Chapter 12 shaft design
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The big picture, you are the designer
12-1 objectives of this chapter
12-2 shaft design procedure
12-3 forces exerted on shafts by machine elements
12-4 stress concentrations in shafts
12-5 design stresses for shafts
12-6 shafts in bending and torsion only
12-7 shaft design example
12-8 recommended basic sizes for shafts
12-9 additional design examples
12-10 spreadsheet aid fro shaft for shaft design
The big picture
• Identify examples of mechanical systems that incorporate
power-transmitting shafts.describe their geometry and the
forces and torques that are exerted on them.
• What kinds of stresses are produced in the shaft?
• How are other elements mounted on the shaft?how does
the shaft geometry accommodate them?
• How is the shaft supported?
• What kinds of bearings are used?
• A shaft is the component of a mechanical device
that transmits rotational motion and power. It is
integral to any mechanical system in which power
is transmitted from a prime mover, such as an
electric motor or an engine, to other rotating parts
of the system.
• Gear-type speed reducers,belt or chain drives,
conveyors, pumps, fans,agitators, and many types
of automation equipment.
• Household appliances, lawn maintenance
equipment, parts of a car, power tools, and
machines around an office or in your workplace.
• Visualize the forces,torques, and bending moments that are
created in the shaft during operation.
• The the process of transmitting power at a given rotational
speed, the shaft is inherently subjected to a torsional
moment, or torque. Torsional shear stress is developed in
the shaft.
• A shaft usually carries power-transmitting components,
such as gears,belt sheaves, or chain sprockets, which exert
forces on the shaft in the transverse direction
(perpendicular to the its axis). These transverse forces
cause bending moments to be developed in the
shaft,requiring analysis of the stress due to bending.
• In fact, most of shafts must be analyzed for combined
stress.
• Describe the specific geometry of shafts from some
types of equipment that you can examine.
• Make a sketch of any variations in geometry that may
occur,such as changes in diameter, to produce shoulders,
grooves, keyseats, or holes.
• How are any power-transmitting elements held in
position along the length of the shaft?
• How are the shafts supported?typically ,bearing…
• What kind of bearings are used?
• You can find much diversity in the design of the
shafts in different kinds of equipment.
• The functions of a shaft have a large influence on
its design.
• Shaft geometry is greatly affected by the mating
elements such as bearings, couplings,gears, chain
sprockets, or other kinds of power-transmitting
elements.
12-1 objectives of this chapter
• Propose reasonable geometries for shafts to carry a variety
of types of power-transmitting elements, providing for the
secure location of each element and the reliable
transmission of power.
• Compute the forces exerted on shafts by gears,belt sheaves,
and chain sprockets.
• Determine the torque distribution on shafts.
• Prepare shearing force and bending moment diagrams for
shafts in two planes.
• Account for stress concentration factors commonly
encountered in shaft design,
• Specify appropriate design stresses for shafts.
• Apply the shaft design procedure recommended
by the standard,ANSI B106.1M-1985, Design of
Transmission Shafting, to determine the required
diameter of shafts at any section to resist the
combination of torsional shear stress and bending
stress.
• Specify reasonable final dimensions for shafts that
satisfy strength requirements and installation
considerations and that are compatible with the
elements mounted on the shafts.
轴的类型
• 根据承受载荷的不同，轴可分为转轴、传动轴和心轴
三种。
• 转轴既承受转矩又承受弯矩，如减速器的转轴；
• 传动轴主要承受转矩，不承受或承受很小的弯矩；如
汽车的传动轴
• 心轴只承受弯矩而不传递转矩；
按轴线的形状轴可分为：直轴，曲轴和挠性轴。
轴的结构设计
• (1)满足制造安装要求，便于加工、方便 装拆；
• (2)满足定位要求；
• (3)满足工艺性要求；
12-2 shaft design procedure
• Because of the simultaneous occurrence of torsional shear
stresses and normal stresses due to bending, the stress
analysis of a shaft virtually always involves the use of a
combined stress approach.
• The recommended approach for shaft design and analysis
is the distortion energy theory of failure.
• Vertical shear stresses and direct normal stresses due to
axial loads also occur at times, but they typically have such
a small effect that can be neglected. On very short shafts or
on portions of shafts where no bending or torsion occurs,
such stresses may be dominant.
