Design of Machine Element: Shaft
Shaft
A rotating Machine member, usually circular
cross section, used to transmit power or
motion.
Shaft, Axle, Spindle
M S Dasgupta BITS Pilani
1
Shafts
Shaft may be integral part of the driver (motor shaft, engine crank
shaft etc.) or may be freestanding shaft connected to another shaft
by coupling.
An axle is a non-rotating member that carries no torque and is used
to support rotating wheels, pulleys etc. (Exception->Automotive
axle)
Main design criteria is either strength or deflection & rigidity.
(Torsional rigidity / Lateral rigidity)
Deflection is not affected by strength, but by stiffness.
Mostly made of low carbon, cold-drawn or hot-rolled steel
Shaft Material
Cold drawn steel is more often used for small (dia<3in) and Hot
rolled steel used for larger sizes
Shafts usually don’t need to be surface hardened unless they serve as
the actual journal of a bearing surface
Typical material choices for surface hardened shaft include
carburizing grades of ANSI 1020, 4320, 4820, and 8620
Hot rolled steel should be machined all over to remove the carburized
outer layer.
Cast iron may be specified if the gears are to be integrally cast with
the shaft
Stainless steel may be appropriate for some environments
Mountings: Keys, Splines, Set Screws
Shaft Layout
• Generally, the geometry of a shaft is a stepped cylinder.
• Each shoulder in the shaft serves a specific purpose
Speed Reducer
Reducing Stress Concentration at shoulder
Suggested techniques for reducing stress concentration at a
shoulder supporting a bearing with a sharp radius.
(a) Large radius undercut into the shoulder.
(b) Large radius relief groove into the back of the shoulder.
(c) Large radius relief groove into the small diameter
Estimating Stress Concentrations
Stress
concentrations
for shoulders
and keyways are
dependent on
size
specifications
that may not be
known the first
time through the
process
But these
elements are
usually of
standard
proportions, it is
possible to
estimate the
stress
concentration
factors for initial
design of the
shaft
These stress
concentrations
are then finetuned in
successive
iterations, once
the details are
known
Estimating Stress Concentrations
First iteration estimates for Stress Concentration Factors Kt
Table 7–1
Design for static load (building up the formulas)
Consider a non rotating shaft subjected to bending Moment M and
Torsional moment T.
x 
32 M
16T
,


xy
d 3
d 3

By Max Shear Stress Theorem,  max 
 x  y
2
2   xy2

16
d 3
M 2 T 2
Now, let the safety factor be N
S yt
 32 N 

2
2
2
 M T 
N
 d  

 max
 S y 

Distrotion Energy Theorem   
When the shaft rotates but
Torsional load remain steady
1
3
 32 N  M 2 T 2 
d  



S y 
   S e
 2   22   1 2
1
3
1
When both2 bending &2Torsional load
2
2




 x
  x

2
  x      x   xy  has
  xy as well
 asxy fluctuatin
 part
3 xy
 steady
   x2 g
 2

