Shaft Diameter Calculator: Torque, Bending, Fatigue & Critical Speed

Torque and drive forces

Mechanical torque is $T_m=P/\omega$, with an explicit rpm-to-angular-speed conversion. The checked range is $T_{min}=T_m-T_a$ to $T_{max}=T_m+T_a$. The input condition requires $0\leq T_a\leq T_m$: torque reversal is rejected because it requires a different gear contact or belt/chain direction model. Zero torque is included when $T_{min}=0$ or the complete range is zero.

For each endpoint, rim force is $F_t=2T/D$. For a spur gear $F_r=F_t\tan\alpha$; for parallel belt or chain runs with tight/slack ratio $r$, the transverse resultant is $F_t(r+1)/(r-1)$. Gear tangential direction is $\beta$, with +y downward and +z lateral. The radial-side selector gives $s=+1$ for $\beta+90^\circ$ or $s=-1$ for $\beta-90^\circ$:
$F_y=F_t\cos\beta-sF_r\sin\beta,\qquad F_z=F_t\sin\beta+sF_r\cos\beta.$
At $\beta=90^\circ$, the two sides therefore produce upward or downward radial force while preserving the same +z tangential force. Select the separating direction from the actual mesh geometry; SDP/SI §16.1 explains tangential and radial gear forces. Use known-force mode for a force system outside this description.

Derived endpoint forces include the same rotor weight at both endpoints. Known-force mode expects total forces including weight; these entered forces stay fixed over the torque range.

Why the endpoints enclose the whole range

With non-reversing torque, each derived force is affine in torque, and gravity is constant. Linear beam moments, displacements, slopes and reaction components are consequently affine in torque at every shaft position. The norm of an affine vector is convex. Therefore its greatest magnitude over this closed interval occurs at an endpoint. Component zero crossings and gravity cancellations inside the interval cannot create a larger magnitude. Both endpoint states are retained; no signed componentwise mixture is treated as a physical load state.

Equilibrium and stiffness

Reactions follow force and moment balance in each transverse plane. At each endpoint the bending resultant is $M=\sqrt{M_y^2+M_z^2}$. The reported bending, deflection, slope and radial-reaction magnitudes are the maxima over the two endpoints. Torque is confined to the shaft segment between the drive stations, or from bearing A to the single station. Euler–Bernoulli point-load solutions plus a uniform self-weight solution give displacement and its derivative, with zero displacement and zero applied end moment at the two bearings. The chart samples 201 equally spaced positions; rotor displacements, bearing slopes and moments at both load stations are evaluated directly. Small slope is reported in mrad. Maximum deflection and its position are sampled, rather than analytic extrema. MIT OCW Lecture 03, attached pp. 11–17, gives the beam relations and superposition assumptions; NPTEL Module 8 Lesson 2, pp. 6–8, gives the stiffness example.

Conservative diameter sizing

For a candidate diameter $d$, use $I=\pi d^4/64$ and the bending envelope $M_b(d)=M_{point,max}+w(d)L^2/8$. Self-weight is recalculated at every candidate diameter. For point loads, the resultant is convex between load stations, so its maximum occurs at a load station. $M_{point,max}$ is the greatest such moment across both torque endpoints. Adding the peak self-weight moment in the worst direction and applying full torque forms a conservative uniform-diameter envelope even when the real moment and torque peaks do not coincide in position or occur at different torque endpoints.

Nominal stresses are $\sigma_b=32M_b/(\pi d^3)$ and $\tau=16T/(\pi d^3)$. Static yield uses $\sigma_{VM}=\sqrt{(K_t\sigma_b)^2+3[K_{ts}(\tau_m+\tau_a)]^2}$.

A fixed endpoint transverse load produces fully reversed bending at a rotating point. For constant or slowly varying transverse loading, the calculation conservatively applies the worst endpoint bending envelope during every revolution: $M_a=M_b$, $M_m=0$. Set $K_f=1+q_b(K_t-1)$ and $K_{fs}=1+q_t(K_{ts}-1)$. DE equivalent stresses are $\sigma'_a=\sqrt{(K_f\sigma_b)^2+3(K_{fs}\tau_a)^2}$ and $\sigma'_m=\sqrt{3}K_{fs}\tau_m$. Modified Goodman gives $1/n_f=\sigma'_a/S_e+\sigma'_m/S_{ut}$. The standard elastic DE-Goodman treatment uses fatigue concentration factors for both mean and alternating components; elastic factors are retained separately for the peak-yield check. No local yielding or mean-stress relaxation is modeled.

