Drawing No. EH–ME–003 // Mechanical Engineering
Bending Stress Calculator
Reviewed August 2026
Find maximum bending stress from an applied bending moment and cross-section shape, for rectangular, solid circular or hollow circular (tube) sections, with an optional yield-strength margin check.
What problem does this solve?
Bending stress is the most fundamental structural and mechanical design check for beams, shafts and any member loaded in bending — it directly determines whether a cross-section is adequate for a given moment. This calculator computes the section's second moment of area and distance to the outer fiber automatically for common shapes, then finds maximum bending stress and, if a yield strength is entered, the resulting factor of safety.
Inputs
Results
Background
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 cross-section's second moment of area about the bending axis. Maximum stress always occurs at the outer fiber, farthest from the neutral axis.
Rectangular: I = bh³/12, c = h/2. Solid circular: I = πd⁴/64, c = d/2. Hollow circular (tube): I = π(Do⁴ − Di⁴)/64, c = Do/2. These match the same section-property relationships used throughout EngineerHub's Civil, Structural & Mechanical formula sheet.
S = I/c combines both section properties into one convenient value — bending stress simplifies to σ = M/S. Section modulus is often the value published directly in standard steel-shape tables, since it's what's needed for a quick capacity check without recalculating I and c separately.
Frequently asked questions
Practical questions about inputs, assumptions and interpretation.
Bending stress is governed by material's distance from the neutral axis — material near the neutral axis contributes little to bending resistance because it experiences very little stress. A tube concentrates material away from the neutral axis (at the outer radius) where it does the most good, which is why hollow sections can achieve a high strength (and stiffness) to weight ratio compared to an equivalent solid section.
This calculator is specifically a yield-strength margin check when that optional field is entered. Do not substitute ultimate strength and interpret the result as a yield factor of safety: allowable stress, ultimate failure, fatigue and code-specific resistance factors are separate design criteria.
No — this calculates pure bending stress only. Most beams and shafts also carry transverse shear stress (τ = VQ/Ib) at the same cross-section, which typically peaks at the neutral axis rather than the outer fiber where bending stress peaks — the two don't simply add at the same point, and a full check considers both separately or via combined-stress theory where they do coincide.