Chapter 12 shaft design

• 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 b

y 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 th

e shaft.• 3. determine the design of the power-transmitting compone

nts 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 power-transmitting 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 di

agrams 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

inchesin gear theofdiameter pitch D inlbin gear on the torqueT

rpmin speed n hpin ed transmittbeingpower p where

2)-(12 )2//(W1)-(12 /)(63000

t

rotational

DTnPT

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

.25or ,,2014.5 typicallyis angle pressure theall.at force total thecompute toneed no is thereand

3)-(12 tan

tr WW

Helical gears• In addition to the tangential and radial forces encountered

with spur gears,helical gears produce an axial force.

load W angle pressure

anglehelix the force; tan W

where4)-(12 cos/tan

r

n

t

radialnormal

gential

WW ntr

load Wwhere5)-(12 tan

x axialWW tx

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

sprocket theofdiameter pitch D where6)-(12 )2//(

DTFc

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

sinF F and cos ccy ccx FF

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 x-direction. 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

8)-(12 )2//(Fed transmitt torque thefrom computed becan

force drivingnet theof magnitude the7)-(12

N

21

DT

FFFN

• But notice that the bending force on the shaft carrying the sheave is dependent on the sum, BFFF 21

But unless the two sprockets are radically different in diameter, little error will result from 21 FFFB

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 5/ 21 FF

• A relationship between FN and FB is derived as following:

11)-(12 FF

C

determined be oconstant tC where10)-(12

21

21

N

B

FFFF

CFF NB

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 flat-belt 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.

13)-(12 )2//(0.20.2 DTFF NB

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 mountin

g 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

runner) (sled 6.1K(profile) 0.2

t tK

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 well-rounded, sharp describes a shoulder with a relatively small fillet radius.

• We use the following values for design for bending:

)fillet (5.1)fillet (5.2

roundedwellKsharpK

t

t

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 sharp-filleted 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

shaft. ain stressshear direct or stress,shear verticalstress,shear sionalsteady torfor value thisusecan we

14)-(12 /)577.0()3/( NsNs yyd

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

)3/4(,considered be toarefactor ion concentrat stress where

section cross theof area A force shearing verticalV

3/4

max

max

AVK

whereAV

t

15)-(12 )(

433.0)4(

)3(0.577SN

gives Nfor

577.03

)4(K

gives ,3/)4(let

/577.0

is thisstress,design a as exp/577.0

577.0s

isshear in strength endurance then the recommed isory energy the distortion the

15)-(5 /

''n

't

max

'd

max'

''sn

max'

Vks

VKA

solvingN

sAV

AVknow

Ns

ressedsN

thens

sN

t

n

t

n

td

n

n

n

sn

• Now, solving for the required area gives

16)-(12 /)(94.2

4/DA

diameter. required thedetermine toshaft theofdesign theis objective usualour

)(31.2433.0

)(

'

2

''

nt

n

t

n

t

SNVKD

buts

NVKSNVK

A

• 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.

19.0s

19.0s

068.0s

068.0s

85.1C

mm)in (D than250mmless diameters C

inches)in (D10in than less diameters )6.7/(C

mm)in (D 50mm than less diameters )3.0/(C

Inches)in (Din 2.0 than lessdiameter

D

forD

forD

forD

for

Table 12-1 reliability factors, CR

Desired reliability 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 400F(200C);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.

5. Compute

• 6. For parts of the shaft subjected to only reversed bending, let the design stress be

Rsnn CCss '

17)-(12 /' Nsnd

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:

1)/3()/sN(k

only, first term in the bendingfor factor ion concentrat stress a int,

1)/3()/s(N

then

ory.energy the distorsion for the 3/s use willwe

18)-(12 1)/()/(

22'nt

22'n

ys

22'

y

y

y

ysn

sN

roducethen

sN

s

ss

T/(2S)therefore

modulussection polar theis 16/ Zwhere

21)-(12 T/Z is stressshear torsionalthe

modulus.section recangular theis 32/DS

where20)-(12 M/S

isM,moment, bending a todue stress bending theshafts,circular solid,rotating,

3p

p

3

D

for

• Substituting these relationships into equation(12-19) gives

24)-(12 4332[

32/Slet

22)-(12 1]2

3[][

3

22

'

3

22'

yn

t

yn

t

sT

sMkND

D

SsNT

SsNMK

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 decimal-inch, 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

TT

T ndP

WT

3

63

2.01055.910 MPa （ 7-13 ）

• 式中， T 为扭转切应力， MPa ； T 为轴所受的扭矩， Nm ；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

333

6

3

6

2.01055.9

2.01055.9

nPA

nP

nPd

TT

式中， TA 3 6 2.0/1055.9 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

22 )( TMM ca （ 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。

MPaW

TMWM ca

ca 1

22 )(

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

SSS

SSSca

22

（ 7-17 ）

仅有法向应力时，应满足 S

KS

ma

1

仅有扭转切应力时，应满足 S

KS

ma

1