Transcript LECTURE # 23

ME 307
Machine
Design I
Dr. A. Aziz Bazoune
King Fahd University of Petroleum & Minerals
Mechanical Engineering Department
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 1
ME 307
Machine
Design I
18-1
18-2
18-3
18-4
18-5
18-6
18-7
18-8
Introduction ……….922
Geometric Constraints ……….927
Strength Constraints ……….933
Strength Constraints – Additional Methods ……….940
Shaft Materials ……….944
Hollow Shafts ……….944
Critical Speeds (Omitted) ……….945
Shaft Design ……….950
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 2
ME 307
Machine
Design I
18-1 Introduction ……….922
18-2 Geometric Constraints ……….927
18-3 Strength Constraints ……….933
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 3
ME 307
Machine
Design I
Fatigue Analysis of shafts
Neglecting axial loads because they are comparatively very small at critical
locations where bending and torsion dominate. Remember the fluctuating
stresses due to bending and torsion are given by
MaC
a  Kf
I
TaC
 a  K fs
J
MmC
m  Kf
I
TmC
 m  K fs
J
Mm: Midrange bending moment,
σm: Midrange bending stress
Ma : alternating bending moment,
σa: alternating bending stress
Tm: Midrange torque,
τm: Midrange shear stress
Ta : alternating torque,
τm: Midrange shear stress
Kf: fatigue stress concentration factor for bending
Kfs: fatigue stress concentration factor for torsion
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 4
ME 307
Machine
Design I
Fatigue Analysis of shafts
For solid shaft with round cross section, appropriate geometry terms can be
introduced for C, I and J resulting in
32 Ma
a  Kf
I
16Ta
 a  K fs
J
32 Mm
m  Kf
I
16Tm
 m  K fs
J
Mm: Midrange bending moment,
σm: Midrange bending stress
Ma : alternating bending moment,
σa: alternating bending stress
Tm: Midrange torque,
τm: Midrange shear stress
Ta : alternating torque,
τm: Midrange shear stress
Kf: fatigue stress concentration factor for bending
Kfs: fatigue stress concentration factor for torsion
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 5
ME 307
Machine
Design I
Fatigue Analysis of shafts
Combining these stresses in accordance with the DE failure theory the vonMises stress for rotating round, solid shaft, neglecting axial loads are given by
 ' a   xa 2  3 2  1/2 16 /  d 3 4(K f Ma )2  3(K fsTa )2  1/2  16 A /  d 3
 ' m   xm2  3 2  1/2 16 /  d 3 4(K f Mm)2  3(K fsTm)2  1/2  16B /  d 3
xya
(18-12)
xym
where A and B are defined by the radicals in Eq. (8-12) as
A   4(K f Ma )2  3(K fsTa )2  1/2
B   4(K f Mm )2  3(K fsTm)2  1/2
The Gerber fatigue failure criterion
2
2
2
Sa  Sm 
n 'a  n 'm 
16nA  16nB 


 3  1
 
 
3
Se  Sut 
Se
 Sut   d Se   d Sut 
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30 Slide 6
CH-18
LEC 30 Slide 6
ME 307
Machine
Design I
Fatigue Analysis of shafts
The critical shaft diameter is given by
2
 
 2BSe  
8nA 
d
1  1  
 
 Se    ASut  

 
1/2
1/3





(18-13)
or, solving for 1/n, the factor of safety is given by
1/2
2
 


 2BSe 
1
8A 




1  1  


3
n  d Se    ASut   
 
 
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
(18-14)
CH-18
LEC 30 Slide 7
CH-18
LEC 30 Slide 7
ME 307
Machine
Design I
Fatigue Analysis of shafts
where
A  4(K f Ma )2  3(K fsTa )2
B  4(K f Mm )  3(K fsTm )
2
2
(18-15)
 'a A
r

 'm B
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 8
ME 307
Machine
Design I
Fatigue Analysis of shafts
Particular Case
For a rotating shaft with constant bending and torsion, the
bending stress is completely reversed and the torsion is
steady. Previous Equations can be simplified by setting Mm
= 0 and Ta = 0, which simply drops out some of the terms.
Critical Shaft
Diameter
Safety Factor
Dr. A. Aziz Bazoune
A  2K f Ma
B  3 KfsTm
2
 
 K fsTmSe  
16nK f Ma 
d
 
1  1  3 
 Se  
K f MaSut  


 
1/2
1/3





1/2
2
 


1 16K f Ma    K fsTmSe   


1  1  

3
n
 d Se    K f MaSut   
 
 
Chapter 18: Axles and Shafts
CH-18
(18-16)
(18-17)
LEC 30
Slide 9
ME 307
Machine
Design I
Fatigue Analysis of shafts
2K f Ma
 'a
r

 'm
3K fsTm
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
(18-18)
CH-18
LEC 30
Slide 10
Shaft Diameter Equation for the DE-Elliptic
Criterion
ME 307
Machine
Design I
Remember
 ' a   xa 2  3 2  1/2 16 /  d 3 4(K f Ma )2  3(K fsTa )2  1/2  16 A /  d 3
 ' m   xm2  3 2  1/2 16 /  d 3 4(K f Mm)2  3(K fsTm)2  1/2  16B /  d 3
xya
(18-12)
xym
where A and B are defined by
A   4(K f Ma )2  3(K fsTa )2  1/2
B   4(K f Mm )2  3(K fsTm)2  1/2
The Elliptic fatigue-failure criterion is defined by
2
2
2
2
2
 Sa   Sm   n 'a   n 'm   16nA   16nB
   3    3
      
  
 Se   Sy   Se   Sy    d Se    d Sy
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
2

