Thumb for Mechanical Engineers 2011 Part 12 pptx

Stress rupture is the continuation of creep to the point where failure takes place.. t = time hours c = empirical parameter relating test specimens to design Creep strength is specified

320 Rules of Thumb for Mechanical Engineers

CREEP RUPTURE

Creep is plastic deformation which increases over time

under sustained loading at generally elevated temperatures

Stress rupture is the continuation of creep to the point where

failure takes place Metallic and nonmetallic materials vary

in their susceptibility to creep, but most common structur-

al materials exhibit creep at stress levels below the propor-

tional limit at elevated temperatures which exceed one-

third to one-half of the melting temperature A few metals,

such as lead and tin, will creep at ordinary temperatures

The typical strain-time diagram in Figure 39 for a ma-

terial subject to creep illustrates the three stages of creep

behavior After the initial elastic deformation, the materi-

al exhibits a relatively short period of primary creep (stage

l), where the plastic strain rises rapidly at first Then the

strain versus time curve flattens out The flatter portion of

the curve is referred to as the secondary or steady state creep

(stage 2) This is the stage of most importance to the en-

I

Figure 39 Three stages of creep behavior

(stage 3) is characterized by an acceleration of the creep rate,

which leads to rupture in a relatively short period of time High stresses and high temperatures have comparable ef- fects Quantitatively, as a function of temperature, a loga- rithmic relationship exists between stress and the creep rate

A number of empirical procedures are available to correlate

stress, temperature, and time for creep in commercial alloys The Larson-Miller parameter (Km) is an example of one of these procedures The general form of the Larson-Miller equation is:

KLM = (O.OOl)(T + 46O)(lOg ct + 20) where

T = temperature in OF

t = time (hours)

c = empirical parameter relating test specimens to design

Creep strength is specified as the stress corresponding to

a given amount of creep deformation over a defined peri-

od of time at a specified temperature, i.e., 0.5% creep in

10,000 hours at 1,200"F The degree of creep that can be

tolerated is a function of the application In gas turbine en- gines, the creep deformation of turbine rotating components must be limited such that contact with the static structure does not occur In such high temperature applications, stress rupture can occur if the combination of temperature and stress is too high and leads to fracture As little as a 20"

to 30°F increase in temperature or a 10% increase in stress

can halve the creep rupture life Murk's Hurzdbook [16] pro-

vides some creep rate information for steels

component

FINITE ELEMENT ANALYSIS Over the last 25 years, the finite element method (FEM)

has become a standard tool for structural analysis Ad-

vances in computer technology and improvements in finite

element analysis (FEA) software have made FEA both af-

fordable and relatively easy to implement Engineers have

access to FEA codes on computers ranging h m mainfram es

to personal computers However, while FEA aids engi- neering judgment by providing a wealth of information, it

is not a substitute

Stress and Strain 321

Overview

FEM has its origins in civil engineering, but the method

first matured and reached a higher state of development in

the aerospace industry The basis of FEM is the represen-

tation of a structure by an assemblage of subdivisions, each

of which has a standardized shape with a finite number of

degrees of freedom These subdivisions are finite elements

Thus the continuum of the structure with an infinite num-

ber of degrees of freedom is approximated by a number of

finite elements The elements are connected at nodes, which

are where the solutions to the problem are calculated FEM

proceeds to a solution through the use of stress and strain

equations to c a l d a t e the deflections in each element pro-

duced by the system of f o m coming from adjacent elements

through the nodal points From the deflections of the nodal points, the strains and stresses are calculated This procedure

is complicated by the fact that the force at each node is de- pendent on the forces at every other node The elements, like

a system of springs, deflect until all the forces balance The solution to the problem requires that a large number of simultaneous equations be solved, hence the need for ma- trix solutions and the computer

Each FEA program has its own library of one-, two-, and three-dimensional elements The elements selected for an analysis should be capable of simulating the deformations

to which the actual structure will be subjected, such as bending, shear, or torsion

