files 2-lectures lec 31 ch-18 shafts and axles

Dr. A. Aziz Bazoune Chapter 18: Axles and Shafts

ME 307 Machine Design I

Dr. A. Aziz BazouneKing Fahd University of Petroleum King Fahd University of Petroleum

& Minerals& MineralsMechanical Engineering Mechanical Engineering

DepartmentDepartment



18-1 18-1 Introduction ……….Introduction ……….92292218-218-2 Geometric Constraints ……….Geometric Constraints ……….92792718-318-3 Strength Constraints ……….Strength Constraints ……….93393318-418-4 Strength Constraints – Additional Methods Strength Constraints – Additional Methods ……….……….94094018-518-5 Shaft Materials ……….Shaft Materials ……….94494418-618-6 Hollow Shafts ……….Hollow Shafts ……….94494418-718-7 Critical Speeds (Omitted) ……….Critical Speeds (Omitted) ……….945945 18-818-8 Shaft Design ……….Shaft Design ……….950950

CH-18 LEC 31 Slide 2



The stress analysis process for fatigue is highly dependent on stress concentrations.

Stress concentrations for shoulders and keyways are dependent on size specifications that are not known the first time through the process.

Fortunately, since 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 will be fine-tuned in successive iterations, once the details are known.

Estimating Stress Estimating Stress ConcentrationConcentration




Shoulders for bearing and gear support should match

the catalog recommendation for the specific bearing or

gear.

A look through bearing catalogs shows that a typical

bearing calls for the ratio of D/d to be between 1.2 and

1.5.

For a first approximation, assume D/d =1.5 can be

assumed.

Fillet radius at the shoulder needs to be sized to avoid

interference with the fillet radius of the mating

component. There is a significant variation in typical

bearings in the ratio of fillet radius r/d versus bore

diameter, with typically ranging from around 0.02 to

0.06.





Figures A-15-8 and A-15-9 show that the stress

concentrations for bending and torsion increase

significantly in this range. For example, with D/d = 1.5

for bending

In most cases the shear and bending moment diagrams

show that bending moments are quite low near the

bearings, since the bending moments from the ground

reaction forces are small.



r/d 0.02 0.05 0.1

Kt 2.7 2.1 1.7



In cases where the shoulder at the bearing is found to be

critical, the designer should plan to select a bearing with

generous fillet radius, or consider providing for a larger fillet

radius on the shaft by relieving it into the base of the shoulder

as shown in Fig. 7-9a.

This effectively creates ahead

zone in the shoulder area that

does not carry the bending

stresses, as shown by the stress

flow lines.



Fig. 7-9a.



A shoulder relief groove as shown in Fig. 7-9b can accomplish

a similar purpose. Another option is to cut a large-radius

relief groove into the small diameter of the shaft, as shown in

Fig. 7-9c.



Fig. 7-9b. Fig. 7-9c.




Figure 7-9Techniques 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.




This has the disadvantage of reducing the cross-sectional

area, but is often used in cases where it is useful to provide a

relief groove before the shoulder to prevent the grinding or

turning operation from having to go all the way to the

shoulder.

For the standard shoulder filletFor the standard shoulder fillet, for estimating Kt values for

the first iteration, an r/d ratio should be selected so Kt values

can be obtained. For the worst end of the spectrum, with r/d

= 0.02 and D/d = 1.5, Kt values from the stress concentration

charts for shoulders indicate 2.7 for bending, 2.2 for torsion,

and 3.0 for axial.




A keyway keyway will produce a stress concentration near a critical

point where the load transmitting component is located. The

stress concentration in an end-milled keyseat is a function of

the ratio of the radius r at the bottom of the groove and the

shaft diameter d. For early stages of the design process, it is

possible to estimate the stress concentration for keyways

regardless of the actual shaft dimensions by assuming a

typical ratio of r/d = 0.02. This gives Kt = 2.2 for bending and

Kts= 3.0 for torsion, assuming the key is in place.




AA keyway keyway will produce a stress concentration near a critical

point where the load transmitting component is located. The

stress concentration in an end-milled keyseat is a function of

the ratio of the radius r at the bottom of the groove and the

shaft diameter d. For early stages of the design process, it is

possible to estimate the stress concentration for keyways

regardless of the actual shaft dimensions by assuming a

typical ratio of r/d = 0.02. This gives Kt = 2.2 for bending and

Kts= 3.0 for torsion, assuming the key is in place.




Table 7-1First iteration estimates for stress concentration factors Kt

Warning: These factors are only estimates for use when actual dimensions are not yet determined. Do not use these once actual dimensions are available.




Fatigue Analysis of ShaftsFatigue Analysis of Shafts

The fatigue strength will be

determined using:

1.Distortion-Energy-Gerber

2 Distortion-Energy-

Elliptic

3

32 a

xa f

MK

d

3

16 m

xym fs

TK

d

Rotating Shaft under stationary bending and torsional moments




Fatigue Analysis of ShaftsFatigue Analysis of Shafts

2' '

1a m

e u

n nS S

2 2' '

1a m

e

n nS Sy

Gerber

ASME-Elliptic

SafetySafety FactorFactor



Problem 18-10 Problem 18-10


A geared industrial roll shown in the figure is driven at 300 rev/min by a force F acting on a 3-in-diameter pitch circle as shown. The roll exerts a normal force of 30 lbf/in of roll length on the material being pulled through. The material passes under the roll. The coefficient of friction is 0.40. Develop the moment and shear diagrams for the shaft modeling the roll force as a concentrated force at the center of the roll,

A geared industrial roll shown in the figure is driven at 300 rev/min by a force F acting on a 3-in-diameter pitch circle as shown. The roll exerts a normal force of 30 lbf/in of roll length on the material being pulled through. The material passes under the roll. The coefficient of friction is 0.40. Develop the moment and shear diagrams for the shaft modeling the roll force as a concentrated force at the center of the roll,




We have a design task of identifying bending moment and torsion diagrams which are preliminary to an industrial roller shaft design.

