = bh³ / 12

c = h / 2

τ_max = 1.5 V/A

Solid Circular

I = πd⁴ / 64

c = d / 2

τ_max = 4V / (3A)

I-Beam (Wide Flange)

I = I_outer − 2I_cutout

c = h_total / 2

τ_max at neutral axis

Summary

Enter beam loading conditions and cross-section dimensions to compute maximum bending stress (σ = Mc/I) and shear stress (τ = VQ/Ib) for rectangular, circular, or I-beam sections.

How it works

Select the cross-section type: rectangular, circular, or I-beam.
Enter the cross-section dimensions (width, height, diameter, or flange/web dimensions).
Enter the applied bending moment M in Newton-meters (N·m).
Enter the applied shear force V in Newtons (N) for shear stress.
Click Calculate to see maximum bending stress, shear stress, moment of inertia, and section modulus.
Use the Reset button to clear all inputs and start a new calculation.

Use cases

Verifying beam sizing in structural steel design using the flexure formula.
Checking bending stress in timber beams for construction projects.
Calculating section modulus for custom I-beam or box sections.
Teaching mechanics of materials and beam theory in engineering courses.
Preliminary beam selection before detailed finite element analysis.
Validating hand calculations against published beam tables.
Designing machine frames and supports for industrial equipment.
Checking stress in shaft design where bending loads apply.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu