Calculate area, moment of inertia, section modulus, neutral axis distance, and radius of gyration for common machine-design shapes. Use these values with beam deflection and bending stress checks when sizing brackets, rails, shafts, tubes, frames, and automation structures.
Select a shape and enter its dimensions. The calculator returns area, moment of inertia, distance to the outer fiber, section modulus, and radius of gyration.
These section properties describe how the shape resists bending. They do not depend on the material strength. Steel, aluminum, and plastic with the same shape have the same section modulus and moment of inertia, but they behave differently because their material properties differ.
Moment of inertia is a geometric stiffness property. Higher I means the shape is harder to bend for a given material, span, and load.
Section modulus is used to estimate bending stress. Higher S means lower bending stress for the same bending moment.
The c value is the distance from the neutral axis to the farthest outside surface where bending stress is highest.
The calculator uses standard geometric formulas for common symmetric sections. The bending direction matters, especially for rectangular bars and rectangular tubes.
Rectangle:
A = b × h
I = b × h³ / 12
c = h / 2
S = I / c
Solid Round:
A = πd² / 4
I = πd⁴ / 64
c = d / 2
S = I / c
Round Tube:
A = π(OD² - ID²) / 4
I = π(OD⁴ - ID⁴) / 64
c = OD / 2
S = I / c
Rectangular Tube:
A = b×h - (b - 2t)(h - 2t)
I = [b×h³ - (b - 2t)(h - 2t)³] / 12
c = h / 2
S = I / c
Radius of Gyration:
r = √(I / A)
Section properties are not the final answer. They are the bridge between shape selection, bending stress, and deflection.
Start with a flat bar, tube, shaft, or structural member that fits your space, mounting, and manufacturing constraints.
Moment of inertia feeds directly into beam deflection checks. If I is too small, the part may sag, bounce, vibrate, or lose alignment.
Section modulus feeds directly into bending stress. If S is too small, the part may be overstressed even if it looks physically large.
If this member supports a shaft, gearbox, conveyor, cylinder, or robot fixture, check the related motion, bearing, and fastening calculations too.
A common mistake is adding material in the wrong direction. For bending, material farther from the neutral axis usually helps more than simply making the part wider.
For a rectangular section, height in the bending direction has a much stronger effect than width because height is cubed in the moment-of-inertia formula.
Solid round sections are common for pins, rollers, shafts, and pivots. They are simple, but may not be the most efficient option for bending stiffness per weight.
Tubes are often efficient because they move material away from the center. This can improve stiffness without making a solid, heavy part.
If a machine component is not stiff enough or has high bending stress, geometry usually matters more than material changes.
Section properties are most useful when paired with load, stress, deflection, bearing, and fastening checks.
Use moment of inertia to estimate how much a beam, rail, bracket, or frame member will bend.
Open Beam Deflection →Use section modulus to estimate bending stress and safety factor against yield.
Open Bending Stress →Check bearing life when shafts, rollers, or rotating supports are part of the design.
Open Bearing Life →Check torque demand when the structural member is part of a moving axis.
Open Servo Torque →Check bolted joints when plates, brackets, frames, or tooling mounts are clamped together.
Open Clamp Load →Return to the full machine design workflow for structures, motion, bearings, and fastening.
Open Machine Design Hub →The right shape can reduce bending stress and deflection more effectively than simply picking a stronger material. Calculate the section first, then check stress and deflection.