One-dimensional Elements

The term one-dimensional does not refer to the spatial

location of the element, but rather indicates that the element

will only respond in one dimension with respect to its

local coordinate system A m s s element is an example of

a one-dimensional element which can only support axial

loads See Figure 40

Figure 40 One-dimensional element

Two-dimensional Elements

A general two-dimensional element can also span three-

dimensional space, but displacements and forces are lim-

ited to two of the three dimensions in its local coordinate

system "bo dimensional elements are categorized as plane

stress, plane strain, or axisymmetric

Plane stress problems assume a small dimension in the

longitudinal direction such as a thin circular plate loaded

in the radial direction As a result the shear and normal stresses in the longitudinal direction are zero Plane strain problems pertain to situations where the longitudinal di-

mension is long and displacements and loads are not a function of this dimension The shear and normal strains in

the longitudinal direction are equal to zero Axisymmetric elements are used to model components which are sym-

metric about their central axes, i.e., a volume of revolution Cylinders with uniform internal or external pressures and turbine disks are examples of axisymmetric problems

Symmetry permits the assumption that there is no variation

in stress or strain in the circumferential direction

Two-dimensional elements may be triangular or quadri- lateral in shape Lower order linear elements have only cor- ner nodes while higher order isoparametric elements may have one or two midsides per edge The additional edge nodes allow the element sides to conform to curved bound- aries in addition to providing a more accurate higher order displacement function See Figure 41

Three-dimensional Elements

Three-dimensional solid elements are used to model structures where forces and deflections act in all three di- rections or when a component has a complex geometry that does not permit two-dimensional analysis Three-dimen- sional elements may be shell, hexahedra (bricks), or tetra- hedra; and depending upon the order may have one or two midside nodes per edge See Figure 42

322 Rules of Thumb for Mechanical Engineers

The choice of elements, element mesh density, bound-

ary conditions, and constraints are critical to the ability of

a model to provide an accurate representation of the phys-

ical part under operating conditions

Element mesh density is a compromise between mak-

ing the mesh coarse enough to minimize the compu-

tation time and fine enough to provide for conver-

gence of the numerical solution Until a “feel” is

developed for the number of elements necessary to

adequately predict stresses, it is often necessary to

modify the mesh density and make additional runs

until solution convergence is achieved Reduction of so-

lution convergence error achieved by reducing ele-

ment size without changing element order is known as

h-convergence

Models intended for stress prediction require more el-

ements than those used for thermal or dynamic analy-

ses Mesh density should be increased near areas of

stress concentration, such as fillets and holes (Figure

43) Abrupt changes in element size should be avoid-

ed, as the mesh density transitions away from the stress

concentration feature

Compared to linear comer noded elements, fewer high-

er order isoparametric elements are required to model

a structure In general, lower order 2D triangular ele-

ments and 3D tetrahedral solid elements are not ade-

quate for structural analysis Some finite element codes

use an automated convergence analysis technique

Figure 45 Increase mesh density near stress concen- trations

known as the p-convergence method This method maintains the same number of elements while in- creasing the order of the elements until solution con- vergence is achieved OT the maximum available element order is reached

Convergence of the maximum principal stress is a

much better indicator than the maximum Von Mises

equivalent stress The equivalent stress is a local mea-

sure and does not converge as smoothly as the maxi-

mum principal stress

Stressandstrain 323

Elements with large aspect ratios should be avoided For

two-dimensional elements, the aspect ratio is the ratio of

the larger dimension to the smaller dimension While an

aspect ratio of one would be ideal, the maximum allow-

able element aspect ratio is really a function of the stress

field in the component Larger aspect ratios with a value

of 10 may be acceptable for models of components such

as cylinders subjected only to an axial load Generally, the

largest aspect ratio should be on the order of 5

Highly distorted elements should be avoided Two-di-

mensional quadrilateral and threedimensional brick el-

ements should have comers which are approximately

right angled and resemble rectangles and cubes re-

spectively as much as possible, particularly in regions

of high stress gradient The angle between adjacent

edges of an element should not exceed 150" or be less

than 30" Many current finite element modeling codes

have built-in options which permit identification of

elements with sufficient distortion to affect the model's

accuracy

Symmetry in a component's geometry and loading

should be considered when constructing a model

Often, only the repeated portion of the component

need be modeled A section of a shaft contains three

equally spaced holes A solid model of the shaft con-

taining one hole or even onehalf of a hole (Figure a),

if the holes are loaded in a symmetric manner, must be

modeled to perform the analysis Appropriate con-

straints which define the hoop continuity of the shaft

Figure 44 Sector model of a shaft cross-section con- taining three holes

must be applied to the nodes on the circumferential boundaries of the model

A number of FEA modeling codes have automated meshing features which, once the solid geometry is de- fined, will create a mesh at the punch of a button This greatly speeds the production of a model, but it cannot

