The shaft diameter calculator above sizes a circular shaft that is twisted and bent at the same time, which is what almost every real transmission shaft actually experiences. A pulley, sprocket or gear does not only apply a torque. It also pulls sideways, and that sideways pull becomes a bending moment which peaks somewhere between the bearings. A shaft sized for torque alone is a shaft sized for half the problem.
Arb Digital publishes free engineering calculators that make the reader supply the material data rather than pulling it from a hidden table. Allowable stress is not a property of steel. It is a decision about steel, made by whoever is responsible for the machine, after they have accounted for surface finish, the keyway, the size effect, corrosion, temperature and the safety factor their work is held to. Every allowable figure on this page is an input for that reason.
What This Shaft Diameter Calculator Does
It takes the torque the shaft transmits, the bending moment at the section you are worried about, the shock and fatigue factors you have chosen, and your two allowable stresses. From those it computes the minimum diameter demanded by each of the two failure theories and reports the larger as the governing answer.
Two theories appear because they can disagree, and which governs depends on the ratio between your two allowable stresses; showing both means the page never quietly picks one. The equivalent twisting moment compresses the combined bending and torsion into a single moment that would produce the same maximum shear stress on its own. Finally, the tool takes the diameter you actually plan to fit — almost always a stock size or a bearing bore rather than the computed minimum — and reports the shear stress that appears in it, so you can see the real margin instead of assuming one.
How to Use It
- Decide where the torque comes from. If you have a motor nameplate, use the power and speed mode. If you have a measured torque, a stall figure or a specified value from the machine builder, switch modes and enter it directly.
- Work out the bending moment yourself. The tool cannot know your bearing spacing or belt pull direction. Draw the shaft, put the reactions in, and take the maximum moment at the section you are sizing.
- Choose the shock and fatigue factors from the code you work to. They separate a steady drive from one that starts against a locked load.
- Enter your own allowable stresses. Both should already include your safety factor and any reduction for a keyway, fillet, press fit or surface condition.
- Compare the answer against the size you can actually buy. Enter the stock diameter in the last field and read the stress that appears in it, so the real margin is visible rather than assumed.
The Formula: How Shaft Diameter Is Calculated
Torque from power is T = P ÷ ω, with ω = 2πN ÷ 60 for a speed N in revolutions per minute. That relation is the rotational form of power equals force times velocity, and OpenStax sets out the rotational dynamics behind it in 10.7 Newton's Second Law for Rotation.
The two loads are combined into an equivalent twisting moment Te = √((KmM)² + (KtT)²). Under the maximum shear stress theory the diameter follows from d³ = 16 Te ÷ (π τallow (1 − k4)), where k is the ratio of bore to outside diameter and is zero for a solid shaft.
Under the maximum normal stress theory the governing quantity is the equivalent bending moment Me = ½(KmM + Te), and the diameter follows from d³ = 32 Me ÷ (π σallow (1 − k4)). Both expressions are the standard circular-section results, which come from dividing the applied moment by the section modulus. The underlying definitions of shear stress, normal stress and elastic modulus are laid out in OpenStax's 12.3 Stress, Strain, and Elastic Modulus.
Work the defaults through by hand. At 15 kW and 1,450 rev/min the angular speed is 2π × 1450 ÷ 60 = 151.84 rad/s, so the torque is 15,000 ÷ 151.84 = 98.79 N·m. With a bending moment of 250 N·m and Km = 1.5, the factored bending term is 375 N·m, or 375,000 N·mm. The factored torque term is 98,786 N·mm. The equivalent twisting moment is √(375,000² + 98,786²) = 387,793 N·mm. Then d³ = 16 × 387,793 ÷ (π × 40) = 49,375 mm³, giving d = 36.69 mm. The normal stress route gives Me = 381,397 N·mm and d³ = 32 × 381,397 ÷ (π × 80) = 48,560 mm³, so d = 36.48 mm. The shear criterion governs by a fraction of a millimetre. In a 40 mm shaft the actual shear stress is 16 × 387,793 ÷ (π × 40³) = 30.9 MPa.
Why the Bending Term Usually Wins
Look at the worked example again. The torque is 99 N·m and the bending moment is 250 N·m before any factor is applied. After the bending factor the two differ by nearly four to one, and because they combine as the root of a sum of squares, the smaller one barely registers. The equivalent twisting moment is 387,793 N·mm against a pure bending term of 375,000 N·mm — the entire torque contribution moved the answer by about three per cent.
That has a practical consequence people miss. Doubling the transmitted power here would not come close to doubling the required diameter, because the torque is not what is sizing it. Moving the pulley 40 mm closer to the bearing cuts the bending moment in direct proportion and shows up immediately.
The reverse case exists. A shaft running between closely spaced bearings with the load between them, or a pure torque tube, has a small bending moment and the torque dominates. There the power rating really does drive the size, and the tool prints both terms so you can see which regime you are in.
What a Keyway Does to the Answer
A keyway is a rectangular slot cut into the shaft surface at exactly the radius where torsional and bending stresses are highest. It removes material and, much worse, introduces two sharp internal corners that concentrate stress well above the nominal value these formulas predict.
The formulas on this page describe a plain circular section. They do not know about your keyway, your circlip groove, your shoulder fillet or your press fit, and none of those effects can be recovered afterwards from the numbers on the screen. The place they belong is in the allowable stress you enter. If your design procedure calls for a reduction at a keyed section, apply it to the allowable figure before you type it in, and the diameter that comes out will already reflect it.
The same logic applies to the fillet where a shaft steps up to a shoulder. A generous radius is one of the cheapest fatigue improvements available on a rotating part. The factor of safety calculator is the right place to track how much margin all of these effects have consumed.