The specimen estimate is $S'_e=\min(0.5S_{ut},700\,\mathrm{MPa})$. Surface, size, reliability, temperature and other modifiers multiply this value; the bending loading factor is one. Bisection finds each diameter at the requested safety factor. MIT Lecture 04, pp. 30–32 and 52–54, supports the endurance estimate, modifiers, Goodman line and notch sensitivity. NPTEL Module 8 Lesson 1, pp. 5–6, gives round-section stress equations; the calculator uses von Mises rather than the lesson's allowable-shear/ASME sizing procedure.

Rayleigh critical speed

Compute the gravity-only sag $y_g(x)$ from rotor weights and distributed shaft self-weight. With $\psi=y_g/y_{g,max}$ and rotor weights $W_i$,

Trapezoidal integration uses the 201-position grid; rotor terms use exact station sag. This extends the concentrated-weight formula in NPTEL Module 8 Lesson 2, p. 7, to include shaft mass through the same energy balance. It is an approximate upper estimate for the ideal first bending frequency. The displayed speed ratio is a preliminary subcritical screen, not a certified separation margin.

The beam superposition, conservative envelope, distributed-mass extension and numerical implementation are derived here from these mechanics relations. The animation shows elastic bending stress, $K_t\sigma_b\cos\theta$, at a material point using the conservative sizing envelope; torsional stress is reported separately.

• One uniform, solid round steel shaft; two ideal bearings at its ends, with no overhang. Both bearings restrain transverse displacement and allow bending rotation. The single-element mode puts the driving coupling at bearing A.
• Rotor masses are point masses without rotary inertia. Density and modulus are uniform. The self-weight switch controls both distributed shaft mass and shaft weight.
• Drive-force directions are stationary in space. Derived forces are checked at both non-reversing torque endpoints, with rotor gravity held fixed. Known forces stay constant. Endpoint envelope values may occur at different states for different positions; they are not a single simultaneous deflected shape.
• Fatigue treats the worst endpoint bending envelope as fully reversed on every revolution, with the entered mean and alternating torsion. This is a constant or slowly varying transverse-load model, not a synchronized load-history fatigue analysis.
• Belt/chain runs are parallel for the sum-of-tensions force approximation. No extra bearing preload or belt pretension is added beyond the entered tension ratio.
• Known forces include rotor weight. Their entered rotor masses still determine gravity sag for critical speed. The user must keep these two descriptions physically consistent.
• Illustrative properties are $S_y=400$ MPa, $S_{ut}=600$ MPa, $E=200$ GPa and density 7850 kg/m³; these do not designate a certified grade. Default notch factors 1.7 and 1.5 are starting assumptions for a rounded shoulder, not universal keyway values.
• Endurance modifiers and notch factors stay fixed during sizing. Re-evaluate the size factor, surface condition and actual shoulder/keyway geometry at the chosen final diameter.
• No finite-life cycle counting or variable-amplitude damage accumulation. A missing safety-factor value at exactly zero stress means no finite stress-based safety factor is needed. With station 2 disabled, its deflection card is blank. Drawings contain explicit SI annotations; numeric cards, charts and tables follow the metric/US display switch.

Use for preliminary sizing and teaching of slender steel shafts under elastic bending and torsion. A span/diameter ratio below 10 and deflections above 1% of span trigger approximation warnings. Deflection and slope targets must come from the application and component suppliers.

The required strength diameter is conservative and does not automatically satisfy stiffness or speed requirements. Increasing diameter changes self-weight and distributed mass. Re-enter the chosen size and review all targets.

Rayleigh assumes rigid supports and ignores bearing/housing flexibility, disk rotary inertia, gyroscopic coupling, damping, imbalance response and higher modes. Its upper estimate can overstate the actual first critical speed. The chosen operating/critical ratio is a user assumption; analyze the complete rotor near resonance or for any supercritical operation.

Torque reversal is excluded: reduce the amplitude to no more than the mean torque. Gear radial directions must follow mesh geometry; use known forces for other load systems. Synchronized force/rotation histories, load-direction changes, finite-life damage accumulation and resonant amplification of changing loads need a separate analysis.

No axial load, thrust, hollow section, stepped flexural rigidity, plastic notch correction, corrosion fatigue, fretting or contact stress is included. Notch factors account only for local stress amplification, not the stiffness of an actual stepped shaft. This is not a DIN 743 verification and embeds no standard tables or charts; check against the standard in force where required.