Reference · Methodology

Shaft Deflection & Critical Speed

Shigley's 11th Ed. Ch. 4 · Timoshenko · API 610

Complete methodology for calculating maximum deflection, bending stress, and critical speed for rotating shafts under point and distributed loads.

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[01]Nomenclature

Symbol	Description	SI Unit	US Unit
L	Shaft length	mm	in
d	Shaft diameter	mm	in
E	Elastic modulus	GPa	psi
I	Moment of inertia	mm⁴	in⁴
P	Point load	N	lbf
w	Distributed load	N/mm	lbf/in
δ_max	Maximum deflection	mm	in
σ_max	Maximum bending stress	MPa	psi
M_max	Maximum bending moment	N·mm	in·lbf
N_cr	Critical speed	RPM	RPM

[02]Equations

Moment of Inertia (Solid Circular Shaft)

I = \frac{\pi d^4}{64}

Second moment of area for a solid circular cross-section. For hollow shafts, use I = π(d_o⁴ − d_i⁴)/64.

Deflection — Simply Supported, Center Point Load

\delta_{\max} = \frac{PL^3}{48EI}

Maximum deflection at midspan for a simply supported beam with a concentrated load at the center.

Deflection — Simply Supported, Uniform Load

\delta_{\max} = \frac{5wL^4}{384EI}

Maximum deflection at midspan for a simply supported beam with a uniformly distributed load.

Deflection — Cantilever, End Point Load

\delta_{\max} = \frac{PL^3}{3EI}

Maximum deflection at the free end of a cantilever beam with a concentrated load at the tip.

Deflection — Cantilever, Uniform Load

\delta_{\max} = \frac{wL^4}{8EI}

Maximum deflection at the free end of a cantilever with a uniformly distributed load.

Maximum Bending Stress

\sigma_{\max} = \frac{M_{\max} \cdot c}{I} = \frac{32 M_{\max}}{\pi d^3}

Maximum bending stress at the outermost fiber, where c = d/2 for a solid circular shaft.

Critical Speed (Rayleigh)

N_{cr} = \frac{30}{\pi}\sqrt{\frac{g}{\delta_{\max}}}

First critical speed (natural frequency) of the shaft. Operation should stay below 0.7·N_cr or above 1.3·N_cr to avoid resonance. Per API 610, critical speed margin should be at least 20% above or below operating speed.

[03]Worked Example

500 mm steel shaft, 25 mm diameter, simply supported, 1 kN center point load.

Step 1 — Moment of Inertia

I = \frac{\pi (25)^4}{64} = 19{,}175 \text{ mm}^4

Step 2 — Maximum Deflection

\delta_{\max} = \frac{1000 \times 500^3}{48 \times 200{,}000 \times 19{,}175} = 0.0679 \text{ mm}

Step 3 — Maximum Bending Moment

M_{\max} = \frac{PL}{4} = \frac{1000 \times 500}{4} = 125{,}000 \text{ N·mm}

Step 4 — Maximum Bending Stress

\sigma_{\max} = \frac{32 \times 125{,}000}{\pi \times 25^3} = 81.49 \text{ MPa}

Step 5 — Critical Speed

N_{cr} = \frac{30}{\pi}\sqrt{\frac{9810}{0.0679}} = 3{,}628 \text{ RPM}

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[04]Assumptions & Limitations

--Assumes linear elastic material behavior (Hooke's law)
--Formulas are for prismatic (constant cross-section) shafts only
--Neglects shear deformation (valid for L/d > 10)
--Critical speed formula assumes a single concentrated mass; multi-mass systems require Dunkerley's method
--Does not account for bearing compliance, keyway stress concentrations, or dynamic loads
--Fixed-fixed support assumes perfectly rigid supports

[05]References

[1]Budynas, R.G. & Nisbett, J.K. — Shigley's Mechanical Engineering Design, 11th Ed., Ch. 4

[2]Timoshenko, S.P. — Strength of Materials, Part I, Ch. 4–5

[3]API 610, 12th Ed. — Centrifugal Pumps for Petroleum, Petrochemical and Natural Gas Industries

[4]Roark, R.J. & Young, W.C. — Roark's Formulas for Stress and Strain, 8th Ed.

Related References

Bolt Torque Reference→Spring Design Reference→

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