mechanical engineering design 8th

BudynasNisbett: Shigleys

Mechanical Engineering

Design, Eighth Edition

I. Basics 3. Load and Stress Analysis110 The McGrawHill

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Load and Stress Analysis 105

313 Stress ConcentrationIn the development of the basic stress equations for tension, compression, bending, andtorsion, it was assumed that no geometric irregularities occurred in the member underconsideration. But it is quite difficult to design a machine without permitting somechanges in the cross sections of the members. Rotating shafts must have shouldersdesigned on them so that the bearings can be properly seated and so that they will takethrust loads; and the shafts must have key slots machined into them for securing pul-leys and gears. A bolt has a head on one end and screw threads on the other end, bothof which account for abrupt changes in the cross section. Other parts require holes, oilgrooves, and notches of various kinds. Any discontinuity in a machine part alters thestress distribution in the neighborhood of the discontinuity so that the elementary stressequations no longer describe the state of stress in the part at these locations. Such dis-continuities are called stress raisers, and the regions in which they occur are calledareas of stress concentration.

The distribution of elastic stress across a section of a member may be uniform asin a bar in tension, linear as a beam in bending, or even rapid and curvaceous as in asharply curved beam. Stress concentrations can arise from some irregularity not inher-ent in the member, such as tool marks, holes, notches, grooves, or threads. The nomi-nal stress is said to exist if the member is free of the stress raiser. This definition is notalways honored, so check the definition on the stress-concentration chart or table youare using.

A theoretical, or geometric, stress-concentration factor Kt or Kts is used to relatethe actual maximum stress at the discontinuity to the nominal stress. The factors aredefined by the equations

Kt =max

0Kts =

max

0(348)

where Kt is used for normal stresses and Kts for shear stresses. The nominal stress 0 or0 is more difficult to define. Generally, it is the stress calculated by using the elemen-tary stress equations and the net area, or net cross section. But sometimes the grosscross section is used instead, and so it is always wise to double check your source of Ktor Kts before calculating the maximum stress.

The subscript t in Kt means that this stress-concentration factor depends for itsvalue only on the geometry of the part. That is, the particular material used has no effecton the value of Kt. This is why it is called a theoretical stress-concentration factor.

The analysis of geometric shapes to determine stress-concentration factors is a dif-ficult problem, and not many solutions can be found. Most stress-concentration factorsare found by using experimental techniques.8 Though the finite-element method hasbeen used, the fact that the elements are indeed finite prevents finding the true maxi-mum stress. Experimental approaches generally used include photoelasticity, gridmethods, brittle-coating methods, and electrical strain-gauge methods. Of course, thegrid and strain-gauge methods both suffer from the same drawback as the finite-elementmethod.

Stress-concentration factors for a variety of geometries may be found inTables A15 and A16.

8The best source book is W. D. Pilkey, Petersons Stress Concentration Factors, 2nd ed., John Wiley &Sons, New York, 1997.




I. Basics 3. Load and Stress Analysis 111 The McGrawHill

Companies, 2008

106 Mechanical Engineering Design

An example is shown in Fig. 329, that of a thin plate loaded in tension where theplate contains a centrally located hole.

In static loading, stress-concentration factors are applied as follows. In ductile( f 0.05) materials, the stress-concentration factor is not usually applied to predict thecritical stress, because plastic strain in the region of the stress is localized andhas a strengthening effect. In brittle materials ( f < 0.05), the geometric stress-concentration factor Kt is applied to the nominal stress before comparing it with strength.Gray cast iron has so many inherent stress raisers that the stress raisers introduced by thedesigner have only a modest (but additive) effect.

