Rules of Thumb for Mechanical Engineers Episode 11 docx

DefinitionMress and Strain The following basic stress quantities am useful in the eval- uation of many simple structures.. E erally used to evaluate yielding in a multiaxial stress fie

the mold, and the molten metal poured into the mold The

metal solidifies and the shell is broken off

Internal passages and other product features can be in-

corporated into the casting using cores Excellent surface

finish and dimensional control can be obtained Complex

turbine blades can be manufactured with this method It is

more expensive than other casting technologies

A specialized form of investment casting is used to

make single crystal and directionally solidified pieces

With these technologies, which are very important for ma-

terials that require long stress rupture and creep properties, the heat is preferentially extracted in a single direction This promotes the growth of a single grain or a single grain ori- entation The grain orientation selected depends on the crystal anisotropy and the property most important for the application

Information about the castability of the various alloys can

be found in Principles of Metal Casting [27] and the ASM Metals Handbook, Vol 15,9th Ed

CASE STUDIES Failure Analysis

Failure analysis entails the systematic investigation of

why or how a component fails Despite the best design, an

improper material selection or a processing sequence can

lead to a premature failure of said component A detailed

history is generally established Temperature, expected en-

vironment, stresses, and strains are all important variables

for the failure analyst to know As one investigates various

failures, documentation of the salient features is required

The methods used include photography, notetaking, videog-

raphy, and the like The examination of the fracture surfaces

optically and electron optically are useful in determining the

type of failure, e.g., brittle or ductile fracture, high or low

cycle fatigue, environmentally assisted fracture, or wear

Two operational failures and fixes will be discussed

Wear is one of the most important causes of failure, although

many factors are usually involved Piston rings, gears, and

bearings are a few of the many parts where resistance to

wear is required Wear is probably the most easily recog-

nized failure mode, as shown in Figure 25 Although wear

may not be prevented, steps can be taken to reduce the rate

and yield a long service life by the proper application of ma-

terials, lubrication, and design

Often, improper application of steels, load distribution,

heat treatment, and inadequate or faulty lubrication result

in excessive wear and poor service life High loads and

speeds are capable of producing very high temperatures

under which metal surfaces may actually melt Friction is

Figure 25 Excessive wear of gear teeth (Reprinted by permission of Republic Steel.)

an important factor in producing temperatures that may cause the breakdown of hardened surfaces, such as those produced by carburizing Therefore, special lubricants for specific applications involving very high unit pressures may be required

The gear wear shown in Figure 25 was corrected by se- lecting a new material that was significantly harder than the

1020 rimmed steel with a Brinell hardness of 116 The worn teeth were driven by rollers in a chain link with a Brinell hardness of 401 An alloy steel with higher hardness was substituted, and the new sprocket was still in service after seven years [33]

Corrosion

The diagram in Figure 26 is a schematic of the lower

end of a tube-and-shell heat exhanger made from mild

steel The unit was designed to heat oil in a chemical

process plant The oil was passed through the small tubes

and the heat was supplied from steam which was inject-

ed into the shell The unit had been in operation for only

2.5 years when one of the tubes perforated When the tubes

were extracted from the shell, it was found that they all

had corroded on the outside over a distance of about 160

mm from the lower tube plate On the worst-affected

areas, attack had occurred to a depth of about 1.5 mm over

regions measuring typically 10 mm by 20 111111 The cor-

roded areas were light brown in color

The heat exchanger was operated on a cyclic basis as fol-

lows First, saturated steam was admitted to the shell at

180°C to heat a new batch of oil The steam condensed on

the surfaces of the tubes and the condensed water trickled

down to the bottom of the shell, where it was drawn off via

the condensate drain When the oil was up to temperature,

the steam supply was cut off and the pressure in the shell

1 6 0 m I

34 nm with

3 m wall

t

Figure 26 Schematic view of the lower end of a tube-

and-shell heat exchanger made from mild steel

was dropped to atmospheric The cycle was repeated when

it was time to heat up a new batch of oil

Based on the above observations and operating cycle, it

is apparent that the carrosion product is red rust, i.e., hydrated Fe203 Of the three forms of iron oxide (FeO, Fe304), and F@O3), the latter has the highest ratio of oxygen to iron It

