Shafting
Transmission elements (e.g. belts, gears) induce
torsional and transverse loads, and possibly axial
loads (e.g. with helical, bevel and worm gears) on
shafts.
The associated deformations are twist, deflection,
and possibly longitudinal strain and buckling.
Although the loads may be steady, the deflection of
the rotating shaft induces completely reversed
bending in the “fibers” of the shaft (with alternating
compression and extension), and bending fatigue
phenomena.
Transverse load
Torsional load
Transverse load Axial load
Twist due to
torsional loads
Torsional load
Deflection due to
transverse loads
Transverse load
1
Axial load
Preliminary shaft considerations
Shaft length determined by
- max size, weight and speed restrictions
Position of elements
- steps (usually in 20% increments), shoulders, fits
- setscrews to secure transmission elements in place
Torque transmission
- use keyways or splines
Bearings
- put them as close as possible to line of action
- shaft deflection and slope must be controlled:
gear mesh separation should be less than 0.13 mm
(0.005 inch), and relative slope at the mesh less than 0.03o.
for rolling element bearings, the slope should be less
than 0.04o unless self aligning bearing are used.
2
Preliminary shaft considerations
Other considerations
- avoid gears machined on shafts because expensive
- size gear or sheave hub width d such that d > 1 ½ D
with steel or d > 2 D with cast iron, where D is shaft
diameter
- all steels have same bending rigidity, go cheap!
Hollow shafts
- allow for weight reductions
- allow fluids to circulate to moving parts
- may not have enough material for shouldering
- may require split rings, tapered hubs
* 3
Experience-based guidelines
1. Try to make the shaft as short, stiff and light as
possible
2. Try to use straddle-mounted bearing support, and
minimize length of overhang
3. Try to place shaft support bearings close to lateral
bending loads
4. Try to avoid more than two support bearings per shaft
5. Try to configure the shaft so that stress concentration
sites do not coincide with high stress regions
6. Check specific manufacture guidelines for constraints
on shaft deflection or slope related to gears, bearings,
sprockets, cams, shaft seals
7. Pay attention to overall assembly!
4
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Design steps to follow
1. Based on requirements, generate conceptual sketch
for the shaft
2. Identify potential failure modes
3. Select tentative shaft material
4. Based on loads and geometry, draw shear force, total
bending moment, axial force and torque diagrams;
calculate (equivalent) stresses at each potential
critical point; include stress concentration factors
5. Select appropriate design safety factor
6. Calculate a tentative shaft diameter for each critical
section, based on strength analysis
7. Check deflection and slopes of tentative design to
assure proper function of gears and bearings
8. Check torsional deflection if necessary
9. Check critical speed if necessary
Notes: Steps 8 to 9 are optional, depending on shaft
geometry and use. Skip them (and possibly step 7) for
stubby shafts.
Ignore axial force first to find shaft diameters as
follows; then include axial force (if necessary) to
verify if diameters are okay.
5
Strength analysis in fatigue
Ductile material + fatigue Modified Goodman criterion
using Von Mises stress for alternating stress
and maximum principal stress for mid-range stress.
Include appropriate stress concentration and column factors
( 1 for tensile axial force).
Consider torque T , bending moment M , and axial force F :
Solid shaft Hollow shaft
16 K fsT 16 K fsTd o
Torsion: xy xy
d3 (d o 4 di 4 )
32 K f M 32 K f Md o
Bending: x x
d3 (d o 4 di 4 )
4 Kf F 4 Kf F
Axial: x x
d2 (d o 2 di 2 )
With a alternating, and m mean,
2 2
ea xa 3 xya
2
xm xm 2
em xym
2 2
Modified Goodman criterion with safety factor n:
ea em 1 6
Sn Su n
Strength analysis in fatigue
If the axial force is ignored or negligible:
For a solid shaft,
16 2 2 16
ea 4 K f Ma 3 K fsTa A
d3 d3
16 2 2 16
em K f Mm K f Mm K fsTm B
d3 d3
The modified Goodman criterion can be recast
to find the shaft diameter d :
16n A B
d 3
Sn Su
1 do
For a hollow shaft, replace 3
by 4 4
.
d (d o di )
7
Deflection and slope
• Perform a beam deflection analysis as required
Torsional deflection
• Perform an analysis as required
• Permissible angle of twist ranges from 0.08 o per foot
for machine tool shafts to 1.0 o per foot for line
shafting
• Torsional rigidity may also be important in cam or
gear-driven mechanisms
Consider torque T , angle of twist , length L and
E
modulus of elasticity in shear G :
2(1 )
Solid shaft Hollow shaft
32TL 32TL
Twist (rad):
Gd 4 G (d o 4 di 4 )
* 8
Critical speed
• Design for critical speed > 2 times operating speed
