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On the probabilistic design of critical engineering components. Agrawal, Avinash Chandra 1971

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ON THE CRITICAL PROBABILISTIC DESIGN OF ENGINEERING COMPONENTS by AVINASH CHANDRA AGRAWAL B.E. (Mechanical), R.E. College, Durgapur, India, 1968 D.I.I.T. (Mechanical), I.I.T. Delhi, India, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept th is thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Apr i l 1971 In presenting th i s thesis in pa r t i a l fu l f i lment of the require-ments for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shal l make i t f ree ly ava i lable for reference and study. I further agree that permission for extensive copying of th i s thesis for scholar ly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publ icat ion of this thesis for f inanc ia l gain shal l not be allowed without my wr i t ten permi ssion. Avinash C. Agrawal Department of Mechanical Engineering The Univers ity of B r i t i s h Columbia Vancouver 8, Canada. Date _M^±J121 ABSTRACT The present study investigates the r e l i a b i l i t y approach to the design of a c r i t i c a l mechanical component and applies th is approach to several design problems. For the design of a c r i t i c a l component the combination of maximum loads and minimum material strength is selected for the design. Under the p r obab i l i s t i c approach, maximum load and minimum material strength values are considered as random variables having Extreme Value density functions of Type I (maximum) and Type III (minimum), respect ive ly. With th i s combination of p robab i l i t y density functions for the material strength and the load, a closed form solut ion does not seem to be feas ib le for e ither the probab i l i ty density function of the safety factor or for the probab i l i ty of f a i l u r e of the design. Consequently, numerical evaluations are made for the probab i l i ty density function of the safety factor V > the probab i l i ty of f a i l u r e Pf and the mean value of the safety factor , V , for a set of parameter values of the density functions for the maximum load and the minimum strength. The e f fect of changing these parameter values on the probab i l i ty of f a i l u r e is studied. An important feature of the design of a c r i t i c a l component from the r e l i a b i l i t y approach, in general, i s that the r e l i a b i l i t y statement implies a s pec i f i c "mission" time of operation for the component. This i s due to the dependence of the value of certa in parameters of E.V. models on the length of time over which the extreme value measurements are taken. Three design models are considered under the r e l i a b i l i t y approach for a given load and material strength, and r e l i a b i l i t y spec i f i ca t i on . i v The parameters defining the extreme value density functions of load and material strength are assumed to be given. In model 1, the problem of designing a s ingle c r i t i c a l mechanical component subjected to purely ax ia l loads i s considered for a given s ingle mater ia l . The f a i l u r e c r i t e r i o n in such a case is assumed to be separation and the output of the design process is a cross section area A of the component with a spec i f ied r e l i a b i l i t y over a corresponding period of operating time. This cross section area is considered as a s t a t i s t i c a l constant in order to avoid addit ional mathematical complexity. In model 2, the design problem of the f i r s t model is extended by considering more than one material avai lable for the design. The design problem thus considered i s one of se lect ing one among various a l te rnat i ve materials on the basis of some design c r i t e r i on such as minimum weight, etc. The method consists of ca lcu lat ing the design cross section area for each material ava i lab le and then ca lcu lat ing the value of the design c r i t e r i o n for each design. The material which optimizes the value of the design c r i t e r i on becomes the choice for that design. It i s observed that for a given load d i s t r i bu t i on and various ava i lable mater ia ls , the design cross section area i s a function of the ra t io of the mean strength to i t s standard deviation and not a function of the mean value of strength alone. It i s , therefore, considered log ica l to take th i s r a t i o of the mean strength to the standard deviation as the measure of the qua l i ty of the material and express the cost of material (dol lars per lb. ) as a function of th is r a t i o . This i s in contrast with the conventional design approach where the cost of material i s considered as a function of a s ing le value of strength of material only. V Model 3 considers the design problem of making a choice from among several materials on the basis of the economic c r i t e r i on of minimum cost. The cost of material i s considered as a function of the ra t i o of mean strength to standard deviat ion, as mentioned e a r l i e r . In the absence of data required to assess th is functional re la t ionsh ip , a l i near re lat ionsh ip is assumed. Another cost factor , the cost of safety factor values, is introduced. This cost is a measure of the margin of safety provided in the design for each component. As the safety factor is a random var iable in the p robab i l i s t i c approach, tools such as s t a t i s t i c a l decision theory and u t i l i t y theory are used to obtain the cost of design for each mater ia l . These cost values are then weighted with respect to the probab i l i ty density function of the safety factor . The expected overal l cost of the design is then evaluated for each material and the given load d i s t r i b u t i o n , such that the desired r e l i a b i l i t y value is attained. The material corresponding to the minimum value of the expected overal l cost is selected as the optimal choice of the designer. TABLE OF CONTENTS Page CHAPTER 1 1.1 Introduction 1-1 1.2 L i terature Survey 1-14 1.3 Present Research Problem 1-32 CHAPTER 2 2.1 Distr ibut ions of the Maximum Load and the Minimum Material Strength 2-1 2.1.1 E.V. Type III (min.)(Weibull) Model for the Minimum Material Strength 2-1 2.1.2 E.V. Type I (maximum) (Gumbel's) Model for the Maximum Load 2-3 2.1.3 Character i s t ic Values of the r .v . ' s Minimum Strength S and the Maximum Load L 2-5 2.2 Probab i l i t y Density Function of the Safety Factor . . 2-6 2.3 Probab i l i ty D i s t r ibut ion Function of the Safety Factor 2-9 2.3.1 P robab i l i t y D i s t r ibut ion Function of the Safety Factor '. . 2-9 2.3.2 Probab i l i t y of Fa i lure 2-11 2.4 Parameters of the Probab i l i ty Density Function of the Safety Factor . . . . . 2-12 v i i Page 2.5 Numerical Results . . 2-17 2.6 Design Methods for a Desired R e l i a b i l i t y Value . . . 2-18 2.6.1 Design of a Component, Given the Load D i s t r ibut ion and a Single Material D i s t r i -bution, for a Desired R e l i a b i l i t y (Model 1) 2-27 2.6.2 Select ion of a Single Material for the Design of the Component for the Given Load D i s t r i -bution and the Design R e l i a b i l i t y on Some Design Cr i te r ion (Model 2) 2-32 CHAPTER 3 3.1 General Decision Problem 3-1 3.2 P robab i l i s t i c Design Model with Economic Decision Cr i te r ion (Model 3) . 3-3 3.3 Cost Analysis 3-5 3.3.1 General Description of U t i l i t y Function . . . 3-5 3.3.2 Cost of Material 3-6 3.3.3 Cost of Realized Values of Safety Factor . . 3-10 3.3.4 Overall Cost of Design 3-13 3.4 Design Procedure 3-16 3.4.1 Decision Inputs 3-16 3.4.2 Decision Process 3-16 CHAPTER 4 Concluding Remarks 4-1 BIBLIOGRAPHY B-1 v i i i Page APPENDIX I - Methods of Obtaining Probab i l i ty Density Function of a Random Var iable, Function of Two Random Variables A - l 1-1 Method of Transformation of Variables A- l 1-2 Me l l i n ' s Transform Method A-4 1-3 Fourier Transform Method A-7 II - Solution for the P robab i l i t y of Fa i lure in Special Case of = 1.0 A- l 1 III - Some Series Solutions for the Probab i l i ty of Fai lure A-l2 IV - S t a t i s t i c a l Decision Theory and U t i l i t y Functions . . A-l3 V - Tables giving Values of Probab i l i ty of Fai lure for Different Values of Parameters, ^ L / ^ , ft and *S/L. A-22 VI - Graph giving Values of Ratio ft. for Di f ferent Values of the Parameter p> A-27 VII - Tables giving Values of Mean Safety Factor V for Different Values of the Parameters (°<\-/X^)} P • C*s/t.) A-29 VIII - I l l u s t r a t i v e Numerical Example A-34 LIST OF FIGURES Figure Page 1-1 Genegral Representation of the P robab i l i t y Density Function of a Random Variable 1-3 1-2 General Relation between the Probab i l i ty Density Function of the Safety Factor V and the Probab i l i ty of Fai lure . 1-7 1-3 S t a t i s t i c a l Representation of the Material Strength in the form of Probab i l i ty Density Function 1-18 1-4 Material Property Distr ibut ions 1-22 1-5 Probabi l i ty-Stress -Cyc le (P-S-N) Curves for a Material Subjected to Fatigue Loading 1-22 l-6(a) Interfer ing Load-Strength Probabi l i ty Density Distr ibut ions 1-26 1-6(b) Interfer ing Load-Strength Probab i l i ty Density Distr ibut ions 1-26 1-7(a) Histogram and Probab i l i ty Density Curve for the Y ie ld Strength of ASTM, A-7 Structural Steel 1-29 l-7(b) Histogram and Probab i l i t y Density Curve for the Ultimate Strength of ASTM, A-7 Structural Steel 1-29 1- 8 Effect of Sample Size ~n on the Location of the Extreme Value Probab i l i t y Density Function 1-36 2- 1 Plot of the Material Strength Values on Weibull P robab i l i ty Paper 2-15 2-2 Gamma Function 2-16 2-3 Probab i l i ty Density Function of the Safety Factor V , j2> = 1.0 to 5.0 ( integer values only) , ( ° V t ) = 0.10 and (*s/t) =2.0 2-19 2-4 Probab i l i t y Density Function of the Safety Factor V , |S = 1.0 to 5.0 (integer values on ly) , ( * i / t ) = 0.10, and ( V t ) = 3.0 2-20 X Figure Page 2-5 Probab i l i ty Density Function of the Safety Factor V , &> = 1.0 to 5.0 ( integer values only) , (°Wt) = 0.10, and («s/t) = 4.0 2-21 2-6 Probab i l i ty Density Function of the Safety Factor V , i& = 1.0 to 5.0 ( integer values only) , («L/Z) = 0.10, and (<*s/t) = 5.0 2-22 2-7 Probab i l i t y of Fa i lure £f vs. (2> (1.0 to 10.0), for values of(.°<s/Z) from 1.0 to 5.0 (only integer values) a n d C ^ L / t ) = 0.10 and 0.15 2-24 2- 8 Relationship between the Parameter fa and ( i ) Mean Value of the Material Strength, ( i i ) Standard Deviation of the Strength 2-26 3- 1 Cost of Material ($/lb.) as a Function of the Parameter 3-9 3-2 Cost of Material ($/lb.) as a Function of the Parameter 3-9 3-3 Cost of the Design Values of Safety Factor vs. V 3-14 VI-1 Relation between the Parameter fi of the Material Strength D i s t r ibut ion and the Ratio ft of the Mean Strength and the Standard Deviation A-28 VIII-1 Probab i l i ty of Fai lure R: vs. Central Safety Factor (°Vt) = V0 A-40 VI11-2 Mean Strength Value IT vs. Central Safety Factor (*s/t) = Vo A " 4 0 ACKNOWLEDGEMENTS The author would l i k e to express his s incere gratitude to Dr. K. V. Bury, who devoted considerable time and gave invaluable advice and guidance throughout a l l stages of the present work. Use of the Computing Centre f a c i l i t i e s at the Univers ity of B r i t i s h Columbia and the f inanc ia l support of the National Research Council of Canada are g ra te fu l l y acknowledged. LIST OF SYMBOLS S Strength of Design L Load V Safety Factor Area of Cross Section of the Material F^ CS) P robab i l i t y Density Function of the Material Strength S PiCL) Probab i l i ty Density Function of the Load l_ |=>(.L>) Probab i l i ty Density Function of the Safety Factor V P\(S) P robab i l i t y D i s t r ibut ion Function of the Material Strength S P2CL) P robab i l i ty D i s t r ibut ion Function of the Load L P(L) P robab i l i t y Density Function of the Safety Factor \) P robab i l i ty of Fai lure R R e l i a b i l i t y Scale Parameter of the Strength D i s t r ibut ion , also the Character i s t ic Value (3 Shape Parameter of the Strength D i s t r ibut ion L Location Parameter of the Load D i s t r i bu t ion , also the Mode Value and the Character i s t ic Value Scale Parameter of the Load D i s t r ibut ion Ratio of Mean Strength to Standard Deviation f^ C-t) Me l l i n ' s Transform of the Random Variable X w. r . t . i t s Probab i l i ty Density Function ^>cx) h 0s Character i s t ic Function (or Fourier Transform) of X w. r . t . x i t s P robab i l i ty Density Function t>(x) e£ (^ x) Fourier Transform of the Random Variable —X w. r . t . * the Probab i l i ty Density Function f=>cx) f"Ct.) Gamma Function of the order -t ^ C A ) Incomplete Gamma Function of Order t and the Argument A Ratio of Scale Parameter to Location Parameter for Load D i s t r ibut ion ( ^ V t ) Ratio of Character i s t ic Values of Minimum Strength and Maximum Load D i s t r ibut ions , the Central Safety Factor M(S) Mean Value of the Strength MOO Mean Value of the Load VQ Central Safety Factor r°i Density of material V Mean Value of Safety Factor V Vc Safety Factor Value as Given in Engineering Codes C'L Cost of the Material in $/lb. C± Minimum Cost of the Avai lable Material ($/lb.) ^ 2 . Maximum,Cost of the Avai lable Material ($/lb.) -^wih Minimum Value of Ratio n fc-max Maximum Value of Ratio /L V± Maximum Cost of Realized Value of Safety Factor at u=io Vo Minimum Cost of Realized Value of Safety Factor at V^VQ, L K LO Cost Associated with a Par t i cu la r Realized Value of Safety Factor U-U,u) Total U t i l i t y (cost) of the Design when Material i s Selected and a Value V i s rea l i zed for the Safety Factor Subscript L Material x i v Abbreviations Used r.v. Random Variable p.d.f. P robab i l i t y Density Function C.S.F. Central Safety Factor S.F. Safety Factor w. r . t . with respect to CHAPTER 1 1.1 INTRODUCTION The design of a mechanical component of a system is generally based on (a) the load for which the component i s to be designed and (b) the relevant material property of the material used. The design load depends upon the mission of the component or the environment in which the component is to operate, while the relevant material property is selected on the basis of a chosen f a i l u r e c r i t e r i o n . For example, i f the f a i l u r e refers to separation, the material property to be considered is ultimate tens i l e strength while in the case of deformation, i t i s y i e l d strength. S i m i l a r i l y , i f fat igue f a i l u r e i s the c r i t e r i o n for design, the relevant material property w i l l be the endurance l i m i t . If several s imi la r test specimens of a pa r t i cu la r material are tested for a spec i f i c material property, under s im i la r test condit ions, i t i s l i k e l y that d i f fe rent property values are obtained. These var iat ions in material properties are, general ly, a resu l t of several factors such as the control over the material production process, the specimen machining process, the constancy of load condit ions, etc. Likewise, measurements of the load for s im i l a r components under s im i la r operating conditions usually resu l t in widely varying load values. In both cases, the d i f fe rent values of the phenomena observed occur randomly. However, the values rea l ized (observed) do not occur in a chaotic manner. The frequency of 1-2 occurrence of the d i f fe rent values of e i ther loads or material property features certa in regu lar i ty patterns. Such a regu la r i ty pattern can be represented by some mathematical funct ion, which i s ca l led the probab i l i ty density function (p.d.f.) of the corresponding random variable ( r . v . ) . In Figure (1.1), the general nature of a p.d.f. i s shown for a random var iable X . {^ Cx-, &0 represents some form of mathematical function which i s indexed by a set of parameters 0 = (QiJ i^^Thus KxjBi) can be considered as a family of p .d . f . ' s . A spec i f i c vector value of 9 defines one par t i cu la r member of the family of p.d.f . ' s ^(XjO) . m * In pa r t i cu l a r , i f the r.v. X has a Normal p.d.f., ' for _oo«c x^oo , -oo<ryO,<oo and o~>C where fx and a- are the parameters of the p.d.f. of X and are defined as ? ju.— mean of the r. v. X o-2-— variance of the r. v. X For a par t i cu la r value of the r. v. X , given by ~}C , the area under the p.d.f., in f i g . (1.1), to the l e f t o f X . , gives the cumulative r e l a t i ve frequency of the values of the random var iable X ^ X . Thus, the cumulative r e l a t i ve frequency of occurrence of a r. v . X is given P C X ) « ( b(X,B)dx (1.2) Page 129 of Reference (1). RELATIVE FREQUENCY OF OCCURENCE o 6 -5 ro •CP CD CO CD rs rs to ro c X 3 rs ro O T3 r r -S _•. n> o v> rs ro rs O r r r r CU - i . O TO rs cu rs o Q- - h O 3 r r rs-< ro cu -s -o -•• "5 CU O cr cr — ' CU ro o-r r E-L 1-4 where PCX) is defined as the probab i l i ty d i s t r i bu t i on function. For eq. (1.1), where X L = -oo P0<) = f K * ; -oo (1.3) From the r e l a t i v e frequency interpretat ion of |^CX;9) , i t follows that j K * ; & ) = (1.4) Each property of each s pec i f i c material gives r i s e to a spec i f i c family of density functions, while the degree of control over production process, such as the percentage of a l l oy ing elements, control of temperature and homogeneity, e t c . , determine the observed var iat ion and thus influences the parameter value 0 . S im i l a r l y , for loads the type of operating conditions w i l l determine the nature of the p.d.f. as well as the values of i t s parameters, which in turn determine the mean and the variance. The conventional design procedure of a mechanical component is based on the formula: ^ - f - (1.5) where S = strength of the material (p . s . i . ) L = load e f fect (stress in p.s. i . ) V = safety factor (dimensionless number). The design, specifying a configuration of the component and i t s dimensions, is carr ied out on the basis of s ingle determinist ic values of the load and the strength of mater ia l . The safety factor 1) provides addit ional safety against a possible change in the value of load or the 1-5 strength of the material used. A par t i cu la r value of \) i s selected from the engineering codes or on the basis of some empirical r e l a t i on . Both the engineering codes and the empirical re lat ions are established on the basis of experience, judgement and knowledge on s im i l a r components used in the past. Whatever sophist icated design theory i s used, th i s determinist ic design procedure cannot be j u s t i f i e d when i t i s recognized that var iat ion occurs in the nature of most engineering phenomena such as applied loads or material properties. The spec i f i ca t i on of a minimum or maximum value of e ither load or material property is meaningful only i f that spec i f i ca t i on is associated with a corresponding probab i l i t y statement on the l i ke l ihood of i t s occurrence. Depending upon a designer 's conservatism, the design might become overweight, i e . , create problems of excessive weight and cost, etc. Moreover, in th i s design method, by choosing a s u f f i c i e n t l y high value of the safety factor , any p o s s i b i l i t y of f a i l u r e is assumed to be eliminated completely. In pract ice, of course, fa i lu res do occur, even for conservatively designed engineering components. This f o r ce fu l l y points to the need of considering loads and material strengths as random var iables. If the random behaviour of load L and material property S i s recognized, a component cannot be designed for a s ingle determinist ic value of the safety factor y . The safety factor given by eq. (1.5), i t s e l f becomes a random var iab le, having a certain p.d.f. \>(V) . The nature of |Xv) w i l l depend upon the nature of the var iat ion of the load L and strength S . The f a i l u r e occurs whenever S < L , i . e . 1/<1 In r e a l i t y , there is always a f i n i t e chance of f a i l u r e , since a l l pos i t ive values of V are possible. A general p.d.f. \>{v) of the safety factor 1-6 (a.r.v.) i s shown in f i g . (1.2). The area under th is p.d.f., to the l e f t of V—1-0 , represents the probab i l i t y that the load w i l l exceed the strength of material , and hence represents the probab i l i ty of f a i l u r e of the design. This probabi l i ty can be obtained using eq. (1.2), as 4 Ff =• J ?(V)cU (T.6) o where Fj; = P robab i l i t y of f a i l u r e . Thus, in the modern approach to design, a f ixed value of i s the input to the design process and corresponds to the input of a f ixed value of 1) in the t rad i t i ona l design process. This input value F£. i s a measure of the r i s k of the design and w i l l never be zero for a loaded component and for a given p.d.f. \>(u) , though i t may be very small. While the values of the random var iable U<.L corresponds to f a i l u r e , any value of th i s random var iable V>.