A shaft is a rotating or stationary component which is normally circular in section.   A shaft is normally designed to transfer torque from a driving device to a driven device.   If the shaft is rotating, it is generally transferring power and if the shaft is operating without rotary motion it is simply transmitting torque and is probably resisting the transfer of power.    A shaft which is not rotating and not transferring a torque is an axle.

Mechanical components directly mounted on shafts include gears, couplings, pulleys, cams, sprockets,links and flywheels.   A shaft is normally supported on bearings.  The torque is normally transmitted to the mounted components using pins, splines, keys, clamping bushes, press fits, bonded joints and sometimes welded connections are used. These components can transfer torque to/from the shaft and they also affect the strength of the shaft an must thereore be considered in the design of the shaft.

Shafts are subject to combined loading including torque ( shear loading ), bending ( tensile & compressive loading ), direct shear loading, tensile loading and compressive loading.  The design of a shaft must include consideration of the combined effect of all these forms of loading.    The design of shafts must include an assessment of increased torque when starting up, inertial loads, fatigue loading and unstable loading when the shaft is rotating at critical speeds (whirling).

Notes:
For information on the bending and shear stresses resulting from side and axial loads refer to Beam Index
For information on the critical speed of shafts refer to webpage Shaft Critical Speeds
For information on the fatigue considerations refer to webpage Fatigue index
For information on the design requirements of keys & splines Keys/splines index

When designing a shaft the following staged procedure is normally adopted

Stage 1 -- Produce a free-body diagram

Stage 2 -- Produce a bending Moment diagram

Stage 3 -- Produce a Torque diagram

The resulting forces on a shaft resulting from different drive components are listed below

The various equations required to evaluate the stresses and strains in a loaded shaft and to arrive at safe operating loads are to be found on other webpages. The simple equations below related to the estimating the torque resulting from a transmitted power and the surface shear stresses resulting from a transmitted torque.

Assuming the shaft is made from a ductile material the failure criteria applicable is the Maximum Shear Stress Theory (MSST) criteria. ref. Failure Criteria


This is very relevant to ductile metals. It is conservative and relatively easy to apply.  It assumes that failure occurs when a maximum shear stress attains a certain value.   This value being the value of shear strength at failure in the tensile test.  In this instance it is appropriate to choose the yield point as practical failure. If the yield point = S_y and this is obtained from a tensile test and thus is the sole principal stress then the maximum shear stress S_sy is easily identified as S_y /2 .

S_sy = S_y /2

Maximum Shear Stress = τ _max = Greatest of      ( σ₁ - σ₂ ) / 2   :    ( σ₂ - σ₃ ) / 2   :    ( σ₁ - σ₃ ) / 2   =     ( σ₁ - σ₃ ) / 2

The factor of safety selected would be

FoS = S_y / ( 2 . τ _max )    =   S_y / ( σ₁ - σ₃ )

For the simple case of a tensile stress σ_x combined with a shear stress τ _xy .   with the principal stress σ₃ = 0. and     σ_y ,σ_z τ _xz , τ _zy = 0. .. (ref to notes on Mohrs circle) .. Mohr's Circle The two non-zero principal stresses are

The resulting FoS =

Or in terms of Torque and bending moments.

Important note:
The above equations are very basic equations taking no account of fatigue conditions, stress concentration factors, etc.. It is also necessary to consider the linear/angular deflection of the shaft to ensure it is within acceptable limits. A variation of the above equation ( based on equation in Machinerys Handbook ) taking into account combined fatigue and shock loading is provided below.

	Type of Load	Km	Kt
	Gradually applied load and steady	1,5	1,0
	Suddenly applied/ minor shocks	1,5-2,0	1,0 - 1,5
	Suddenly applied heavy shocks	2,0 -3,0	1,5 - 3,0

It is important that a shaft should not only have the strength to transmit the specified torque and load without the risk of failure but also that shaft deflections due to bending and torque are within acceptable limits.    The limiting of the angular /radial deflection is required by the need to reduce vibration and ensure reliable operation of the related components including gears, splines bearings etc.    The design procedure for a shaft should therefore include an appraisal of the resulting deflections when the shaft is loaded. If it is deemed necessary to calculate the angular and linear deflections due to bending at critical points then various methods are available.. some of these methods are identified on webpage Shaft Deflections. Notes on torsional deflections are provided below

Angular Deflection_Due to bending

Radial deflection due to bending.

The maximum radial deflection at the teeth for spur gearing = 0.01 * m and for bevel and worm wheels = 0.005 * m (m = module of gear wheel (mm)). The suggested maximum deflection (not at gears) for general engineering = 0.0003 * L and for machine tools = = 0.0002 * L .(L...distance between bearings).

Torsional Angular Deflection

The angular deflection of a shaft is obtained by the equation

In Machinerys handbook 27th ed a safe maximum deflection of 0,26 degrees per m length (0,0045 rads/m) is recommended. Therefore

This table is provided to allow comparison between shafts and is based on very simplistic assumptions with no allowance for fatigue, additional stresses to Bending Moments etc etc etc

τ = The skin torsion stress of a solid round shaft : T= The torque transmitted by the shaft : T = τ.π.D³/16
The table is based on a torsion stress level of 25 N/mm²
Power transmitted by a shaft P = 2 * π * T * N (N = Revs /sec)
Table power based on pure torque values