Find the second moment of area, the section modulus, and the maximum bending stress for a rectangular or circular cross-section under an applied bending moment. The calculator applies the flexure formula σ = M·c/I, giving the peak fiber stress you compare against the material's allowable stress.
When a beam bends, the material on the outer (convex) face stretches and the material on the inner (concave) face compresses, with the highest stress at the surfaces farthest from the neutral axis. The flexure formula links the applied bending moment to that peak stress through two geometric properties of the cross-section: the second moment of area and the section modulus. This calculator evaluates both for rectangular and solid circular sections and returns the maximum bending stress you check against the allowable stress of your material.
σ = M·c / I, which can also be written σ = M / S because the section modulus is defined as S = I / c.
M is the bending moment at the section, c is the distance from the neutral axis to the extreme fiber, I is the second moment of area, and S is the section modulus. The stress varies linearly from zero at the neutral axis to its maximum at the surface. This calculator works in millimetres: with M entered in N·m it converts to N·mm (×1000), so the resulting stress comes out directly in N/mm², which equals MPa.
The second moment of area I measures how the cross-sectional area is distributed about the neutral axis — area placed farther from the axis contributes far more because the distance is squared. For a rectangle bent about its horizontal centroidal axis, I = b·h³/12; for a solid circle, I = π·d⁴/64. The cubic and quartic dependence on depth is why a beam oriented with its long dimension vertical is dramatically stiffer and stronger in bending than the same beam laid flat.
The section modulus S = I/c bundles the geometry into a single number that, divided into the moment, gives the maximum stress directly: σ_max = M/S. A larger S means a lower stress for the same moment, so designers size beams by selecting a section whose S exceeds M/σ_allow. For a rectangle S = b·h²/6 and for a solid circle S = π·d³/32. Steel section tables list S for every standard shape precisely because it is the property that controls bending strength.
The neutral axis passes through the centroid of the section; it is the line of zero bending stress and zero longitudinal strain. Because stress grows with distance from this axis and stiffness grows with the square of that distance, putting material far from the neutral axis is enormously efficient. Doubling a rectangular beam's depth multiplies its moment of inertia by eight and its section modulus by four. This principle drives the design of I-beams, which concentrate material in flanges far from the neutral axis.
The second moment of area (often loosely called moment of inertia), I, governs deflection and stiffness and has units of length to the fourth power. The section modulus S = I/c governs maximum bending stress and has units of length cubed. Use I when calculating how much a beam deflects; use S when checking the peak bending stress.
All lengths are in millimetres and the moment is converted to N·mm, so I is in mm⁴, S is in mm³, and σ = M/S comes out in N/mm². One newton per square millimetre is exactly one megapascal, so the bending stress reads directly in MPa with no further conversion.
At the extreme fibers — the points of the cross-section farthest from the neutral axis, a distance c away. For a symmetric rectangle or circle, the top and bottom surfaces are equidistant from the neutral axis, so the maximum tensile and compressive stresses are equal in magnitude and occur at those surfaces.
No. The flexure formula gives only the normal (bending) stress due to the moment. Beams also carry transverse shear, which produces a separate shear-stress distribution that is maximum at the neutral axis. For most slender beams bending stress dominates, but short, deep beams should also be checked for shear.
Compute the maximum bending moment in your beam, then require the section modulus to satisfy S ≥ M / σ_allow, where σ_allow is the allowable stress (yield strength divided by a factor of safety). Pick a cross-section whose S meets or exceeds that value, then verify deflection and shear separately.