-i implies that the design is strong enough to take the load and hence w i l l not f a i l . This probab i l i ty of the design carrying the load without f a i l u r e is then a measure of the r e l i a b i l i t y of the design. Thus, the r e l i a b i l i t y R of the design is given by R = Pn,iV^±) = j (1.7) 1 Using the general property of a p.d.f., given by eq. (1.4), the r e l i a b i l i t y R and the probab i l i ty of f a i l u r e fx are related as, * 0 The problem of design is thus reduced to the analysis of the data on loads and various material properties under various conditions (1.8) > RANDOM VARIABLE V Figure 1.-2 - General Relation between the Probab i l i ty Density Function of the Safety Factor D and the Probab i l i ty of Fa i lure p£ 1-8 of use, in terms of s t a t i s t i c a l random variables and the i r p .d . f . ' s . Depending upon the nature of the var iat ion of the load and the material strength, d i f fe rent p.d.f . ' s can be used to represent the i r random nature. In the l a s t two decades, several combinations of the p.d.f . ' s for S and L have been used to obtain the p.d.f. of the safety factor" V , and then to calculate the r e l i a b i l i t y of the design, using eq. (1.8) (see sec. 1.2). However, some component of a system may be c r i t i c a l in the sense that the success of the ent i re system depends upon the successful operation of such a component. It appears log ica l that such a component be designed for the worst combination of the Toad and the material strength, i e . for the maximum load and the minimum material strength. These extreme values of S and L are also random variables and can be defined by certa in extreme value density functions, (see section 1.3). Thus, i f the p.d.f. ' s of the extreme values of the load and the strength are known, the design of a c r i t i c a l component can be carr ied out, using eq. (1.5) and eq. (1.8), where V is now considered as a r.v.. Such a concept of the design of c r i t i c a l components has received no consideration in the l i t e r a t u r e and has been selected for the present work (see sec. 1.3). The procedure of designing an engineering component for the given load ( L ) and the material ( S ) , or making a choice among several materials ava i lable for the design, changes completely, i f these inputs to the design problem are defined as r . v . ' s . A. Conventional Method: ( i ) A load L , given as a s ing le determinist ic value, and a s ing le spec i f i c value of the safety factor V being selected from the engineering 1-9 codes, the strength of the component is determined. With th i s s ingle strength value, and the design material being known, the cross section of the component can be calculated for a spec i f ied conf igurat ion. This is the well known determinist ic approach to the design of an engineering component. ( i i ) For a given load L , the se lect ion between d i f f e ren t materials can be made on the basis of some design c r i t e r i o n , such as minimum weight, etc. Thus the procedure ( i ) i s followed for each mater ia l , resu l t ing in design cross section Ai for the -c mater ia l . Given the material densit ies , the optimum material can be selected to give the minimum weight. This is a log ica l extension of the procedure ( i ) given above. ( i i i ) If an economic decision is to be made on the basis of cost considerations, se lect ing (a) a deta i led design, (b) a mater ia l , (c) an input value i)*, then apart from the cost of the material ($/ lb . ) , the cost of the safety margin as a function of the decision var iable \) i s needed. An optimum decis ion is made by,choosing that material and the value of the safety factor V , for which the combined cost of material and safety margin, i s a minimum. The choosen material and the optimum value V* of the safety factor thus imply a spec i f i c detai led design. This i s an immediate appl icat ion of (determinist ic) decision theory to the design problem of se lect ing (a), (b) and (c ) . Since the premise of the present work is a p r obab i l i s t i c approach to design, th i s determinist ic decision model w i l l not be elaborated further. B. Modern Approach: ( i ) With the load L and the strength S considered as random var iables, which are associated with certa in p.d.f . ' s and the i r respective 1-10 parameter values, the safety factor it also becomes a r.v. The design procedure changes from specify ing a value for the safety factor D , to specifying a r e l i a b i l i t y value R for the design. Dimensions of the required design configuration are determined for the spec i f ied r e l i a b i l i t y value. This design process (model 1 )w i l l be outl ined in some deta i l in section 2.6.1 of this thes i s . ( i i ) The se lect ion among a l te rnat i ve materials on the basis of minimum weight or some other design c r i t e r i o n , not involving economic considerations, can be made by solving model 1 for each material and the speci f ied r e l i a b i l i t y value R . That material is selected which features the best performance, in terms of the decision c r i t e r i o n . This i s the model 2 of the design process and is also discussed in some deta i l in section 2.6.2. ( i i i ) I f the decision is to be based on the cost, with the r i sk of the design spec i f ied quant i tat ive ly in terms of a r e l i a b i l i t y value R , the cost of the design has to include a component which relates to the v a r i a b i l i t y of the safety factor l) (see chapter 3) along with the cost of the mater ia l . This cost of the safety factor \l needs to be known as a function of l) (as in the case ( i i i ) previous ly), since I) i s an i m p l i c i t r.v. under the p robab i l i s t i c approach to the design. S t a t i s t i c a l decision theory has to be used to specify an optimal decision on the choice of the mater ia l , given an input value of the r e l i a b i l i t y R . This i s model 3 of the design process and w i l l be analysed in chapter 3 of the thes i s . This modern approach to the problem of design can be used for indiv idual components and systems. A system can be synthesised as a combination of a number of components in series or pa ra l l e l only. If the 1-11 components, are in se r ie s , the system f a i l s when any one of the components f a i l s and the probab i l i ty o f ' t he system f a i l u r e is equal to the probab i l i ty of f a i l u r e of the weakest component. In the case of pa ra l l e l (redundant) component systems,•the system works as long as only a s ing le component works. The probab i l i t y of f a i l u r e of the system i s given by the product of the values for each component. In such a case, an assumption is involved that the f a i l u r e of a s ing le component does not e f fect the load d i s t r i bu t i on of the remaining components. This means that the probab i l i ty of f a i l u r e of each component is independent of the f a i l u r e of other component. Such.a r e l i a b i l i t y design of a complete system has not been considered in the scope of the present work but is a necessary requirement for system design. With the addit ional information concerning the topology of the constituent components, the design models discussed in th is thesis can be applied d i r e c t l y to system design. Two basic improvements resu l t from the s t a t i s t i c a l interpretat ion of the loads and the material propert ies, and the consequent design for a r e l i a b i l i t y value: (a) Reduce possible wastage of mater ia l , which might occur because of the (usually) conservative value of the chosen S.F. V (eq. 1.5). (b) Provide a quant itat ive statement of the r i sk which is associated with the design. Th is . r i sk of design is expressed in terms of the prob-a b i l i t y of f a i l u r e f^ . or the r e l i a b i l i t y R , for which the design has been carr ied out. • The l a t t e r r e s u l t - i s more important; refer r ing to the design (?) of a system, i t has been stated that, "This p r inc ip le neither impl ies, nor s p e c i f i c a l l y advocates, a reduction of safety in re la t i on to the 1-12 conventional approach, in which any r i s k of f a i l u r e is inadmissible. The p r inc ip le only attempts to place the concept of s t ructura l safety in the realm of physical r e a l i t y , in which such absolutes as minimum strength values do not ex i s t , and knowledge i s not perfect. The r i s k , or the probab i l i t y of f a i l u r e , is introduced as a quant i tat ive measure, in terms of which the safety and r e l i a b i l i t y of various parts of a system, can be defined, compared and a uniform r e l i a b i l i t y or safety of the complete system assured." Lack of actual data, both for material properties and loads, has been a serious problem in assessing the exact nature of the p.d.f . ' s for these var iab les . Materials are produced under widely varying con-d i t ions and the properties of materials should be avai lable in s t a t i s t i c a l form for a l l these condit ions. Novel projects give r i s e to new load patterns for which no data are ava i lab le from which p.d. f . ' s can be constructed. This lack of data a v a i l a b i l i t y is more severe in the extreme value regions, i e . for the var ia t ion in the maximum load values and minimum values of strength. However, the general nature of these extreme value random var iab les , as discussed in (1.3), has been used to se lect the E.V. 1 vnax (Gumbel's model) p.d.f. for maximum loads and the E-V.^ (Weibul l ' s model) p.d.f. for minimum strength of mater ia l . Various mathematical methods (Appendix I) are ava i lab le to obtain the p.d.f. of the safety factor \) , when the p.d. f . ' s of the r .v. ' s S and L are known. For some combination of the p.d.f . ' s for S and L , none of these methods may provide an ana ly t i ca l form of the p.d.f. for the; r.v. V . In such a case, numerical results are obtained, using a d i g i t a l computer to determine the nature of {^O) , as well as 1-13 (see eq. (1.6)). Such results are i n f e r i o r to even an approximate ana-l y t i c a l so lut ion for the p.d.f. of the r.v. V and the of the compo-nent, since no ins ight is gained into the cause-effect re lat ionsh ip among the various parameters which influence the function )p>{u) or The purpose of the present invest igat ion is to analyse models 1, 2 and 3 for the design process of a s ingle c r i t i c a l engineering compo-nent. This design problem, has been solved with various s impl i fy ing assumptions such as (a) the manner of loading does not e f fect the material property ( S and L are independent r . v . ' s ) , (b) the area of cross section A is a s t a t i s t i c a l constant and (c) the component i s subjected to pirely tens i l e loads. Such assumptions imply that the proposed design models correspond to approximations of real world condit ions. However, these assumptions can be removed at the cost of complexity. For example, the area of cross section A can be considered as a random var iable which is associated with a certa in p.d.f., and combined loads (bending and tens i le ) can be used for design. The proposed design models apply equally we l l , with su itable modif icat ions, to various other time-independent f a i l u r e c r i t e r i a and to the design of systems as well as to that of components. Different f a i l u r e c r i t e r i a merely imply d i f fe rent relevant material properties as well as d i f fe rent configurations for the component under consideration, while system design requires the addit ional considerations of the component topology of the system. 1-14 1.2 LITERATURE SURVEY The f i r s t comprehensive study on the importance of considering the random behaviour of loads and material strength, introducing the s t a t i s t i c a l concept of r e l i a b i l i t y to the design of mechanical components, was published by Freudentha1 v ' in 1947. This work drew the attention of many researchers in the f i e l d of design and was followed by a number of discussions of the problem as well as the f e a s i b i l i t y of such a study. The problem, since then, has been of continuing concern to researchers and pract i t ioners in the f i e l d of design. A great deal of work has been done in th i s f i e l d and i s ava i lab le in the l i t e r a t u r e , as can be seen from the appended bibl iography. As most of the o r i g ina l concepts of the s t a t i s -t i c a l r e l i a b i l i t y of design were originated by Freudenthal, much of the fol lowing summarises his work. 1. Conventional Design: The basic steps in the conventional design of any mechanical component are ( i ) the determination of the load ( L lbs.) on the component, and ( i i ) the comparison of a maximum load value with a rather f i c t i t i o u s resistance charac te r i s t i c ( 5 lbs.) ca l led the permissible or allowable strength value, which is a f ract ion of the strength of the respective mater ia l . The conventional concept of comparison implies that some margin of safety has to be provided between the strength of the material and the maximum load. This margin is expressed by eq. (1.5) which was given on page 1-4 as w S 1-15 The value 1) is defined as a safety factor by which the strength of a given design is higher than a l l the loads ant ic ipated during the mission of the component. The introduction of th i s safety factor (a s ingle determinist ic number) is assumed to take care of unusual conditions of load and material properties such as higher load and/or lower strength, l imited knowledge of the designer, and arb i t rar iness of his assumptions regarding loading conditions and material propert ies, etc. For a par t i cu la r design of a component, for a given load, and for a par t i cu la r mater ia l , the value of the safety factor V i s usually taken from engineering codes or chosen on the basis of empirical formulae. These code values, or empirical formulae, are established on the basis of past experience with the design of components which are made of s im i la r materials and are used s imi la r environmental and loading condit ions. 2. The Nature of Design Inputs: Modern design problems involve the design of a component which is l i k e l y to be used under widely varying loading condit ions. As an example, a s ingle compressive or tens i le component may be used as a part of some equipment which is exposed to contro l led temperatures and s t a t i c loads, or i t may be part of a large airframe structure which is exposed to widely varying temperatures and dynamic loads. The recent development in the f i e l d of material science has resulted in the production of new materials with properties which are d i f fe rent from the properties of materials used before. If code values for new materials are not establ ished, the designer must make a subjective choice based on his judgement of design conditions. A designer is thus 1-16 free to use his conservatism, judgement and bias for the se lect ion of a su i table value of safety factor V . Obviously such a design procedure might resu l t in rather a rb i t rary values for V , ranging perhaps from (4) ' ' 1.5 to 4.0 or even higher. ' In conventional design pract i ce , the material property as well as loads are considered to be determinist ic and defined by s ingle s pec i f i c values for each. For example, a maximum design load of 4000 lbs . and a minimum strength of material of 38,000 p . s . i . may be selected for a par t i cu la r design. In a real case, absolute maxima or minima do not occur with cer ta inty . There are d i f fe rent factors which cause these phenomena (3) to vary randomly. These factors have been summarised by Freudenthalv ' (5) and Mittenbergs. v ' (3) Ba s i ca l l y , the safety factor provides a safety margin f o r v (a) Ignorance:- Imperfection of i n t e l l e c t ua l concepts such as exact know-ledge of load and material property re lat ionsh ips , the f a i l u r e mechanism, the knowledge pf the underlying production process of the material spec i f i ed , etc. (b) Chance causes:- While the var iat ions due to ignorance can be largely eliminated by increasing the perfection of the design process or by i n s t i t u t i n g better controls over material production processes and environmental load var ia t ions , there are certa in chance factors which are d i f f i c u l t to el iminate completely. For example, a var ia t ion in the quantity of a l loys mixed, the temperature control during the production of mater ia ls , a var iat ion in the homogeneity of material produced, e t c . , are some of the chance causes for var iat ion in material properties. In the case of loads, as an example, a change in the environmental temperature w i l l change the dimensions of the component and so the load e f fect w i l l change. 1-17 It should in addition be noted that the values of load and material property are obtained as a resu l t of human observation and measurement. There are always some l im i ta t ions on the precis ion of measuring devices and human perceptions. Thus, neither the material property nor the load can be r e a l i s -t i c a l l y represented by a s ingle value for e i ther . Both strength S and load I— have to be recognised as random var iab les , which are characterised by certain patterns of the frequency of occurrence of the i r respective values. If this fact is recognised, the relevant information on S and L is spec i f ied by certa in probab i l i ty density functions and the i r respective parameter values. For example, the ultimate strength of certa in material may be represented by a Log-Normal p.d.f. given by eq. (1.9) and values for the parameters fx and o~~ of th i s p.d.f.: for s ^ O , jx. < t O } cr^O 2. where fx. = mean, and c r = variance. For a par t i cu la r mater ia l , parameters might be spec i f ied as, fx- = 3.0,000 p . s . i . , <r = 1,000 p . s . i . , for example. Looking at f i g . (1.3) where th i s density function is p lot ted, a minimum value S of the strength can be reasonably spec i f ied only with an associated probab i l i ty statement. The relevant probab i l i t y statement is provided by the area under the p.d.f., to the r ight side ofiS> , and is interpreted as the probab i l i ty that the strength S (a random variable) is equal to or greather than, the given value 3 • S im i l a r l y , the loads .6 0 10 20 30 40 50 60 X10 MATERIAL STRENGTH S (psi) Figure 1-3 S t a t i s t i c a l Representation of the Material Strength in the form of P robab i l i t y Density Function 1-19 can be expressed by certa in p.d.f . ' s and the spec i f i ca t i on of the maximum value of a load must be associated with a certain probab i l i t y statement. The fol lowing factors influence the var iat ion in loads/material properties/both: Group A: Causes of Fluctuations in load L : (I) Uncertainty and var ia t ion of loading conditions (a) Gravity loads: var iat ion occurs because of the var iat ion in the density of the mater ia l , dimensions, etc. (b) Applied loads: for example, in a simple turning process, the var iat ion in the depth of cut or the angle of cutt ing w i l l change the cutt ing force. In a c i v i l engineering structure, the applied loads refer to l i v e loads such as the load of a vehicle moving over a bridge. (II) Uncertainty and v a r i a b i l i t y of external conditions that are independent of applied loads such as, a change in temperature, uncertainty of the behaviour of subso i l , etc. (I I I) Uncertainty resu l t ing from load computations, such as, (a) Variations of r i g i d i t y (b) Imperfection of computational methods and shortcomings of assumptions: ( i ) Accuracy of method and tolerances of numerical computation; ( i i ) Inadequacy of assumptions concerning i n i t i a l and boundary condit ions, stress concentrations and secondary s t ra ins . Group B: Causes of Fluctuation of the Resistance to Applied Loads: This resistance depends on the configuration and dimensions of the design, as well as on the material strength S • 1-20 (IV) Uncertainty and inaccuracy of the assumed mechanism of res istance, (a) Inaccuracy or inadequacy of conceived mechanism. For example, the assumed s t i f fnes s conditions of a s t ructura l j o i n t may be un rea l i s t i c . (b) V a r i a b i l i t y of resistance l im i t s of mater ia l . These depend upon the rate of s t r a i n , the duration of s t r a i n , and the amplitude of the s t ra in cyc le, etc . (V) Variat ion of s t ructura l dimensions, surface condit ions, homo-geneity, etc. (VI) Time-dependence of material propert ies, depending on aging processes. (VII) Cumulative damage phenomena such as fat igue, creep. The nature of the var iat ion of loads and material strength is a combined ef fect of the s t a t i s t i c a l behaviour of the above factors . No data are ava i l ab le , ind icat ing how each of these factors effects the (4) f i na l random behaviour of e i ther load or strength. Again, Freudenthal v ' has made a theoret ical study of par t i cu la r influences. The f a i l u r e of a component may be time-dependent or independent. Basic f a i l u r e modes of s t ructura l elements have been summarised by Mit ten-(6) bergs v ' as ( i ) deformation ( i i ) f racture and ( i i i ) i n s t a b i l i t y such as e l a s t i c or p l a s t i c buckling. These f a i l u r e modes are effected by several factors such as: (a) The magnitude, d i rect ion and manner of loading; eg. steady, intermittent, c y c l i c , impact, or imposed by thermal gradients. (b) The load environment; eg. high and low temperatures, r e l a t i ve 1-21 vacuum, high pressure, various atmospheres. (c) The s ize and shape of the part and the conditions of i t s surface. (d) The material and i t s propert ies, including various material treatments and material responses to loads and environments. M i t t e n b e r g s ^ has shown ( f i g . 