AtlasCalcsUtility Calculation Suite
2025-08-21 14 min

Bending Stress Calculator

Calculate maximum bending stress in beams using sigma = My/I.

Calculator

Unit System
Bending Moment (M)(N*m)
N*m
Distance to Fiber (y)(mm) Help
mm

From neutral axis to outer fiber

Moment of Inertia (I)(mm^4)
mm^4
Section Width (b)(mm)
mm
Section Height (h)(mm)
mm
Decimal Precision
Results update live as you type

Results

Normal

Max Bending Stress

0.13MPa

Section Modulus (S)

198019.80mm^3

Visualizations

Results Overview

0.1MPaMax Bending Stress
198,020MPaSection Modulus

Formula

Bending stress equals bending moment times distance from neutral axis divided by moment of inertia.

sigma = My / I

Alternative Form

sigma = M / S where S = I/y

Variables

SymbolNameUnitDescription
sigmaBending StressMPaMaximum normal stress at extreme fiber
MBending MomentN*mInternal bending moment at the section
yDistance to FibermmDistance from neutral axis to extreme fiber
IMoment of Inertiamm^4Area moment of inertia about bending axis
SSection Modulusmm^3S = I/y, geometric measure of bending strength

Explanation

Bending stress varies linearly across the section. Maximum stress at the fiber farthest from the neutral axis.

Derivation

From Euler-Bernoulli beam theory: epsilon = -y/r. With Hooke law sigma = E*epsilon gives sigma = My/I.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Determine the bending moment M at the critical section

Result: M known
Step 2

Calculate moment of inertia I for the cross-section

Result: I computed
Step 3

Find distance y from neutral axis to extreme fiber

Result: y determined
Step 4

Calculate sigma = My/I

Result: Maximum bending stress computed
Step 5

Compare against allowable stress with safety factor

Result: Design check
Final Answer
0.13

Worked Examples

See how the calculator works with different scenarios.

Scenario

A W200x22 steel I-beam (I = 20.0E6 mm^4, c = 101 mm) carries M = 25 kN*m.

Inputs

Bending Moment:25000 N*m
Distance y:101 mm
Moment of Inertia:20000000 mm^4

Calculation

sigma = 25000*1000*101/20000000 = 126.3 MPa.

Result

sigma = 126.3 MPa

Interpretation

Within yield strength of structural steel (250 MPa). Safety factor ~2.0.

Understanding Your Result

Your Result: Max Bending Stress: 0.13 MPa

Typical range: Steel: 100-200 MPa. Wood: 5-12 MPa. Aluminum: 50-150 MPa.

What This Means

Bending stress is the normal stress resisting bending deformation. One face in tension, the opposite in compression.

Why It Matters

Exceeding yield causes permanent deformation; exceeding ultimate causes fracture.

Professional Interpretation

Compare against allowable stress (yield/safety factor). Verify deflection limits.

Recommendations

  • •Use I-beams or hollow sections for efficiency
  • •Check both strength and deflection
  • •Apply stress concentrations at notches

Educational Content

Introduction

Bending is one of the most common loading conditions in structural and mechanical engineering.

Concept Explanation

sigma = My/I assumes plane sections remain plane (Euler-Bernoulli theory). Stress varies linearly from neutral axis.

Applications

▸Building floor beams
▸Bridge girders
▸Machine frames
▸Vehicle chassis
▸Aircraft wing spars

Benefits

  • •Simple and widely validated formula
  • •Applicable to any symmetric cross-section
  • •Foundation for complex analyses

Limitations

  • •Valid only for elastic range
  • •Does not account for shear lag
  • •Not valid for very short or deep beams

Best Practices

  • •Calculate I accurately
  • •Use correct y to extreme fiber
  • •Include stress concentrations

Common Misconceptions

Myth: Bending stress is uniform across the section
Truth: It varies linearly from zero at the neutral axis to maximum at the extreme fiber.

Practical Tips

  • •I-beams are most efficient for bending
  • •Deep sections reduce stress but increase deflection
  • •Check stress at maximum moment points
  • •For asymmetric sections check both tensile and compressive stresses

Common Mistakes

Using wrong axis for I
Use I about the axis of bending.
Confusing y with full depth
y is distance from neutral axis to fiber of interest.

Assumptions

This calculator makes the following assumptions to provide estimates:

Linear Elastic Material: Stress is proportional to strain within elastic limit.
Plane Sections: Cross-sections remain plane after bending.

Limitations

Be aware of these limitations when using this calculator:

Elastic Only: Not valid beyond yield stress.
Small Deflections: Assumes small deflection geometry.

Frequently Asked Questions

Answers to common questions about this calculator.