Gear

Roller





This approach over-estimates the bending moment at C, torque at C but not at A.





1. Using a 1035 hot rolled steel, estimate the necessary diameter at the locations of peak bending moment using a design factor of 2. These are likely to be fillets at both ends of the right hand bearing seat, where the bending moment is slightly less than the local extreme.

2. Estimating the fatigue stress-concentration factor as 2, and using a design factor of 2, what is the approximate necessary diameter of the bearing seat using the DE-elliptic fatigue failure criterion in Problem 18-10?




1/3

1/22 2164 3 0.6 in

y

nd M T

S

From static Analysis





Problem 18-Problem 18-11 11




As an example equation 18-21 is modified to take into account the hollow shaft case:

where di and do are respectively the inner and outer diameters of the shaft.

With this, one can consider that the stress-strength analysis is completed. You have obtained the minimum diameter at the critical section that can withstand the applied loads.

1/31/222

4

164 3

(1 )

f a fs mo

e yi o

K M K Tnd

S Sd d

Hollow ShaftsHollow Shafts




One approach is (See Lab Handbook):

1. Selecting a material (usually steel)

2. Drawing a free body diagram of the shaft

3. Performing static equilibrium analysis and

4. Locating the critical area

5. Performing static stress analysis to find a starting diameter size, d’.

6. Using the value of d’ in calculating the endurance limit (a trial diameter can also be used)

7. Estimating the critical value of the diameter, d, using DE-Gerber or DE-ASME-elliptic methods

8. Repeat step 6 if d different from d’.

9. Building the rest of the shaft by considering the machine parts to be mounted on the shaft (bearings, gears, pulleys, …)

10.Performing deflection analysis

11.Performing Dynamic analysis

Shaft DesignShaft Design




Shafts are usually made of ductile materials.

Small shafts with diameters less than 3.5 in (90 mm)

are usually made of Cold Drawn carbon steel (AISI

1018-1050).

Larger diameter shafts are machined from Hot

Rolled steel.

Heat treated steels are also used when higher

strengths are necessary.

Shaft MaterialsShaft Materials




Shaft

Design?

Find critical diameter, d

Shaft rotating?N

Static AnalysisEq. 6-42 or 6-44


d = Critical shaftdiameter

d. NE. d’

Y

N

find safety Factor, n

Y N


n

Fatigue Analysis

Y

N Y

Shaft rotating?

n

Reversedbending & steady

torque?

N

Eq. 18-17 or 18-22 Eq. 18-14 or 18-20

Y Reversedbending & steady

torque?

N

Eq. 18-16 or 18-21 Eq. 18-13 or 18-19

d d'

Fatigue Analysis

d

N

Complete shaftgeometry & perform

Deformation analysis

Y




Geometric ConstraintsGeometric Constraints

Unlike stress, which is a function of local geometry and load, deflection is a function of the geometry everywhere. Thus, The task of deflection and rigidity analyses can be started only when the entire geometry of the shaft is determined.

However the approach described in section 18-2, which is based on bearing slope constraints as limiting, may be used first assuming a uniform diameter shaft and using equations 18-1 and 18-2 to find the diameters at the bearings.




Shoulders

ShouldersSled runner keyseat

ShoulderGroove

Profile keyseat

Stress Concentrations and Shaft Stress Concentrations and Shaft Geometry Geometry

Shaft shoulders are used to position and provide necessary thrust supports for elements such as bearings, gears, pulleys,… Provisions must be made for torque-transfer elements such as keys, splines, pins The theoretical stress concentration factors for shoulders, grooves and transverse holes can be obtained from appendix [A15+]. Others are

Kt = 2.0 for profile key seatsKt = 1.6 for sled runner keyseats




Shaft Geometry Shaft Geometry

To determine the entire geometry of the shaft one has to rely on existing models. Some of these models are given in figures 18-1 through 18-8 of the Textbook. More shaft configurations can be found in the FAG handbooks of the Design of Rolling Bearing Mountings.




The transverse deflection of the elastic curve of the shaft

can be determined by any one of the methods studied in

Chapter 5.

The superposition method, which utilizes Appendix A-9, is

recommended. For complex shaft geometry the numerical

integration or computer program may be used.

Geometric ConstraintsGeometric Constraints: : Shaft Deflection Shaft Deflection and Slopesand Slopes




1. The slope at ball bearings should be limited at 0.25 deg the

slope at roller bearings and long journal bearings should be a

lot less. For details on acceptable slopes refer to FAG and

SKF catalogs.

2. For machinery shafting, the deflection should be no greater

than 0.001 in/ft (0.075 mm/m) of shaft length between

bearing supports.

3. For shafts mounting good quality spur gears, the deflection

at the gear mesh should not exceed 0.005 in. (0.125 mm) or

F/200 (F is the gear face width in inches) and the slope

should be limited 0.0286 deg.

4. For shafts mounting good quality bevel gears, the deflection

at the gear mesh should not exceed 0.003 in. (0.076 mm).

Geometric ConstraintsGeometric Constraints: : Shaft Deflection Shaft Deflection and Slopesand Slopes

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