1.4) that a material property varies ( i ) among components which are taken (a) from various production lots and (b) from components produced from the same heat, and ( i i ) even at d i f fe rent cross sections of a s ing le component. It i s c lear that the var iat ion for case (a) is larger than for e i ther (b) or ( i i ) , while the var iat ion for case (b) is larger than for case ( i i ) . Thus i f the components are designed for materials from d i f fe rent sources, the var iat ion in the same property is l i k e l y to be quite appreciable. While the deta i l s of the e f fect of these factors upon the f a i l u re mode are avai lable in Mittenbergs p a p e r , ^ i t may be noticed that most of these factors e f fect ing f a i l u r e modes may change with time, and therefore a l l the f a i l u r e modes may be time-dependent from the re -l i a b i l i t y view point. For example, a component made of a certa in material may loose i t s strength because of corrosion, and have a s t a t i c tens i le f a i l u r e when the strength becomes less than the load. The f a i l u r e modes that are time-dependent are creep deformation, creep rupture, creep buckl ing, fat igue, and f a i l u r e occurring in one of the basic modes after the material has been subjected to creep or fat igue. The phenomenon of fatigue f a i l u r e i s an important one and has been studied by W e i b u l l ^ , Freudenthal and G u m b e l ^ , Pope et a l ^ , Armitage^ 1 0 ^, Hel ler et a l ^ 1 1 ^ , W h i t e ^ 1 2 \ Simon^ 1 3^, Forrester et a l ^ 1 4 \ etc. 1-MATERIAL PROPERTY Figure 1-4 - Material Property Distr ibut ions 10 5 10 6 10 7 io ; NUMBER OF CYCLES TO FAILURE , N Figure 1-5 - P robab i l i t y - s t re s s -cyc le (P-S-N) Curves for a Material Subjected to Fatigue Loadinn 1-23 Apart from studying the physics of the phenomenon of fat igue, the most common approach to the problem of fatigue has been to draw p - S - M curves for a material subjected to fatigue loading, as shown in f i g . (1.5) where, P = Probabi l i ty of Survival in percent, = Stress Amplitude, N = Number of cycles to f a i l u r e for these values of P and £ . (6) I t has been observed^ ' that the scatter for time or cycle dependent properties of materials i s usually considerably higher than that of s t a t i c propert ies. From Fig. (1.5), i t can be seen that for a given l i f e time (number of cyc les ) , the var iat ion in stress amplitude is considerably less than the var ia t ion in the l i f e time for a f ixed stress amplitude. The l a t t e r var ia t ion is r e l a t i v e l y high for the lower values of stress amplitude ( )> where the curve becomes almost horizontal This means that the var iat ion in actual l i f e times, for components which are designed for large l i f e times, is r e l a t i v e l y high. The concept of (8) ' f a i l safe des igns ' , introducing redundancy, was discussed by Pope et a l v 1 (2 1110 29) and has been analysed by many other researchers. ' ' ' ' The anlys is of loads and material properties under s t a t i c loading conditions and time-independent modes has received considerably more attent ion. The reason might be that r e l a t i v e l y more research has been applied to once-through projects. "For many appl icat ions , which involve steady loads and known environments, the changes occurring in materials and, as a consequence, in the character i s t ic s of the part, are r e l a t i v e l y small over the time period of interest and therefore can 1-24 (6) be neglected." K ' This statement can be j u s t i f i e d espec ia l ly in case of once-through operations such as the launching of a spacecraft and soforth. Furthermore, in most of the l i t e r a t u r e the random variables S (Material strength) and L (Applied load) are considered to be independent. This implies that the design is one for a r i g i d structure, carr ied out on the basis of an undeformed shape. If these two r .v . ' s are considered to be dependent, the r a t i o V becomes the ra t i o of funct iona l ly dependent r .v. ' s and cannot be derived unless the functional dependence is con-sidered in the s t a t i s t i c a l analys is . The former assumption s imp l i f i e s the problem to a great extent and simple s t a t i s t i c a l methods can be used. 3. Modern Design The load ( L ) and the strength of material ( S ) being random var iables, associated with certa in p .d . f . ' s , the concept of safety factor V , given by eg. (1.5), as a s ingle spec i f i c number, can no longer be j u s t i f i e d . It is a well known fact in s t a t i s t i c s that the ra t i o of two r .v. ' s i s another r.v.. The safety factor V thus must be considered as a r.v., which may give r i s e to observed values of V anywhere on the real + ve l i n e (0 to oo ). The frequency pattern of the r.v. i s defined by a certain p.d.f., the form and parameter values of which w i l l depend upon the p.d.f . ' s of S and L and the respective values of the i r parameters. It is c lear that a design on the basis of a s ingle value of i) i s not rat ional in the sense that the observable var iat ion in S and U i s ignored. Conventional design thus constitutes a f i r s t approximation to the more rat ional p robab i l i s t i c approach to design. The degree of approximation depends upon the actual var iat ion in S and/or L_ . 1-25 Furthermore, the r i s k associated with design for a given material cannot be ascertained under the conventional approach. In f i g . (1.6a), the p.d.f. ' s of S and L are shown by |\(,s) and jb 2 CL) •» respect ive ly. In the conventional procedure, the safety factor can be defined as the ra t i o of mean values of S and L . For a certa in value of strength ( S ), the f a i l u r e w i l l occur i f the load value ( L_ ) i s greater than this value of S . Thus the area which is common to both density functions refers to such a p o s s i b i l i t y . This area corresponds to probab i l i ty of f a i l u r e and the larger th i s area i s , the larger w i l l be the probab i l i ty of f a i l u r e . Also, however large the r a t i o of the mean values of S and L might be, th i s area w i l l never be zero and so the probab i l i ty of f a i l u r e of any design, even with a high value of safety factor , cannot be zero. In f i g . (1.6b), for the same load d i s t r i bu t i on , a material with d i f fe rent p.d.f. (lower mean strength and lower value of dispersion) is used. It can be seen that although the safety factor value has decreased, the area common to both the density functions, and hence the probab i l i t y of f a i l u r e , has remained the same as in case (a), because of the decrease in the v a r i a b i l i t y of F^CS) . Hence the design r e l i a b i l i t y R is the same for both mater ia l s , even though the conventional safety factor is higher for the f i r s t mater ia l . It has been suggested that in a rat ional design the analysis should be done on the basis of the probab i l i ty of f a i l u r e , which can be obtained as Probab i l i ty of f a i l u r e Pr — P/tCs<L) = fo.(s/L4.1) 1 = Pi(y<l) == J>(u)di; ( i . i o ) 1-26 MEAN'L MEAN'S' P^S) R E G I O N O F FAILURE Figure 1-6(a) - Interfer ing Load-Strength Probab i l i ty Density Distr ibut ions > CO LU Q > CQ < C D O DC Q_ • - R E G I O N O F FAILURE Figure l-6(b) - Interfer ing Load-Strength Probab i l i ty Density Distr ibutions 1-27 If the p.d.f. of the safety factor V is unimodal (having only one peak), values near the mode have the maximum frequency of occur-rence and so are more l i k e l y than other values removed from the mode. Such a value of V can be used as a "measure of l ocat ion " of the central range of the p.d.f. of the safety factor \J . A conveniently selected (2) "measure of l ocat ion " has been defined by Freudenthar ' as the "central safety factor y0 ," representing the ra t i o (S>/L0) of the ' cent ra l values' 5 0 and L e • The se lect ion of these central values w i l l depend upon the character of the functions {^CS) and CL.) • For example, in the case of a Normal p.d.f. the central value is the mean, in case of the Log-Normal model, the chosen central value is the mean of O03S ) and (log L ) which conveniently becomes the median of s and L , respec-t i v e l y , at the probab i l i ty value 0.5. In the case of extreme value density (15) functions, Gumbelv ' has found that convenient central values are the " cha rac te r i s t i c " values, equal to or close to the mode values, which are given by the probab i l i t y value (1/e) for the E.V-ivnxx P-d.f. and (1-1/e) for the E.V!jj W J M p.d.f. The nature of these density functions w i l l be discussed b r i e f l y in chapter 2. Various forms of p.d.f. ' s have been used to represent the random nature of S and L . The Normal p.d.f. has been used most e x t e n s i v e l y ^ t o 22 27) ' ' because of the s imp l i c i t y of ca lcu lat ion of the probab i l i t y of f a i l u r e . For a random var iable X , in general, the p.d.f. for a Normally d i s t r ibuted r.v. is given by eg. (1.1). The probab i l i ty of f a i l u r e (eg. 1.10) can be re-written as Pr ,. Prc(M.^) *KS--L<C) (1.11) 1-28 If S and L are Normal r . v . ' s , ( 5 - L ) w i l l also be a Normal r.v., with i t s parameters given b y , ^ Eg. (1.11) then reduces to: where <^> is the Standard Normal D i s t r ibut ion Function which is avai lable in tabulated f o r m . ^ (16) Jul ian^ ' plotted the histograms for the y i e l d strength and ultimate strength of Structural Steel (ASTM-A.7), as shown in figures 1.7(a) and 1.7(b). These histograms were plotted on the basis of actual test data ava i lab le . It is seen that these material properties are skewed to the r ight s ide. Moreover, material strength values are always pos i t ive and, for a given f a i l u r e mode, load values are always pos i t ive as w e l l . These two facts together make the Normal p.d.f. (-oo < x <LOO) a questionable choice for the s t a t i s t i c a l model of e ither S or L . Along with J u l i a n ^ 1 6 \ Freudenthal( 2 > 1 7 > 1 8 ) has used the Log-Normal p.d.f. given by eg. (1.9), to represent both S and L . This p.d.f. s a t i s f i e s the requirements of ( i ) only pos i t ive values of the r.v., and ( i i ) a density function which is skewed in the pos i t ive d i rec t i on . The safety factor V then also has a Log-Normal p.d.f. s ince, using eg. (1.5), ttxiV = io<j, (S/L) = £ n.c._f.^L . ( i . i 4 ) and, as S and L have Log-Normal p .d . f . ' s , logarithms of these r .v. ' s have Normal p .d . f . ' s . The var iable h>'^l) is therefore Normally d i s t r ibuted 62-L 1-30 and hence V has a log-Normal p.d.f. Both in the case of Normal density functions and Log-Normal density functions, the ef fect of a change of parameters, such as the mean /»-and the variance cr , has been studied and is ava i lab le in the papers of Freudenthal ^ , J u l i a n ^ ^ , M i t t e n b e r g s ^ , etc. Other combinations of p.d.f . ' s have been considered to represent the r .v. ' s S and L as fol lows: Load Strength Reference E - V - I , m a x . E ' V - I I I , m i n . <2> Weibull Weibull (21) Normal Weibull (21) However, in none of these combinations, an ana ly t i ca l closed form solut ion was found to represent the p.d.f. of V and therefore numerical methods were recommended to be used. The conclusions derived by the r e l i a b i l i t y approach to design can be summarised as fol lows: ( i ) A narrow strength d i s t r i bu t i on (large value of the ra t i o of mean to standard deviation) results in higher r e l i a b i l i t y than does a widely dispersed d i s t r i bu t i on for the same mean strength, for given loading condit ions. ( i i ) A narrow d i s t r i bu t i on ( l i t t l e var iat ion) permits the use of a material with a lower mean strength (less expensive material/less weight) for the same r e l i a b i l i t y . ( i i i ) A s imi la r reduction in the var iat ion of loads increases the r e l i a b i l i t y of the design, or, for the same r e l i a b i l i t y , a material with 1-31 a lower strength can be used. Thus a new design approach has been suggested by several authors (18 23 25 28) ? ' ' . This approach recognizes e x p l i c i t l y the random nature of material strength S and applied load L , and allows the quant itat ive assessment of the design r e l i a b i l i t y R as a measure of r i s k . For the design of a s ingle component, the approach has been to change the values of the parameters of the p.d.f . ' s for >^ and L which changes the values of the parameters of the p.d.f, of V and hence the value of the r e l i a b i l i t y P> (or the probab i l i ty of f a i l u r e Ff ). In the case of a system, the problem has been approached with the help of dynamic pro-g r a m m i n g , ^ 2 4 ' 2 6 ' 2 7 ^ by using r e l i a b i l i t y and cost constra ints. In a l l these methods, the main stress has been to change the concept of design from a determinist ic safety factor to one which is recognised as a random var iab le, and to design the component or the ent i re system for a certa in minimum but known value of r e l i a b i l i t y . The problem of se lect ing a mater ia l , with or without economic considerations, does not appear to have been treated in the l i t e r a t u r e . However, s t a t i s t i c a l decision models, in general, have been discussed in detai l .. (35 to 37) i l , by S c h l a i f f e r ^ 3 2 t o 3 4 ^ , Pratt et a l ^ 3 0 , 3 1 ^ , other authors This s t a t i s t i c a l decision theory has been applied to many other areas, including some engineering problems such as the design of (38) gear t ra in s , etc., by Bury. v ; 1-32 1.3 PRESENT RESEARCH PROBLEM In the l i t e r a t u r e survey i t was established that under a more r e a l i s t i c approach to the design problem the load L a n d the material strength S should be treated as s t a t i s t i c a l random variables and not as determinist ic f ixed numbers. As outl ined in the section on the l i t e r a t u r e survey, various combinations of p.d. f . ' s for S and L have been considered for the design pf a s ingle component. However, i f a system of components is considered, i t i s often the case that one component is c r i t i c a l in the sense that f a i l u r e of th i s par t i cu la r component precip itates f a i l u r e of the ent i re system. As an example, consider the case of a turbine wheel which f a i l s when a s ingle blade f a i l s . This means that the wheel (system) Should be designed for the required r e l i a b i l i t y of the weakest blade (component), for the maximum load which might operate on that blade. Thus, in th i s example, the weakest turbine blade is a c r i t i c a l component. Such a concept is not r e a l i s t i c under the conventional approach where load and strength are considered to be determin i s t ic , and a large value of V implies that the chance of f a i l u r e is zero. The maximum value of the load l _ for a spec i f i c design is rare ly a constant. S imi la r l y the minimum strength of a pa r t i cu la r material cannot be speci f ied by a s ingle value. For example, i f a number of sets of samples of the same material are tested under the same loading conditions for some character i s t i c strength property, the minimum strength value in each set is l i k e l y to vary. Thus not only are the general loads and material properties recognised as r .v. ' s (see discussions in 1-33 " l i t e r a t u r e survey"), but also the i r respective maximum and minimum values are properly recognised as r . v . ' s . These r .v. ' s can be defined in terms of certa in p.d.f. ' s and the i r respective parameter values. Although there is a lack of data to ascertain the exact nature of these p .d . f . ' s , i t can be seen thdt, ( i ) maximum values of loads may occasional ly have very high values, so that for the purpose of the analysis i t may be assumed that the values of the r.v. L are unbounded in the d i rect ion of the extreme of interest (upper extreme, or maximum value), ( i i ) Also i t i s found empir ica l ly that the rate of decrease of the p.d.f . ' s for general load values is at least as great as a simple exponen-t i a l for those p.d.f . ' s which are usually associated with load d i s t r i -butions such as Normal, Log-Normal, Gamma, Raleigh, Weibul l , e t c . , ( i i i ) The minimum values of the strength of material cannot be less than zero and so the values of the r.v. 5 are bounded in the d i rect ion of the extreme of interest (lower extreme, or minimum value). (15) GumbeV ' has analysed the behaviour of the extreme value r .v. ' s in general. Conditions ( i ) , ( i i ) and ( i i i ) above, on the behaviour of the general r .v. ' s load 1_ and material strength S , imply the fol lowing select ion of the p.d.f . ' s for the maximum value of ]__ and the minimum value of S (a) E . V - j -,-rnw p.d.f. (Gumbel's Type I model) for the maximum values of the load L ; (b) E.V.jn ;vnlv» p.d.f. (Gumbel's Type I I I, or Weibul l ' s model) for the minimum values of the strength S • These models are the correct asymptotes with respect to the 1-34 underlying sample s i ze . The appl icat ion of the E.V.-X(-wax model for maximum loads L implies that the underlying loads, which produce the maximum value under consideration, const i tute a r e l a t i v e l y large sample. S im i l a r l y , the appl icat ion of E .V . H T ^ m o d e l for the minimum values of the material properties S implies that the number of the c r i t i c a l components, each of which may cause system f a i l u r e , is r e l a t i v e l y large. In pract ice, these assumptions have been found to hold quite we l l . For example, in the case of the turbine wheel design, mentioned e a r l i e r , the number of indiv idual gas impluses on each rotat ing blade is very large during the overhaul l i f e of the engine, so that the E.V.i;»vo< model for the maximum loads should hold exceedingly we l l . S im i l a r l y , each turbine blade i s a c r i t i c a l component, enough of which are assembled to a typ ica l turbine wheel, thus making the E.Ym>y^hx model an appropriate choice for the material strength of the blade mater ia l . Hence, th is combination of the probab i l i ty models has been selected for the prob-a b i l i s t i c design of a s ingle c r i t i c a l component, for the design models ( i ) , ( i i ) and ( i i i ) (see section 1.1). Several addit ional assumptions are made for the purpose of avoiding mathematical complexity, which might ar i se in an attempt to obtain the p.d.f. of the safety factor V (eq. 1.5) or the probab i l i ty of f a i l u r e . These assumptions are the same as given in section 1.2 and can be summarised as: ( i ) the load L and the material s t r e n g t h s are independent r . v . ' s ; ( i i ) the area of cross section of the design, A , i s a s t a t i s t i c a l constant; 1-35 ( i i i ) the component i s subjected only to purely ax ia l loads, the f a i l u r e c r i t e r i o n being separation. Even though a time-independent f a i l u r e c r i t e r i o n (fracture due to purely ten s i l e loads) may be considered for the design when designing a c r i t i c a l component, a certa in "mission l i f e " for the design has to be speci f ied when using this combination of p .d . f . ' s . It has been shown (15)* r-by Gumbelv ' that the parameter va lues ' fo r the E-V-i, model are a function of the number of observations of the extreme values. Referring to f i g . (1.8) i t i s seen that the p.d.f. of the upper extreme values sh i f t s in the pos i t ive d i rect ion with an increase in the number of obser-vations. A spec i f i c value of the scale parameter H of the p.d.f. for maximum loads (chapter 2, eq. 2.7), therefore, corresponds to a certa in number of observations or, equivalent ly,to a certain time period during which the component operates. Any design based on th i s value of the parameter L i s meaningful only i f th i s time period is spec i f ied as the mission l i f e of the design. This i s conceptually d i f fe rent from the p robab i l i s t i c design which is based on other p .d . f . ' s , where mission l i f e i s not required to be spec i f ied when time-independent f a i l u r e c r i t e r i a are considered. S im i l a r l y , the scale parameter of the E-VJU,^;^ (Weibull) model depends on the sample s ize from which the minimum value of the material property i s considered. However, t y p i c a l l y a Weibull d i s t r i bu t i on is d i r e c t l y f i t t e d to the data on the material propert ies, and the extreme value model is constructed d i r e c t l y from th is measurement model by considering Page 166-175 of Reference (15). > CO z UJ Q > 1 . P.d.f. of a r.v. 2 Rd.f. of upper extreme value of the r.v. for sample size fl-j 3 P.d.f. of upper extreme value of the r.v. for sample size PI > IT 2 1 CQ < CQ O or Q _ RANDOM VARIABLE Figure 1-8 - Effect of Sample Size T> on the Location of the Extreme Value Probabi l i ty Density Function 1-37 the number of components among which the component under consideration is c r i t i c a l : eg. i f a Wei bu l l (e< s,j3 ) model (see eq. (2.1)) i s f i t t e d to a relevant strength property of the turbine blades, the E. V. model of the weakest blade is given by the Weibull ( — ^ ) model, where-"n is the number of blades per d i sc . With th i s combination of p.d.f . ' s for the maximum load L and the minimum material strength S - the design of a c r i t i c a l component i s considered for: ( i ) Model 1; Given the p.d.f. of the maximum load L. and the minimum material strength S :. design the component (/rea A ) for the given' design value of the r e l i a b i l i t y R . ( i i ) Model 2; Selection of a material among a given number of a l t e r -natives, for a given p.d.f. of the maximum load 1_ and a given design r e l i a b i l i t y value R , on the basis of some design c r i t e r i on not involving cost, such as minimum weight, etc. ( i i i ) Model 3; Selection of a material among a given number of a l t e r -natives, for a given p.d.f. of the maximum load L and a given design value of r e l i a b i l i t y R , on the basis of economic c r i t e r i a (minimum cost) , using s t a t i s t i c a l decision theory. CHAPTER 2 As discussed in the section (1.1), the fol lowing probab i l i t y density functions are selected to represent the s t a t i s t i a l random behaviour . of the minimum strength of material and the maximum values of the load, respect ively: ( i ) Extreme Value Type III (minimum) (Weibull) Model for the minimum material strength values, ( i i ) Extreme Value Type I (maximum) (Gumbel's) Model for the maximum values of the load. 2.1.1 Extreme Value Type III (minimum) (Weibull) Model ( E - V - I n m l n ^ ' ' ^ * ' The general form of the p.d.f. for the E . V . J J J m i - n (Weibull) model, selected fo r the minimum strength S , is given as where S ^ s , s <>0 , ° < s ^ O a n d /6>0 S = Location Parameter = Scale Parameter P = Shape Parameter. In the case where the locat ion parameter's is zero, the p.d.f. of the r .v.S takes the form, *Page 277-281 of reference (15). 