Shaft Diameter Calculator
Calculate Smarter. Work Faster.
Calculate the minimum diameter of a solid or hollow shaft from torque, power and rpm, bending moment, combined or fluctuating loads, or a permissible angle of twist — with every formula and unit handled for you.
Shaft Design Inputs
Pick a design basis, enter the loads and allowable stresses, and get the minimum shaft diameter instantly.
Enter values and hit calculate
Select a design basis to see which stress check governs the diameter.
Inner diameter is derived from the constant dᵢ/dₒ ratio you entered — it is not treated as a fixed bore, so it scales with the outer diameter.
Enter values above to see a step-by-step breakdown.
How Shaft Diameter Is Calculated
A shaft is a rotating member that transmits power between machine elements through a twisting moment. Because almost every shaft has a circular cross-section, shaft design effectively means calculating the shaft diameter that keeps every stress in the shaft inside its permissible limit. Three stresses matter:
- Shear stress from the torque (torsion) the shaft carries.
- Bending stress from mounted gears, pulleys and sprockets, plus the shaft's own weight.
- Combined stress where twisting and bending act together — the normal case in real machinery.
So before you can calculate shaft size, you have to decide what the shaft is actually loaded by. This calculator covers five common shaft-design bases used in machine design:
- Shaft transmitting twisting moment only.
- Shaft under bending moment only.
- Shaft under a combination of twisting and bending moments.
- Shaft under fluctuating twisting and bending moments (shock and fatigue factors).
- Shaft designed for a desired torsional rigidity (permissible angle of twist).
Each basis has its own formula, and each has a solid-shaft and a hollow-shaft form. Every diameter the calculator returns is a minimum — the next step in practice is always rounding up to a listed stock size your supplier actually carries.
Units used internally: torque and bending moment in N·mm, stresses in N/mm² (MPa) and lengths in mm, which returns the diameter directly in millimetres. You can enter data in any unit you like from the dropdowns; the calculator converts before computing.
Shaft Diameter for Twisting Moment Only
Some shafts carry torque alone, with any other load negligible — a propeller shaft or a plain line shaft is a typical example. These are sized directly from the torsion equation:
where T is the twisting moment, J is the polar moment of inertia about the axis of rotation, τ is the torsional shear stress and r is the distance from the neutral axis to the outermost fibre, which equals the shaft radius d/2 for a circular section.
Solid shaft. For a solid circular shaft, J = πd⁴/32 and r = d/2. Substituting both into the torsion equation gives the working formula:
Hollow shaft. For a hollow shaft, J = (π/32)(dₒ⁴ − dᵢ⁴) and r = dₒ/2. Writing k = dᵢ/dₒ for the ratio of inner to outer diameter simplifies the result to:
So a hollow shaft needs one extra input beyond torque and allowable shear stress: the diameter ratio k. Once the outer diameter is known, the bore follows as dᵢ = k × dₒ.
Shaft Diameter for Bending Moment Only
Shafts are rarely designed for bending alone — that case belongs more to axles — but it applies where torsional load is negligible. The permissible stress then comes from the bending equation:
where M is the bending moment, I is the moment of inertia of the cross-section, σ_b is the bending stress and y is the distance from the neutral axis to the outermost fibre.
Solid shaft. With I = πd⁴/64 and y = d/2:
Hollow shaft. With I = (π/64)(dₒ⁴ − dᵢ⁴), y = dₒ/2 and k = dᵢ/dₒ:
Note the difference from the torsion case: the constant is 32 instead of 16, because the section modulus in bending is half the polar section modulus in torsion for the same circular section.
Combined Twisting and Bending Moments
This is the realistic case for most machine shafts — anything carrying a gear, pulley, sprocket or overhung coupling twists and bends at the same time. Two classical theories are used, and the shaft is sized for whichever gives the larger diameter.
Maximum shear stress theory (Guest's theory) gives the maximum shear stress in the shaft as τ_max = ½√(σ_b² + 4τ²). Substituting the solid-shaft expressions τ = 16T/πd³ and σ_b = 32M/πd³ produces the equivalent twisting moment T_e:
Maximum normal stress theory (Rankine's theory) gives σ_b(max) = ½[σ_b + √(σ_b² + 4τ²)], which produces the equivalent bending moment M_e:
You therefore get two candidate diameters — one from T_e with the allowable shear stress, one from M_e with the allowable bending stress. Always take the larger of the two so the shaft survives both checks. The calculator computes both and tells you which one governs.
