Shaft Sizing Load Fatigue Deflection

8/22/2019 Shaft Sizing Load Fatigue Deflection

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Chapter 11: Shafts and Associated Parts

Design issues to learn here:Loading:

TorqueBending moment

Fatigue strengthSizing

DeflectionLinearSlope at bearings and gears

Keys

Whirling


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Loading: Needed first in calculations

Text Reference:

Figure 11.1, page425

(a) Gear forces

P1 andP2;

(b) free-body diagram with

torque and forces

(c) moment diagram in x-z and x-y planes

(d) torque diagram.

Min x-y plane Min x-z plane

Torsion

Combined bending

moment atx22xzxyx MMM +=

(Direction is somewherebetweenx-y andx-z planes.)

(11.1)

a

p

N

hT

63025= (4.41)

horsepower

rpm

ForcerTP /=

)unitsSI(

PowerT=

radius


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Example 11.2 (Moment diagrams only)

Figure 11.3 Example 11.2. (a) Assembly drawing; (b) free-body diagram.Text Reference: Figure 11.3, page 430

Design for static strength is almost never relevant.Problem: Given all forces, draw the bending momentdiagrams (2 planes) and torque diagram.


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Example 11.2 (cont.) Text Reference: Figure 11.3, page 430

(c)Min x-y plane (d)Min x-z plane

Figure 11.3

(e) torque diagram.

Note: Almost all shaft illustrations inHamrocks book are unrealistically long.

Real-world shafts are designed as shortas possible, often with no space

between gears and bearings.Advantages of shorter shafts:

lower bending moments andstressesSmaller deflectionsLower cost.


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Stresses

332dM

=Bending stress ( =Mr/I) : (11.5)

3

16

d

T

=

Torsional stress ( =Tr/J) : (11.6)

Axial stress (=P/A) : 24

d

P

=

Assume stress concentrationKfmultiplies the alternating part ofbending stress.

Assume stress concentrationKfsmultiplies the alternating part oftorsional stress.


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Fluctuating Stresses in shaft

Text Reference: Figure 11.4, page 432

(b) on oblique plane at angle .

Figure 11.4 Fluctuating stresses. (a)

on rectangular element

Given stresses on rectangularelement, determine stresses on

plane at an angle .

Force equilibrium

( ) 0sincossinsincoscos

=++

+++

AK

AKAKA

afm

afsmafsm


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Figure 11.5

Soderberg line for

shear stress.


SoderbergLine for ShearStress


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Fatigue

calculation

Figure 11.6 Eq. (11.29).



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Shaft fatigue formulae: MSST

Safety factor against fatigue: see next page.

From Mohr circle

3

32

d

Mx

=

3

16

d

Tyx

=


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Fatigue safety factor versus diameter

according to MSST

22

3

32

++

+

=

afse

y

mafe

y

m

ys

TKS

STMK

S

SM

Sdn

(11.34)

31

2232

++

+= afs

e

ymaf

e

ym

y

s TKS

STMK

S

SM

S

nd

(11.35)

Safety factor against fatigue (based on MSST):

To calculate diameter for a given safety factor, use


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Fatigue safety factor versus diameter

according to DET

22

3

++

+= afs

e

ymaf

e

ym

s

yK

S

SK

S

S

n

S

(11.36)

22

3

4

332

++

+

=

afse

ymaf

e

ym

ys

TKS

STMK

S

SM

Sdn (11.37)

31

22

4

332

++

+= afs

e

ymaf

e

ym

y

s TK

S

STMK

S

SM

S

nd

(11.38)

Safety factor against fatigue (based on DET):

To calculate diameter for a given safety factor, use

For DET, replace Eq. 11.31 with:

Derivation similar to MSST results in


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Diameter calculation: notes

31

2232

++

+= afs

e

ymaf

e

ym

y

s TK

S

STMK

S

SM

S

nd

(11.35)

Recall diameter for a given safety factor (MSST):

Stress concentration factors KfandKfs depend on size

factor ks, which depends on size d. Need iteration.

For a shaft that torques and rotates in only one direction:

Mean bending momentMm = 0.

Alternating torsion Ta = 0.

31

22

32

+

= maf

e

y

y

s TMK

S

S

S

nd

(11.35a)


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Example 11.4

Figure 11.7, page 438

31

2232

++

+= afs

e

ymaf

e

ym

y

s TKS

STMK

S

SM

S

nd

0 0


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Example 11.4


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Example 11.4


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Example 11.4

Shaft Sizing Load Fatigue Deflection

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