2-2 hcsjas = (4)<4)Mex|,I" (-4/]ds (2.2) where S>>0 , c<s^ >o and f>>0 When th i s p.d.f. i s selected to represent the minimum value of the material strength ( p . s . i ) , the parameter <=<$ ' i a s the units of the strength ( p . s . i . ) , while p , the shape parameter of the Weibull model, i s a dimensionless number. The Probab i l i ty d i s t r i bu t i on function of the r.v. 5 in this case is given as p i ( s ) = / ( 4 X £ ) ^ « + K 4 / j * •••<2.3> For such a r.v. S , having a Weibull p.d.f., Mean (S) = ° < s c*s.T(i+d//a) r „, -I/, I (2.5) s t d . Dev. (s) = * s [ _ r2C^j] J The r a t i o of the mean strength of the material to i t s standard deviation i s denoted by It. and expressed as, = 1/coeff ic ient of var iat ion This ra t io At i s , thus, a function of the shape parameter £ only. The values' of the parameter Ac are computed for a range of values of the parameter ~> (see sec. 2.2) and given ir) Appendix VI. While the obvious advantage of th i s parameter ft- over e i ther the mean value or the standard deviation of the r.v. is that Ac is a function of only one (2.6) 2-3 parameter ( ^ ), the usefulness of such a parameter in design problems is discussed in Chapter 3 of th i s thes i s . Once again i t may be mentioned (see section (T.3)) that the p.d.f. given by eq. (2.1) is the correct asymptote w.r . t . the underlying sample s i ze . In the context of the present problem, this means that the number of the c r i t i c a l components, each of which may cause system f a i l u r e , i s r e l a t i v e l y large. This extreme value model implies that underlying measurements (from among which the maximum i s being considered) are bounded at the o r i g i n . Again, i t was shown in section (1.3) that th i s assumption is s a t i s f i ed in the case of the material strength 5 which cannot be less than zero. Therefore, the E-V.JJJ m i - n model can be used, in general, to represent the d i s t r i bu t i on of the minimum strength of mater ia l . 2.1.2 Extreme Value Type I (maximum) (Gumbel's) Model (E.V.j m a x ) : ^ 1 5 ^ The E.V.J m a x model, selected to represent the p.d.f. of the maximum values of the load i s also ca l led Gumbel's model. The p.d.f. for th is model is given as, fe(L)dL = ^ ^ { - ( ^ j - e x K - (2.7) where - oo< L - ^ Q O , — oo<&<oo and °^L>0 L ~ Location Parameter = Scale Parameter The nature; of the parameter L and i t s dependence upon the number of observations from which the maximum value is selected is d i s -cussed in section (1.3). When the load L i s speci f ied in l b s . , the units 2-4 of both the parameters L and o<L are l b s . , since Mode value of the r.v. L = L Std. Dev. of the r.v. L = ^ L J (2.8) 46 The probab i l i ty d i s t r i bu t i on function for the random var iable L i s MCL) - CX^(-G4>C^)} (2.9) The area under the p.d.f.(eq. (2.7)), in the negative range of the r.v. L , gives the probabi l i ty of L having a negative value as, P z O O l " 0 = -exK^AO) (2.10) From eq. (2.8), Mode (2.11) C^O =1.28 x s t d > D e v > In the case where the r.v. 1_ represents the extreme values of the load, the var ia t ion in such extreme load values i s , general ly, small compared to the mode values which are, in general, large ( f i g . 1.8). Thus the r a t i o of the mode value of the maximum load to i t s standard deviation is generally a large number. For example, even in a case where the r a t i o given by eq. 2.11 is as low as 4.0, the area under the negative range of the p.d.f., as given by eQ-2.7, i s given as, L=0 PzCO l L _ ^ - ^ 4 > { - ^ ( 4 0 ) j (2.12) .— 0,2 X 10 ^ 0 This area i t s e l f i s neg l i g ib le for pract i ca l purposes in the context of the present problem and w i l l be s t i l l less for values of (2.11) greater than 4.0. Therefore, the probab i l i ty of the r.v. L having negative values can be safely assumed to be zero. This makes the se lect ion of the E.V.T model appropriate to represent the maximum values of the load i , ma x as these values are generally not negative. In reference to the present problem, |>2.(L) w i l l be defined only for pos i t ive values of L , and the error implied w i l l be assumed to be neg l i g ib le . As for the Weibull model (previous sect ion), the p.d.f. given by eq. (2.7) i s the correct asymptote w. r . t . the underlying sample s i ze . This means that the underlying loads, which produce the maximum value under consideration, const i tute a r e l a t i v e l y large sample. The underlying assumption is that ( i ) the measurement var iable i s unbounded in the d i rect ion of the extremes of interest (upper extreme in the present case), and ( i i ) the rate of decrease of the p.d.f. for the measurement var iable is at least as great as that of a simple exponential. In the case of loads, i t was observed (see section 1.3) that the p.d.f . ' s representing the load d i s t r i b u t i o n , in general, such as Normal, Log-Normal, Raleigh, Gamma, e t c . , sa t i s f y these requirements, and so the E.V.T m 3 V can be used to represent the d i s t r i bu t i on of the maximum load values. 2.1.3 Character i s t ic Values of the r .v. ' s Minimum Strength Sand  Maximum Load L : In the case of the extreme value density functions, a new measure of the average of the extremes i s introduced which is analogous to quantiles. These are ca l led the characte r i s t i c largest and the character i s t i c smallest value for the E.V.T ^ and E.V. T T T .„ p .d . f . ' s , I, max I I I , min r respect ively. In the case of the E.V., 3 U p.d.f. th is character i s t i c i , max value is that value of the o r i g ina l ( i n i t i a l ) measurement var iable (load) for which there is only one larger observation. S im i l a r l y , for the E . V . J J J m i - n p.d.f. th i s is the value of the or ig ina l ( i n i t i a l ) measure-ment var iable having only one s ingle smaller observation. These charac-t e r i s t i c values are given by that value of the r.v. for which the p.d.f. has the value i^/e) for maximum extremes and for the minimum extremes, respect ive ly. At the cha rac te r i s t i c value L c the E . V . T m ^ v d i s t r i bu t i on i , max becomes - e ^ [ - ^ (- = V e (2.13) or -ext> f - b d r l - i Hence, L c = L (2.14) At the charac te r i s t i c value S C the E . V . T T T . d i s t r i bu t i on becomes I I I, min \ ~ ^ (- C S c / ^ i ' ^ - (2-15) or ^K-CS-As/]-' / e Hence, . 5 C = K s (2.16) The r a t i o of these charac te r i s t i c values ° < s for the minimum material strength and t for the maximum load i s used to provide a measure of the central locat ion for tiie p.d.f. for the safety factor V (given in section (1.2)). The ra t i o C^ s/tO 1 S also defined as the "central  safety factor " V0 • 2.2 Probab i l i ty Density Function of the Safety Factor It : From eq. (1.5), the safety factor V i s given as, v=. S / L where S = Strength of material ( lbs.) L = Load applied ( l b s . ) , V = Safety factor . At th i s stage, i t may be noted that the safety factor V i s always a dimensionless number. On the other hand, the d i s t r i bu t i on of the material strength is given in units of p . s . i . By introducing a scal ing factor A (which w i l l correspond to the design cross-section in sq. inch), the material strength d i s t r i bu t i on can be changed to have the units of lbs. Eq. (1.5) can then be rewritten as, u s.A _ 'S_ fjb^_\ Note that the p.d.f. of the new r.v. ( S - A ) w i l l remain the s a m e , ^ since the scal ing factor A is assumed to be a s t a t i s t i c a l constant. Only the value of the scale parameter^5 wi11 be mul t ip l ied by a factor A and given as The units of o<^ w i l l be lbs . now. The value of (5 w i l l remain uneffected (and dimensionless). The deta i led procedure for obtaining the p.d.f. of a r.v. which is a quotient of two independent r .v. ' s i s given in Appendix I. As S and L are assumed to be independent r .v. ' s in the design analys i s , the j o i n t p.d.f. of S and L is the product of the p.d.f . ' s of S and L , respect ively, and given as f 3 ( s , L ) c*s<=U- — foes)- fczCL) els. A- (2.17) From eq. (1.5) 2-8 When the eq. (2.17) is integrated over the range of S , the var iable L is constant and so, = L . ^ (2.18) Subst itut ing this in eq. (2.17), \>(T)}L.)«U. = k,(VL).k<L) L.e!v>cU_ (2.19) which i s the j o i n t p.d.f. I f the r .v. ' s V and L . \ If eq.(2.19) i s integrated over the range of L (0 to oo) CO ]pM cH =J & k ( L ) L c t L d v , (2.20) which is the p.d.f. of the r.v. 1/ (the safety f ac to r ) . Subst i tut ing eq. (2.1) and eq. (2.7) (the p.d.f . ' s of S and L , respect ive ly ) , the p.d.f. of the r.v. V is given as, Subst itut ing L-ty and CLL — c\_ch^ (2.23) gives, \>(y)Jii) = ^  0 ( c ^ + ^ - t A e ^ ) J ^ - ^ (2.24) where, (2.25) The p.d.f. of the safety factor ]) , therefore, has three groups of parameters involving the four basic parameters o(LX. , «^ s p , in the form ofp, -V\Q , and ("tyC )=-VO- . These three parameters of the p.d.f. of the safety factor ^ are dimensionless. The physical s ign i f icance of the parameter combinations ^ , , /^co "is discussed in section (2.4) of this chapter. I t can be seen immediately that the tota l number of parameters has been reduced by one. It was not found possible to express the integral given by eq. (2.24) in a closed form so lut ion. The numerical method is therefore used to obtain the p.d.f. of the r.v. )) by evaluating eq. (2.24) for d i f fe rent values of V and for a set of values of the parameters >^ , , , '/^ . Two other methods of obtaining the p.d.f. ( ( i ) Me l l i n ' s transform and ( i i ) Laplace transform method) were also t r i ed for a possible so lu t ion . They too could not y i e l d a closed form solut ion for the p.d.f. of the r.v.V The deta i l s of these methods, and the results obtained using these methods, are given in Appendix I. 2.3.1 Probab i l i t y D i s t r ibut ion Function of the Random V a r i a b l e ^ : The probab i l i t y d i s t r i bu t i on function of the r . v . V is given by, V pfr) = j \>b>)di) (2.26) i . e . by evaluating the area under the p.d.f. of the r.v. V from the lower l i m i t of the function to a certa in desired value V . Eq. (2.26) can be rewritten as, P(V) ^  J / ^CVL)f)2a)L.^L^ ( 2 - 2 7) 2-10 As the l im i t s of integrat ion are independent of the variables (the upper l i m i t of the outer integral i s a par t i cu la r value of V and not the random var iable V i t s e l f ) , changing the order of integrat ion Wi l l not change the re su l t , and eq. (2.27) can be written as oo D = J f L h C ^ M ^ v ^ L ( 2 28) Let ^ (2.29) Then, L--0 or, oo L= 0 CO , / PiCvD^CL) (2.30) Substitut ing the expressions for F^(VL) and \>z ( l ) in eq. (2.30) gives = j i * + { - 0 ^ ) - ^ ' ^ ) J and therefore P M = i-4Hl-[^)^^ + ^ K-L-*)]]^  (2.32) 2-11 Again, a closed form solut ion for th is integral was not found and numerical computations were carr ied out to ca lcu late the values of P(i>) , for d i f fe rent values of the r.v. \) and the parameters of the integral given by eq. (2.32). 2.3.2 Probab i l i ty of Fai lure The probab i l i ty of f a i l u r e i s defined as, Putting V = 1 in eq. (2.32), Pf i s given as, oo Using the symbols given by eq.(2.25), CO r {- [( HdsJ+ k+ A • exJP (r L/*i.)]] • , (2.34) and subst i tut ing ^Uur^f a n d ^ - - ^ n c ^ - , the probab i l i ty of f a i l u r e Pf i s given as P f = 1 - A J ^{-[(V+^  + ^ ^ J ] - ^ ( 2 ' 3 5 ) A closed form solut ion could not be found for th i s i n teg ra l . Numerical methods were used to obtain the values for the given para-meter values of the load (E.V., ) and the material strength ( E .V . T T T . ) I, max i l l mm d i s t r i bu t i on s . Though th i s integral can be replaced by an i n f i n i t e series solut ion (see Appendix I I I ) , th is series needs to be summed on the computer, requir ing much more tine for computation than d i rec t numerical integrat ion. 2.3.3 Special Case: ft = 1.0 In the special case of |3> = 1.0, the Weibull d i s t r i bu t i on 2-12 reduces to a simple exponential p.d.f. In such a case, a closed form solut ion is ava i lab le for the probab i l i t y of f a i l u r e F| and given as (see Appendix I I): Pf ==. l - ^ K - ^ A s ^ b f i t A ) , (2.36) where A and a are given in eq. (2.25) and, \ HCA) = jV+1H < f » ^ (2.37) = Incomplete Gamma Function of order ( b 4 l ) and argument A . For very large values of A , i e . when the mode value of the load L i s very large as compared to the standard deviat ion, the incomplete gamma function approaches the gamma funct ion. However, both the incomplete (44 45) and complete gamma functions are ava i lab le in tabulated fo rm/ ' 2.4 Parameters of the Probab i l i ty Density Function of the Safety Factor V : Ihe probab i l i ty density function of the r.v. V , the safety factor, is given by eq. (2.24). There are three parameters for this p.d.f., given as (see section 2.2): ( i ) f> = Shape parameter of the Weibull d i s t r i bu t i on for the material strength S ; M i l °<y^ = J/a> = Std. Dev. of L u i ; / L ' 1.28 Mode Value pf L ; ( i i i ) ° s^/~ = Vb/ - cha rac te r i s t i c value of the strength d i s t r i bu t i on bCs) ' / L " charac te r i s t i c value of the load d i s t r i bu t i on ^(L) = "Central safety f a c to r . " = V0 ( i ) The f i r s t parameter, (5 , i s the shape parameter of the p.d.f. of the 2-13 material strength and i s dimensionless. The value of f> effects the value of the mean strength as well as the variance of the strength (see eq. (2.5)). In the case when has the value equal to 1, the Weibull density function i s reduced to an exponential p.d.f. given by hCs) * s = 4 ^ {-CS/*s)} , (2.38) for S > 0 , °<s>0 . Both the mean and the standard deviation of this d i s t r i bu t i on are equal to ois . For values of (3 less than one, the value of the coe f f i c i en t /C , given by eq. (2.6), i s less than one. This means that for the values of £<"f-0, the var iat ion in the strength of material i s larger than the mean value of strength and hence for p ract ica l appl icat ions, values of ! A can be expected to be greater than 1. A lack of data ava i lab le to determine the s t a t i s t i c a l behaviour of the material properties (strength), spec ia l l y in the extreme value region, makes i t d i f f i c u l t to estimate the range of values of fi> which might occur in pract ice. Some ava i lab le data^ ; for the y i e l d strength and the ultimate tens i l e strength of ASTM, A-7 s t ructura l steel are plotted on Weibull p robab i l i t y paper and are shown in f i g . (2.1). This places the value of in the range of 6.0 to 7.0. These plots on the Weibull paper were s t ra ight l ines which confirmed the absence of a possible locat ion parameter 's given in eq. (2.1) for the p.d.f. of the material strength. Though the values of j2> obtained on the basis of those few data hardly prqvide a s o l i d basis for deciding the range of values of p> , i t was decided for the purpose of generating design tables 2-14 (Appendix V) to vary jS between 1.0 and 10.0, the lower l i m i t corresponding to an exponential p.d.f. for the material strength S • ( i i ) The second parameter of the p.d.f. of V is a function of the two parameters of Gumbel's model, given byeq. (2.7), for the maximum extreme values of the load L . For a component to be designed, i t is a measure of the r a t i o of the var iat ion in extreme values of the load to the charac-t e r i s t i c load value H which is the mode value. Unless subjected to extra o rd ina r i l y wide f luctuat ing loads, th i s value can be assumed to be always less than 0.30. On the lower s ide, theo re t i ca l l y i t may be zero, which w i l l refer to a case of the determin i s t ic loads where the var ia t ion is zero as compared to the central mode value. However, in the case of the extreme values of loads, the var iat ion is l i k e l y to be small as compared to a large mode value of the maximum operating load, and the lower l i m i t of ( °0L/t ) i s taken as 0.05. These values were chosen for the purpose of providing design tables for a p ract i ca l l i k e l y range of th i s parameter. The i m p l i c i t assumption i s , therefore, that i f the var iat ion in the load is less than about 5 p.c. of the mode value, the designer might as well assume a f ixed maximum load value in his design, as in the conventional pract ice. ( i i i ) The th i rd parameter is the r a t i o pf the cha rac te r i s t i c value of the strength d i s t r i bu t i on (Weibull) to the cha rac te r i s t i c value of the load d i s t r i bu t i on (Gumbel's). This i s defined as the central safety factor ])Q . Depending upon the character i s t i c s of the strength d i s t r i bu t i on of the given material ( c*s JJ.A/L.) and the load applied (T Iks. ), th i s parameter can have d i f fe rent pos i t ive values for d i f fe rent values of the scal ing factor A » the area of cross section of the design (sq. inch). 2-16 The charac te r i s t i c value <*(s of the material strength and the mean value of S are related by the equation, MCs) = o^.rc^-L ) ^s.rcp') (2.39) For A > ± , which was assumed to be the lower l i m i t of the range of (3 , the value |S/ w i l l be 2 when = 1 and w i l l be 1 when /S> = ° ° . For these two values of ^ , R ^ O w i l l be equal to 1 (see f i g . (2.2)). For a l l other values of £> , the value of ^ w i l l be in the range \^^<Z. For a l l these values of §>' , the gamma function w i l l have values <£-1.0 which means that, 0 1 2 3 4 X Figure 2-2 - Gamma Function to 1 2-17 The mean value of Gumbel's d i s t r i bu t i on i s given by M ( L ) = t 4 0 . 5 7 7 2 . * L (2.41) Thus when the value of the central safety factor VQ is equal 4 . « - y r > _^ 1±M__ — ^aJ) (2.42) for 6> i and <=^ _>0 • This refers to a case where the mean of minimum strength values is less than the mean of maximum load values for which the design is carr ied out. I t , therefore, appears l og i ca l to assume that, in pract ice, the value of the c s . F . is l i k e l y to be greater than one. Any higher value of th is parameter i s feas ib le depending upon ( i ) the requirement of the r e l i a b i l i t y , ( i i ) the loading, and ( i i i ) the material avai lable for the design. In the present research work, to generate the design tables, values of the parameter ( ^s/H ) were chosen to range from 1.25 to 5.0. 