Bending Stress Calculator
Find the maximum bending stress at the extreme fiber of a beam from the bending moment, distance from the neutral axis, and moment of inertia.
📏 What is Bending Stress?
Bending stress is the internal normal stress that develops within a beam's cross-section as it resists an applied bending moment. It is zero at the neutral axis, the line within the section that experiences no change in length under pure bending, and grows linearly with distance from that axis, reaching its maximum value at the extreme fiber, the point farthest from the neutral axis. The governing relationship is the flexure formula, sigma = Mc/I.
Structural and mechanical engineers use bending stress checks constantly: sizing floor joists and roof beams against a design moment, verifying a steel I-beam will not yield under a crane load, checking a shaft under bending from a gear or pulley, and confirming a bracket or cantilever arm stays within its material's allowable stress. Every one of these checks reduces to the same formula once the moment M, distance c, and moment of inertia I are known.
A common point of confusion is mixing up bending stress with shear stress. Bending stress (sigma = Mc/I) is a normal stress acting perpendicular to the cross-section, caused by the bending moment. Shear stress (tau = VQ/Ib) is caused by the shear force and acts parallel to the cross-section. A complete beam design checks both separately, along with deflection.
This calculator computes the flexure formula directly from your bending moment, distance from the neutral axis, and moment of inertia, showing the result in both MPa and ksi with full step-by-step working, so you can immediately compare it against your material's allowable stress.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Timber Floor Joist
A joist under a 50 kN·m design moment, with c = 200 mm and I = 300,000,000 mm⁴
Example 2 — Steel Crane Girder
A crane girder under a 120 kN·m design moment, with c = 250 mm and I = 500,000,000 mm⁴
Example 3 — Small Aluminum Bracket
A bracket arm under a 10 kN·m moment, with c = 100 mm and I = 50,000,000 mm⁴
❓ Frequently Asked Questions
🔗 Related Calculators
What is bending stress?
Bending stress is the internal normal stress that develops in a beam cross-section as it resists an applied bending moment. It is zero at the neutral axis and reaches its maximum at the extreme fiber, the point farthest from the neutral axis, following sigma = Mc/I.
What is the formula for bending stress?
sigma = Mc/I, where M is the bending moment, c is the distance from the neutral axis to the point of interest (usually the extreme fiber), and I is the second moment of area of the cross-section about the bending axis.
What is the relationship between bending stress and section modulus?
Section modulus Z is defined as Z = I/c, so the flexure formula simplifies to sigma = M/Z. Using Z directly avoids computing I and c separately once Z is already known for a standard section.
Why is bending stress zero at the neutral axis?
The neutral axis is defined as the line within the cross-section that experiences no change in length under pure bending, meaning zero strain and therefore zero stress. Stress increases linearly with distance from this axis, following sigma = Mc/I.
What units does this calculator use?
Bending moment is entered in kilonewton-meters (kN-m), distance from the neutral axis in millimeters (mm), and moment of inertia in millimeters to the fourth power (mm^4). The result is shown in both megapascals (MPa) and kilopounds per square inch (ksi).
How do I find the moment of inertia I for my cross-section?
Use the Second Moment of Area Calculator for common shapes, a rectangle (I = bh^3/12), a solid circle (I = pi d^4/64), or a hollow circular tube (I = pi(D^4 - d^4)/64), then bring that value directly into this calculator.
What happens if the bending stress exceeds the material's yield stress?
The beam will yield (deform permanently) at the extreme fiber once bending stress reaches the yield stress. Structural design applies a safety factor, keeping allowable bending stress well below yield, so the section should be checked against the allowable stress, not just the yield stress.
Does this formula apply to any beam cross-section?
The flexure formula sigma = Mc/I applies to any cross-section as long as the beam is made of a linear elastic material, the section is symmetric about the plane of bending (or the correct axis properties are used), and deflections stay small (elastic beam theory).
Is bending stress the same as shear stress in a beam?
No. Bending stress (sigma = Mc/I) is a normal stress caused by the bending moment, acting perpendicular to the cross-section. Shear stress (tau = VQ/Ib) is caused by the shear force and acts parallel to the cross-section. Both are checked separately in a complete beam design.
What is the maximum allowable bending stress for steel?
It depends on the steel grade and design code. For example, ASTM A36 steel has a yield stress of about 250 MPa (36 ksi), and design codes such as AISC apply a safety factor or resistance factor to that value to set the allowable or factored design stress, always check the applicable code for the exact allowable value.