Procedure for design of a shaft
• 1. Determine the rotational speed of the shaft.
• 2.determine the power or the torque to be transmitted by
the shaft.
• 3. determine the design of the power-transmitting
components or other devices that will be mounted on the
shaft, and specify the required location of each device.
• 4.specify the location of bearings to support the shaft. The
reactions on bearings supporting radial loads are assumed
to act at the midpoint of the bearings.
• Another important concept is that normally two and only
two bearings are used to support a shaft.they should be
placed on either side of the power-transmitting elements if
possible to provide stable support for the shaft and to
produce reasonably well-balanced loading of the bearings.
The bearings should be placed close to the powertransmitting elements to minimize bending moments.
• The overall length of the shaft should be kept small to keep
deflections at reasonable levels.
• Propose the general form of the geometry for the
shaft,considering how each element on the shaft will be
held in position axially and how power transmission from
each element to the shaft is to take place.
• Determine the magnitude of torque that the shaft sees at all
points. It is recommended that a torque diagram be
prepared.
• Determine the forces that are exerted on the shaft, both
radially and axially.
• Resolve the radial forces into components in
perpendicular directions,usually vertically and horizontally.
• Solve for the reactions on all support bearings in each
plane.
• Produce the complete shearing force and bending moment
diagrams to determine the distribution of bending moments
in the shaft.
• Select the material from which the shaft will be made, and
specify its condition:cold-drawn,heat-treated, and so on.
• Determine an appropriate design stress,considering the
manner of loading.
• Analyze each critical point of the shaft to determine the
minimum acceptable diameter of the shaft at that point in
order to ensure safety under the loading at that point. In
general, the critical points are several and include those
where a change of diameter takes place, where the higher
values of torque and bending moment occur, and where stress
concentrations occur.
• Specify the final dimensions for each point on the shaft.
Design details such as tolerances, fillet radii, shoulder heights,
and keyseat dimensions must also be specified. Sometimes
the size and tolerance for a shaft diameter are dictated by the
element to be mounted there.
12-3 forces exerted on shafts by machine elements
• Gears, belt sheaves, chain sprockets, and other
elements typically carried by shafts exert forces on
the shaft that cause bending moments.
Spur gears
T  63000( P) / n
(12 - 1)
Wt  T /( D / 2)
(12 - 2)
where p  power being transmitt ed in hp
n  rotational speed in rpm
T  torque on the gear in lb  in
D  pitch diameter of the gear in inches
The angle between the total force and the tangential component
is equal to the pressure angle,,of the tooth form. Thus, if the
tangential force is known, the radial force can be computed
directly from
Wr  Wt tan 
(12 - 3)
and there is no need to compute the total force at all.
the pressure angle is typically 14.5,20, or 25.
Helical gears
• In addition to the tangential and radial forces encountered
with spur gears,helical gears produce an axial force.
Wr  Wt tan  n / cos
(12 - 4)
where
Wt  tan gential force;
  the helix angle
 n  normal pressure angle
Wr  radial load
Wx  Wt tan 
where Wx  axial load
(12 - 5)
Bevel gears
• Refer to chapter 10 to review the formulas for the
three components of the total force on bevel gear
teeth in the tangential, radial, and axial directions.
Worms and wormgear
• Chapter 10 also gives the formulas for computing
the forces on worms and wormgears in the
tangential, radial, and axial directions.
Chain sprockets
The upper part of the chain is in tension,referred to the tight side,
and produces the torque on either sprocket.
The lower part of the chain, referred to as the slack side, exerts no
force on either sprocket.
Forces in chain
Fc  T /( D / 2)
(12 - 6)
where D  pitch diameter of the sprocket
Fc acts along the direction of the tight side of the chain.
Because of the size difference between the two sprockets, that
direction is at some angle from the centerline between the
shaft centers. A precise analysis would call for the force,Fc, to
be resolved into components parallel to the centerline and
perpendicular to it. That is
Fcx  Fc cos 
and
Fcy  Fc sin 
Where the angle  is the angle of inclination of the tight side of the
chain with respect to the x-direction
• These two components of the force would cause bending
in both the x-direction and the y-direction. Alternatively,
the analysis could be carried out in the direction of the
force Fc, in which single plane bending occurs.