be assumed that the model that is created will be free

of distorted elements Auto mesh programs are prone

to creating an excessive number of elements in areas where the stress field is fairly uniform and such mesh density is unwarranted The analyst must use available mesh controls and diagnostic tools to minimize these potential problems

Generally, the finer the element mesh, the more accu-

rate the analysis However, this also assumes that the

model is loaded appropriately to mimic the load conditions

to which the part is exposed It is always advisable to

ground the analysis with actual test results Once an ini-

tial correlation between the model and test is established,

then subsequent modifications can be implemented in the model with relative confidence In many instances, the

FEA results predict relative changes in deflection and stress between design iterations much better than they predict absolute deflections and stresses

324 Rules of Thumb for Mechanical Engineers

CENTROIDS AND MOMENTS OF INERTIA FOR COMMON SHAPES

Key to table notation: A = area (in.2); II = moment of inertia about axis 1-1 (in."; J, = polar moment of inertia (in.4); c-denotes centroid location; a and p are measured in radians

BEAMS: SHEAR, MOMENT, AND DEFLECTION FORMULAS FOR COMMON END CONDITIONS

Key to table notation: P = concentrated load (lb.); W = uniform load (lbhn.); M = moment (in, lb.); V = shear (lb); R = reaction (lb.); y = de-

flection (in.); 0 = end slope (radians); E = modulus (psi); I = moment of inertia (in?) Loads are positive upward Moments which produce com-

pression in the upper surface of beam are positive

1 Cantilever - End LMd

c -

4 Cantilever - End Moment

5 End Supports - Intermediate Load

0 =

WL2

7 One End Supported and One End Fixed - Intermediate Load

9 Both Ends Fmed - Intermediate Load

328 Rules of Thumb for Mechanical Engineers

1 Dept of Defense and Federal Aviation Administra-

tion, Mil-Hdbk-5D, Metallic Materials and Elements for

Aerospace Vehicle Structures, Vol 1-2, Philadelphia:

Naval Publications and Forms Center, 1983

2 Aerospace Structural Metals Handbook Vol 1-5,1994

ed W Brown Jr., H Mindlin, and C Y Ho @Is.) West

Lafayette, IN: CINDAS / USAF CRDA Handbooks Op-

erations Fkudue University

3 Wang, C., Applied Elasticity New York: McGraw-Hill

Book Co., 1953, pp 3&3 1

4 Young, W C., Roark 's Formulas for Stress and Strain,

6th Ed New York: McGraw-Hill Book Co., 1989

5 Hsu, T H., Stress and Strain Data Han&ook Houston:

Gulf Publishing Co., 1986, pp 364-366

6 Seely, F B and Smith, J O., Advanced Mechanics of

Materials, 2nd Ed New York John Wiley & Sons,

Inc., 1952, p 415

7 Peterson, R E., Stress Concentration Factors New

York John Wiley & Sons, Inc., 1974

8 Hsu, T H., Stuctural Engineering &Applied Mechan-

ics Data Handbook, Volume I : Beam Houston: Gulf

Publishing Co., 1988

9 Higdon, A., Ohlsen, E H., Stiles, W B., and Weese,

J A., Mechanics of Materials, 2nd Ed New York

John Wiley & Sons, Inc., 1967, p 236

10 Perry, D J and Azar, J J., Aircraft Structures, 2nd Ed

New York McGraw-Hill Book Co., 1982, p 313

11 Shigley, J E., Mechanical Engineering Design, 3rd

Ed New York McGraw-Hill Book Co., 1977, p 208

12 Gleason Works, Gleason Curvic@ Coupling Design Manual Rochester, Ny: Gleason Works, 1973

13 Machine Design 1993 Basics of Design Engineering

Reference Volme, Vol 65, No 13, June 1993, p 271

14 Dann, R T 'Wow Much Preload for Fasteners?" Ma-

chine Design, Aug 21, 1975, pp 66-69

15 Franm, I? R "Are Your Fasteners Really Reliable?" Ma- chine Design, Dec 10, 1992, pp.66-70

16 MacGregor, C W and Symonds, J "Mechanical Prop-

erties of Materials" in Marks ' Standard Handbook for Mechanical Engineers, 8th ed T Baumeister, E A Avallone, and T Bameister 111 (Eds.) New Yak: Mc-