Hollow shafts use the same two equations with the (1 − k⁴) factor applied:
Fluctuating Twisting and Bending Moments
In service, torque and bending moment almost never stay constant. Starting, stopping, load swings and shock all raise the effective load well above the nominal figure, so the ASME approach multiplies each moment by a combined shock and fatigue factor before the equivalent moments are formed:
where K_m is the factor for bending and K_t the factor for torsion. These equivalent moments then go into exactly the same diameter equations as the steady-load case, for solid or hollow shafts.
| Load type (rotating shafts) | Km | Kt |
|---|---|---|
| Gradually applied or steady load | 1.5 | 1.0 |
| Suddenly applied load, minor shocks | 1.5 – 2.0 | 1.5 – 2.0 |
| Suddenly applied load, heavy shocks | 2.0 – 3.0 | 1.5 – 3.0 |
Choosing K values is a judgement call, so err on the higher side for crushers, mills, reciprocating compressors, and any drive with frequent direct-on-line starts or reversals. For a smoothly loaded fan or pump drive, the steady-load row is normally adequate.
Shaft Diameter for Torsional Rigidity
Some shafts are limited by twist rather than stress. A camshaft is the classic example: once the angle of twist grows beyond the limit set for that design — a figure such as 0.25° per metre is often quoted, though the actual limit depends on the engine and the manufacturer's requirements — valve timing shifts. Machine-tool lead shafts, long line shafts and indexing drives have similar limits. Here the torsion equation is used in its rigidity form:
where G is the modulus of rigidity (about 79.3 GPa for steel), θ is the permissible angle of twist in radians, and L is the shaft length. Substituting the polar moment of inertia:
Note the fourth root rather than the cube root — rigidity sizing scales with d⁴, so a small diameter increase buys a large reduction in twist. Enter θ in degrees in the calculator; the conversion to radians is handled internally.
Always check both. A shaft sized for rigidity may still fail on stress, and a shaft sized for stress may twist too much. For precision drives, run both bases and take the larger diameter.
Calculating Shaft Diameter from Power and RPM
On site you usually know the motor rating and speed, not the torque. Torque follows from the power relation:
Switch the torque input to From power & rpm in the calculator and both steps run together: power and speed become torque, and torque becomes diameter. For a motor, use the shaft speed at the point you are sizing — after a gearbox the torque rises in proportion to the reduction ratio, so a shaft on the slow-speed side of a 20:1 reducer carries roughly twenty times the motor torque and needs sizing accordingly.
If you only need the torque figure itself, the dedicated Shaft Torque Calculator does that conversion on its own.
Allowable Stress Values for Common Shaft Materials
| Material / basis | Allowable shear τ (MPa) | Allowable bending σb (MPa) | G (GPa) |
|---|---|---|---|
| ASME transmission-shaft practice — no keyway | 56 | 112 | 79.3 |
| ASME transmission-shaft practice — with keyway | 42 | 84 | 79.3 |
| C45 / EN8 — typical shafting steel (medium-carbon) | 40 – 60 | 70 – 90 | 79.3 |
| Alloy steel EN19 / EN24 | 70 – 90 | 120 – 150 | 79.3 |
| Stainless steel 304 | 30 – 40 | 55 – 70 | 77.0 |
| Cast iron | 25 – 35 | 45 – 60 | 41.0 |
| Aluminium 6061-T6 | 35 – 45 | 60 – 75 | 26.0 |
These are typical working values that already include a design margin — they are not raw yield strengths. The ASME rows are the figures most often quoted for transmission shafting in machine-design references, and the drop from 56 MPa to 42 MPa is that method's own allowance for a keyway — which is why adding a further diameter allowance for the same keyway would count it twice. Ranges for the other materials vary with grade, heat treatment and surface finish, so confirm the figure against your material certificate or design code before finalising a drawing.