2.5 Numerical Results Numerical methods are used'to evaluate eq. (2.24) for d i f fe rent values of the r.v. V and a set of values of the parameters , H L / £ ) , {^s/t) , in the ranges described in the previous sect ion. S im i l a r l y , the probab i l i t y of f a i l u r e values, Fjj: , are also evaluated by evaluating the integral given by eq. (2.35) for d i f fe rent values of three parameters. Some of these results are given in Appendix V. (A complete compilation of tables w i l l be done under separate cover, subsequent to th is thesis work.) 2-18 The results obtained by evaluating eq. (2.24) are presented in the form of p.d. f . ' s of the r.v. V . Some of these p.d.f. ' s are shown in f i g . (2.3) to f i g . (2.6) for d i f fe rent values of parameters p , (fVC) , i^s/i) . These curves show how the p.d.f. of the safety factor changes with a change in the value of any one of the three parameters mentioned e a r l i e r . It i s seen from these curves that the function f>(X) i s also skewed to the r i gh t , as in the case with functions |p<(s) and J^CL) . These density functions can be used to provide the probab i l i t y of f a i l u r e values by ca lcu lat ing the area under these curves for O^V^i . The value of the safety factor V corresponding to the peak of the density function gives the mode value of the safety factor (the mode value i s that value of the r.v. having the highest frequency of occurrence). The values of the probab i l i t y of f a i l u r e for d i f fe rent values of the three parameters are presented in a tabular form. Some typ ica l results are shown in f i g . (2.7) for the purpose of invest igat ing the e f f ec t , and i t s physical i n te rpretat ion , on V^. values with a change in any one of the parameters. (a) When the parameters p and ^s/^ are kept constant, a reduction in the value of the parameter ^ L / f results in a lower value of the probab i l i ty of f a i l u r e . This reduction in the value of ( ^L /C ) might be a resu l t of one of the fo l lowing: ( i ) a decrease in the value of oi\_ alone, ( i i ) a decrease in the value of as well as L such that ^L/f) decreases and a concomitant decrease in °<s to keep ( ^S/L ) unchanged, ( i i i ) an increase in the value of and T , such that (°^-/L ) decreases and an increase in the value of <*5 , keeping ^s/C unchanged. 2-19 •9 RANDOM VARIABLE V Figure 2-3 Probab i l i ty Density Function of the Safety Factor V , £> = 1.0 to 5.0 (integer values only) , l^/L) = 0.10, C s^/t) =2.0 0 1 2 3 4 5 6 7 8 RANDOM VARIABLE Figure 2-4 - Probabi l i ty Density Function of the Safety Factor V f & = 1.0 to 5.0 (integer values only) , O^i/V) = 0.10, (f<s/C> 3.0 ' J K 3 O 2-21 0 1 2 3 4 5 6 7 8 R A N D O M V A R I A B L E Figure 2-5 - Probabi l i ty Density Function of the Safety Factor V fy,<*Z 1 - 0 t 0 5 , 0 ( i n t e 9 e r values only), =0.10 0 3 4 5 6 RANDOM VARIABLE 8 Figure 2-6 Probab i l i ty Density Function of the Safety Factor l) P = 1.0 to 5.0 (integer values on ly) , = 0.10, K/L> 5.0 y ro i 2-23 The f i r s t case implies that, although the mode value of the maximum load remains the same, the standard deviation given by eq. (2.8) is reduced, which means less f luctuat ion in the operating conditions for the component to be designed. Thus, a t i ghter control over the var iat ions in operating condit ions, for the same material used for the design, reduces the probab i l i t y of f a i l u r e (or increases r e l i a b i l i t y ) , as would be expected. In the second case, the var iat ion in the load as well as i t s mode value is reduced such that the value of the i r r a t i o is reduced. The central safety factor value is held constant by using a d i f fe rent material having a smaller value of °< s (a con s tan t^ means that though both the mean value and the standard deviation of the material strength decrease, the i r r a t i o is constant, see eq. (2.5) and e c i - (2.6)). Thus i t means that for a decreased load value with a r e l a t i v e l y large reduction in i t s standard deviat ion, even the use of a weaker material may resu l t in a higher r e l i a b i l i t y value ( i e . lower ). The th i rd case is ju s t thevreverse of the second case. Both the var iat ion and the mode value of the load increase in such a way that the i r r a t i o decreases. If the CSF value is kept constant by using a stronger cross section of the mater ia l , the probab i l i ty of f a i l u r e w i l l go down. (b) Analysing the e f fect of a change in the value of the CSF (^s/C ), i t i s seen that for the same values of the other two parameters, the f^. value decreases with an increase in the CSF value. This increase in the value of the CSF may be the resu l t of, ( i ) an increase in the value of only, which means a material 2-24 Va = 0.10 1/a = 0 . 1 5 V0 = 1 . 0 BETA Figure 2-7 Probab i l i ty of Fai lure Pf vs. fo (1.0 to 10.0), for values of (p<s/tD from 1.0 to 5.0 (only integer values) and QXLAL) = 0.10 and 0.15 2-25 of higher mean strength is used. ( i i ) a r e l a t i v e l y lower value of the mode value of the operating loads reducing the value of the parameter X. . In th is case, when t is reduced, i t i s implied that <^ i_ i s also reduced such that (<*L/L ) remains unchanged. (c) An increase in the value of fi alone i s found to reduce the value of . The evaluation of the mean of S for d i f fe rent values of |3 , and the same value of the parameter v<s , shows that the mean value of the strength decreases for increasing values of ,4<2-i7 (see f i g . 2.8). For other higher values of >^ , the mean strength w i l l increase with an increase in the value of jS . This i s indicated by the property of the gamma function which decreases between 1.0 and 1.46 and then increases again (see f i g . 2.2). The standard deviation of the minimum strength S is found to decrease for an increase of a l l pos i t ive values of (see f i g . 2.8). For values of ^ <£2 . 17 , th is decrease is greather than the decrease in the value of the mean strength. The ra t i o of the mean strength and the standard deviation for the Weibull model i s given by eq. (2.6). It may be noted that th i s r a t i o Jz is a function of only one parameter |2> of the Weibull model. On the basis of the above discussions (also see Appendix VI), i t i s seen that an increase in the value of ^ w i l l resu l t in the increase in the value of the r a t i o J~L . This, therefore, implies that, even though the mean value of the mean minimum strength may decrease with an increase in the value of ^ , the ra t i o / L w i l l increase, which w i l l reduce the F|- value. Thus, the decrease in f|_ values for increasing |& values can be understood in terms of the ra t io h. which bears physical in terpretat ion. 2-26 .85 I i l I 1 i • i i I 1 2 3 4 5 6 7 8 9 10 BETA Figure 2-8 - Relationship between the Parameter (3 and ( i ) Mean Value of the Material Strength, ( i i ) Standard Deviation { of the Strength 2-27 The ra t i o i s a function of only one parameter, >^ , of the material strength d i s t r i b u t i o n , and can be used as a measure of the qual i ty of a mater ia l . The higher the value of the parameter te. i s , the better i s the corresponding material since the p£ value goes down with an increase in the value of A. , a l l other parameters being constant. This property of the ra t i o /L i s used in Chapter 3 for the determination of the cost of mater ia l . In a more general case, a l l three parameters of the p.d.f. of the safety factor V may change simultaneously, thus changing the r e l i a b i l i t y of the design. It w i l l be shown in the next section how a design is carr ied out for the given load and material strength for a desired r e l i a b i l i t y , from the p robab i l i s t i c approach, using the tables given in.Appendix V. 2.6 Design Method For a Desired R e l i a b i l i t y Value As mentioned e a r l i e r , one of the advantages of the p r obab i l i s t i c approach to the design problem i s to provide a more rat iona l and quant i -ta t i ve statement for the safety of the design in terms of i t s r e l i a b i l i t y rather than in terms of a conventional safety factor . The design methods are outl ined here for model 1 and model 2 as given in section (1.1) and are compared with the conventional design approach for each model. The various s impl i fy ing assumptions for the design, as described in the same section are made for these design models. 2.6.1 Model 1 Given the load I_ and the material strength S , the problem is to design the component, i e . to specify the area of cross section A . 2-28 (A) Conventional Method: In th i s method both the ant ic ipated actual load |_ 0 ( lbs.) and the design strength S D ( lbs.) are assumed to be determinist ic values. The f ixed value of the design load i _ 0 i s given. Depending upon the past experience avai lable for the operating conditions for s im i la r components and the material used, the conservatism of the designer and his judge-ment, a value of the safety factor V is se lected. Thus, V is a decision var iab le , the value of which the designer must choose. When the value of V can be related to an economic object ive, a simple decision model can be constructed to f a c i l i t a t e the determination of a rat ional (economic) decis ion. However, since V is a determinist ic value, the r i sk of a decision cannot be computed. (See Appendix IV for the d i s -cussion of s t a t i s t i c a l decis ion models in general.) Nowadays, this value of the safety factor can, in many design problems, be taken from various engineering codes which specify a par t i cu la r value of the safety factor Vo for the component, which i s to be used under certa in operating conditions and made of a certa in class of mater ia ls . Using eq. (1.5), the required design strength is given as, S 0 , the design strength, i s again a s ingle number ( l b s . ) . The subsequent steps in the design process are: ( i ) Selection of a mater ia l : This spec i f ies relevant material properties such as ultimate strength, y i e l d strength, etc. The resu l t i s a s ingle number M (p . s . i . ) as a representative value of the material property for the f a i l u r e c r i t e r i on considered. 2-29 ( i i ) Selection of a conf igurat ion: In a simple design problem, this corresponds to the se lect ion of a su i table cross section which may be a c i r c u l a r , square, I sect ion, channel or some other shape. The select ion of the configuration is generally related to the par t i cu la r material selected for the design because, general ly, material properties are spec i f ied on the basis of tests conducted on a par t i cu la r type of section of the tested material specimen. However, sometimes the design requirements decide the nature of cross sect ion. In that case, the value of the relevant material property is corrected appropriately by using an empirical adjustment factor . At th is stage of the design process, two numbers S Q ( lbs.) and M (p.s . i ) are ava i lab le . The area of cross section required i s then determined as, A = -M^o <2-43' Once the area A i s determined, depending upon the conf igurat ion, the dimensions of the cross section are obtained. For example, in the case of a simple design of a tens i l e member of c i r cu l a r cross sect ion, A = 2 . D2" (2.44) And thus eq. 2.43 and 2.44 completely specify the design which can withstand a load L 0 at a safety factor value Vo . Such a design implies the assumption of i n f i n i t e l i f e with a probab i l i ty of f a i l u r e equal to zero, since S0 and l _ 0 are assumed to be constants. The concept of " r i s k " of the design has therefore, no meaning under the conventional approach to the design problem. 2-30 (B) P r obab i l i s t i c Approach: In the p robab i l i s t i c approach to such a design problem, both the load L ( lbs. ) and the material property S (p.s . i . ) are considered as r . v . ' s . In the present problem, the load d i s t r i bu t i on i s given by E . V . j m a x ( 1" j < L^ ), while the d i s t r i bu t i on of the material strength is E . V . J J J M1-N ( °<s, )• These p.d.f . ' s are selected to design the component which is c r i t i c a l with respect to the performance of the whole system. (a) The fol lowing information is assumed to be provided as the input to the design process: ( i ) the desired ' r e l i a b i l i t y ' value R of the component, which is the probab i l i ty of the design surviving a random load for a contemplated "mission t ime." I t i s a s ingle number denoted by ^ - F or R . ( i i ) p.d. f . ' s of the maximum load L ( lbs. ) and the material strength S (p . s . i . ) are spec i f i ed . This means that the values of the parameter-; ( L^L^ ^ S y f i ) are ava i lable on the basis of a s t a t i s t i c a l analysis of data on extreme loads and minimum material propert ies. As discussed in the section (1.3), the spec i f i ca t ion of a contemplated mission time is important when extreme value d i s t r ibut ions are considered. (b) The step of se lect ing a design conf igurat ion, or the cross section shape, for the component and the material used, is s im i la r to that for the conventional design method. (c) ( i ) For the given load d i s t r i bu t i on , i e . values of the parameters L and &!L , the parameter ( °<L./¥ ) is obtained as a dimension-less number. 2-31 ( i i ) For the desired r e l i a b i l i t y R , the p robab i l i t y of f a i l u r e is calculated a:. Rp =1—R . Using the tables as given in Appendix V, for the calculated value of the parameter (^L/L ) and the given value of the parameter (5 for the material strength d i s t r i b u t i o n , the value of the th i rd parameter (^s/Z T is se lected, such that the F^ . value is equal to the desired value. As the units of are in p . s . i . while the parameter H i s spec i f ied in l b s . , a sca l ing factor A having the units of sq. inch is introduced as, ( TW IT ) - (2.45) where represents a pa r t i cu la r value of th i s parameter selected from the tables so that Ff =1-R has the design value. Knowing the values of L and (^ s/C) > the design area A is obtained from eq.(2.45). Thus, for a given design conf igurat ion, the area spec i f ies the design completely. Par t i cu la r rea l i za t ions of th i s design (actual components) w i l l each feature a s ing le rea l i zed (actual) value of the random material strength S . When subjected to the random variable load L , these components w i l l survive the spec i f ied mission time with a r e l i a b i l i t y R = i - g ; i e . R per cent of the components w i l l survive the mission time. This procedure, thus, completely spec i f ies the design of a c r i t i c a l component, for the given material d i s t r i bu t i on and the given load d i s t r i b u t i on , featuring the speci f ied r e l i a b i l i t y R over the con-templated mission time. In this design analys i s , the cross section area A is considered to be a s t a t i s t i c a l constant. In pract ice there are, however, var iat ions 2-32 in the manufacturing tolerances, e t c . , thus rendering the area of cross section A , a r.v.. If th is random behaviour of the design area is considered by specifying some p.d.f. |D(A) , the e f fect i ve strength d i s -t r i bu t ion ( in lbs. units) w i l l be the product of the r.v. S ( E .V . T T T . ) 111,mm and the r.v. A (perhaps a Normal p.d.f. models the s t a t i s t i c a l nature of A ). This ref ined probab i l i ty model would evidently be excessively complicated because the nature of the r.v. ( S .A ) is un l ike ly to be simple. The p robab i l i s t i c design model 1 is further explained in terms of an i l l u s t r a t i v e numerical example in Appendix VII I. 2.6.2 Model 2 Selection of a material among a given number of a l te rnat i ve mater ia ls , for a given load, on the basis of some design c r i t e r i on such as minimum weight, etc. In pract ice, the designer i s often faced with the problem of choosing one among a l ternat ive materials having d i f fe rent properties. In such a case, a choice of a spec i f i c material needs to be made on the basis of some design c r i t e r i o n . This design c r i t e r i o n may be one of: minimum weight, minimum def lec t i on , e t c . , depending upon the overal l objective of the design. Such a problem of design is considered in the fol lowing: (A) Conventional Method: Using the method given for model 1 for the conventional approach, the cross section area A^ is determined for each material ( ) and the s ingle given load value L 0 . The value of the design c r i t e r i o n is then calculated for each material ( M i ) for the cross section area 2-33 ( ) by using the relat ions ava i lab le to ca lcu late the value of the design c r i t e r i o n . For example, in a simple case where the design c r i t e r i o n relates to the minimum weight M of the design, the weight i s calculated from the given density fi and design area Al as, K c - A ; - f x and the material ( X ) corresponding to the minimum value of k/£ is selected for the design. The value of other design c r i t e r i a such as min. de f lec t i on , e t c . , can also be calculated for each material and the se lect ion of a par t i cu la r material can be made in an analogous fashion. (B) P robab i l i s t i c Method: When the area of cross section A i s considered as a s t a t i s t i c a l constant and other material properties such as the density -p are assumed to have a s ingle determinist ic value for each mater ia l , the se lect ion of a par t i cu la r material under the p robab i l i s t i c approach is s im i la r to that for the conventional approach. For the given load d i s t r i bu t i on (t)°<L- ) and each material (c< S ) ft ) I > the area of cross section Al i s determined for the spec i f ied r e l i a b i l i t y value R , using the design procedure given in model 1 for the p robab i l i s t i c approach. The procedure of se lect ing one spec i f i c mater ia l , then, in order to optimize the value of the design c r i t e r i on becomes s im i l a r to that for the conventional approach. However, i f the area of cross section and the material properties such as the density f l for each material are considered to be r .v. ' s with spec i f ied p .d . f . ' s , the problem is l i k e l y to become far more complicated than is warranted for the l i k e l y gains that might be obtained from the refinement, 2-34 which i s , therefore, not considered in the present analys i s . In the special case of the design problem, when the se lect ion of a par t i cu la r material i s to be based on a decision c r i t e r i o n such as minimum cost, the second model cannot be e f f e c t i ve l y used. Apart from the cost of each material used, a cost value should be associated with each possible value of the safety factor . Such a cost consideration makes i t necessary to use decision theory for the se lect ion of a par t i cu la r material for the design. This design problem (model 3) i s considered separately in the next chapter. CHAPTER 3 3.1 Decision Problem It was mentioned in section 1.1 that, i f a design problem i s to be solved on the basis of economic c r i t e r i a , not only the cost of the material must be taken into consideration, but also the cost of the safety margin which is to be designed into the product. This add i -t ional cost component, which can be associated with a par t i cu la r value of the design safety factor , may be explained in the fol lowing manner: Consider a component designed, under the conventional design pract ice, for certain values of the load and the material strength, and a value of the safety factor which i s selected by the designer. The r i s k of such a design cannot be spec i f ied e x p l i c i t l y . However, the designer knows that whatever high value of design safety factor he may choose to guard the design against eventual i t ies such as adverse var iat ions in the load L and the strength of material S , the p o s s i b i l i t y of a f a i l u r e can never be ruled out. When such a component, designed on the basis of a certain design safety factor value, w i l l be tested, some value of the material strength and the load w i l l be observed ( rea l i zed ) , giving the designer a rea l i zed value of the safety factor for that par t i cu la r component. If th i s rea l i zed value of the safety factor is less than, or equal to, one, that par t i cu la r component specimen w i l l f a i l . In such a case of f a i l u r e , the designer might have to pay a certa in penalty which may be considered inversely proportional to the margin of safety for which the component i s designed. That i s , i f the design safety factor is equal to a well established value which i s commonly used for the type of component in question, the cost to the designer of a f a i l e d component ( rea l i zed value of the safety factor i ^ l ) may be considered minimal. On the other hand, in the hypothetical case of the design safety factor being equal to one, an actual f a i l u r e would presumably enta i l some maximum cost to the designer. Thus, i t appears reasonable to assume that the cost of an actual f a i l u r e decreases with an increase of the safety margin designed into the component. The cost, in general, w i l l depend upon the amount of the overal l damage that w i l l re su l t because of the f a i l u r e of th is pa r t i cu la r component and may take into consideration factors such as the r i sk to human l i f e , the delay in the mission achievement and the loss of the sales because of the bad reputation earned. This cost of safety factor values, when plotted as a function of the various safety factor values, w i l l decrease with an increase in the safety factor value. In the conventional approach, where the safety factor is a s ing le determinist ic number, with such a cost consideration the resu l t of the design process w i l l be a s ingle cross-section area hi of the material and a safety factor value VQ for v the given load L D such that the cost of the design is optimized. However, with the proposed p robab i l i s t i c approach to the design problem, a s ing le value of the safety factor such as V0 i s not meaningful. Though the function for the cost of the safety factor , as described for the conventional approach, can be used, the s ign i f i cance of th is cost function changes. Also, in the conventional design method, the cost of the material i s expressed only as a function of a s ing le strength value. It i s more log ica l to assume that with the var iat ion in material properties recognized, th is cost should be a function of some parameter which takes into account both the central value as well as the var iat ion in the material strength. 