Graw-Hill Book Co., 1978, pp, 5-11

Fatigue

J Edward Pope Ph.D., Senior Project Engineer Allison Advanced Development Company

Introduction 330

Design Approaches to Fatigue 331

Residual Stresses 332

Notches 332

Real World Loadings 335

Temperature Interpolation 337

Material Scatter 338

Estimating Fatigue Properties 338

339 Stages of Fatigue 330

Crack Initiation Analysis 331

Crack Propagation Analysis 338

K-The Stress Intensity Factor

Crack Propagation Calculations 342

Creep Crack Growth 344

Inspection Techniques 345

Fluorescent Penetrant Inspection ( P I ) 345

Magnetic Particle Inspection (MPI) 345

Radiography 345

Ultrasonic Inspection 346

Eddy-Current Znspection 347

Evaluation of Failed Parts 347

Nonmetallic Materials 348

Fatigue T ~ n g 349

Liabhty Issues 350

References 350

329

330 Rules of Thumb for Mechanical Engineers

INTRODUCTION

Fatigue is the failure of a component due to repeated ap-

plications of load, which are referred to as cycles An ex-

ample of fatigue failure can be generated using a paper clip

Bending it back and forth will cause failure in only a few

cycles It has been estimated that up to 90% of all design-

related failures are due to fatigue This is because most de-

sign problems are worked out in the development stage of

a product, but fatigue problems may not appear until many

cycles have been applied By this time, the product may al-

ready be in service

In the 1840s, the railroad industry pushed the limits of

engineering design, much as the aerospace industry does

today It was noted by those in the field that axles on rail-

road cars failed after repeated loadings At this time, the con-

cept of ultimate stress was well understood, but this type

of failure was clearly something new and puzzling The phe-

nomenon was termed fatigue because it appeared that the

material simply became tired and failed August Wohler, a German railroad engineer, performed the first thorough investigation of fatigue in the 1850s and 1860s He showed

that fatigue life was related to the applied load

The basic principles discovered by Wohler are still valid today, although much additional knowledge has been gained Many of these lessons have been learned the hard way Some of the more notable fatigue problems include:

World War II liberty ships, which sometimes broke in half Two Comet aircraft, the world’s first passenger jet, lost due to fatigue failure which originated at the cor- ner of a window

Several USAF F-111 aircraft, lost in the 1960s due to

the unforgiving nature of titanium

Fatigue failure generally consists of three stages (see Fig-

ure 1):

1 Crack initiation (may be multiple initiation sites)

2 Stable crack growth

3 Unstable crack growth (fast fracture)

Figure 1 Typical fatigue fracture surface (Courtesy of

A E Grandt, JL )

Although cracks may be created during manufacturing, they generally do not initiate until after a considerable pe- riod of usage Cracks commonly form at metallurgical de- fects such as voids or inclusions, or at design features such

as fillets, screw threads, or bolt holes A crack can initiate

at any highly stressed location

After a crack has initiated, it will grow for a while in a stable manner During this stage, the crack will grow a very short distance during each load cycle This creates pat- terns known as “beach marks” because they resemble the patterns left in sand by wave action along a beach As the crack becomes larger, it usually grows at an increasingly rapid rate

Final failure occurs very quickly For small components,

it happens when the cross-sectional area has been reduced

by the crack so much that the applied stress exceeds the ul- timate strength of the material In larger components, fast fracture occurs when the fracture toughness of the mater- ial has been exceeded, even though the remaining cross-sec- tional area is still large enough to keep the applied stress well below the ultimate strength This will be explained in the section on crack propagation

