Bending Stress Calculator

Bending stress is a normal stress that develops within a beam when it is subjected to a bending moment. As the beam bends, material on one side experiences tension while the other side experiences compression. The maximum bending stress occurs at the outermost fibers of the cross-section, farthest from the neutral axis.

The maximum bending stress is calculated using the flexure formula: σ = M × c / I, where M is the applied bending moment, c is the distance from the neutral axis to the outermost fiber, and I is the second moment of area (moment of inertia) of the cross-section. This can also be written as σ = M / S, where S = I/c is the section modulus.

For a square beam with side length a, the moment of inertia is I = a⁴/12 and the distance to the outermost fiber is c = a/2. Substituting into σ = M × c / I gives σ = M × (a/2) / (a⁴/12) = 6M / a³. Simply enter the side length and bending moment in this calculator and select the Square cross-section type.

For a 200 mm wide × 300 mm deep rectangular beam: I = (200 × 300³) / 12 = 450,000,000 mm⁴ and c = 300/2 = 150 mm. For a bending moment of 5,000,000 N·mm, the bending stress σ = 5,000,000 × 150 / 450,000,000 ≈ 1.667 MPa. Use this calculator with those dimensions to get the result automatically.

Bending stress (normal stress) acts perpendicular to the cross-section of the beam and results from the bending moment. It is maximum at the top and bottom fibers and zero at the neutral axis. Shear stress, on the other hand, acts parallel to the cross-section and results from the transverse shear force — it is maximum at the neutral axis and zero at the outer fibers.

The section modulus S = I/c combines the moment of inertia and the distance to the outermost fiber into a single property of the cross-section. A higher section modulus means the beam can resist more bending moment for the same allowable stress. It is a key parameter in beam design: σ = M/S, so a larger S leads to lower bending stress.

The shape of a beam's cross-section significantly affects its ability to resist bending. I-beams and T-sections concentrate material far from the neutral axis, giving a high moment of inertia relative to their area, which reduces bending stress compared to solid rectangular or square sections of equal weight. Hollow sections also improve efficiency by removing material near the neutral axis where stress is lowest.

This calculator uses millimeters (mm) for all dimensions and Newton-millimeters (N·mm) for the bending moment, giving bending stress results in megapascals (MPa = N/mm²). To convert from other units: 1 kN·m = 1,000,000 N·mm; 1 N·m = 1,000 N·mm. Always ensure consistent units throughout your calculation.