Figure 329

Thin plate in tension or simplecompression with a transversecentral hole. The net tensileforce is F = wt, where t isthe thickness of the plate. Thenominal stress is given by

0 =F

(w d )t =w

(w d)

02.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2.2

2.4

2.6

2.8

3.0

d/w

Kt

d

w

EXAMPLE 313 Be Alert to ViewpointOn a spade rod end (or lug) a load is transferred through a pin to a rectangular-cross-section rod or strap. The theoretical or geometric stress-concentration factor for thisgeometry is known as follows, on the basis of the net area A = (w d)t as shown inFig. 330.

d/w 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Kt 7.4 5.4 4.6 3.7 3.2 2.8 2.6 2.45

As presented in the table, Kt is a decreasing monotone. This rod end is similar to thesquare-ended lug depicted in Fig. A15-12 of appendix A.

max = Kt0 (a)max =

Kt FA

= KtF

(w d)t (b)

It is insightful to base the stress concentration factor on the unnotched area, wt . Let

max = K tFwt

(c)

By equating Eqs. (b) and (c) and solving for K t we obtain

K t =wt

FKt

F(w d)t =

Kt1 d/w (d )




I. Basics 3. Load and Stress Analysis112 The McGrawHill

Companies, 2008

Load and Stress Analysis 107

A power regression curve-fit for the data in the above table in the form Kt = a(d/w)bgives the result a = exp(0.204 521 2) = 1.227, b = 0.935, and r2 = 0.9947. Thus

Kt = 1.227(

dw

)0.935(e)

which is a decreasing monotone (and unexciting). However, from Eq. (d ),

K t =1.227

1 d/w(

dw

)0.935( f )

Form another table from Eq. ( f ):d/w 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

K t 8.507 6.907 5.980 5.403 5.038 4.817 4.707 4.692 4.769 4.946

which shows a stationary-point minimum for K t . This can be found by differentiatingEq. ( f ) with respect to d/w and setting it equal to zero:

d K td(d/w)

= (1 d/w)ab(d/w)b1 + a(d/w)b

[1 (d/w)]2 = 0

where b = 0.935, from which(dw

)= b

b 1 =0.935

0.935 1 = 0.483

with a corresponding K t of 4.687. Knowing the section w t lets the designer specify thestrongest lug immediately by specifying a pin diameter of 0.483w (or, as a rule of thumb,of half the width). The theoretical Kt data in the original form, or a plot based on the datausing net area, would not suggest this. The right viewpoint can suggest valuable insights.

314 Stresses in Pressurized CylindersCylindrical pressure vessels, hydraulic cylinders, gun barrels, and pipes carrying fluidsat high pressures develop both radial and tangential stresses with values that dependupon the radius of the element under consideration. In determining the radial stress rand the tangential stress t , we make use of the assumption that the longitudinalelongation is constant around the circumference of the cylinder. In other words, a rightsection of the cylinder remains plane after stressing.

Referring to Fig. 331, we designate the inside radius of the cylinder by ri, the out-side radius by ro, the internal pressure by pi, and the external pressure by po. Then it canbe shown that tangential and radial stresses exist whose magnitudes are9

t =pir2i por2o r2i r2o (po pi )/r2

r2o r2i

r =pir2i por2o + r2i r2o (po pi )/r2

r2o r2i

(349)

9See Richard G. Budynas, Advanced Strength and Applied Stress Analysis, 2nd ed., McGraw-Hill, NewYork, 1999, pp. 348352.

po

r

dr

ri ro

pi

Figure 331

A cylinder subjected to bothinternal and external pressure.

d

t

A B

F

F

w

Figure 330

A round-ended lug end to arectangular cross-section rod.The maximum tensile stress inthe lug occurs at locations Aand B. The net areaA = (w d) t is used in thedefinition of K t , but there is anadvantage to using the totalarea wt.




Back Matter Appendix A: Useful Tables 1001 The McGrawHill

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Table A15

Charts of Theoretical Stress-Concentration Factors K*t

Figure A151

Bar in tension or simplecompression with a transversehole. 0 = F/A, whereA = (w d )t and t is thethickness.

Kt

d

d/w0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2.0

2.2

2.4

2.6

2.8

3.0

w

Figure A152

Rectangular bar with atransverse hole in bending.0 = Mc/I, whereI = (w d )h3/12.

Kt

d

d/w0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.0

1.4

1.8

2.2

2.6

3.0

w

MM0.25

1.0

2.0

d /h = 0

0.5h

Kt

r

r /d0

1.5

1.2

1.1

1.05

1.0

1.4

1.8

2.2

2.6

3.0

dww /d = 3

0.05 0.10 0.15 0.20 0.25 0.30

Figure A153

Notched rectangular bar intension or simple compression.0 = F/A, where A = dt and tis the thickness.