Choosing a stronger material lets a smaller shaft carry the same torque, but the diameter often ends up dictated by the bearing bore, coupling bore or keyway anyway, so the strength gain is not always usable. Check what the mating components need before paying for an alloy upgrade.
Standard Shaft Sizes and Rounding Practice
| Calculated d | Round up to | Commonly stocked sizes (mm) |
|---|---|---|
| up to 20 mm | nearest 1–2 mm step | 8, 10, 12, 14, 15, 16, 17, 18, 19, 20 |
| 20 – 50 mm | nearest 2–5 mm step | 22, 24, 25, 28, 30, 32, 35, 38, 40, 42, 45, 48, 50 |
| 50 – 100 mm | nearest 5 mm step | 55, 60, 63, 65, 70, 75, 80, 85, 90, 95, 100 |
| 100 – 400 mm | nearest 10–20 mm step | 110, 120, 125, 130, 140, 150, 160, 170, 180, 190, 200, 220, 240, 250, 260, 280, 300, 320, 340, 360, 380, 400 |
| above 400 mm | nearest 20 mm step | 420, 440, 460, 480, 500… |
These are exactly the sizes the calculator rounds to, drawn from a common metric stock series, not a universal standard — what is actually available depends on your supplier, the material grade and the standard series your market follows, so always check a real stock list before committing. Bar stock, ready-made shafting and standard bearing bores exist only in discrete sizes, and the calculated minimum almost never lands exactly on one. Always round up, never down, and check the chosen size against the bore of every bearing, coupling and pulley the shaft has to mate with — in the 20–60 mm range typical of industrial drives, the bearing bore frequently dictates the shaft diameter before the torsion calculation does.
It is also worth confirming the size against your supplier's actual stocked diameters in your chosen grade. A size that needs a special mill order instead of a stock bar can cost more in lead time than the few millimetres are worth.
Shaft Diameter Calculation Examples
Example 1 — Shaft size to transmit 20 kW at 200 rpm
Torque first: T = 60P / (2πN) = (60 × 20,000) / (2π × 200) = 955 N·m = 955,000 N·mm. Taking an allowable shear stress of 42 MPa (ASME transmission shaft with a keyway) and using T = πτd³/16: d = (16 × 955,000 / (π × 42))1/3 ≈ 48.7 mm, rounded up to a 50 mm listed stock size.
Example 2 — Same shaft, but hollow with k = 0.5
The (1 − k⁴) factor is 1 − 0.5⁴ = 0.9375, so dₒ = (16 × 955,000 / (π × 42 × 0.9375))1/3 ≈ 49.8 mm with a bore of dᵢ = 0.5 × 49.8 ≈ 24.9 mm. The outer diameter grows only about 2%, while roughly a quarter of the cross-sectional area — and its weight — is removed. That trade is why weight-critical drive shafts are bored out.
Example 3 — Combined 100 N·m torque and 150 N·m bending moment
Equivalent twisting moment T_e = √(150² + 100²) = 180.3 N·m, giving d = (16 × 180,300 / (π × 42))1/3 ≈ 28.0 mm. Equivalent bending moment M_e = ½[150 + 180.3] = 165.1 N·m, giving d = (32 × 165,100 / (π × 84))1/3 ≈ 27.2 mm. The shear check governs, so the design diameter is 28.0 mm → use a 28 mm listed stock size. Sizing the same shaft for its 100 N·m torque alone would have given only 23.0 mm — a clear illustration of why the bending load cannot be ignored.
Example 4 — Camshaft sized for torsional rigidity
A 1 m long steel shaft (G = 79.3 GPa = 79,300 MPa) carrying 100 N·m with a permissible twist of 0.25° (0.004363 rad): d = (32 × 100,000 × 1000 / (π × 79,300 × 0.004363))1/4 ≈ 41.4 mm, rounded up to 42 mm. The same shaft sized for stress alone at 42 MPa would need only about 23 mm — proof that for a rigidity-limited shaft, twist, not strength, sets the size.
This calculator gives preliminary design diameters. It does not cover axial (thrust) loading, detailed fatigue life against an endurance limit, stress concentration at shoulders and fillets, critical (whirling) speed, or torsional vibration. For any safety-critical or high-value shaft, have the design verified by a qualified mechanical engineer against the applicable code before manufacture.