• If the angle  is small, little error will result from the
assumption that the entire force, Fc, acts along the xdirection. Unless stated otherwise, this book will use this
assumption.
V-belt sheaves
• Both sides of the V-belt are in tension. The tight side
tension,F1, is greater than the “slack side” tension,F2, and
thus there is a net driving force on the sheaves equal to
FN  F1  F2
(12 - 7)
the magnitude of the net driving force
can be computed from the torque transmitt ed
FN  T /( D / 2)
(12 - 8)
• But notice that the bending force on the shaft carrying the
sheave is dependent on the sum, F1  F2  FB
But unless the two sprockets are radically different in diameter,
little error will result from
FB  F1  F2
To determine the bending force,FB, a second equation involving
the two forces F1 and F2 is needed. This is provided by assuming a
ratio of the tight side tension to the slack side tension. For v-belt
drives, the ratio is normally taken to be F / F  5
1
2
• A relationship between FN and FB is derived as following:
FB  CFN
(12 - 10)
where C  constant t o be determined
F
F  F2
C B  1
(12 - 11)
FN F1  F2
From equation (12-9), then
C=1.5
FB=1.5FN=1.5T/(D/2)
(12-12)
Flat-belt pulleys
• The analysis of the bending force exerted on shafts by flatbelt pulleys is identical to that for V-belt sheaves except
that the ratio of the tight side to the slack side tension is
typically taken to be 3 instead of 5. Using the same logic as
with V-belt sheaves, we can compute the constant C to be
2.0, then for flat-belt drives.
FB  2.0 FN  2.0T /( D / 2)
(12 - 13)
Flexible coupling
• More detailed discussion of flexible coupling was
presented in chapter11.
• A flexible coupling is used to transmit power
between shafts while accommodating minor
misalignment in the radial, angular, or axial
directions. Thus, the shafts adjacent to the
couplings are subjected to torsion, but the
misalignment cause no axial or bending loads.
12-4 stress concentrations in shafts
• In order to mount and locate the several types of machine
elements on shafts properly, a final design typically
contains several diameter,keyseats,ring grooves, and other
geometry discontinuities that create stress concentrations.
• These stress concentrations must be taken into account
during the design analysis. But a problem exists because
the true design values of the stress concentration factors,Kt,
are known at the start of the design process. Most of the
values are dependent on the diameters of the shaft and on
the fillet and groove geometries, and these are the
objectives of the design.
• You can overcome this dilemma by establishing a
set of preliminary design values for commonly
encountered stress concentration factors, which can
be used to produce initial estimates for the
minimum acceptable shaft diameters.
• Then you can analyze the final geometry to
determine the real values for stress concentration
factors after the refined dimensions are selected.
• Comparing the final values with the preliminary
values will enable you to judge the acceptability of
the design.Figures A15-1 and A15-4.
preliminary design values for Kt
• Considered here are the types of geometric discontinuities
most often found in power-transmitting shafts: keyseats,
shoulder fillets, and retaining ring grooves. In each case, a
suggested design value is relatively high in order to
produce a conservative result for the first approximation to
the design.
• The final design should be checked for safety. That is , if
the final value is less than the original design value, the
design is still safe. Conversely, the stress analysis must be
checked.
keyseats
• A keyseat is a longitudinal groove cut into a shaft for the
mounting of a key,permitting the transfer of torque from
the shaft to a power-transmitting element, or vice versa.
• Two types of keyseats are most frequently used: profile
and sled runner.
• The profile keyseat is milled into the shaft, using an end
mill having a diameter equal to the width of the key. The
resulting groove is flat-bottomed and has a sharp, square
corner at its end. The sled runner keyseat is produced by a
circular cutter having a width equal to the width of the key.
As the cutter begins or ends the keyseat, it produces a
smooth radius. For this reason, the stress concentration
factor for the sled runner keyseat is lower than that for the
profile keyseat.
• Normally used design values are
K t  2.0
(profile)
K t  1.6
(sled runner)
Each of these is to be applied to the bending stress calculation for
the shaft, using the full diameter of the shaft. The factors take into
account both the reduction in cross section and the effect of the
discontinuity.