Back Matter Appendix A: Useful Tables1002 The McGrawHill

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Useful Tables 1007

Table A15

Charts of Theoretical Stress-Concentration Factors K*t (Continued)

1.5

1.10

1.05

1.02

w/d =

Kt

r

r /d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

3.0

dw MM

1.02

Kt

r/d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

3.0

r

dD

D/d = 1.50

1.05

1.10

Kt

r/d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

3.0

r

dD

D/d = 1.02

3

1.31.1

1.05 MM

Figure A154

Notched rectangular bar inbending. 0 = Mc/I, wherec = d/2, I = td 3/12, and t isthe thickness.

Figure A155

Rectangular filleted bar intension or simple compression.0 = F/A, where A = dt and tis the thickness.

Figure A156

Rectangular filleted bar inbending. 0 = Mc/I, wherec = d/2, I = td3/12, t is thethickness.

*Factors from R. E. Peterson, Design Factors for Stress Concentration, Machine Design, vol. 23, no. 2, February 1951, p. 169; no. 3, March 1951, p. 161, no. 5, May 1951, p. 159; no. 6, June1951, p. 173; no. 7, July 1951, p. 155. Reprinted with permission from Machine Design, a Penton Media Inc. publication.

(continued)





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Table A15


Figure A157

Round shaft with shoulder filletin tension. 0 = F/A, whereA = d 2/4.

Figure A158

Round shaft with shoulder filletin torsion. 0 = Tc/J, wherec = d/2 and J = d4/32.

Figure A159

Round shaft with shoulder filletin bending. 0 = Mc/I, wherec = d/2 and I = d4/64.

Kt

r/d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

r

1.05

1.02

1.10

D/d = 1.50

dD

Kts

r/d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

3.0

D/d = 2

1.09

1.20 1.33

r

TTD d

Kt

r/d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

3.0

D/d = 3

1.02

1.51.10

1.05

r

MD dM





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Useful Tables 1009

Table A15


Figure A1510

Round shaft in torsion withtransverse hole.

Figure A1511

Round shaft in bending witha transverse hole. 0 =M/[(D3/32) (dD2/6)],approximately.

Kts

d /D0 0.05 0.10 0.15 0.20 0.25 0.30

2.4

2.8

3.2

3.6

4.0

Jc

TB

d

D316

dD26= (approx)

AD

Kts, A

Kts, B

Kt

d /D0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

3.0d

D

MM

Figure A1512

Plate loaded in tension by apin through a hole. 0 = F/A,where A = (w d)t . Whenclearance exists, increase Kt35 to 50 percent. (M. M.Frocht and H. N. Hill, StressConcentration Factors arounda Central Circular Hole in aPlate Loaded through a Pin inHole, J. Appl. Mechanics,vol. 7, no. 1, March 1940,p. A-5.)

dh

t

Kt

d /w0 0.1 0.2 0.3 0.4 0.60.5 0.80.7

1

3

5

7

9

11

w

h/w = 0.35

h/w 1.0

h/w = 0.50

(continued)






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Table A15




Figure A1513

Grooved round bar in tension.0 = F/A, whereA = d 2/4.

Figure A1514

Grooved round bar inbending. 0 = Mc/l, wherec = d/2 and I = d4/64.

Figure A1515

Grooved round bar in torsion.0 = Tc/J, where c = d/2and J = d4/32.

Kt

r /d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

3.0

D/d = 1.50

1.05

1.02

1.15

d

r

D

Kt

r /d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

3.0

D/d = 1.501.02

1.05

d

r

D MM

Kts

r /d0 0.05 0.10 0.15 0.20 0.25 0.30

1.0

1.4

1.8

2.2

2.6

D/d = 1.30

1.02

1.05

d

r

D

TT





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Useful Tables 1011

Table A15


Figure A1516

Round shaft with

flat-bottom groove in

bending and/or tension.