Fatigue, Not Static Strength, Is What Breaks Rotating Shafts
A shaft that rotates under a constant bending moment does not experience a constant stress. Every point on its surface travels from full tension to full compression and back on every revolution. At 1,450 rev/min that is more than two million fully reversed cycles a day. The relevant material property is therefore a fatigue strength at the required number of cycles, not the yield or ultimate strength quoted on a datasheet.
This is the single most important reason the allowable stress here is an input. A static calculation using a yield-derived allowable sizes a shaft that is comfortably strong on the first revolution and cracked after a few months. The shock and fatigue factors bridge that gap in the classical method, but only as well as the source you took them from.
The torsional component is usually steady while the bending component reverses, so the two are not equivalent in fatigue even when their magnitudes match. The classical combined-moment approach here is a preliminary sizing method, not the last line of a calculation.
Hollow Shafts and Where the Material Actually Works
Both torsion and bending put the highest stress at the outside surface and nothing at all at the centre. Material near the axis contributes almost nothing to strength while contributing its full share of weight, which is why hollow shafts are so effective. The strength penalty is captured by the factor (1 − k4), and because it is a fourth power it stays close to one for a long time.
A bore ratio of 0.5 leaves 93.75 per cent of the strength while removing a quarter of the mass, and 0.6 still leaves 87 per cent, which is why driveshafts and aircraft structures are tubes. The catch is that thin-walled tubes fail in ways solid shafts do not. Torsional buckling of the wall, local denting and joint design at the ends all become live concerns, and none of them appear in the strength arithmetic. If you are pushing the bore ratio high, the wall thickness has become a separate design problem. The polar moment of inertia calculator gives the underlying section property for both solid and hollow circular sections, and the section modulus calculator does the same for the bending side.
Deflection and Critical Speed Often Govern Before Stress
A shaft can be perfectly adequate on stress and still be the wrong size. A shaft that deflects too much at a gear mesh runs noisily and wears unevenly regardless of how far it is from failing, and bearings have their own limits on angular misalignment at the journal.
Deflection scales with the fourth power of diameter while stress scales with the third, so on long shafts with precision components it is common for the deflection limit to set the diameter and for the stress margin to end up embarrassingly large. The beam deflection calculator handles that side of the check with the same load and support geometry.
The third constraint is dynamic. Every shaft has a critical speed at which its natural bending frequency coincides with the rotational frequency, and long slender shafts can reach it inside the operating range. Passing one of these three checks says nothing about the other two.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Sizing for torque alone — on most real shafts the bending moment is the larger term, and ignoring it can undersize the shaft by a wide margin.
- Entering yield strength as the allowable stress — a rotating shaft under bending is a fatigue problem, and a yield-derived allowable with no reduction and no safety factor is not a design value.
- Forgetting the keyway — the slot sits exactly where the stress is highest and concentrates it further. Its effect belongs in the allowable stress you enter, and there is no way to add it afterwards.
- Using the same shock factor for every drive — a directly coupled centrifugal load and a reciprocating compressor started against pressure are not the same duty, and the factors that separate them come from the code you are working to.
- Stopping at the stress check — deflection at the gear mesh, angular misalignment at the bearings and critical speed are separate limits, and any one of them can govern the diameter on its own.
Related Free Tools From Arb Digital
The torque calculator converts between power, speed and torque in whatever units your nameplate uses, and the shear stress calculator works the stress in a section you have already chosen. For a combined-stress check on a section under several load components at once, the von Mises stress calculator applies the distortion energy criterion. On the drive side, the gear ratio calculator gives the torque at each shaft in the train, and bolt torque calculator covers the fasteners holding the assembly together. Everything Arb Digital publishes sits on the free online tools hub.
Frequently Asked Questions
Because allowable stress is not a material property. It is a design decision that depends on the safety factor your work is held to, the fatigue life you need, whether the section carries a keyway or a fillet, and the consequences of failure. The allowable stresses on this page are inputs so that the responsibility for them stays where it belongs.
The tool reports the larger diameter as the governing answer, which is the conservative reading and normal engineering practice. Which theory produces it depends on the ratio of your two allowable stresses; ductile materials are usually assessed on the shear criterion. Showing both means you can see whether the choice mattered.
They are combined shock and fatigue factors that multiply the bending moment and the torque respectively in the classical shaft design method, accounting for loading that is not steady. Their values come from the code you are working to, and different codes give different figures for what sounds like the same duty.
Less than most people expect, because the strength factor is (1 minus k to the fourth power) where k is the ratio of bore to outside diameter. A bore of half the outside diameter leaves about 94 per cent of the strength while removing a quarter of the mass. The separate concern with thin walls is buckling and end joint design.
No. The formulas describe a plain circular section. A keyway removes material and, more importantly, adds two sharp internal corners that concentrate stress locally. If your design procedure calls for a reduction at a keyed section, apply it to the allowable stress before entering it, so the diameter that comes out already reflects it.
Because a rotating shaft under a constant bending moment sees fully reversed stress at every point on its surface once per revolution. At normal machine speeds that is millions of cycles a day. The material property that matters is fatigue strength at the required life, not yield or ultimate strength. A shaft sized on a static basis can be comfortably strong on its first turn and cracked within months.
No. Deflection at a gear mesh, angular misalignment at the bearing journals and critical speed are separate limits, and any one of them can require a larger diameter than the stress calculation does. Deflection scales with the fourth power of diameter while stress scales with the third, so stiffness requirements grow more demanding faster as loads increase.
This tool is provided for educational and preliminary engineering use only. It applies classical combined-stress shaft formulas to figures you supply, and it takes no responsibility for material selection, allowable stress, fatigue assessment, stress concentration or code compliance. A shaft is a load-bearing rotating component whose failure can injure people, and its design must be reviewed and signed off by a qualified mechanical engineer working to the applicable standard.
