Calculate bending stress from bending moment and beam dimensions or M, y, and I, with section modulus and inertia results in common units.

Bending Stress Calculator

Use the beam-shape mode for the main workflow, or switch to the direct My/I stress mode for a quick section-property check.

Beam Shape Calculator
My/I Stress Calculator
Rectangle uses strong-axis bending about the section height, so stress is based on width and height.

Bending Stress Formula

The basic bending stress equation relates bending moment, distance from the neutral axis, and the second moment of area.

\sigma = \frac{M y}{I}
\sigma = \frac{M c}{I} = \frac{M}{S}
S = \frac{I}{c}
  • σ = bending stress
  • M = bending moment
  • y = distance from the neutral axis to the point where stress is being calculated
  • c = distance from the neutral axis to the extreme outer fiber of the section
  • I = second moment of area, also called area moment of inertia
  • S = section modulus

In the My/I stress mode, you enter M, y, and I directly. The calculator applies σ = My/I after converting the selected units to base units.

In the beam shape mode, you enter a common cross-section shape and its dimensions. The calculator finds I, sets c as the distance to the outermost fiber, computes S = I/c, and then calculates stress from σ = M/S.

The section property formulas used for the built-in shapes are:

I_{rectangle} = \frac{b h^3}{12}
I_{square} = \frac{a^4}{12}
I_{solid\ round} = \frac{\pi d^4}{64}
I_{hollow\ round} = \frac{\pi (D^4 - d^4)}{64}
I_{rectangular\ tube} = \frac{B H^3 - (B - 2t)(H - 2t)^3}{12}
  • b = rectangle width
  • h = rectangle height
  • a = square side length
  • d = diameter, or inner diameter for a hollow round section
  • D = outer diameter of a hollow round section
  • B = outer width of a rectangular tube
  • H = outer height of a rectangular tube
  • t = wall thickness of a rectangular tube

Common Bending Stress Units and Conversions

These conversions help you check whether the output unit is in the range you expect.

Stress Unit Equivalent Common Use
1 MPa 1 N/mm² Metric beam and material stress calculations
1 GPa 1000 MPa Large stress or modulus values
1 ksi 1000 psi US customary structural calculations
1 MPa 145.038 psi Comparing metric and US stress values

Shape Inputs Used for Section Properties

Shape Required Inputs Extreme Fiber Distance
Rectangle Width b, height h c = h/2
Square Side a c = a/2
Solid round Diameter d c = d/2
Hollow round Outer diameter D, inner diameter d c = D/2
Rectangular tube Outer width B, outer height H, wall thickness t c = H/2

Example Bending Stress Calculations

Example 1: Rectangular beam section

You have a rectangular section with a bending moment of 500 Nm, width 50 mm, and height 100 mm.

  • I = bh³/12 = 50 × 100³ / 12 = 4,166,666.67 mm⁴
  • c = h/2 = 50 mm
  • S = I/c = 4,166,666.67 / 50 = 83,333.33 mm³
  • M = 500 Nm = 500,000 Nmm
  • σ = M/S = 500,000 / 83,333.33 = 6 MPa

Example 2: Direct My/I calculation

You have a bending moment of 2 kNm, a distance from the neutral axis of 25 mm, and a moment of inertia of 8,000,000 mm⁴.

  • M = 2 kNm = 2,000,000 Nmm
  • σ = My/I = 2,000,000 × 25 / 8,000,000
  • σ = 6.25 MPa

Bending Stress Calculator FAQ

What is the difference between y and c?

y is any distance from the neutral axis where you want to find the bending stress. c is the maximum distance from the neutral axis to the outermost fiber of the cross-section. If you use c, the result is the maximum bending stress in that section.

Why does increasing height reduce bending stress so much?

For rectangular and tube sections, the moment of inertia depends strongly on height. A solid rectangle has I = bh³/12, so height is cubed. Increasing the section height usually increases stiffness and section modulus much more than increasing width by the same amount.

Does the calculator check whether the beam material is safe?

No. The result is the bending stress for the entered moment and cross-section. To judge safety, compare the calculated stress with an allowable stress for the material and design code you are using. You may also need to check shear, deflection, buckling, stress concentrations, load combinations, and connection details.