See reference 5
Shoulder fillets (轴肩圆角)
• When a change in diameter occurs in a shaft to create a
shoulder against which to locate a machine element, a
stress concentration dependent on the ratio of the two
diameters and on the radius in the fillet is produced. It is
recommended that the fillet radius be as large as possible
to minimize the stress concentration.
• We can classify fillets into two categories: sharp and wellrounded, sharp describes a shoulder with a relatively small
fillet radius.
• We use the following values for design for bending:
K t  2.5( sharp fillet )
K t  1.5( well  rounded fillet )
Referring to the chart for stress concentration factors in figure
A15-1,these values correspond to ratio of r/d of
approximately 0.03 for the sharp fillet case and 0.17 for the
well-rounded for a D/d ratio of 1.50。
Retaining ring grooves
• Retaining rings are used for many types of locating tasks in
shaft applications. The rings are installed in grooves in the
shaft after the element to be retained is in place. The
geometry of the groove is dictated by the ring manufacturer.
Its usual configuration is a shallow groove with straight
side walls and bottom and a small fillet at the base of the
groove. The behavior of the shaft in the vicinity of the
groove can be approximated by considering two sharpfilleted shoulders positioned close together. Thus, the stress
concentration factor for a groove is fairly high.
• When bending exists, we will use Kt =3.0 for
preliminary design as an estimate to account for
the fillets and the reduction in diameter at the
groove to determine the nominal shaft diameter
before the groove is cut.
When torsion exists along with bending, or when only
torsion exists at a section of interest, the stress
concentration factor is not applied to the torsional
shear stress component because it is steady.
• To account for the decrease in diameter at the
groove,increase the resulting computed diameter
by approximately 6%, a typical value for
commercial retaining ring grooves. But after the
final shaft diameter and groove geometry are
specified, the stress in the groove should be
computed with the approximate stress
concentration factor for the groove geometry.
12-5 design stresses for shafts
• In a given shaft, several different stress conditions can exist at
the same time. For any part of the shaft that transmits power,
there will be a torsional shear stress, while bending stress is
usually present on the same parts. There may be other parts
where only bending stresses occur. Some points may not be
subjected to either bending or torsion but will experience
vertical shearing stress. Axial tensile or compressive stresses
may be superimposed on the other stresses. Then there may be
some points where no significant stresses at all are created.
• The decision of what design stress to use depends
on the particular situation at the point of interest.
• In many shaft design and analysis projects,
computations must be done at several points to
account completely for the variety of loading and
geometry conditions that exist.
• Several cases discussed in chapter 5 for computing
design factors,N, are useful for determining design
stresses for shaft design. The bending stresses will
be assumed to be completely reversed and
repeated because of the rotation of the shaft.
• Ductile materials perform better under such loads.
Design shear stress-steady torque
• The best predictor of failure in ductile materials
due to a steady shear stress was the distortion
energy theory in which the design shear stress in
computed from
 d  s y /( N 3 )  (0.577 s y ) / N
(12 - 14)
we can use this value for steady tor sional shear stress,
vertical shear stress, or direct shear stress in a shaft.
Design shear stress-reversed vertical shear
• Points on a shaft where no torque is applied and where the
bending moments are zero or very low are often subjected
to significant vertical shearing forces which then govern
the design analysis. This typically occurs where a bearing
supports as end of a shaft and where no torque is
transmitted in that part of the shaft.
• The stress decreases in a roughly parabolic
manner to zero at the outer surface for the special
case of a solid circular cross section can be
computed from
 max  4V / 3 A
where V  vertical shearing force
A  area of the cross section
where stress concentrat ion factor are to be considered ,
 max  K t (4V / 3 A)
'
N  s sn
/  max
(5 - 15)
the distortion energy the ory is recommed
then the endurance strength in shear is
'
s sn
 0.577 s n'
then
N  0.577 s n' /  max
exp ressed as a design stress, this is
 d  0.577 s n' / N
now let
 max   d  k t (4V ) / 3 A, gives
K t (4V ) 0.577 s n'

3A
N
solving for N gives
0.577S 'n (3 A) 0.433s n'
N

K t (4V )
k t (V )
(12 - 15)
• Now, solving for the required area gives
K t (V ) N 2.31K t (V ) N
A

'
0.433S n
s n'
but our usual objective is the design of the shaft
to determine the required diameter.