is the favored oxide in an oxygen-rich environment When the oxygen concentration is low, the corrosion product con- sists of hydrated Fq04 (magnetite), which is black But thm

was no evidence that this was present as a corrosion prod- uct There is evidence, however, of oxygen in the conden- sate which presumably came from air dissolved in the make-up feed water to the boiler This would have provid-

ed the oxygen needed for the cathodic reaction

The design of the unit allows condensate to build up to the level of the drain It is interesting that corrosion has only occurred in, or just above, the pool of condensate; it has not

taken place farther up the tubes even though they would have

been dripping with condensed steam A likely scenario is that when the shell was let down to atmosphere, the water at the bottom of the shell was boiled off by the residual heat in the tube plate This would have left either a concentrated solu- tion or a solid residue containing most of the impurities that were originally dissolved in the condensate pool With each cycle of operation, the cotlcentration of impurities in the pool would have increased A prime suspect is the carbonic acid, derived h m carbon dioxide dissolved in the feed water This would have made the liquid in the pool very acidic and given

it a high ionic conductivity, both of which would have re- sulted in rapid attack It can be seen from the electrochem- ical equilibrium diagram for iron [39], iron does not form

a surface f l in acid waters Finally, the temperature is el- evated so the rates of thermally activated corrosion process-

es should be high as well

A simple design modification of moving the condensate drain from the side to the lowest point of the shell would prevent water from accumulating in the bottom of the shell [34]

1 Bolz, R E and Tuve, G L (Eds.), Handbook of Tables

for Applied Engineering Science, 2nd Ed Boca Raton:

CRC Press, 1984

2 ASM Metals Handbook: Properties and Selection-

Irons and Steels, Vol 1, 9th Ed., ASM International, Ma-

terials Park, OH, 1978

3 Callister, W D., Jr., Materials Science and Engineer-

ing, An Introduction New York: John Wiley & Sons,

Inc., 1985

4 Dieter, G E., Mechanical Metallurm New York Mc-

Graw-Hill, 1986

5 Hertzberg, R W., Deformation and Fracture Mechan-

ics of Engineering Materials, 2nd Ed New York: John

Wiley & Sons, 1983

6 Schackelford, J F., Introduction to Materials Science

for Engineers, 2nd Ed New York Macmillan Pub-

lishing, 1988

7 Askeland, D R., The Science and Engineering of Ma-

terials Belmont, C A Wadsworth, 1984

8 Van Vlack, L H., Materials Science for Engineers

Redding, M A : Addison Wesley, 1970

9 Uhlig, H H and Revie, R W., Corrosion and Corro-

sion Control and Introduction to Comsion Science

and Engineering, 3rd Ed New York John Wiley &

Sons, Inc., 1985

10 Fontana, M G., Corrosion Engineering New York Mc-

Graw-Hill, 1986

11 McCrum, N G., Buckley, C P., and Bucknall, C B.,

Principles of Polymer Engineering New York Ox-

ford University Press, 1988

12 Powder Metallurgy Design Solutions Metal Powder In-

dustries Federation, Princeton, NJ, 1993

13 German, R M., Powder Metallurgy Science Metal

Powder Industries Federation, Princeton, NJ, 1984

14 “Amdry MCrAlY Thermal Spray Powders Specially

Formulated and Customized Alloys Provide Oxida-

tion and Corrosion Resistance at Elevated Tempera-

tures,,, Amdry Product Bulletin 961,970,995, Alloy

Metals, Inc., 1984

15 Engineered Materials Handbook, Vol 4: Ceramics and

Glasses S J Schneider, Jr., Volume Chairman, ASM

International, Materials Park, OH, 1991

16 Davis, J R (Ed.), ASM Materials Engineering Dic-

tionary ASM International, Metals Park, OH, 1992

17 Craig, B D (Ed.), Handbook of Corrosion Data ASM

International, Materials Park, OH, 1989,

18 McEvily, A J (Ed.), Atlas of Stress-Corrosion and

Corrosion Fatigue Curves ASM International, Mate-

rials Park, OH, 1990

19 Coburn, S K (Ed.), Corrosion Source Book ASM In- ternational, Materials Park, OH, 1984

20 Sedriks, A J (Ed.), corrosion of Stainless Steels New

York John Wiley & Sons, Inc., 1979

21 Uhlig, H H., Corrosion Handbook New York John Wiley & Sons, Inc., 1948

22 ASM Metals Handbook: Properties and Selection-

Nonferrous Alloys and Pure Metals, Vol 2, 9th Ed.,

ASM International, Materials Park, OH, 1979

23 Massalski, T B., Okamoto, H., Subramanian , P R., and Kacprzak, L (Eds.), Binary Alloy Phase Diagrams,

2nd Ed., ASM International, Materials Park, OH, 1990

24 Haynes International, Product Bulletin H-1064Dy 1993

25 Inco Alloys International, Product Handbook, 1988

26 Sims, C T., Stoloff, N S., and Hagel, W C (Eds.), Su-

peralloys IZ High Temperature Materials for Aero- space and Industrial Powel: New York John Wiley & Sons, Inc., 1987

27 Heine, R W., Loper, C R., and Rosenthal, P C., Prin-

ciples of Metal Casting, 2nd Ed St Louis: McGraw- Hill, 1967

28 Birks, N and Meier, G H., Introduction to High Tem- perature Oxidation of Metals Great Britain: Edward

Arnold, 1983

29 ASTM E112, Standard Method for Average Grain Size

of Metallic Materials, Volume 03.01 , Metals-Mechan- ical Testing; Elevated and Low Temperature Test; Met- allography, ASTM, 1992

30 A S M E18, Standard Test Methods for Rockwell Hard- ness and Rockwell Superjkial Harrbzess of Metallic Ma-

terials, Volume 03.01 , Metals-Mechanical Testing; El- evated and Low Temperature Test; Metallography, ASTM, 1992

3 1 ASTM El 0, Standard Test Method for Brinell Hardness

of Metallic Materials, Volume 03.01, Metals-Mechan- ical Testing; Elevated and Low Tempera- Test; Met- allography, ASTM, 1992

32 ASTM E92 Standard Test Method for vickers Hardness

of Metallic Materials, Volume 03.01, Metals-Me-

chanical Testing; Elevated and Low Temperature Test; Metallography, ASTM, 1992

33 “Analysis of Service Failures,” Republic Alloy Steels

Handbook Adv 1099R, Republic Steel Corporation,

1974

34 Jones, D R H., Engineering Materials 3, Materials

Failure Analysis, Case Studies and Design Implications

New York Pergamon Press, 1993

35 Aurrecoechea, J M., “Gas Turbine Hot Section Coat-

ing Technology,” Solar Turbines Incorporated, 1995

36 ASM Metals Handbook: Welding, Brazing, and Sol-

dering, Vol 6., 9th Ed ASM International, Materials

Park, OH

37 Harper, C A (Ed.), Handbook of Plastics and Elas-

tomers New York: McGraw-Hill, Inc., 1975

38 ASM Metals Handbook, Vol 15,9th Ed., ASM Inter-

national, Materials Park, OH, 1988

39 Pourbaix, M., Atlas of Electrochemical Equilibria in Aqueous Solutions, National Association of Corrosion

Engineers (NACE), Houston, TX, 1974

40 ASM Metals Handbook: Properties and Selection- Stainless Steels, Tool Materials, and Special Purpose Metals, Vol 3,% Ed., ASM International, Materials Park, OH, 1980