Common Mistakes When Calculating Shaft Diameter
1. Sizing for torsion only when the shaft also bends. Any shaft carrying a pulley, gear, sprocket or overhung coupling bends as well as twists. Pure-torsion sizing undersizes it — use the combined basis with both M and T.
2. Taking only one of the two combined-load answers. Guest's theory (T_e) and Rankine's theory (M_e) each give a diameter. The design value is the larger of the two, not whichever you calculated first.
3. Mixing N·m and N·mm. With τ in N/mm², torque must be in N·mm. Forgetting the ×1000 understates the required diameter by a factor of ten in the cube root's input.
4. Using the raw shear yield strength as the allowable stress. The allowable value already carries the design margin. Feeding in yield strength produces a shaft with essentially no factor of safety.
5. Applying the solid-shaft formula to a hollow shaft. Without the (1 − k⁴) factor, a hollow shaft's torque capacity is overstated. The higher the k, the bigger the error.
6. Ignoring the keyway. A keyway removes material and adds a stress raiser. Either increase the diameter by roughly 6–10% at that section or drop to the code's keyway allowable stress of 42 MPa.
7. Rounding down to a convenient stock size. The calculated figure is a minimum. Rounding down deletes the margin the whole calculation was built on.
8. Skipping the shock and fatigue factors on a shock-loaded drive. Crushers, mills, reciprocating compressors and reversing drives see peak torque far above nameplate. Use the fluctuating basis with realistic Km and Kt.
9. Forgetting the critical speed check on long, slender shafts. A shaft can be strong enough in torsion and still resonate destructively near its whirling speed. That is a separate dynamics check this calculator does not perform.
10. Sizing from motor torque on the slow side of a gearbox. Torque multiplies by the reduction ratio. Always size each shaft for the torque it actually sees at its own speed.
Frequently Asked Questions
Shaft diameter, shaft sizing and shaft design — answered
What is the formula for minimum shaft diameter under pure torsion? +
d = (16T / (π × τ_allow))^(1/3), where d is the shaft diameter, T is the twisting moment and τ_allow is the allowable shear stress. Keeping T in N·mm and τ_allow in N/mm² (MPa) gives d directly in millimetres. For a hollow shaft it becomes dₒ = (16T / (π × τ_allow × (1 − k⁴)))^(1/3), where k is the ratio of inner to outer diameter.
What shaft size is required to transmit 20 kW at 200 rpm? +
About 48.7 mm using an allowable shear stress of 42 MPa (ASME transmission-shaft practice with a keyway). First find torque: T = 60P / (2πN) = (60 × 20,000) / (2π × 200) = 955 N·m. Then T = πτd³/16 gives d = 48.7 mm, which is rounded up to a 50 mm listed stock size. A higher allowable stress — 56 MPa for the same shaft without a keyway — brings the required diameter down to about 44.2 mm.
How do I calculate shaft diameter from power and rpm? +
Convert power to torque first using T (N·m) = 60 × P (W) / (2π × N), or the shortcut T = 9550 × P(kW) / N(rpm), then substitute that torque into d = (16T / (πτ))^(1/3). Switching the torque input above to "From power & rpm" runs both steps automatically.
How is shaft diameter calculated for combined bending and torsion? +
Two diameters are calculated and the larger is used. Maximum shear stress theory gives the equivalent twisting moment T_e = √(M² + T²) = πτ_max·d³/16, and maximum normal stress theory gives the equivalent bending moment M_e = ½[M + √(M² + T²)] = πσ_b·d³/32. Whichever produces the bigger diameter governs the design.
What are the shock and fatigue allowance factors Km and Kt? +
They are allowance factors applied to the bending moment and the twisting moment respectively when loads fluctuate. For a rotating shaft with a gradually applied steady load, Km ≈ 1.5 and Kt ≈ 1.0. Suddenly applied loads with minor shock use roughly 1.5–2.0 for both; heavy shock uses Km 2.0–3.0 and Kt 1.5–3.0. Note that scaling the full moments this way is a preliminary design allowance, not a fatigue analysis — it does not separate mean and alternating stress or apply endurance-limit, size, surface-finish and notch factors, which a fatigue-critical shaft still needs.
