Round Beam Section Formulas for Bending and Torsion

A solid circular section is common in shafts, pins, rods, and round beams. Its bending and torsional properties are compact to calculate, but the load case and the kind of stress must be identified first. This table collects the section properties, elementary stress equations, and two standard elastic deflection cases.

Round section reference table

MECHANITEC / ENGINEERING REFERENCE

Round beam section formulas

Elementary section and load formulas for a solid circular beam or shaft.

1–7 of 7 matching rows (7 total)

Round beam section formulas — page 1 of 1
Quantity or load case	Formula for a solid circular section	Applicability
Centroidal second moment of area	I = pi d^4/64	Bending about a centroidal axis; d is the section diameter
Elastic section modulus	S = I/(d/2) = pi d^3/32	Use sigma_max = M/S for elastic bending
Polar moment for torsion	J = pi d^4/32	Solid circular section under elastic torsion
Extreme-fiber bending stress	sigma_max = 32 M/(pi d^3)	Linear-elastic beam theory; excludes local stress concentration
Maximum torsional shear stress	tau_max = 16 T/(pi d^3)	Elastic Saint-Venant torsion for a solid circular shaft
Cantilever with end point load P	delta_max = P L^3/(3 E I)	Ideal support, prismatic beam, small deflection
Simply supported beam with center point load P	delta_max = P L^3/(48 E I)	Ideal simple supports, prismatic beam, small deflection

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Symbols and unit handling

d is the solid diameter, I is the area moment of inertia about a centroidal bending axis, J is the polar second moment for torsion, and S is the elastic section modulus. For a solid circle, I = πd⁴/64, J = πd⁴/32, and S = I/(d/2) = πd³/32. Diameter enters to the third or fourth power, so even a modest size or unit error can cause a large result error.

In a consistent N-mm system, moments and torques are in N·mm, dimensions in mm, and elastic modulus E in N/mm² (MPa). The resulting bending and torsional stresses are MPa. Convert all quantities before substitution; do not mix meters with millimeters.

Bending, torsion, and deflection

For a circular section under elastic bending, the extreme-fiber normal stress is σ = M/S = 32M/(πd³). Under elastic torsion, the maximum shear stress is τ = Tc/J = 16T/(πd³), where T is torque and c = d/2. These are different load modes and stress components. A combined bending-and-torsion design needs an appropriate failure criterion and any required factors; neither equation alone establishes capacity.

The reference includes the tip deflection of a cantilever with a point load, δ = PL³/(3EI), and the center deflection of a simply supported span carrying a centered point load, δ = PL³/(48EI). In both expressions P is force and L is the unsupported length. The support arrangement changes the answer substantially, so do not use one row for a different boundary condition or load position.

As a quick scale check, increasing the diameter by 10% increases I by about 46% because I is proportional to d⁴. The same diameter increase lowers the calculated bending stress for the same moment by about 25% because stress is proportional to 1/d³. These relationships are useful for catching a mistaken exponent in a hand calculation.

Assumptions and limits

The formulas assume a prismatic, homogeneous, linearly elastic member with small deflection and idealized supports. They do not include stress concentrations, keyways, threads, shoulders, shear deformation, yielding, fatigue, residual stress, vibration, or instability. Deflection equations also assume the stated load and support model. Use a more complete shaft or beam analysis when the geometry or loading departs from these assumptions.

Sources

The section-property and stress relations follow standard mechanics of materials. See MIT OpenCourseWare’s introductory civil engineering design notes and MIT OpenCourseWare’s Solid Mechanics beam-deflection notes. Verify boundary conditions and design criteria for the actual component.