 2

 2
  2
 32 N 32 N
Safety factor N 
 d  d   M 2

 S y 
S yt

1
3
M

 3T 2M
   a  m
Sy
 4 S e

2

Ta
T
 m
Se
Sy

2



1
3
Shaft Design for Strength
Design is for critical section. Axial stresses on shafts due to helical
gears or tapered roller bearings is almost always negligible as
compared to the stress due to bending moment or torsion
For Solid Shaft:
Where
Mm and Ma are the midrange and alternating bending moments,
Tm and Ta are the midrange and alternating torques, and
Kf and Kf s are the fatigue stress concentration factors for bending and torsion
Shaft Design for Stress
Combining these and for Von Mises stresses for rotating round,
solid shafts, neglecting axial loads, are given by
These equivalent alternating and midrange stresses can be
evaluated using an appropriate failure curve i.e Mod-Goodman,
Soderberg, Gerber’s or ASME elliptic
Shaft Design for Stress
For example, the fatigue failure criteria for the
modified Goodman line expressed as
Modified Goodman
Shaft Design for Stress
Similar expressions can be obtained for any of the
common failure criteria
ASME Elliptic
Soderberg
Shaft Design for Stress
Gerber
Problem:
A 50 mm diameter solid shaft is transmitting a power of 20
kW at a steady speed of 300 rpm is subjected to a
completely reverse bending moment of 16000 N-mm. The
shaft is made of forged steel (Sy = 480 MPa and Sut = 580
MPa) and surfaces are machined. The theoretical stress
concentration factor and notch sensitivity are 2.4, 0.78
respectively for bending and 2.2, 0.85 respectively for
torsion. Determine the factor of safety based on Goodman
theory.
Problem:
The rotating solid steel shaft is simply supported by bearings at points B and C
and is driven by a gear (not shown) which meshes with the spur gear at D, which
has a 150-mm pitch diameter. The force F from the drive gear acts at a pressure
angle of 20o. The shaft transmits a torque to point A of TA = 340 N-m. The shaft is
machined from steel with Sy = 420 MPa and Sut = 560 MPa. The notch
sensitivities are 0.8 and 0.9 for bending and torsion, respectively. Using a factor
of safety of 2.5, determine the minimum allowable diameter of the shaft based on
(a) a static yield analysis with fatigue stress concentration factors using the
distortion energy theory and (b) a fatigue-failure analysis. Assume sharp fillet
radii at the bearing shoulders for estimating stress concentration factors.
d 
d 
Ft    T  F cos    T
2
2
Problem:
The shaft shown in the figure is driven by a gear at the right keyway,
drives a fan at the left keyway, and is supported by two deep-groove
ball bearings. The shaft is made from AISI 1020 cold-drawn steel. At
steady-state speed, the gear transmits a radial load of 1.1 kN and a
tangential load of 3 kN at a pitch diameter of 200 mm. Determine
fatigue factors of safety at gear key way and fan key way locations.
For bending and torsion, the critical theoretical stress concentration
factors at both the key ways are 1.9 and 1.32 and notch sensitivities
are 0.75 and 0.92 respectively.
Problem:
A transmission shaft, supported by two bearings B1 and B2, rotating at 720 rpm and
transmitting power from the pulley to the spur gear G is shown in fig. The tension
and Gear teeth forces are P1= 475 N, P2= 155 N, Pt= 480 N and Pr= 200 N. The
weight of the pulley is 100 N. The diameter of the pulley is 120 mm and pith
diameter of gear is 80 mm. The fatigue stress concentration factors for bending
and torsion are 1.9 and 1.3 respectively. The shaft is made of AISI 1040 CD steel.
Using a factor of safety of 2, determine the minimum allowable diameter of the
shaft based on (a) a static yield analysis with fatigue stress concentration factors
using the distortion energy theory and (b) a fatigue-failure analysis.
Circlips / Snap rings
Design of Keys and Pins
Square Key
Round pin
Round Key
Taper pin
Round pin
Split tubular
spring pin
Design of Keys and Pins
Two special types of keys:
(a) Gib-head key, and
(b) Woodruff key
Design of Keys and Pins
Dimensions of rectangular and square keys: Table 7-6 and
Woodruff Keys: Table 7-7 and 7-8.
Most often the keys are subjected, as shown, to direct shear
along ab and crushing. Here the shear area is given by
A=l*t
Here l is the length of the hub.
T = P/(2N), where P is the power in Watts and N is the
speed of the shaft in RPS.
F = T/r, T= Torque being transferred.
F S sy
Direct shear : 
tl
n
Sy
F
Crushing :

tl 2 n
Loose Pins:
The loose pin mostly used for transmitting axial load
It is a double shear case, where there are two areas supporting the total shear
load. Hence shear stress = (1/2)(4V/2A)