41 ASM Met& Handbook: Corrosion, Vol 13, 9th Ed.,

ASM International, Materials Park, OH, 1987

13

Stress and Strain

Marlin W Reimer Development Engineer Structural Mechanics Dept., Allison Engine Company

Fundamentals of Stress and Strain 295

Introduction 295

Definitions-Stress and Strain 295

Equilibrium 297

Compatibility 297

Saint-Venant’s Principle 297

Superposition 298

Plane StressPlane Strain 298

Thermal Stresses 298

Stress Concentrations 299

Determination of Stress Concentration Factors 300

Design Criteria for Structural Analysis 305

General Guidelines for Effective Criteria 305

Strength Design Factors 305

Beam Analysis 306

Limitations of General Beam Bending Equations 307

Short Beams 307

Plastic Bending 307

Torsion 308

Pressure Vessels 309

Thick-walled Cylinders 309

Press Fits Between Cylinders 310

Thin-walled Cylinders 309

Rotating Equipment 310

Rotating Disks 310

Rotating Shafts 313

Flange Analysis 315

Flush Flanges 315

Undercut Flanges 316

Mechanical Fasteners 316

Threaded Fasteners 317

Pins 318

Rivets 318

Welded and Brazed Joints 319

Creep Rupture 320

Finite Element Analysis 320

Overview 321

The Elements 321

Modeling Techniques 322

Advantages and Limitations of FEM 323

Centroids and Moments of Inertia for Common Shapes 324

Beams: Shear, Moment, and Deflection Formulas for Common End Conditions 325

References 328

294

~~~ ~

Introduction

Stress is a defined quantity that cannot be directly ob-

served or measured, but it is the cause of most failures in

manufactured products Stress is defined as the force per

unit area (0) with English units of pounds per square inch

(psi) or metric units of megapascals (mpa) The type of load,

Le., duration of load application, coupled with thermal

conditions affects the ability of a structural component to

resist failure at a particular magnitude of stress Gas turbine

airfoils under sustained rotating loads at high temperature

may fail in creep rupture Components subjected to cyclic

loading may fail in fatigue High speed rotating disks al-

lowed to overspeed will burst when the average stress ex-

ceeds the rupture strength which is a function of the duc-

tility and the ultimate strength of the material

Conversely, strain is a measurable quantity When the size

or shape of a component is altered, the deformation in any

dimension can be characterized by the deformation per

unit length or strain (E) Strain is proportional to stress at

or below the proportional l i t of the material Hook's law

in one dimension relates stress to strain by the modulus of

elasticity (E) Typical values for E at 70°F are listed in Table

1 At elevated temperatures, the modulus will decrease for

the materials listed Note that the ratio of modulus to den- sity for the selected materials is relatively constant, i.e., E/(p/g) = lo8:

Material Modulus (E) psi Density (p/g) Ib/h2

Aluminum alloys 10.0 - 11.2 x 108 0.1 0 cobalt alloys 32.6 - 35.0 x 108 0.33 Magnesium alloys 6.4 - 6.5 x 10' 0.065 Nickel alloys 28.0 - 31.5 x 10' 0.30

Steel-stainless 28.5 - 31.8 x 1 0' 0.28

Titanium alloys 15.5 - 17.9 x 10' 0.1 6

30.7 - 31 O x 10'

Sources: Mil-Hdbk-5D fl], Aerospace Structural Metals Handbook p]

DefinitionMress and Strain

The following basic stress quantities am useful in the eval-

uation of many simple structures They are depicted in

Figures 1 through 4 Today, complex components with

rapid changes in cross-section, multiple load paths, and

stress concentrations are analyzed using finite element

models However, the basic equations supplemented by

handbook solutions should be employed for prelhinary cal-

culations and to check finite element model results

Basic Stress Quantfties

where P=load

A = area

Bending Stress: <r = Mc/I

where M = moment

I = area moment of inertia

c = distance from neutral surface

Ttansverse Shear Stress: z = VQ/It where V = shear force

Q = first moment of the area

I = area moment of inertia

t = thickness of cross-section

Torsional Shear Stress: T = TfIJ

where T = torque

r' = distance from axis of shaft,

J = polar moment of inertia

other For an incompressible material, v = 0.5 Since actu-

al materials are compressible, Poisson’s ratio must be less than OS-typically 0.25 I v 50.30 for most metals

Hooke’s Law in three dimensions for normal stresses [3]:

where G is the modulus of elasticity in shear

The relationship between the shearing and tensile mod- uli of elasticity for an elastic material:

Figure 5 Transverse shear stress E

erally used to evaluate yielding in a multiaxial stress field, allowing the comparison of a multiaxial stress state with the

The proportionality of load to deflection in one dimen-

sion is written as:

OX

Q = E& (for normal stress Q and strain E) dz

z = Gy (for shearing stress z and strain y) az &

a

J

Poisson’s ratio (v) is the constant for stresses below the

proportional limit that relates strain in one dimension to an- Figure 5 Three-dimensional normal stresses

uniaxial material data If the nominal equivalent stress is

less than the yield strength, no gross yielding will occur

Note that equivalent stresses are always positive If the sum

of the principal stresses ox, oy, and o, is positive, the equivalent stress is considered tensile in nature A negative sum denotes a compressive stress

To successfully analyze a structural component, it is

necessary to defme the force balance on the part A free

body diagram of the part will assist in determining the

path which various loads take through a structure For ex-

ample, in a gas turbine engine it is necessary to determine

the separating force at axial splitline flanges between the

engine cases to ensure the proper number of bolts and size

the flange thicknesses A free body diagram helps to iso-

late the various loads on the static structure connected to

the case The compressor case drawing in Figure 6 shows

the axial vane and flange loads on the case The pressure

differential across the case wall would also contribute to the

axial force balance i f the case was conical in shape

The internal pressure inaeases from the F1 totheF9vanes

at flange mating

Axial gas loads on vanes

Figure 0 Free body diagram of a compressor case

from a gas turbine engine

Compatibility

Compatibility refers to the concept that strains must be 100 lb

compatible within a continuum, i.e., the adjacent deformed

elements must fit together (see Figure 7) Boundary equa-

tions, strain-displacement, and stress equilibrium equa-

tions must be defined for the complete solution of a gen-

Figure 7 Compatibility

Saint-Venant’s Principle

Saint-Venant’s principle states that if the forces acting on

a local section of an elastic body are replaced by a statical-

ly equivalent system of forces on the same section of the body,

the effect upon the stresses in the body is negligible except

in the immediate area affected by the applied forces The

stress field remains unchanged in areas of the body which

are relatively distant from the surfaces upon which the farces are changed “Statically equivalent systems” implies that the two distributions of forces have the same resultant force and moment Saint-Venant’s principle allows simplification

of boundary condition application to many problems as long

as the system of applied forces is statically equivalent

Superposition

The principle of superposition states that the stresses at

a point in a body that are caused by different loads may be

calculated independently and then added together, as long

as the sum of the stresses remains below the proportional

limit and remains stable Application of this principle al- lows the engineer to break a more complex problem down into a number of fundamental load conditions, the solutions

of which can be found in many engineering handbooks

Plane stiss/Plane Strain

Often, for many problems of practical interest, it is possi-

ble to make simplifying assumptions with respect to the

stress or strain distributions For example, a spinning disk

which is relatively thin (see Figure 8) is in a state of plane

stress The centrifugal body force is large with respect to grav-

ity No normal or tangential loads act on either the top or bot-

tom of the disk 4, T ~ , and 2eZ are zero on these surfaces Since

the disk is thin, these stresses do not build up to significant

values in the interior of the disk Plane stress assumptions are

valid for thin plates and disks that are loaded parallel to their

long dimension Thin plates containing holes, notches, and

other stress concentrations, as well as deep beams subject to