  1

CH-18
LEC 30 Slide 11
CH-18
LEC 30 Slide 11
ME 307
Machine
Design I
Shaft Diameter Equation for the DE-Elliptic
Criterion
Substituting for A and B gives expressions for d, 1/n and r:
Critical Shaft Diameter
2
2

2
  K M 2





K
T
K
M
K
T


16n 
d
4  f a   3  fs a   4  f m   3  fs m  
 S 
 S  

S
S

e
e
y







 y  


1/2
1/3





(18-19)
Safety Factor

 K f Mm 
 K fsTm 
 K fsTa 
1 16   K f Ma 
 3 4
  3 

  3
  4 
n  d   Se 
Se 
Sy 
Sy 




2
Dr. A. Aziz Bazoune
2
Chapter 18: Axles and Shafts
2
CH-18
2




1/2
LEC 30
(18-20)
Slide 12
ME 307
Machine
Design I
Shaft Diameter Equation for the DE-Elliptic
Criterion
4  K f Ma   3  K fsTa 
 'a A
r
 
2
2
 'm B
4  K f Mm   3  K fsTm 
2
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
2
CH-18
LEC 30
Slide 13
ME 307
Machine
Design I
Shaft Diameter Equation for the DE-Elliptic
Criterion
Particular Case
A  2K f Ma
For a rotating shaft with constant bending and torsion,
the bending stress is completely reversed and the
torsion is steady. Previous Equations can be simplified
by setting Mm = 0 and Ta = 0, which simply drops out
some of the terms.
Critical Shaft Diameter
2

  K M 2



K
T
1
6
n

 4  f a   3  fs m  
d
 S  
    Se 
 y  


 K fsTm 
1 16   K f Ma 
 3 4

  3 
n  d   Se 
Sy 


2
Safety Factor
Dr. A. Aziz Bazoune
B  3 KfsTm
Chapter 18: Axles and Shafts
2
1/2
1/3





(18-21)
1/2




(18-22)
CH-18
LEC 30
Slide 14
ME 307
Machine
Design I
Shaft Diameter Equation for the DE-Elliptic
Criterion
4  K f Ma 
2K f Ma
 'a A
r
 

2
 'm B
3K fsTm
3  K fsTm 
2




At a shoulder Figs. A-15-8 and A-15-9 provide information about Kt and
Kts.
For a hole in a solid shaft, Figs. A-15-10 and A-15-11 provide about Kt and
Kts .
For a hole in a solid shaft, use Table A-16
For grooves use Figs. A-15-14 and A-15-15
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 15
ME 307
Machine
Design I
Shaft Diameter Equation for the DE-Elliptic
Criterion
 The value of slope at which the load line intersects the junction of
the failure curves is designated rcrit.
 It tells whether the threat is from fatigue or first cycle yielding
 If r > rcrit, the threat is from fatigue
 If r < rcrit, the threat is from first cycle yielding.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 16
ME 307
Machine
Design I
Shaft Diameter Equation for the DE-Elliptic
Criterion
 For the Gerber-Langer intersection the strength components Sa and
Sm are given in Table 7-10 as
2

 2Se   Sy
S 
Sm 
1 1 
1


2Se 
 Sut   Se

Sa  Sy  Sm
2
ut
rcrit
Dr. A. Aziz Bazoune





(18-23)
Sa Sy  Sm Sy



1
Sm
Sm
Sm
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 17
ME 307
Machine
Design I
Shaft Diameter Equation for the DE-Elliptic
Criterion
 For the DE-Elliptic-Langer intersection the strength components Sa
and Sm are given by
Sa 
2Sy Se2
S S
2
e
2
y
Sm  Sy  Sa
rcrit
Dr. A. Aziz Bazoune
(18-24)
Sa
Sa


Sm Sy  Sa
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 18
ME 307
Machine
Design I
Note that in an analysis situation in which the diameter is known and the
factor of safety is desired, as an alternative to using the specialized equations
above, it is always still valid to calculate the alternating and mid-range
stresses using the following Eqs.
and substitute them into the one of the equations for the failure criteria ,
Eqs. (7-48) to (7-51) and solve directly for n.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 19
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 20
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 21
 In a design situation, however, having the equations pre-solved for
ME 307
Machine
Design I
diameter is quite helpful.
 It is always necessary to consider the possibility of static failure in
the first load cycle.

The Soderberg criteria inherently guards against yielding, as can
be seen by noting that its failure curve is conservatively within the
yield (Langer) line on Fig. 7–27, p. 348.
 The ASME Elliptic also takes yielding into account, but is not
entirely conservative throughout its range. This is evident by
noting that it crosses the yield line in Fig. 7–27.
 The Gerber and modified Goodman criteria do not guard against
yielding, requiring a separate check for yielding. A von Mises
maximum stress is calculated for this purpose.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 22
ME 307
Machine
Design I

To check for yielding, this von Mises maximum stress is compared to the
yield strength, as usual

For a quick, conservative check, an estimate for σ’max can be obtained by
simply adding σa and σm . (σa + σm ) will always be greater than or equal
to σ’max, and will therefore be conservative.
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 23
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 24
ME 307
Machine
Design I
Example
Solution
720
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
720
CH-18
LEC 30
Slide 25
ME 307
Machine
Design I
Figure 7-20
Figure 7-21
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 26
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 27
ME 307
Machine
Design I
1822
1814
Dr. A. Aziz Bazoune
1824
Sold
erbe
rg
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 28
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 29
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 30
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 31
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 32
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 33
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 34
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 35
ME 307
Machine
Design I
Dr. A. Aziz Bazoune
Chapter 18: Axles and Shafts
CH-18
LEC 30
Slide 36