3-3 In such a case where uncertainty exists regarding the strength of material and the load, and the safety factor as a decision var iable i s not deter-m in i s t i c , s t a t i s t i c a l decision theory must be used to make the se lect ion of a par t i cu la r material on the basis of economic c r i t e r i a . S t a t i s t i c a l decision theory has been studied in deta i l by Pratt et a l ^ 3 1 ^ , S c h l a i f f e r ^ 3 2 t 0 3 4 ^ , Degroo l 3 7 ^, etc. The various elements of th is decision theory and how to use i t in making rat ional decisions under uncertainty, in general, are given in some deta i l in Appendix IV. In b r i e f , the common elements of decision theory are; ( i ) a decision var iable ^ , considered as a r.v., ( i i ) a set of a l ternat ive actions ava i lab le for the decis ion maker, ( i i i ) a preference scale ( u t i l i t y function) which measures the value of each action taken for a pa r t i cu la r value of the decis ion var iable rea l i zed , ( iv) the state of knowledge w. r . t . the unknown state of nature of the decis ion var iable Cj- . Since ^ i s a r.v., a p.d.f. ^C^) i s speci f ied as the knowledge about the random nature of the decision v a r i -able g . 3.2 Model 3 In the present analys i s , the r.v. l) , the safety factor , i s considered as a decision var iab le. This decision var iable U i s the ra t i o of two random measurement variables S and L . Depending upon the par t i cu la r values observed for the load L and the material strength 3 5 there w i l l be a rea l i zed (actual) value of the safety factor V for a par t i cu la r component. The set of a l te rnat i ve actions ava i lable to the decis ion maker are the choice of a number of a l te rnat i ve materials -C , from which one par t i cu la r material i s to be selected for the design. The common scale of preference is a cost function (or the u t i l i t y funct ion), given as U(ijL>) which expresses the cost of the design when a material L i s selected for the design and a spec i f i c value of the safety factor is being rea l i zed . The decis ion problem i s to choose that material L which w i l l optimize the value \J(i}v) . The fourth element of the decis ion theory is the p.d.f. of the r.v. V , given as p-C.^ ) in eq. (2.24). This p.d.f. [>CLO gives the information avai lable regarding the random behaviour of the decis ion var iab le. Thus the decision problem related to model 3 of the p robab i l i s t i c design methods can be summarised as, (1) Decision Constraint: the minimum value of the r e l i a b i l i t y i s provided as a decision constra int. Sometimes, an addit ional constraint such as a certa in weight l i m i t , e t c . , may also be spec i f i ed . (2) A L i s t of A l ternat ive Materials that can be used, each with a p.d.f. of i t s minimum strength as E. V .M (<=<s y ft) 'L , the values o(s and p being defined for each one of these m a t e r i a l s ! . (3) State of Nature: The maximum load p.d.f. j \ C - ) i s given as E-V-x max^L/°^L) W l t ' 1 ^ e v a l u e s °f t n e parameters L and and the minimum material strength i s given as E. V-j£ f (pi% (i) ^ • The state of knowledge w. r . t . the uncertaint ies pertaining to the r .v. ' s S and L i s summarized by the p.d.f. |?(v) on the decision var iable V . (4) Objective of the Decision Process: to se lect one spec i f i c material -L from various a l ternat ives JL , which under the spec i f ied load d i s t r i bu t i on w i l l resu l t in a design with at least the desired r e l i a b i l i t y value over the contemplated l i f e time at minimum cost. 3-5 3.3.1 U t i l i t y Function The u t i l i t y of se lect ing a par t i cu la r material for the design of the component for a given load d i s t r i bu t i on and the spec i f ied r e l i a b i l i t y requirement, can be expressed for any material on a common scale of pre-ference in terms of cost. This means that the u t i l i t y function i s measured in do l lars of cost, and optimization implies minimizing the u t i l i t y funct ion. The two costs considered are, ( i ) cost of the mater ia l : The cost of the material required for the design i s d i f fe rent for the a l te rnat i ve materials depending upon the qual i ty of each material required and the cost of each material in $ per l b . of the mater ia l . The quantit ies of d i f fe rent materials required for the design can be calculated by using model 1 and model 2 for each material ava i l ab le . The cost of material i s conventionally given as a function of a s ingle strength value with the cost generally increasing with increase in the strength value for the same class of mater ia ls. However, considering the material strength as a r.v. and not as a s ing le determinist ic value, i t i s more log ica l to consider th i s cost coe f f i c i en t as a function of some parameter which considers both the central value of the strength and the var iat ion in i t s value. This cost function is discussed in some deta i l in next pages of th is chapter. ( i i ) cost of the safety factor value rea l i zed : In the case of conventional design pract ice, i t was mentioned that a designer might associate some cost value in do l lars with the each value of the safety factor , which might be chosen for the design purpose. A s imi la r cost function is va l i d in the p robab i l i s t i c approach too. For a par t i cu la r actual component, a certain value of the material strength S is rea l ized 3-6 so that for a s pec i f i c load L the safety factor U has a s pec i f i c rea l ized  value. The cost associated with th i s value of the safety factor under the conventional approach can be associated with th i s rea l i zed value of i> . For d i f fe rent components made of the same mater ia l , tested under the same load d i s t r i b u t i on , d i f fe rent values of ^ ( 0 to oo theoret i ca l l y ) w i l l be observed. With each rea l ized value of the safety factor a cost w i l l be associated, and the expected cost for a l l the possible values of the safety factor can be obtained by using the p.d.f. as the weighting factor for these cost values. The procedure for drawing such a cost function is given in section 3.3.3. 3.3.2 Cost of the Material The cost of any material per l b . of weight, in one par t i cu la r class (such as the carbon base a l l o y s , etc.) should take into consideration the central value of the material strength as well as the var ia t ion in strength values. A meaningful function of both the central value and the var ia t ion of the material strength is the ra t io of the mean strength to the standard deviation of the p.d.f. of the pa r t i cu la r mater ia l . An increase in the value of th i s parameter X (see eq. 2.6) may be a re su l t of ( i ) an increase in the value of mean strength only ( i i ) a decrease in the standard deviation or the var iat ion in the material strength, ( i i i ) an increase in the mean strength and a decrease in the standard dev iat ion, or ( iv) an increase in the mean strength along with an increase in the standard deviation such that the r a t i o of the two increases. Any one of these changes w i l l , obviously, be a resu l t of one or more of the fo l lowing: (a) An improvement of the material strength as a re su l t of advances in the material sciences. For example,-a composition using a l loys of 3-7 better qua l i ty may be used to increase the strength of material produced. In general, such improvements are cost ly and so is the process of material production, when high strength materials are manufactured. The resu l t i s that the mean value of the minimum strength S increases. (b) Improving the qual i ty control of material production, which can be accomplished by using the cost ly but more precise production equipments and methods. This w i l l decrease the variat ions in material propert ies, i e . , the standard deviation of the minimum strength S w i l l decrease. Therefore, any method adopted to increase the ra t i o of the mean strength to the standard deviation is l i k e l y to resu l t in higher cost of material production per l b . I f the cost of the mater ia l , in general, i s given by C, i t may be stated that, C <X h. C*s,£) (3-D where h, , the ra t i o of the mean strength and the standard deviation^,wi 11 be a function of the parameters (^s,^ ) for the Weibull d i s t r i bu t i on for the material strength S . For a par t i cu la r class of mater ia ls , the exact nature of the re lat ionsh ip between the cost of material per l b . and the ra t i o h. can be determined only by obtaining the data for several materials in s t a t i s t i c a l form ( i e . determining the values °^ s,£> for each material) and the i r respective costs. There exists a lack of such information and in order to proceed further, the s impl i fy ing assumption of a l inear re lat ionsh ip between the cost C ($/lb.) and the ra t i o \c is made. In the case where the material strength has the Weibull p.d.f. given by eq. (2.1), Mean (S) = ° < s r^V^>) Std. Deviation (S) = °<sLrc±+?p " r #+ 0^ from eq. (2.5) 3 - 8 rci+4) and h. = •—- —71/2. from eq. (2.6) [ r a + 2 / £ ) - r V W and thus, the r a t i o h, i s a function of only one para-meter (the shape parameter), of the material strength p.d.f. The cost re lat ionsh ip given by eq. (3.1), then, becomes, C = -f c/o = - f l C ^ (3.2) and therefore, when the material cost i s related to the ra t i o k. , the cost i s dependent on only one parameter, namely . Generally, there is a minimum qual i ty of material (minimum value of the ra t io h~ ) that can be produced using ordinary equipments, controls , raw materials, etc. Let -h-vnw = r a t i o of the mean strength to the standard deviation for th is minimum cost . Let f^yr\<xv.~ maximum value of rt. that can be obtained by using the best equipments ava i lab le for producing mater ia ls , best technology avai lable within the maximum allowable cost of the material Cq_ . If the l i n e a r i t y assumption holds (see f i g . (3.1)), Slope = t< JL = — C ± - (3.3) and the cost of the mater ia l/ lb . , r c 1+ vf) [rci +2^)_ r^ 1 +y A)]V£ Ci-cj. N r c u V A ) i ( 3 . 4 ) 3-9 Figure 3-2 Cost of Material ($/lb.) as a Function of the Parameter A 3-10 P lo t t ing eq. (3.4), a curve as shown in f i g . (3.2) can be obtained showing the re lat ionsh ip between the parameter |3 of the p.d.f. of the material strength and the cost of material C in $ per l b . of the mater ia l . Normally, the values of the ra t i o /t , for various a l ternat ive materials co'nsidered/may not d i f f e r by a large number (say from 5.0 to 10.0). Even though the exact re lat ionsh ip between the cost of the material and the r a t i o ft may not be l i nea r , i t is expected that the l i near cost function w i l l serve as an adequate f i r s t approximation to some non-l inear, but unknown, cost funct ion. 3.3.3 Cost of the Safety Factor V : I t was mentioned e a r l i e r that a decision maker may associate some cost value with each rea l i zed value of the r.v. V by taking into consideration the cost value of the margin of safety which is implied by that determinist ic value of the safety factor under the conventional approach. The procedure of obtaining such a cost curve for various rea l ized values o f the safety factor V i s outl ined here. For most engineering designs, a certain value of the safety factor is spec i f ied in the engineering codes for a pa r t i cu la r class of materials and loading condit ions. Under the conventional design procedure, the designer might assume that i f a component designed for th i s value \ )^ of the safety factor f a i l s because the actual value of for that pa r t i cu la r component turned out to be less than i . , the money lo s t w i l l be a certain amount V 0 . This amount V£ , then, can be considered as the cost of the design safety factor Vc when the real ized value of the r.v. V i s less than one. When the actual value of V i s 3-11 greater than ± , the design w i l l not f a i l and, of course, no cost i s involved. The frequency of any actual value of i) i s spec i f ied by the p.d.f. }>(>>) which i s determined by the random nature of S and l _ (see sec. 2.2). I t may appear log ica l that, in general, th i s amount Vo w i l l be zero since the design code value y c precludes legal r e spons ib i l i t y for the f a i l u r e , i e . the cost associated with the design safety factor V = Vc w i l l be zero, unless the loss of designer 's reputation on future busipess i s quant i f ied in terms of a cost value. On the other hand, the minimum value of the safety factor that can be selected by the designer in the conventional approach w i l l be equal to 1, implying that no margin of safety is incorporated in the design. The amount Vj_ which w i l l be l o s t i f f a i l u r e occurs for a design with this value of the design safety factor can be equated to the highest possible cost to the decis ion maker for a f a i l e d design. Thus, the cost associated with values of the design safety factor \) i s expressed in terms of the "safety margin" of the design in the difference between the design value of V and the real ized value of V&L . Once these two points are f ixed for the desired cost curve, the designer may be able to s im i l a r l y assess the cost assoc i -ated with other design values of the safety factor . However, since i t makes no sense to design for a " safety" factor <1 , the corresponding cost can be considered a very large quantity and so the cost curve w i l l be va l i d for design values of \ J o n l y . For the intermediate design values of the safety factor , the designer may f ind the corresponding cost value by using the u t i l i t y theory given in Appendix IV by asking himself the question: "For a par t i cu la r value of the safety factor U such that i./LV^V^ , what is the value of 3-12 ^ {0<fy<± ) such that I am ind i f fe rent between the money I have to pay for the corresponding safety margin and a reference gamble which has a chance ^ °f loosing a n Q l a chance (1- 0^ ) of loosing Vo ." For a par t i cu la r value of the S.F. \) , the cost is then given as, = (as V0-=o) (3-5) For d i f fe rent decision makers, the value Cj/will be d i f fe rent giving d i f fe rent functional re lat ionsh ip between the cost \J(V) and the r.v. D . However, a l inear approximation of such a cost function may be considered to give reasonable re su l t s . This l i n e a r i t y assumption implies (see Appendix IV for a more complete discuss ion): ( i ) the decis ion maker is ' r i s k neut ra l ' and w i l l not assess r e l a t i v e l y high values of for small values of the safety factor . The v a l i d i t y of such an assumption depends completely upon the nature of the decision maker and may be assumed to hold at least as a f i r s t approximation. ( i i ) the decis ion problem is terminal. This requirement is completely s a t i s f i ed as one of the given material has to be selected. Once a par t i cu la r material i s selected, there is no other decision to be made in the future. The nature of the cost function for the design values of the safety factor is considered to be l i near and with the quantity being decided, a curve as shown in f i g . (3.3) can now be obtained. Let UCV) = u t i l i t y (cost) of the design for the design value \) . In general, from f i g . (3.3), UQJ) = V ± - ( y - i ) 3 (3-6) 3-13 where 1<2 = slope of the cost l i ne at V—VQ, » where = code value of the safety factor , U(V c)^0 = V - L-C^c-i)^ (3.7) or k„ — P\ ~ slope of the cost funct ion. (3.8) For any feas ib le value of the safety factor , U E U ) = TCT0^ (3-9) Thus, once the decision maker's u t i l i t y curve i s drawn, the cost associated with any design value of V can be obtained. 3.3.4 The Overall U t i l i t y Function The overal l cost of designing for the given load d i s t r i bu t i on and for a pa r t i cu la r material JL , for a given value of the r e l i a b i l i t y R i s obtained as the sum of the cost of material used and the cost of the safety factor U realized,and can be given as, U C i > V ) = 7 ^ 1 ) ^ ) 4- Q . ^ (3.10) where = cost of the material ( $ / l b . ) / = weight of the design using ^ mater ia l . Eq. (3.10) can be rewritten as, Where = length of component in inches. Since D i s a r.v. having the p.d.f. \>(y) given by eq. (2.24) th i s state of knowledge about the random behaviour of V can be taken into account by weighting the cost U(/C,y) w. r . t . the p.d.f. j x v ) • The expected cost w. r . t . random var iable \) i s given as, (3.11) — ! V| - k CO /SLOPE = V l o \ / Vc-1 o ^ \ v o L. I I I X I S A F E T Y FACTOR V Figure 3-3 - Cost of the Design Values of Safety Factor UQ>)vs. V 3-15 U a , P ) = - J " [ C O K S L W * - _ 3d;)] (3.12) CN&r<xX\ ValaLti. -hV^&^Al^\^ (3-13) (Dc-1) xfc r v^ y - / t w i u CPc-l) (3.14) where V = mean value of the r.v.L> (3.15) As the p.d.f. f>(L>) features parameters which are functions of the parameters of the load d i s t r i bu t i on ( L ; ) and of the material strength d i s t r i bu t i on ( j jS )> there w i l l be a spec i f i c value of U for each combination of load and mater ia l . These values of V are computed for the range of values of parameters fi , (°<L/t ), and (°^s/|& ) defined ea r l i e r in chapter 2. However, only a sample output is shown in tabulated form in Appendix VII. Once the various coe f f i c ient s involved in the overal l u t i l i t y funct ion, given by eq. (3.14),a re evaluated, the decision process reduces to ca lcu lat ing the value of TJ (^/D) for each material d i s t r i bu t i on given and se lect ing the material which minimizes the value of th i s cost funct ion, i e . TJ^U* ,~V2_*) = Lifij] (3.16) 3-16 3.4.1 Decision Inputs: Defining values of various inputs as, ( i ) the r e l i a b i l i t y R ( i i ) the load d i s t r i bu t i on ( ) ( i i i ) the d i s t r i bu t i on ( o(s } £ ) < of the relevant material property for each a l ternat ive material L , and ( iv) the decision parameters V± , VQ, ( for the cost of the design safety factor value V , f i g . (3.3)), and C Cn • h ->• 5 2 5 y ^ W i A j - " w a x (for the cost of mater ia l , f i g . (3.1)). 3.4.2 Decision Process: ( i ) for the given load d i s t r i bu t i on ( < L^ ) a n a " the desired r e l i a b i l i t y R ca lcu late the weight of the component V4^— Aj;P^ for each mater ia l , using model 1, given in sec. (2.6). ( i i ) ca lcu late the r a t i o h-i for each material x (a function of i?