Fatigue 331

DESIGN APPROACHES TO FATIGUE

Fatigue is dealt with in different ways, depending on the

application:

1 Infinite life design

2 Safe life design

3 Fail safe design

4 Damage tolerant design

In Wohler’s original work on railroad axles, he noted that

there is a stress below which failure will not occur This

stress level is referred to as the fatigue strength or en-

durance limit The simplest and most conservative design

approach is to keep the stress below this level and is called

infinite life design For some applications, the cycles ac-

cumulate so rapidly that this is virtually the only approach

A gear tooth undergoes one cycle each time it meshes with

another gear If the gear rotates at 4,000 rpm, each tooth wi@

experience nearly a quarter million cycles during every hour

of operation Vibratory stresses must also be kept below the

endurance limit, since these cycles mount up even faster

This type of loading is referred to as high cyclefatigue, or

HCF The term Zow cyclef&gzie, or LCF, is used to describe

applications in which the load is applied more slowly, such

as in steam turbines One cycle is applied when the engine

is started and stopped, and the engine may run continuously

for months at a time

In the aerospace business, the excessive weight required

to design for infinite life is prohibitive With the safe life de-

sign approach, a life is calculated which will cause a small

percentage of the parts (typically 1 out of 10,000) to initi- ate a crack All parts are removed from service when they reach the design life, even though the vast majority show no evidence of cracking This approach has been used in the air-

craft and turbine engine industries When the design life is calculated, the analysis must account for significant scatter

in the applied loads and fatigue properties of the materials

In some instances, design precautions can be taken such that the failure of a particular component will not be cata- strophic This is known asfail safe design After failure, the

component can be replaced This often involves redun- dant systems and multiple load paths An obvious exam- ple of this approach is a multi-engine plane If one engine fails, the others can still provide power to keep the plane flying In the design of the aircraft and engine, it is neces- sary to ensure that debris from the failure of one engine will not take out vital systems In one airline accident, fragments from a turbine wheel burst in one engine knocked out all three hydraulic systems which were placed closely to- gether at one point along the fuselage

D a m g e tolerant design assumes that newly manufac-

tured parts may have cracks already in them The design life

is based on the crack growth life of the largest crack that may escape detection during inspection This approach has been championed by the U.S Air Force for many years It puts a greater emphasis on the crack growth prop- erties of the material, while the safe life approach empha- sizes the crack initiation properties It also requires good inspection capability

CRACK INITIATION ANALYSIS

The first step in calculating crack intitiation life is to de-

termine ths stresses in a component Life is related to the

range of stress, as shown in Figure 2 (Life can also be re-

lated to strain range) These are known as S-N curves, and

are plotted on log-log paper While the alternating stress is

the major factor in determining life, the ratio of the mini-

mum stress to the maximum stress, also known as the R

ratio, is a secondary factor For a given alternating stress,

increasing the R ratio will decrease the crack initiation

life For example, a component with stresses varying from

50 to 100 ksi will have a lower life than a component with

stresses varying from 0 to 50 ksi

-b

5

N Figure 2 Typical S-N (log stress versus log life) plot [14]

(Reprinted with permission of John Wiley & Sons, lnc.)

332 Rules of Thumb for Mechanical Engineers

Residual Stresses

I

The designer can sometimes use the R ratio effect to his

advantage Surface treatments such as shot-peening and car-

burizing create residual stresses Residual stresses are

sometimes referred to as self-stresses Figure 3 shows how

the residual stress varies below the surface One seldom gets

something for nothing, and residual stresses are no excep-

tion Although the stress at the surface is significantly

compressive, it is counterbalanced by tensile stress below

the surface Fortunately, this condition is generally a very

good trade because most cracks initiate at the surface

Residual stresses can also reduce crack hitiation life Im-

proper machining, such as grinding burns, can cause large

tensile residual stresses While in service, a component

may be exposed to a compressive stress that is large enough

to cause yielding, leaving a tensile residual stress Tensile

residual stresses increase the R ratio, and therefore lower

crack initiation life

Depth below surface

Figure 3 Typical distribution of residual stress under a shot-peened surface D41 (Reprinted with permission

of John Wiley & Sons, he.)