A  D 2 / 4
D  2.94 K t (V ) N / S n'
(12 - 16)
• This equation should be used to compute the required
diameter for a shaft where a vertical shearing force V is the
only significant loading present. In most shafts, the
resulting diameter will be much smaller than that required
at other parts of the shaft where significant values of
torque and bending moment occur.
• Also, practical considerations may require that shaft be
somewhat larger than the computed minimum to
accommodate a reasonable bearing at the place where the
shearing force is equal to the radial load on the bearing.
But values for vertical shearing stress are rarely reported.
As an approximation, we will use the values for Kt for
torsional shear stress when using these equations.
Design normal stress-fatigue loading
• For the repeated,reversed bending in a shaft caused
by transverse loads applied to the rotating shaft, the
design stress is related to the endurance strength of
the shaft material. The actual conditions under which
the shaft is manufactured and operated should be
considered when specifying the design stress.
Procedure for computing estimated endurance
strength and design stress
• 1.determine the ultimate tensile strength of the
material,su,from test results,supplier specifications, or
published data. The most accurate and reliable data
available should be used. When there is doubt about the
accuracy of the data, larger than-average design factors
should be used.
• Determine the estimated endurance strength,sn, of the
material from figure 5-9, recall that the data in this figure
consider the manner in which the shaft is produced in
addition to the relationship between the basic endurance
strength and the ultimate strength. If the ultimate strength
is higher than the limit of figure 5-9, use the values for
su=220Ksi.
• 3. Apply a size factor Cs to account for the stress
gradient within the material and for the probability
of a given section having a damaging occlusion that
can serve as a place for initiation of a fatigue crack.
for diameter less than 2.0 in (D in Inches)
C s  ( D / 0.3) 0.068
for diameters less than 50mm (D in mm)
C s  ( D / 7.6) 0.068
for diameters less than 10in (D in inches)
C s  D 0.19
for diameters less than250mm (D in mm)
C s  1.85 D 0.19
Table 12-1
Desired reliability
reliability factors, CR
Reliability factor,CR
0.50
1.00
0.90
0.90
0.99
0.81
0.999
0.75
• 4. Apply a reliability factor,CR. Endurance strength data
typically reported are average values over many tests, thus
implying a reliability of 0.50 (50%). Assuming that the actual
failure data follow a normal distribution, the factors from table
12-1 can be used to adjust for higher levels of reliability.
Any stress concentration factor will be accounted for in the design
equation developed later. Other factors, not consideration here, that
could have an adverse effect on the endurance strength of the shaft
material, and therefore on the design stress, are temperatures above
approximately 400F(200C);variation in peak stress levels above
the nominal endurance strength for some periods of
time;vibration;residual stresses; case hardening; interference fits;
corrosion; thermal cycling; plating or surface coating; and stresses
not accounted for in the basic stress analysis.
'
s
5. Compute n  s n C s C R
• 6. For parts of the shaft subjected to only reversed
bending, let the design stress be
d  s / N
'
n
(12 - 17)
Design factor,N
• Under typical industrial conditions, the design
factor of N=3 is recommended. If the application
is very smooth, a value as low as N=2 may be
justified. Under conditions of shock or
impact,N=4 or higher should be used, and careful
testing is advised.
12-6 shafts in bending and torsion only
• Examples of shafts subjected to bending and torsion only
are those carrying spur gears, v-belt sheaves, or chain
sprockets. The power being transmitted causes the torsion,
and the transverse forces on the elements cause bending. In
the general case, the transverse forces do not all act in the
same plane. In such cases, the bending moment diagrams
for two perpendicular planes are prepared first. Then the
resultant bending moment at each point of interest is
determined.
• A design equation is now developed based on the
assumption that the bending stress in the shaft is
repeated and reversed as the shaft rotates, but that
the torsional shear stress is nearly uniform. The
design equation is based on the principle shown
graphically in figure 12-10 in which the vertical
axis is the ratio of the reversed bending stress to
the endurance strength of the material. The
horizontal axis is the ratio of the torsional shear
stress to the yield strength of the material in shear.