How do you calculate hollow shaft diameter? +
Use the solid-shaft equations with an extra (1 − k⁴) factor, where k = dᵢ/dₒ. For torsion, T = (π/16) × τ × dₒ³ × (1 − k⁴); for bending, M = (π/32) × σ_b × dₒ³ × (1 − k⁴). Once dₒ is known, the bore is simply dᵢ = k × dₒ.
How is shaft diameter calculated for torsional rigidity? +
From T/J = Gθ/L, a solid shaft gives d = (32TL / (πGθ))^(1/4), with θ in radians, L the shaft length and G the modulus of rigidity (≈79.3 GPa for steel). A hollow shaft adds (1 − k⁴) in the denominator. This basis is used for camshafts and long line shafts where twist, not stress, is the limiting factor.
What is the maximum permissible shear stress for a transmission shaft? +
ASME transmission-shaft practice, as commonly published in machine-design references, takes 56 MPa where no keyway allowance is needed and 42 MPa where a keyway is present, with matching permissible bending stresses of 112 MPa and 84 MPa. Treat these as typical design figures and confirm them against the edition of the code or standard your project actually works to.
What allowable shear stress should I use for mild steel? +
For general-purpose shafting steels, a commonly used allowable shear stress falls in the 40–60 MPa range, already including a reasonable factor of safety against yield. Note that the usual shafting grades C45 and EN8 are medium-carbon steels rather than true low-carbon mild steel, and the two designations are broadly comparable rather than exact equivalents — so check the shear yield strength of your specific grade and apply your own required safety factor instead of assuming one universal figure.
What is the difference between a shaft and an axle? +
A shaft rotates and transmits torque and power between machine elements, so it is designed mainly for torsion or combined torsion and bending. An axle is normally stationary and supports a rotating body such as a car wheel, so it is designed mainly for bending moment.
Does a keyway reduce shaft strength? +
Yes. A keyway removes material and creates a stress concentration, so common practice either raises the calculated plain-shaft diameter by roughly 6–10% at that section (a rule of thumb, not a stress-concentration calculation — the real factor depends on keyway geometry, fillet radius and notch sensitivity) or drops to the keyway allowable stress of 42 MPa — one or the other, not both, otherwise the same keyway is allowed for twice. The calculator above uses the keyway allowable stress by default and warns you if you switch on the extra diameter allowance as well.
Should I round the calculated shaft diameter up to a standard size? +
Yes — always up, never down, to the nearest standard stock diameter that also suits your bearing and coupling bores. Rounding up only adds margin; rounding down removes the safety margin the calculation was based on.
Why is torsional rigidity important when designing camshafts? +
A camshaft controls valve timing, so once the twist along its length exceeds the permissible limit for that design, valve events shift out of position. A figure such as 0.25° per metre is often quoted as a typical criterion, but the actual limit is set by the application and the manufacturer's requirements. Either way, such shafts are sized for a permissible angle of twist rather than for stress alone.
Is a hollow shaft stronger than a solid shaft? +
For the same outer diameter, no — removing the core removes load-carrying material. For the same weight of material, yes — a hollow shaft with a larger outer diameter carries more torque than a smaller solid one, because material far from the axis contributes most to torsional strength. That is why weight-critical drive shafts are usually hollow.
Does this calculator cover fatigue life and critical speed? +
No. The fluctuating-load basis applies ASME shock and fatigue factors, which is a design allowance, not a full endurance-limit fatigue analysis with surface finish, size and notch factors. Critical (whirling) speed and torsional vibration are separate dynamic checks, essential for long or slender shafts, and are not covered here.
Explore More Categories
Electrical Calculators
Cable size, transformer size, motor current, DG size, solar sizing & more.
Browse all →Mechanical Calculators
Belt length, bearing life, cooling tower efficiency & more.
Browse all →Financial Calculators
EPF, PPF, SIP, gratuity, income tax, CAGR & more.
Browse all →Blog & Guides
Maintenance guides & engineering articles.
Browse all →