> ^  only), using eq. (2.6). This can be done by using f i g . (VI.1) given in Appendix V i . ( i i i ) ca lcu late the expected value ~V of the safety factor V , using eq. (3.15) which can be evaluated for given load parameters ( X }°<L ), material strength parameters ( ) \, for each a l ternat ive material JL j and the expression for j ^ C ^ (see eq. (2.24)). To f a c i l i t a t e th is cumbersome step, tables are prepared (see Appendix VII for sample r e su l t s ) , which can be entered v ia the parameter $> and the two parameter rat ios CVL_/£) and &s/Z) . ( iv) for the given load and each mater ia l , subst i tute the values of 1^1 L (step i ) , jzi (step i i ) and D (step i i i ) , along with the system inputs, into eq. (3.14). This determines the tota l expected cost of the 3-17 design using the material x. , for the given load d i s t r i bu t i on and achieving the design r e l i a b i l i t y R over a contemplated l i f e time. (v) select that material -i which minimizes the tota l expected cost of the design. Hence, the output of such a decision process w i l l be a design cross section for the material with strength d i s t r i bu t i on ( ^< S ) & } and the load d i s t r i bu t i on ( L , °(\_ ), achieving the design r e l i a b i l i t y value R over a contemplated mission time at a minimum tota l expected cost This design model is further i l l u s t r a t e d in Appendix VIII by solving a numerical example. CHAPTER 4 Closing Comments As mentioned e a r l i e r in the section 1.1, the aim of the present study was to develop design procedures for the important problem of designing a c r i t i c a l engineering component using the modern p robab i l i s t i c approach rather than to the conventional design method. The modern design method considers the v a r i a b i l i t y both in maximum load as well as minimum strength values and spec i f ies the design for a required value of r e l i a b i l i t y rather than a certa in determinist ic value of the safety factor. Three spec i f i c design problems, given as model 1, model 2 and model 3 are solved and f a c i l i t a t e d by various tables (Appendix V and VII) and plots (Appendix VI). The conclusion drawn from the design method^outlined in model 1^is that for a spec i f ied value of r e l i a b i l i t y , the design cross section area depends upon the values of the ra t i o of the central value to the var iat ion for loads and the material strength. It i s further shown with the help of an i l l u s t r a t i v e example that though the mean value of the material strength might decrease from one material to the next, an increase in the ra t i o of the mean strength to the standard deviation corresponds to a reduction in the cross section area for the same r e l i a b i l i t y value. However, the outstanding feature of th is proposed design method l i e s in the fact that a quant itat ive statement can be made for the r i s k of the design in terms of the probab i l i ty of f a i l u r e or the r e l i a b i l i t y which was not possible when conventional design pract ice was used. Subsequently, the ra t i o H. of the mean value of the strength of material to i t s standard deviation becomes an important parameter, used as a measure of the qua l i ty of the mater ia l . In the case of the minimum strength represented by a Weibull d i s t r i b u t i o n , th i s r a t i o k. i s a function of only one parameter (3 (the shape parameter). In making a choice among several materials ava i l ab le , with th i s modern design approach, the cost of material i s then considered as a function of th is parameter Ac . Because of the p robab i l i s t i c nature of the problem, s t a t i s t i c a l decision theory is used to develop the design model 3. Another cost factor, considered in evaluating the cost of the design for a certa in mater ia l , was the cost associated with the design values of the safety factor which was properly interpreted as a r.v. in the present analys i s . I t c4.n be seen from the example given in Appendix VIII, that with the introduction of such a cost factor , a material having higher cost of the quantity of material used may be selected as a choice when the design c r i t e r i on is minimum cost. However, as compared to the p robab i l i s t i c design approach for p.d.f . ' s other than E. V. models, used to represent the d i s t r ibut ions of the material strength and the load, the r e l i a b i l i t y statement is va l id only for a certain spec i f ied operating time which may be defined as the mission period. This feature of the design i s due to the fact that the locat ion parameter of the E.V.T m = v i , max model for maximum loads is a function of the number of observations in a sample from which the extreme value is taken. Throughout the present analys i s , various assumptions were involved such as the independence of the applied load L and the material strength 5 , the non - s ta t i s t i ca l nature of the area of cross section and the 4-3 material density, and the purely ax ia l nature of loads. These assumptions avoid the complexity in mathematical analysis which may ar i se otherwise. Depending upon the nature of design problem, in pa r t i cu l a r , and the extent to which these assumptions take the analysis away from the real s i t ua t i on , one or more of these assumptions can be removed at the cost of increased mathematical complexity. The present analysis also presupposes a v a i l a b i l i t y of data to establ i sh the properties of the p.d.f.'s for the maximum load and the minimum material strength. These data are often ava i lab le but are, usual ly, not published (for example, only a mean value of a certa in material property i s commonly given in material handbooks, e t c . ) . It i s expected that an emphasis on the modern approach to design w i l l help to generate such necessary data. It may be recommended that the present work may be extended by redefining model 3 such that the decision process speci f ies a par t i cu la r material and an optimum value of the probab i l i t y of f a i l u r e . This may be achieved by adding another cost, as a function of the probab i l i t y of f a i l u r e , to the overal l cost of the design. In such a case the probab i l i ty of f a i l u r e w i l l be another decis ion var iab le, rather than an assumed decision input. The present cost function for model 3 may also be changed by adding another cost as the weight penalty. I t i s generally seen that i f the weight of the design is an important feature, a certa in cost value may be associated with the weight above some maximum spec i f ied l i m i t . For example, in the case of an airplane structure a heavy component i s highly undesirable since i t implies a penalty in terms of reduced payload. Thus the consideration of such a cost factor for the optimum design on the basis of economic c r i t e r i a w i l l avoid components of large weights from being selected for designs. BIBLIOGRAPHY (A) L i s t of Various Journals Surveyed: 1 Applied Mechanics Review, Yr. 1950-1970. 2 Journal of Structural D iv i s ion, ASCE, Yr. 1947-1970. 3 IRE Transactions on R e l i a b i l i t y and Quality Control, Yr. 1953-1962. 4 IEEE Transactions on R e l i a b i l i t y and Quality Control, Yr. 1963-1970. 5 IRE Proceedings, Yr. 1957-1962. 6 IRE National Symposium on R e l i a b i l i t y and Quality Control, Yr. 1954-1965. 7 Proceedings of the Symposium on R e l i a b i l i t y , Annals of Assurance  Science, IEEE, Yr. 1966-1970. 8 SAE Annals of R e l i a b i l i t y and Ma in ta inab i l i t y , Yr. 1965-1970. (B-l) L i s t of the A r t i c l e s and Books Referred to in th i s Thesis: 1 Freund, J . E., Mathematical S t a t i s t i c s , Prent ice-Hal l Inc., Engle-wood C l i f f s , N. J . , 1962. 2 Freudenthal, A. M., "The Safety of Structures, " Journal of Structural  D iv i s ion, Proc. of ASCE, Part I, Vol. 87 (St-3), March 1961, pp. 1-16. 3 Freudenthal, A. M., " The Safety of Structures, " Am. Soc. of C i v i l  Engrs., Transactions, No. 112, 1947, pp. 125-185. 4 "Structural Appl icat ion of Steel and Light Weight A l l o y s , " A Symposium, Transactions, ASCE, Vol. 102, 1937, pp. 1207-1208. 5 Mittenbergs, A. A., "Fundamental Aspects of Mechanical R e l i a b i l i t y , " Mechanical R e l i a b i l i t y Concepts, ASME., Design Conference, 1965, pp. 17-34. B-2 6 Mittenbergs, A. A., "The Material Problem in Structural Analys i s , " Annals on R e l i a b i l i t y , SAE , 1966, pp. 148-158. 7 Weibul l, W., "A S t a t i s t i c a l Representation of Fatigue Fai lure in So l id s , " Transactions, Royal Inst, of Technology, Stockholm, #27, • 1949. 8 Freudenthal, A. M., Gumbel, E. J . , "On the S t a t i s t i c a l Interpretation of Fatigue Tests," Proceedings, Royal Soc. of London, Series A, Vol. 216, 1953, pp. 309-9 Pope, J . A., Foster, B. K., Bloomer, N. T., "Limited L i fe Design, A Survey of the Problem," Engineering, August 23-30, 1957, No. 112, pp. 236-241. 10 Armitage, P. H., " S t a t i s t i c a l Aspect of Fatigue," Meta l lurg ica l  Review, Vol. 6, No. 23, 1961, pp. 353-385. 11 He l le r , R. A., He l le r , A. S., "Fatigue L i f e and R e l i a b i l i t y of a Redundant Structure, " SAE Annals on R e l i a b i l i t y and Ma in ta inab i l i t y , 1964, pp. 881-896. 12 White, R. T., "Some Studies into the Fatigue Properties of 2024-T3 Sheet Aluminum," M.A.Sc. Thesis, M.E. Dept., U.B.C., 1965. 13 Simon, R., "Dimensionless Parameter R e l i a b i l i t y Analysis and Appl ication to Mechanical Creep," Transactions of ASME., Series D, Journal of  Basic Engineering, Vol . 88, No. 1, March 1966, pp. 87-92. 14 Forrester and Thevenow, "Designing for Expected Fatigue L i f e , " Annals of SAE, 1968, pp. 511-519. 15 Gumbel, E. J . , S t a t i s t i c s of Extremes, Columbia Univers ity Press, New York, N.Y., 1958. 16 J u l i a n , 0. G., "Synopsis of F i r s t Progress Report of Committee on Factors of Safety," J . of St. Div., Proc. of ASCE., paper 1316, Vol. 83, ST. 4, July 1957, pp. 1-22. 17 Freudenthal, A. M., "Safety and Probab i l i t y of Structural Fa i l u re , " Transactions of ASCE., Vol. 121 , 1956, pp. 1337-1375. 18 Freudenthal, A. M., Garre l t s , J . M., Shinozuka, M., Journal of  Structural Div., Proc. ASCE, Vol. 92, St. 1, Feb. 1966, pp. 267-325. 19 Haugen, E. B. " S t a t i s t i c a l Methods for Structural R e l i a b i l i t y Analys i s , " IEEE., Tenth National Symposium on R e l i a b i l i t y and Quality Control, 1964, Washington, D.C., pp. 97-121. 20 Cable, C. W., Virene, E. P., "Structural R e l i a b i l i t y with Normally Distr ibuted S ta t i c and Dynamic Loads and Strength," Proceedings, IEEE., 1967, pp. 329-336. B-3 21 Disney, R. L., Sheth, N. J . , "The Determination of Probab i l i ty of Fai lure by Strength/Stress Interference Theory," Proceedings 1968 Annual Symposium on R e l i a b i l i t y , Boston, Massachusetts, pp. 417-422, January 16-18, 1968. 22 Kececioglu, D., Haugen, E. B., "A Unif ied Look at Design Safety Factor, Safety Margin, and Measure of R e l i a b i l i t y , " Annals of  R e l i a b i l i t y and Ma in ta inab i l i t y , 1968, pp. 520-530. 23 Benjamin, J . R., " P r obab i l i s t i c Structural Analysis and Design," Journal of Structural D iv i s ion, Proc. of ASCE., July 1968, pp. 1665-1679. 24 Feigen, M., and H i l t on , H. H., "Minimum Weight Analysis Based on Structural R e l i a b i l i t y , " J . of Aerospace Sciences, Vol. 27, No. 9, September 1960, pp. 641-652. 25 Broding, N. C , Diedrich, F. W., and Parker, P. S., "Structural Optimization and Design Based on a R e l i a b i l i t y Design C r i t e r i o n , " J . of Spacecraft and Rockets, Vol. 1, No. 1, Jan.-Feb. 64, pp. 56-61. 26 Kalba, R., "Design of Minimum Weight Structures for Given R e l i a b i l i t y and Cost," Journal of Aerospace Science, Vol. 29, 43, March 1962, pp. 355-56. 27 Ghista, D. N., "Minimum Weight Structural Design with Probab i l i ty of FAilure Constraint, " Journal of Science and Engineering Research, I.I.T., Vol. II, No. I, 1967, pp. 19-30. 28 Shinozuka, M. and Hanai, M., "Structural R e l i a b i l i t y of a Simple Rigid Frame," SAE Annals on R e l i a b i l i t y , 1967, pp. 63-67. 29 Shinozuka, M., "On the R e l i a b i l i t y of Redundant Structures, " SAE  Annals on R e l i a b i l i t y , 1966, pp. 611-617. 30 Prat t , J . W., Ra i f fa , H., and S ch l a i f f e r , R., "The Foundations of Decision Under Uncertainty , An Elementary Expos it ion, " Journal of  ASA, Vol. 59, 1964, pp. 353-375. 31 Prat t , J . W., Ra i f f a , H., and S ch l a i f f e r , R., Introduction to S t a t i s - t i c a l Decision Theory, McGraw-Hill Book Company, Inc., N. Y., 1965. 32 S ch l a i f f e r , R., Probab i l i ty and S t a t i s t i c s for Business Decisions, McGraw-Hill Book Company, Inc., 1959. 33 S ch l a i f f e r , R., Introduction to S t a t i s t i c s for Business Decisions, McGraw-Hill Book Company, Inc., 1961. 34 S ch l a i f f e r , R., Analysis of Decisions Under Uncertainty, McGraw-Hill Book Company, Inc., 1969. B-4 35 Myron, T., Rational Descriptions, Decisions and Designs, Pergamon Press, Year 1969. 36 Weiss, L., S t a t i s t i c a l Decision Theory, McGraw-Hill Book Company, Inc., 1961. 37 De Groot, Morris, Optimal S t a t i s t i c a l Decisions, McGraw H i l l Book Company, Inc., 1970. 38 Bury, K. V., "Bayesian Decision Method Applied to a Problem in Gear Train Design," International Journal of Production Research, to be published in 1971. 39 Huntington, E. V., "Frequency D i s t r ibut ion of Product and Quotient," Annals of Mathematical S t a t i s t i c s , Vol. 10, 1939, pp.195-198. 40 Cur t i s , J . H., "D i s t r ibut ion of a Quotient," Annals of Mathematical  S t a t i s t i c s , Vol. 12, 1941, pp. 409-421. 41 Epstein, B., "Appl icat ion of Mel 1 i n ' s Transform," Annals of Mathematical  S t a t i s t i c s , Vol. 19, 1948, pp. 370-379. 42 Lucas, E., Character i s t ic Functions, G r i f f i n , London, 1960. 43 Dyckman, T. R., Smidt, S., and McAdams, A. K., Management Decision  Making under Uncertainty, The McMillan Company, London, 1969. 44 Handbook of Mathematical Functions, Edited by M. Abramowitz and I. A. Stegun, New York, 1965, pp. 267-273. 45 Pearson, K., Tables of the Incomplete Gamma Function, Computed by the Staf f of the Dept. of Applied S t a t i s t i c s , University of London, University College, London, 1951. (B-2) L i s t of Related A r t i c l e s perused: 46 Ang, Amin M., " R e l i a b i l i t y of Structures and Structural Systems," Journal of Engineering Mechanics D iv i s ion, Proc. of ASCE, Apr i l 1968, pp. 671-691. 47 Brat t , M. J . , Truscott, H. A., and Weber, G. W., " P r obab i l i s t i c Strength Mapping, A R e l i a b i l i t y vs. L i f e Predict ion Too l , " SAE Annals on R e l i a b i l i t y and Ma in ta inab i l i t y , 1968, pp. 501-510. 48 Broding, W. C:, Diedrich, F. W. and Parker, P. S., "Structural Optimization and Design Based on R e l i a b i l i t y Design C r i t e r i a , " Journal of Spacecraft, Jan. - Feb. 1964, Vol. 1, No. 1, pp. 56-61. B-5 49 Corne l l , C. A., "Bounds on the R e l i a b i l i t y of Structural System," Journal of the St. Pi v., Proc. of ASCE, Feb. 1967, pp. 171-200. 50 Dick, C , and Wilson, S., "Structural R e l i a b i l i t y - The General Engineering Approach," Eleventh Annual Symposium on R e l i a b i l i t y , IEEE, pp. 169-181. 51 Haugen, E. B., "Implementing Structural R e l i a b i l i t y Program," Eleventh Annual Symposium on R e l i a b i l i t y , IEEE, pp. 158-168. 52 Ingram, G. E. "A Basic Approach For Structural R e l i a b i l i t y , " Eleventh Annual Symposium on R e l i a b i l i t y , IEEE, 19, pp. 154-157. 53 Mesloh, R., " R e l i a b i l i t y Design Cr i te r ion for Mechanical Creep," SAE Annals on R e l i a b i l i t y and Ma in ta inab i l i t y , 1966, pp. 590-597. 54 Moses, F., "Analysis of Structural R e l i a b i l i t y , " Journal of St. P iv . , Proc. of ASCE, Oct. 1967, pp. 147-164. 55 Pugsley, A. G., "Concepts of Safety in Structural Engineering," Proceedings, In s t i tute of C i v i l Engineers, London, Vol. 29, 1951. 56 Simon, R., "Pimensionless Parameter R e l i a b i l i t y Analysis and App l i -cation to Mechanical Creep," Journal of Basic Engineering, Transactions ASME, March 1966, pp. 87.92. A-l APPENDIX I METHODS FOR OBTAINING THE PROBABILITY DENSITY FUNCTION OF A RANDOM  VARIABLE, SUM OR QUOTIENT OF TWO INDEPENDENT RANDOM VARIABLES (a) Method of Transformation of Variables. ' Let X and Y be two independent continuous random var iables. The p.d.f . ' s of X and y a r e given by {^OO and I^CY), respect ive ly. PlC*) and P 2 ( Y ) are the probab i l i ty d i s t r i bu t i on functions of the r .v. ' s X and Y . As X and Y are independent random var iables, the j o i n t p.d.f. of these two r . v . ' s , JXVY'),, wi l 1 be given as; f><XY) - W y = h^.^OO.cbt.d-Y (1.1) Introducing the quotient -2 — ^/y y i e l d s , X - S - Y (T.2) and d i f f e ren t i a t i n g X w.r . t . 2 , ( ^ / a £ ) = Y d.3) Equation (1.1), then, becomes, ^2Y)>2CY)- (||)^ z.^ Y = hCz.Y).feCyj Y-^z.^Y (1.4) This is the j o i n t p.d.f. bCZjV) of the q u o t i e n t s (r.v.) and the random v a r i a b l e V . On integrat ing th i s expression over the entire range of the values of r . v . Y (generally 0 t o c o ) , the uncertainty A-2 due to the r.v. Y 1 S eliminated and the p.d.f. of the q u o t i e n t ^ is obtained as a marginal density, CO b t z ) ^ = J fcteY) ^Cr ) . Y ^ Y o«i . (1.5) Y=o In the case where the r.v. Y has a range-oo^Y^"? 0 > -foe? - J ^ V ) . ^ Y J Y . d . Y . ^ . (1.6) The probab i l i ty d i s t r i bu t i on function P(z) of the quotient Z i s obtained as, where -b = a value of the r.v."2:(constant). Substituting eq.(1.6) for the p.d.f. of •£ » "t oo P&)L - f [ / ^ Z V ) . ^ C Y ) Y ^ Y J ^ . (i.7) The independence of the l im i t s of integrat ion of i? and Y f r o m the respective variables can be used to change the order of integrat ion, and y i e l d s , PC2))._ = / / Y^CVJ.hC^YJotz.cLY (1.8) 2 = t Y=0 2 = 0 0 0 (1.9) where P ^ - Y ) = / Y I^L^CLZ. . Equations (1.5) and (1.9) can be used to obtain the p.d.f. and the probab i l i t y d i s t r i bu t i on function of the quotient of two indepen-dent r .v. ' s X and Y ; °y subst i tut ing the respective of p.d. f . ' s of X and Y • This method was used to obtain the p.d.f. and the probab i l i t y d i s t r i bu t i on function of the r.v. V , the safety factor , which i s a ra t i o of two independent r . v . ' s , strength of material S and load 1_ (see eq. 1.5). Eqn. (2.24) and eqn. (2.32) correspond to eqn. (1.5) and eqn. (1.9) respect ively. However, the integrals could not be solved to provide ana ly t i ca l so lut ions, and results were obtained by numerical computation techniques. When 2 i s the dif ference of two r .v. ' s X and Y » 2 ^ X - Y X = Z 4 - Y (I on pa r t i a l d i f f e ren t i a t i on of X w. r . t . , Ovaz) = i (i This can be substituted in eqn. (1.1), to obtain the j o i n t p.d.f. of r .v. ' s 2 and Y • Then the p.d.f. and the probab i l i ty d i s t r i -bution function of 2 > the d i f ference of two r .v . ' s "X and Y > c a n be obtained by going through steps s imi la r to those involved in eqn. (1.4) to eqn. (1.9). In the present problem, as an ana lyt ica l form of solut ion was not ava i lable for the quotient of the material strength and the load, defined as the safety factor D , another factor M was defined as the A-4 margin of safety, given as M .= S - L (1.12) The p.d.f. of the r.v. M was obtained as, k M ) a . M = k ' JCM + L ) ~ ^ { ( M + L ) P - ^ - ^ ( - L ^ U L , (1.13) where k' = exp f - ( 4 ) ? _ £ . , ! - , (1.14) s and CO PCM ) = [- C(^ -LA ^  + « 4 (-1^)]] JL .(I.15) The probab i l i ty of f a i l u r e Pp is given as, L - 6 J While the expression for Pf. i s the same as eqn. (2.35), the expressions for the p.d.f. and the probab i l i ty density function are rather more complex as compared to eqn. (2.24) and eqn. (2.32). As no ana ly t i ca l solut ion seems to be ava i lab le for integrals given by eqn. (1.13) and eqn. (1.15), they are assumed to be of no further use. (b) The Use of Me l l i n ' s Transform Method to Determine the Probab i l i ty  Density Function of the Product (or the Quotient of  Two Independent Random V a r i a b l e s : ^ ^ Let X and y be two independent r .v. ' s with the i r respective p.d.f. ' s ^Cx) and ixy.) • The Me l l in transforms of these r .v. ' s X and y are given as, A-5 o f°tl ' ( I - 1 7 ) In other words, i t m e a n s , ^ kCt) - EC** " 1 ) . (i.is) If X and y are independent, ELCV-Y) 1" 1] = E C x ^ . E C Y ^ 1 ) = P x a ) - ^ ) (1.19) and, again, for a given Mel 1 in transform {^(.-t) , boo ^ j ^ >,ct).dt. (1.20) c - too The Mel Tin transform method, thus, can be used in the fol lowing steps; ( i ) obtain the Mel Tin transforms for the p.d.f . ' s of both r .v. ' s X and Y (use eqn. (1.17)), ( i i ) Mu l t ip ly the two transforms, ( i i i ) Invert the product (1.19) as per eqn. (1.20), which, in turn, gives the p.d.f. of the product of two independent r . v . ' s . In a special case, l e t 2- - Y^ then, t^(t) E.C2Lfc"') = E (V^"*) ~ ^ Y 1 ) (1.21) i f <K=-i. , 2 = Vy , |Pyy0fc) - t> Y ( -±+2) . (1.22) A-6 Thus, the method of Mel! in transforms can be used to obtain the p.d.f. of the quotient of two independent r . v . ' s . For ( i ) E-V-in^>nivi (Weibull 's)model: The p.d.f. of the r.v. S i s given by eg. (2.1) as, where S>^o y ^s^O fr>0 Then, the Mel 1 in transform i s given by, o = A*-'(4)(4)N-K-C4f}^. (I-23) o Subst i tut ing ( S / ^ ) ^ - ^ - ; Dr. S O , 00 -y 1 . ^ rH-t " t - 1 ^ where = Gamma Function, evaluated at 2 ( i i ) E . V . , (Gumbel's) model, The p.d.f. of the r.v. L is given by equation (2.7) as, where L ^ o , t >. o cinX <*i_>0 (1.25) A-7 The Mel 1 in transform for th i s function <XL. i s given as, 0 7 putting k ^ - - ^ or L , ; and o<L.<:J.^  ^ and so = J .€x|p[-^ -^  ^ . ( i . 2 7 ) It seems that th i s integral cannot be solved a n a l y t i c a l l y , and so the Mel 1 in transform for the E . V . T model cannot be obtained. I, max In such a case, th is method cannot be used to obtain the p.d.f. of the quotient of two r .v. ' s having E . V . J J J m - j n and E . V . j m a x density functions. (1.26) (c) The Fourier Transform Method to Obtain the p.d.f. of the Sum (or  Difference) of Two Independent Random V a r i a b l e s : ^ ) Let X be a r.v. with the p.d.f. fcCx) and the probab i l i ty d i s t r i bu t i on function P(>.) . The charac te r i s t i c f u n c t i o n ^ ) 0 f is the function c£> defined for real A by •+cO , <£ XCA) ^ / ^ A K jpuydx (1.28a) S -00 = ^ (A ) -f ^t>(A) (1.29b) CO -OO ~f°° I -oo (1.30) A-8 This cha rac te r i s t i c function i s also known as Four ier ' s Trans-form in Mathematics. In the case where the r.v. i s (-X) rather than x, the cha rac te r i s t i c function is given as, < £ _ v O ) = L i O ) - ! l K A ) ( i . 