• The points having the value of 1.0 on these axes indicate
impending failure in pure bending or pure torsion,
respectively.experimental data show that failure under
combinations of bending and torsion roughly follows the
curve connecting these two points, which obeys the
following equation:
( / s n' ) 2  ( / s ys ) 2  1
(12 - 18)
we will use s ys  s y / 3 for the distorsion energy the ory.
then
(N/s 'n ) 2  ( N 3 / s y ) 2  1
then, int roduce a stress concentrat ion factor for bending in the first term only,
(k t N/s 'n ) 2  ( N 3 / s y ) 2  1
for rotating, solid, circular shafts,
the bending stress due to a bending moment, M, is
  M/S
(12 - 20)
where
S  D 3 / 32 is the recangular section modulus.
the torsional shear stress is
  T/Z p
(12 - 21)
where Z p  D 3 / 16 is the polar section modulus
therefore
  T/(2S)
• Substituting these relationships into equation(12-19) gives
K t NM 2
NT 3 2
[
] [
] 1
'
2Ss y
Ss n
let
D [
(12 - 22)
S  D 3 / 32
32 N

 kt M 
3T 
 '    
4  s y 
 sn 
2
3
2
(12 - 24)
12-8 recommended basic shafts
• In U.S. Customary Unit System,diameter are usually
specified to be common fractions or their decimal (十进制)
equivalents.
• Appendix 2 lists the preferred basic sizes that you can use
for dimensions over which you have control in decimalinch, fractional-inch, and metric units.
轴的强度计算
• 在工程设计中，轴的强度计算主要有三种方法：
扭转法、弯扭组合法和精确校核法；
1。按扭转强度计算
• 1．按扭转强度计算
• 对于仅承受扭矩或主要承受扭矩的传动轴，可用此法
计算。对承受弯扭复合作用的轴，通常用这种方法初
步估算轴径。对于不太重要的轴，也可作为最后计算
结果。
• 轴的扭转强度条件为
T
9.55P  10 6
3
T 
 10 
  T
3
WT
0.2d n
MPa （7-13）
• 式中，T为扭转切应力，MPa；T为轴所受的扭矩，
Nm；WT为轴的抗扭截面系数，mm3；n为轴的转速，
rpm；P为轴传递的功率，kW；d为计算截面处轴的直
径，mm；为许用扭转切应力，MPa，见附表8-1。
由上式可得轴的直径
9.55P  10 6
9.55  10 6 3 P
P
d 3
3

 A3
0.2 T  n
0.2 T
n
n
式中， A  3 9.55  10 6 / 0.2 T
可根据不同材料查附表8-1
在计算中，当弯矩相对扭矩很小或只受扭矩时，取较大
值，A取较小值；反之，取较小值，A取较大值。当轴上
有键槽时，会削弱轴的强度。因此，开一个键槽，轴径应增
大3%；开两个键槽，轴径应增大7%
• 对于空心轴，可将式（7-14）中的换成。
其中，=d1/d为空心轴的内径d1与外径d
之比。
2．按弯扭组合强度计算
• 根据已作出的总弯矩图和扭矩图（见图4-31），
求出计算弯矩Mca，其计算公式为
M ca  M  (T )
2
2
（7-15）
式中，为扭矩应力特性系数。当扭矩应力为静应力时，取为
0.3；为脉动循环应力时，取为0.6；为对称循环应力时，取为
1.0。
（7）校核轴的强度
• 已知轴的计算弯矩Mca后，即可针对某些危险截面作强
度校核计算。按第三强度理论计算弯曲应力。
 ca
M ca


W
M 2  (T ) 2
  1 
W
MPa
式中，W为轴的抗弯截面系数，mm3，计算公式见附表8-2；
为轴的许用弯曲应力。
• 3．精确校核
• 按弯扭组合计算适用于一般转轴的强度计算，而且是
偏于安全的。当需要精确评定轴的安全性时，应按精
确校核计算对轴的危险截面进行安全性判断。精确校
核计算包括疲劳强度和静强度校核计算两项内容。
• （1）按疲劳强度精确校核
• 在已知轴的外径、尺寸及载荷的基础上，通过分析确
定出一个或几个危险截面，求出计算安全系数Sca并使
其大于或至少等于设计安全系数S，即
S ca 
S  S
S2  S2
S
（7-17）
仅有法向应力时，应满足
S 
 1
S
K  a     m
仅有扭转切应力时，应满足
 1
S 
S
K  a     m