0 =4P

d 2+ 32M

d 3

Source: W. D. Pilkey, PetersonsStress Concentration Factors,2nd ed. John Wiley & Sons,New York, 1997, p. 115

Kt

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1.00

0.5 0.6 0.7 0.8 0.91.0 2.0 3.0 4.0 5.0 6.01.0

a/t

0.03

0.04

0.05

0.07

0.15

0.60

d

ra

r

DM

Pt

MP

rt

0.10

0.20

0.40





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Table A15


Figure A1517

Round shaft with flat-

bottom groove in torsion.

0 =16T

d 3

Source: W. D. Pilkey, PetersonsStress Concentration Factors,2nd ed. John Wiley & Sons,New York, 1997, p. 133

0.03

0.04

0.06

0.10

0.20

rt

0.5 0.6 0.7 0.8 0.91.0 2.01.0

2.0

3.0

4.0

5.0

6.0

3.0 4.0 5.0 6.0

d

ra

r

D T

t

Kts

a/t





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Useful Tables 1013

Table A16

Approximate Stress-

Concentration Factor Ktfor Bending of a Round

Bar or Tube with a

Transverse Round HoleSource: R. E. Peterson, StressConcentration Factors, Wiley,New York, 1974, pp. 146,235.

The nominal bending stress is 0 = M/Znet where Znet is a reduced valueof the section modulus and is defined by

Znet = A32D

(D4 d4)

Values of A are listed in the table. Use d = 0 for a solid bar

d/D

0.9 0.6 0

a/D A Kt A Kt A Kt

0.050 0.92 2.63 0.91 2.55 0.88 2.42

0.075 0.89 2.55 0.88 2.43 0.86 2.35

0.10 0.86 2.49 0.85 2.36 0.83 2.27

0.125 0.82 2.41 0.82 2.32 0.80 2.20

0.15 0.79 2.39 0.79 2.29 0.76 2.15

0.175 0.76 2.38 0.75 2.26 0.72 2.10

0.20 0.73 2.39 0.72 2.23 0.68 2.07

0.225 0.69 2.40 0.68 2.21 0.65 2.04

0.25 0.67 2.42 0.64 2.18 0.61 2.00

0.275 0.66 2.48 0.61 2.16 0.58 1.97

0.30 0.64 2.52 0.58 2.14 0.54 1.94

M M

D d

a

(continued)





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Table A16 (Continued)

Approximate Stress-Concentration Factors Kts for a Round Bar or Tube Having a Transverse Round Hole and

Loaded in Torsion Source: R. E. Peterson, Stress Concentration Factors, Wiley, New York, 1974, pp. 148, 244.

TTD

a d

The maximum stress occurs on the inside of the hole, slightly below the shaft surface. The nominal shear stress is 0 = T D/2Jnet ,where Jnet is a reduced value of the second polar moment of area and is defined by

Jnet = A(D4 d4)

32

Values of A are listed in the table. Use d = 0 for a solid bar.

d/D

0.9 0.8 0.6 0.4 0

a/D A Kts A Kts A Kts A Kts A Kts

0.05 0.96 1.78 0.95 1.77

0.075 0.95 1.82 0.93 1.71

0.10 0.94 1.76 0.93 1.74 0.92 1.72 0.92 1.70 0.92 1.68

0.125 0.91 1.76 0.91 1.74 0.90 1.70 0.90 1.67 0.89 1.64

0.15 0.90 1.77 0.89 1.75 0.87 1.69 0.87 1.65 0.87 1.62

0.175 0.89 1.81 0.88 1.76 0.87 1.69 0.86 1.64 0.85 1.60

0.20 0.88 1.96 0.86 1.79 0.85 1.70 0.84 1.63 0.83 1.58

0.25 0.87 2.00 0.82 1.86 0.81 1.72 0.80 1.63 0.79 1.54

0.30 0.80 2.18 0.78 1.97 0.77 1.76 0.75 1.63 0.74 1.51

0.35 0.77 2.41 0.75 2.09 0.72 1.81 0.69 1.63 0.68 1.47

0.40 0.72 2.67 0.71 2.25 0.68 1.89 0.64 1.63 0.63 1.44

mechanical engineering design 8th

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