bending, can be analyzed as plane stress problems

Another simplification can be made for long cylinders

or pipes of any uniform cross-section which are loaded lat-

erally by forces that do not vary appreciably in the longi-

tudinal direction If a long cylinder (see Figure 9) is sub- jected to a uniformly applied lateral load along its length and is constrained axially at both ends, the axial deflection

(6,) at both ends is zero By symmetry, the axial deflection

at the center of the cylinder is also zero and the approximate assumption that S, is zero along the entire length of the cylin- der can be made The deformation of a large portion of the body away from the ends is independent of the axial coor- dinate z The lateral and vertical displacements are a func- tion of the x and y coordinates only The strain components

E,, 'yxz, and y are equal to zero and the cylinder is in a state

Y?

of plane strain A pipe carrying fluid under pressure is an

example of plane strain

Figure 8 Thin spinning disk-an example of plane dress Figure 9 Pipe 1in-n example of plane strain

Thermal Stresses

Thermal stresses are induced in a body when it is subjected

to heating or cooling and is restrained such that it cannot ex-

pand or contract The body may be restrained by external

forces, or different parts of the body may expand or contract

in an incompatible fashion due to temperature gradients

within the body A straight bar of uniform cross-section, r e strained at each end and subjected to a temperature change

AT, will experience an axial compressive stress per unit

length of EaAT a is the coefficient of thermal expansion

A flat plate of uniform section that is restrained at the edges

and subjected to a uniform temperature increase AT will de-

velop a compressive stress qual to W T / ( l - v) Additional

miscellaneous cases for thermally induced stresses in plates,

disks, and cylinders, are listed in Young [4] and Hsu [5] Typ-

ical values for the coefficient of thermal expansion (a) for

several common materials are listed in Table 2

Design Hints

If thermally induced stresses in a member exceed the

capability of the material, increasing the cross-sec-

tional area of the member will generally not solve the

problem Additional cross-section will increase the

stiffness, and the thermally induced loads will increase

almost as rapidly as the section properties Often, the

flexibility of the structure must be increased such that

the thermal deflections can be accommodated without

building up large stresses

Thermal stress problems can be minimized by match-

ing the thermal growths of mating components through

appropriate material selection

In situations where transient thermal gradients cause peak stresses, changes in the mass of the component,

changes in the conduction path, addition of cooling flow, and shielding from the heat source may reduce the transient thermal gradients

Table 2 Range of Coefficient of Thermal Expansion for Common

Titanium alloys 400-1,OOO 5.05.4 x 1 @ 5.5-5.6 x 10-8

Aluminum alloys 300-600 13.0-1 4.2 x 1 p6 -

Magnesium alloys 300-600 15.5-1 5.7 x 1 0-8 -

and IOW alloy 45&1,000 7.1-7.3 x 10-6 7.7-8.3 X 1 W6

Sources: Mil-Hdbk-5D [l], Aerospace Structural Metals Handbook p]

STRESS CONCENTRATIONS

The basic stress quantities used in design assume a con-

stant or gradual change in cross-section The presence of

holes, shoulder fillets, notches, grooves, keyways, splines,

threads, and other discontinuities cause locally high stress-

es in structural members Stress concentration factors as-

sociated with the aforementioned changes in geometry

have been evaluated mathematically and experimentally

with tools such as finite element models and photoelastic

studies, respectively

The ratio of true maximum stress to the stress calculat-

ed by the basic formulas of mechanics, using the net sec-

tion but ignoring the changed distribution of stress, is the

factor of stress concentration (KJ A concentrated stress is

not significant for cases involving static loading (steady

stress) of a ductile material, as the material will yield in-

elastically in the local region of high stress and redistrib-

ute However, a concentrated stress is important in cases

where the load is repeated, as it may lead to the fatigue fail-