Notches

In most cases, cracks initiate at some kind of notch or

stress concentrator Typical examples include:

Figure 4 shows how the negative effects of notches can be

lessened These strategies can be summed up as follows:

Allow the stress to flow smoothly through the com-

Provide generous fillets and avoid sharp comers

Increase the cross-section where the notch occurs The

stress concentration factor will be just as high, but the

nominals tress it is applied to will be lowered

ponent (think of stresses as flowing water)

The first example in 4 may seem strange at first How can

the stress concentration effect of a hole be lowered by

drilling more holes? The two smaller holes provide for a

smoother flow of stress around the larger hole Figure 5

shows that this could significantly lower the stress concen-

tration factor Because, as a rule of thumb, a 10% decrease

in stress doubles the crack hitiation life, this could lead to

a dramatic improvement in the durability of the component

A similar effect could be achieved by creating an elliptical hole instead of a round one It should be pointed out that nei- ther strategy should be applied unless the single round hole results in insufficient crack initiation life Extra holes mean extra expense, and no one wants to driu elliptically shaped holes The best design is the one that meets the criteria at the lowest cost Keep the design simple whenever possible

Fatlgue Notch Factor

The stress concentration factor is normally represented

For crack initiation life calculations, the fatigue notch fac- tor (Kf) is applied to the nominal stress rather than the

stress concentration factor These two factors are related by

the equation:

Fatigue 333

Figure 4 Good and bad design practices [15]

Improved Fatigue Strength

Large fillet radius

E k l

.Undercut radiused fillets

Stress- relieving grooves

Enlarged section at hole

Central hole diameter-to-plate width ratio, c/w

Figure 5 Stress concentration factors with and without

auxiliary holes [16] (Reprinted with permission of Soci- ety for Experimental Mechanics.)

334 Rules of Thumb for Mechanical Engineers

The notch sensitivity factor “q” is a material property

which varies with temperature Its value ranges from 1

(fully notch sensitive) to 0 (notch insensitive) The value

of q should be assumed to be 1 if it is not known This will

give a conservative estimate of crack initiation life

localized Yielding

Cracks generally initiate at a notch where there is some

localized plastic yielding Typically, the designer has only

elastic stresses from a finite element model Fortunately,

elastic stresses are sufficient to make a good approximation

of the true stresses as long as the yielding is localized

There are two methods for making this approximation:

Neuber method

Glinka method

Figure 6 graphically compares the two methods Each di-

agram shows a plot of stress versus strain For both meth-

ods, the area under the elastic stress-strain curve (Al) is cal-

culated With the Neuber method [l] a point on the true

stress-strain curve is found such that the triangular area A2

equals Al With the Glinka method [2], a point on the true

stress-strain curveis found such that the area under that

curve (A3) equals Al The Neuber method is more com-

monly used in industry, and has an advantage in that it can

(A) Stress ,,,,,qz::asticalh/ Calculated Stress

Figure 6 Neuber (A) and Glinka (B) methods of com-

puting true stresses from elastically calculated variables

be solved directly The Glinka method is slightly more ac-

curate, but the calculation is a little more difficult because

the area under the stms strain curve must be integmted Both

methods q u i r e an iterative solution The user should always remember that these methods are limited to cases where thm

is only localized yielding For situations involving hgescale yielding, plastic finite element analysis is required The Ramberg-Osgood equation can be used to define the rela- tionship between true stress and true strain:

where: he = true strain

a = true stress

E = modulus of elasticity

K = monotonic strength coefficient

n = strain hardening exponent The parameters K and n have different values for the ini- tial monotonic loading and the stabilized cyclic loading The monotonic and cyclic behaviors may be very different, as Figure 7 illustrates In general, if the ratio of the ultimate strength to the 2% yield strength is high (>1.4),the mate- rial will cyclically harden (waspaloy in Figure 7) If the ratio

of the ultimate strength to the .2% yield strength is low (c1.2), the material will cyclically soften (SAE 4340 in Fig-

ure 7) [3] For most crack initiation analysis, adequate re- sults can be obtained by using the monotonic K and n on

the initial loading, and the cyclic values for all subsequent loadings The transitional behavior from monotonic to cyclic is seldom significant

Monotonic

Man-Ten Steel 7075-T6