What is bending stress?
Bending stress is the normal stress (tensile or compressive) that develops in a beam when subjected to a bending moment. It varies linearly from zero at the neutral axis to maximum at the outermost fibers.
What is the bending stress formula?
The flexure formula is sigma = My/I, where M is bending moment, y is distance from neutral axis, and I is area moment of inertia about the bending axis.
What is the neutral axis?
The neutral axis is the line through the cross-section where bending stress is zero. It passes through the centroid for symmetric sections.
What is the section modulus?
Section modulus S = I/y_max determines bending strength. A larger S means lower stress for the same moment: sigma = M/S.
How does cross-section shape affect bending stress?
I-beams and hollow sections are most efficient because they place material far from the neutral axis, maximizing I and S while minimizing weight.
What is the area moment of inertia?
I = integral(y^2 dA) quantifies how area is distributed relative to the neutral axis. Higher I means greater resistance to bending.
How do you calculate I for a rectangle?
For a rectangular section: I = bh^3/12, where b is width and h is height parallel to bending direction.
What is the maximum bending stress in a simply supported beam?
For a central point load P: M_max = PL/4, so sigma_max = PLc/(4I). For UDL: M_max = wL^2/8.
What is the difference between bending and direct stress?
Bending stress varies across the section while direct (axial) stress is uniform. Combined: sigma = P/A + My/I.
How does bending stress relate to deflection?
Stress depends on M/I while deflection depends on 1/(EI). A beam can have low stress but high deflection if it is long.
What is the bending stress in a cantilever beam?
For end load P: M_max = PL at the fixed end. sigma_max = PL*c/I.
What is the allowable bending stress?
Steel: 0.66*Fy (AISC). Aluminum varies by alloy. Wood: 800-1200 psi for common species. Always apply safety factors.
How does temperature affect bending stress?
Temperature gradients create thermal bending stresses. Elevated temperature reduces yield strength, lowering allowable bending stress.
What is residual bending stress?
Residual stresses remain after loads are removed, from manufacturing like rolling or welding. They affect fatigue life.
What is the bending stress in a composite beam?
Use the transformed section method to account for different moduli of bonded materials.
How do you find the maximum bending moment?
Draw the shear diagram and find where shear is zero or changes sign. M is maximum at these points.
What is the relationship between shear and bending moment?
V = dM/dx. The bending moment is the integral of shear force. Where V=0, M has a maximum or minimum.
How do holes affect bending stress?
Holes reduce I and S, increasing stress. Stress concentrations around holes can increase local stress by 2-3 times.
What is the bending stress in a curved beam?
Curved beams have non-linear stress distribution. Use the Winkler-Bach formula instead of sigma = My/I.
What is maximum bending stress in a circular section?
I = pi*d^4/64, S = pi*d^3/32. So sigma = 32*M/(pi*d^3).
How does bending stress affect fatigue life?
Cyclic bending is the most common fatigue cause. Use S-N curves and Goodman diagrams for analysis.
What is the bending stress in a T-beam?
Locate the neutral axis (centroid), calculate I about it. Max tensile and compressive stresses may differ.
What is pure bending?
Constant bending moment with zero shear force. The beam deflects into a circular arc. Achieved in four-point bending.
How is bending stress measured experimentally?
Using strain gauges bonded to the surface. Convert strain to stress via Hooke law: sigma = E*epsilon.
What is the bending stress in a pipe?
I = pi*(Do^4-Di^4)/64, c = Do/2. sigma = M*c/I.
What is the plastic modulus?
Plastic modulus Z gives fully plastic moment M_p = F_y * Z. Shape factor Z/S is 1.1-1.5.
How do you reduce bending stress?
Increase S (deeper beams, I-beams, hollow sections), reduce span, add supports, or reduce load.
What is the bending stress in a stepped beam?
Calculate I for each section. Maximum stress occurs at highest M/S ratio.
What is the bending stress distribution?
Linear across depth: zero at neutral axis, maximum at extreme fibers. Valid for elastic analysis only.
How does bending differ from torsional stress?
Bending produces normal stress varying from neutral axis. Torsion produces shear stress varying from center.
What is a cantilever bending moment diagram?
For end load P, moment varies linearly from 0 at the free end to PL at the fixed end. Maximum stress is at the fixed support.
How do you size a beam for bending?
Calculate required S = M_allow/sigma_allow. Select a standard section with S >= required S. Verify deflection.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Bending Stress
Normal stress due to bending moment: sigma = My/I.
Neutral Axis
Line where bending stress is zero.
Section Modulus
S = I/y_max, geometric measure of bending strength.
Moment of Inertia
I = integral(y^2 dA), resistance to bending deflection.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

AISC 360Eurocode 3NDS for Wood

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Page Information

Last Updated:

2025-08-21

Version:

2.0.0

License:

Free to use for educational and professional purposes.