ure of the component Often components are subject to a

combination of a steady stress (0,) due to a constant load

and an alternating stress (GJ due to a fluctuating load such

that the stresses cycle up and down without passing through zero (see Figure 10) Note that the steady stress and the mean stress (om) may not have the same value The steady stress can have any value between the maximum and minimum stress values The damaging effect of a stress concentration

is only associated with the alternating portion of the stress cycle Hence, it is common practice to apply only any ex- isting stress concentration to the alternating stress [6] A

good example of this situation is a shaft transmitting a steady state torque that is also subject to a vibratory torsional

Figure I O Fluctuating stress

load which may be 6% of the steady state torque For stress

concentration features such as shoulder fillets, the & would

be applied to the vibratory or alternating stress

Design Hint

Eliminate unnecessary stress concentrations Avoid

abrupt changes in section where stress concentrations

cannot be relieved by a tolerable degree of local plas-

tic deformation All fillet radii should be made as gen-

erous as is practicable

When possible, keep hole locations away from areas of

high nominal stress For example, in high speed rotat-

ing disks such as turbine wheels (see Figure Il), holes

near the bore will be in a region of high hoop stress

Peak stresses at holes in the web of a rotating disk

m a y also be affected by the radial stresses due to ther-

mal gradients, rotational speed, and bending in the

web due to eccentric loads on the rim of the disk If web

holes are unavoidable, try to locate the holes in the most

biaxial stress field in the web, i.e., where the tensile hoop

stress and the tensile radial stress are nearly equal It

may also be necessary to increase the thickness of the

section around the holes to compensate for the stress

concentration

The use of corrosion-resistant materials helps prevent

stress concentrations caused by the pitting that may ac-

company typical corrosive attack

In certain situations, the clever removal of material reduces the effect of stress concentrations such as flange bolt holes Scalloping flanges as shown in Fig- ure 12 cut the hoop stress path, thus decreasing the ef- fect of the holes on the peak stress

Figure 11 Axisymmetric cross-section of a turbine wheel

f

1 7 Section"-"

Figure 12 Scalloped flange

Determination of Stress Concentration Factors

A first approximation for the stress concentration due to

a single s m a l l hole in a plate (Figure 13) subjected to a uni-

axial stress field is J$ = 3 In a biaxial stress field with equal

stresses (oo) of the same sign, the same hole (Figure 14)

would cause a maximum stress equal to twice the nominal

stress (% = 2) Conversely, for a biaxial stress field with

equal stresses of opposite sign, Kt = 4 This latter situation

would occur at a small hole in a thin cylinder subjected to pure torsion where o equals four times the nominal tor- sional shear stress (2) Stress Concentration Factors by R

E Peterson [7] is the best source of numerical values of &

for grooves, notches, shoulder filets, holes, and certain other miscellaneous design elements

Figure 13 Small hole in a plate subject to a uniaxial

stress field

Figure 14 Small hole in a plate subject to a biaxial

stress field

Example: For the hollow shaft in Figure 15, determine the

maximum equivalent stress at the shoulder fillet The shaft

is subjected to an axial tensile load and torque

thus Khollow = 1.53 Determine the axial K, from Figure 18:

rld = 0.07 Dld = 2.14 thus Kl = 2.22 Determine the nonconcentrated axial and torsional shear stresses:

oequivalent =

[2 (2.22 (-11,072))' + 6 (1.53 (28,650))2 -Os

= 79,803 psi

Figure 15 Hollow shaft subject to axial load and toque

(text coiitinued on page 305)

Figure 16 Stress concentration factor for torsion of a shaft with a shoulder fillet (From Stress Concentration Factors

by R E Peterson m Reprinted by permission of John Wiley & Sons, Inc.)

Figure 17 Effect of axial hole on stress concentration factor of a torsion shaft with a shoulder fillet (From Stress Concentration Factors by R E Peterson m Reprinted by permission of John Wiley & Sons, Inc.)