3 i ) If there are two independent r .v. ' s X and Y with p.d.f.s K x ) and fcty) respect ive ly, = E { > * J . E ( < ^ ) ( I 3 2 ) 4-00 — CO but and Let "2 be another r.v. defined as, ^ =- * + Y . Then, from eqn. (1.32), <f-zC>0 = # X CA) (I-33a) •If X—Y> e c l n - (1.33a) i s modified as, ^ C A j - ^ ( A V £ Y ( A ) ; (1.33b) where ^ y ( A ) " i s defined by eqn. (1.31). From the theorem of uniqueness of cha rac te r i s t i c functions, i t can be stated that the product of the cha rac te r i s t i c functions of the two A-9 independent r .v. ' s i s the cha rac te r i s t i c function of the sum of those two independent r . v . ' s . I f K )^ denotes the p.d.f. of the r.v. , from the inversion formula for the Four ier ' s Transforms, •+00 i f f ! n £ " - ^ O O ^ J ^ - (1.34) Thus, the method can be used to obtain the p.d.f. of the sum (or the difference) of two independent r.v.'s using fol lowing steps; ( i ) Obtain the Fourier Transforms of the r.v. ' s X and Y w. r . t . the i r respective p .d . f . ' s , using eqn. (1.28) ( i i ) Mult ip ly the two transforms. In the case of the difference of two independent r . v . ' s , use eqn. (1.31) and eqn. (1.33b). ( i i i ) Invert the product to obtain the p.d.f. of the sum (or the difference) of the r.v. ' s X and Y » as the case may be. ( i ) E.V. _j- V y ^ ^ (Gumbel's) model: where L>^o . t>^o oW- *L>o . Then, §LtX) = | V > " L . i . ^ { - k = £ ( - t § i.)j^ L . (,.35) Subst itut ing I |_ _ i . . , . in eqn. (I.35), obtains 4>L(s\) c U L | ^ A ^ . ^ Q - e x ^ ) ) . ^ ; ( I > 3 6 ) again, put - e " ^ — zt and — -e^.ckq, — A-10 Eqn. (1.36) then reduces to , - • V c l _ i A ^ CA) ; where, A — -Gxf> (^AL) ; and ^ (A) = Incomplete Gamma Function of the order evaluated at the argument A = -oxp C L /c< L ) . X A L Hence £ L ( A ) ^ $ _ L ( A ) = -e ^CA) , (1.38) (ii) E-y-m}yyiln ( W e i b u 1 1 ) m o d e 1 : where s >> o ^ c<s^o , The Fourier Transform i?sCx) i s given as, — <33 f^ AS(4)C4f1-K-CI;/j^  (i.39) 0 For eqn. (1.39) a closed form Solution seems unavailable, except when $ = 1 or = 2. While = 1 represents the exponential density function for the material strength S , £ = 2 (Raleigh 's p.d.f.) is also not of much interest as far as the present problem is concerned. Under such circumstances, the method of Fourier Transforms cannot be used to obtain the p.d.f. of the difference of S and L , APPENDIX II PROBABILITY OF FAILURE IN THE SPECIAL CASE, WHEN fi =1.0 as, The probab i l i ty of f a i l u r e , in general, i s given by eqn. (2.35) p f = I - A J % > { - l i l ^ f * ^ F A . < r * l W . When & =1.0 Subst i tut ing in eqn. ( I I . l ) and d i f f e ren t i a t i n g both sides separately, Eqn. ( I I . l ) then reduces to, where A b »k «*F> C ^ s ) . ^ Thus. 1- -e*|> (- LAs) J ( I I . l ) (II.2) (II.3) (H.4 ) (II.5) APPENDIX III SOME SERIES-SOLUTIONS FOR THE PROBABILITY OF FAILURE Tfr (a) For integer values of parameter ft ; where / r°> i ,.. .. = >n^ non central moment of the r.v. L subst i tut ing the expression for fiz^L) Pr £ — 2. ' i i i ( - i ) > » a ^ l - ' c i ^ t f e J ( i n . 2 ) >1=l (b) For a l l +ve values of ft : D;^±.'.)... 1 n n 3) The parameter A i s defined in Appendix II (eqn. I I.3). The series given by eqn. (III.2) as well as by eqn. (III.3) are convergent and converge rapid ly for large values of A • However, since the computer time involved in calculat ions i s much larger in this case as compared to the d i r ec t evaluation of the integral given by eqn. (2.35), these solutions are not of much importance. APPENDIX IV Decision Theory: Decision theory is concerned with the problem of se lect ing a spec i f i c action among feas ib le a l te rnat i ves , such that a quant i tat ive objective function is optimized. Each of the acts is re lated to a set of possible consequences in terms of the objective funct ion. If there is a known re la t i on between the various possible acts and the corresponding consequences and i f these consequences are known with cer ta in ty , the problem of decision making i s one of choice and not chance, i e . the act which optimizes the objective function i s the rat iona l choice. Thus, in such a determinist ic case, the process of decis ion making i s reduced to evaluating the objective function for the d i f fe rent acts, and comparing these values on some common scale of preference. The correctness of a decision w i l l depend only on the correctness of the judgement of the decision maker with respect to the scale of preference. In those cases where uncertainty exists regarding the re la t i on between the acts and possible consequences, s t a t i s t i c a l decision theory has to be used to make a rat ional decis ion. The problem of decision making may be, (a) a s ingle stage decision problem, in which case the optimal action is selected at a s ingle point in time. The state of nature, which is unknown at the time the decision is made, and the action selected by the decision maker, j o i n t l y determine the outcome and thereby the value A-14 of the decis ion in terms of the given preference scale. (b) Multi-Stage Decision Problem: Sometimes mult ip le decisions are to be made' sequential ly before a problem s i tuat ion is resolved and frequently the ent i re set of decisions cannot be made at the time an i n i t i a l decision is required. In some of the multi-stage problems, the outcome of one of the stages may be treated as having no impact on future choice, and thus each stage can be treated as a s ingle stage problem. In other cases, however, an immediate decision can be made only a f te r consideration of the events and the decisions that might fo l low. These types of problems can be solved by working backward from the l a s t stage to obtain the optimal i n i t i a l act ion. The method used in such a problem i s that of dynamic programming. The present study (Chapter 3) involves a s ingle stage decision problem and so s t a t i s t i c a l decision theory and i t s elements are discussed in that context only. The basic elements of a decision theory are, (1.) a decision var iable ^- '» the (unknown) value of cj. must be assessed by the decision maker i f a rat ional decision is to be made. In general, is a r.v. In the decision problem investigated in Chapter 3, th is decision var iable ^ is the safety factor V (a r . v . ) , which is the ra t io of two random measurement variables S and L . Thus, <^  pertains to the (unknown) future "state of nature." (2.) a set of a l ternat ive actions deD ; th i s refers to the decision choices related to the problem environment. For example, in the present problem, a set of a l ternat ive actions i s the choice of one of various materials {i. j which can be used for the design, the decis ion problem A- l 5 being to choose one pa r t i cu la r mater ia l . (3.) Consequences of the choice i and (future) rea l i zed state of nature ^ ; these consequences must be evaluated in terms of a common preference scale of the decision maker. The quant i tat ive expression of such a preference scale i s a u t i l i t y function lx-^ci-,i) which evaluates the u t i l i t y of a value CJ. observed for the decision var iable when the decision d was taken. In the design problem considered, the scale of preference is the cost given as u.(.i,i)) . In general, the c r i t e r i o n of choosing a par t i cu la r decision cl (here a par t i cu la r material ) i s to optimize the value of the u t i l i t y function U, (minimize the cost of the design). (4.) the state of knowledge w. r . t . the unknown "state of nature" of the decision var iable ^ . Since Cfr i s a r.v., th i s knowledge is a p.d.f. over the r.v.^ccp. In the problem of decis ion investigated in model 3, such knowledge is provided by the p.d.f. of the safety factor i) , given by eq. (2.24), \>(V) . This p.d.f. \>(\>) has parameters which are functions of the parameters ( L ,<*,_^  0 < s ^ ^ and i s , thus, condit ional on the values of these four parameters. It i s also seen in Chapter 3 that various other input values are required such as Ci ,C 2. / /Z.m;* , f L ^ ^ x , f i > , Vc , etc. These values are assessed by the decision maker and, thus, an optimal decision is condit ional on these various system input values as we l l . Process of Decision Making : ^ 3 5 t 0 Consider a par t i cu la r future "experiment" which w i l l give as i t s outcome a possible value of the decision var iable <^  in i t s space SI (generally 0 to. <=o ). For example, th i s experiment might be the test ing of a component designed for a pa r t i cu la r material L , and depending upon A-16 the observed values of S and |_ a value of u ( in i t s range o t o ° o ) w i l l be rea l i zed . In the context of the design problems considered in Chapter 3, a decision has to be made now without knowing the future outcome value of ^ . The consequence of the decision cL w i l l depend upon the future outcome V of the "experiment." Let D be a set of possible a l t e r -native dec i s ions (cL ] and l e t R be the space of a l l the possible rewards {.Jt-j , each related to a par t i cu la r decision cL and a pa r t i cu la r value of the decision v a r i ab l e^ - . This reward ( u t i l i t y ) i s denoted by UU.,^.) , when the decision taken is cL and the outcome i s <^  . For each f ixed decis ion cL £ D , the function u c ^ c j - ) w i l l have a d i s t i n c t real value for a s ingle value of the decis ion var iable <^  . As the decision var iable Cj i s a r.v. associated with a p.d.f. {x<p , the u t i l i t y U tcL^ ) can be weighted, w. r . t . p-tg) for a pa r t i cu la r decision cL , to cover the ent i re space of possible future outcomes Si . In s t a t i s t i c a l terms, such a weighting corresponds to forming the expec-tat ion of ^.(^9) w. r . t . the r.v. . In such a case, This integral gives the expected u t i l i t y value of the decision cL . In order to make a " r a t i o n a l " dec is ion, that decision cL which corresponds to the maximum expected value of the u t i l i t y should be selected. "Rat iona l " implies that the decision <X w i l l maximize the average value of the u t i l i t y U. for many such decision instances. Sometimes the decis ion c r i t e r i o n i s to minimise the loss (or the cost ) . In th i s case, the loss function is defined by the fol lowing equation, A-17 L ( c L / 9 ) = - U . C i / 9 ) (IV.2) and the decision G(_ which corresponds to the minimum value of the expected loss (cost) in the "best" decis ion. U t i l i t y Function: The se lect ion of a par t i cu la r u t i l i t y function is the choice of the decision maker. It re f lec t s the att i tude of the decis ion maker towards r i s k and the nature of actions cL in D . The u t i l i t y function U.(.d-/9) is often assumed to be l inear in 9 , in the absence of s u f f i c i e n t information to determine i t s exact nature. Thus, for two possible decisions d ( and dq_ , the two l inear functions might be given as (IV.3) These functions evaluate the worth of the decisions d.^ and dL<}_ in terms of the unknown value of the decision var iable Cj With the l i n e a r i t y assumption, the weighted u t i l i t y of a par t i cu la r decision oL , given by eq. ( IV.3) , becomes E f u - C ^ g ) / ^ ) } - j C c< + c ^ ) h 9 ) d - 9 (IV.4) (IV.5) or where g = E | , ( 9 ) i s the mean value of g w. r . t . i t s A-18 probab i l i ty density function p>(9) . In general, + a 9 The optimum rat ional action cL i s the one for which u. i s maximum, i e . t i c a l decision problem to the case of a determinist ic decision problem, with the mean value i) replacing the determinist ic value V . This con-s t i tu tes a s i gn i f i c an t reduction in the complexity of the problem. The assumption of l i near u t i l i t y function has two s i gn i f i c an t i m p l i c a t i o n s , ^ ) (1) The f i r s t impl icat ion relates to the concept of the decision maker being ' r i s k neu t ra l . ' Generally, u t i l i t y functions for each action cC in a set C> are constructed on the basis of reference gambles. The reference gambles help the decision maker to evaluate his u t i l i t i e s in a s e l f -consistent manner for each cL and a f ixed value of the decis ion var iable *<£j' . This method consists of considering the u t i l i t y ordinate as a probab i l i ty scale ^ (ranging from 0 to 1) and converting the abcizza for the decision var iable Cj. into a scale of monetary "certa inty equivalents" V which re la te to values of ^ . Two reference values of V are selected: V0 i s the lowest monetary value associated with a reasonable lower l i m i t of cj , and is the highest monetary value related to a reason-able upper l i m i t of ^ . A reference gamble i s defined as one for which the decision maker is i nd i f fe rent between the certa in cash receipt V > and a gamble with a de f in i te chance ^ to win and a complimentary Thus, i t i s seen that 1inear u t i l i t y functions reduce the s t a t i s -A-19 chance to win V0 . Vo w i l l thus correspond to Cj, = o while to •= £ . V ? and V j _ being determined, next an assessment is made of the values corresponding to a number of de f i n i t e values of o < ^ l s u c h as % = ]_ , J_ and 3_. The decision maker is asked for instance, "What cash 4 2 4 receipt v / would you prefer to a gamble which gives you a chance of 1_ 4 of winning Vj_ and a chance 3_ of winning V 0 ? What would the certa in cash receipt V have to be so you would prefer that gamble to v " ?" The range between V ; and \j" i s successively narrowed unt i l the certa inty equivalent V is reached for which the decis ion maker is i nd i f fe rent between certain cash receipt V and the reference gamble with chance <^  = 1_ . Thus f i ve points are obtained on the u t i l i t y graph for vs. V . A smooth curve through these points i s the true u t i l i t y curve for the decision maker who made the evaluation. I t has been experienced that in the case where the difference between and V 0 i s close to the decision maker's widest accustomed monetary range of decision making r e spon s i b i l i t y , his u t i l i t y curve tends to level o f f towards Vj_ . This implies that for high monetary values ( YL ) the increase in u t i l i t y ( ^ ) i s less for a given increament ZW . This behaviour of decision maker i s ca l led ' r i s k avers ion. 1 For example, though a decision maker may be i nd i f fe rent between $10,000 cash and a 50-50 gamble on $0 and $20,000, he i s l i k e l y to prefer $1 m i l l i o n cash to a 50-50 gamble on $0 and $2 m i l l i o n . On the other hand for a small difference between and V 0 , the u t i l i t y function can be expected to be a s t ra ight l i ne exactly. This w i l l imply that the decision maker is r i sk neutral (neither r i sk averse nor r i sk seeking). A test of the (8) v a l i d i t y of the l i n e a r i t y approximation has been given by Sch la i fer v , on page 29. A-20 (2) The second impl icat ion of the l i n e a r i t y assumption relates to the nature of action <± in D . I f the decision maker is r i s k neut ra l , a l i n e a r i t y assumption is va l i d provided the action taken i s s t r i c t l y terminal. On the other hand, i f the action <*/ in D implies a subsequent act ion, the l i n e a r i t y assumption for the u t i l i t y of <*! may not be j u s t i f i e d necessar i ly. In such a case the sequence of future decisions affects the u t i l i t y of the present action dl , and so the evaluation of u t i l i t y of action <*! becomes complicated. Certain considerations which help in this s i tuat ion are: (a) an assumption that the p.d.f. of the decision var iable ^ does not change from one decision point to another; (b) the decision maker l im i t s his time horizon up to which he evaluates his decis ions, consequences, etc. As an example, in an acceptance or re ject ion problem of a production l o t , while (a) means that i f a l o t is rejected, the next l o t w i l l be drawn from the same s t a t i s t i c a l d i s t r i b tu i on as the f i r s t one, condition (b) means that we cannot keep our decision pending t i l l i n de f i n i t e l y in the future. This seems relevant since most of the time decisions must be taken before certa in project completion dates. However, in the case of acceptance of the l o t , the l i n e a r i t y assumption is v a l i d , since acceptance is a terminal action and no further test ing i s necessary. Last ly , even in cases of s t r i c t l y non-l inear u t i l i t y functions, the range of the decision var iable which is associated with s i g n i f i c an t l y non-zero p robab i l i t i e s of occurrence, is often small as compared to (v{-Vo)} and the l i n e a r i t y assumption may appear quite appropriate. It may be noted that, in the context of the thesis problem, A-21 the u t i l i t y function (constraints V A , ) i s a given input, along with inputs <X S /^L, ^ u , c±,C2/km<x.y. and . The decision analysis of the present work is therefore condit ional on these input values. The various elements of s t a t i s t i c a l decision theory, as discussed in th is appendix, and the problem of constructing a l inear u t i l i t y funct ion, are discussed in Chapter 3 of the thes i s , in re la t i on to the problem under study. APPENDIX V TABLES FOR PROBABILITY OF FAILURE VALUES B A- 23 (°VTI) = 0 . 1 0 0 PROBABILITY OF FAILURE (3 ^ s / t = 1 .25 1.50 1.75 2. 00 1.0 0.5688 0. 5043 0.4522 0.4096 1.1 0.5626 0.4921 0 . 4 3 5 8 0.3901 1.2 0.5565 0.4802 0.4198 0.3714 1.3 0.55C3 0.4684 0.4042 0 . 3 5 3 2 1.4 G.5443 0.4568 0 . 3 8 9 0 0 . 3 3 5 8 1.5 0.53 82 0.44 54 0.3742 0 . 3 1 9 0 1.6 0.5322 0.4342 0.3599 0.3029 1.7 0.52 63 0. 42 32 0.3460 0.2875 1.8 0.5204 0 . 4 1 2 4 0 . 3 3 2 5 0.2728 1. 9 0.5146 0.4019 0.3194 0.2587 2.0 0.5088 0.3915 0.3068 0.2452 2.1 0.5030 0.3814 0.2946 0.2324 2.2 0.4974 0.3715 0.2828 0.2201 2.3 0.4917 0.3618 C.2715 0.20B5 2.4 0.4862 0.3523 0.2605 0.1974 2.5 0.480 7 0.34 30 0.2500 0. 1869 2.6 0.4752 0.3340 0.2398 0.1769 2.7 0.4699 0. 32 51 0.2301 0. 1674 2.8 0.4645 0.3165 0.2207 0 . 1 5 8 4 2.9 0.4593 0. 30 81 0.2116 0.1498 3.0 0.4 541 0.29 99 0.20 29 0.1417 3.1 0.4489 G.2920 0.19 46 0.1341 3.2 0.4439 0.2842 0. 1866 0.1268 3.3 0.4389 0.2766 0.1789 0.1199 3.4 0.43 39 0.2692 0.1715 0.1 134 3.5 0.4290 0.2621 0.1644 0. 10 73 3.6 0.4242 0.2551 0.1577 0.1014 3.7 0.4195 0. 2483 0.. 1512 0.0959 3.8 0.4148 0.2417 0. 1449 0.0907 3. 9 0.4101 0.2353 0. 1390 0.0858 4.0 0.4056 0.22 90 0.1332 0.0312 4. 1 0. 4011 0.2229 0.1278 0.0768 4.2 0.3966 0.2170 0.1225 0. 0726 4.3 0.3923 0.2113 0.1175 0.0687 4.4 0.3879 C. 2057 0.1127 0.0650 4.5 0.3837 0.200 3 0. 1081 0.0615 4.6 0.3795 0. 19 51 0.1037 0.0582 4.7 G.3754 C. 1930 0.0994 0.0551 4.8 0.3713 0 . 1850 0.C954 0 . 0 5 2 1 4. 9 0.36 73 0. 1802 0.C915 0.0493 5.0 0.3633 0.1755 0.0878 0 . 0 4 6 7 A-24 (°Vt) =0.100 PR0BA8IL I TV OF FAILURE °<s/Z=. 2.25 2.50 2.75 3.00 1.0 0.3741 0.3442 0.3186 0.2 965 1.1 0.3525 0.3211 0.2945 0.2717 1.2 0.3319 0. 2992 0.2719 0.2487 1.3 0.3122 0.2786 0.2507 0.2273 1.4 0.2934 0.2591 0.2309 G.2 075 1.5 0.2756 0.2408 0.2125 0.1892 1.6 0.2586 0.2236 0. 1954 0.1724 1.7 0.2426 0.20 75 0.1796 0.1570 1.8 0.2275 0.1924 0.1649 0.1429 1.9 0.2131 0. 17 84 0.1513 0.1299 2.0 0.1996 0. 1653 0.1388 0. 1181 2.1 0.1869 0.1530 0.1273 0.1073 2.2 0.1750 0. 1417 0.1166 0.0974 2.3 0.1637 0. 1311 0.1069 0.0885 2.4 0.1531 0.L2 13 0.0979 0.0803 2.5 0.1432 0.1122 0. 0896 0.0728 2.6 C.13 39 0.1037 0.C821 0.0661 2.7 0.1252 0.09 59 0.0751 0.0599 2.8 0.1170 0.0837 0.0687 0.0544 2.9 0.1093 0. 0820 0.0629 0.049 3 3.0 0.1022 0.07 57 0.0576 C.0447 3.1 0.0955 0.0700 0.0527 0.0405 3.2 0.0892 C.0647 0.0482 0.0367 3.3 0.0833 0.0598 0.0441 0.0333 3.4 0.0778 0.0552 0.0403 0.0302 3.5 0.0727 0.0510 0.0 369 0.0274 3.6 0.0679 0.0472 0.C338 0.0248 3.7 0.0635 0.0436 0.C309 0.0225 3.8 0.C593 0.0403 0.0283 0.0204 3.9 0.0554 0. 03 72 0.0259 0.0185 4.0 0.0518 0.0344 0.G2 37 0.0168 4. 1 0. 04 84 0.0318 0.0217 0.0153 4.2 0.0452 C.0294 0.0198 0.0138 4.3 0.0422 0.0272 0.0182 0. 0126 4.4 0.0395 0.0251 0.0166 0.0114 4.5 0.0369 C.02 32 0. 0152 0.0104 4.6 0.0 345 0.0215 0.C140 0.0094 4. 7 0.0323 0.0199 0.0128 0.0085 4.8 C.03C2 0. 01 84 0.0117 0.0078 4.9 0.0282 0.0170 0.C107 0.0071 5.0 0.02 64 0.0157 0.0098 0.0064 A-25 PROBABILITY OF FAILURE p «s/r^  3.25 3.50 3.75 4. 00 1.0 0.2773 0.2604 0.2454 0.2320 1.1 0 .2521 0.23 49 0. 2198 0.2065 1.2 0.22 88 0.21 16 0.1966 0.1834 1.3 0.2 0 74 0.1903 0.1756 0.1627 1.4 0.1878 0.1710 0.1566 0.1441 1.5 0.1698 0.1535 C.1395 0. 1275 1.6 0.1535 0. 1376 0.1242 0.1128 1.7 C.1386 0. 1233 0.1105 0.0996 1.8 0.1250 0.1104 0.C982 0.0879 1.9 0.1127 0. 0987 0.C872 0.0776 2.0 G.1016 0.08 8 3 0.0774 0.0684 2.1 0.0915 0.0789 0.0687 0.0603 2.2 0.0824 0.0705 0.C609 C. 05 31 2.3 0.0742 0.0630 0.0540 0.0468 2.4 0.0668 0. 05 6 3 0.0479 0.0412 2.5 0.0601 0.0502 0.C425 0.036 3 2.6 0.0541 0.044 8 0.0376 0.0319 2.7 0.G486 0.0400 0.C3 34 0.0281 2.8 0.0437 0.0357 0.0296 0.0247 2.9 0.0 3 93 0.0319 0.0262 0.0218 3.0 0.03 54 0.0284 0.0232 0.0192 3.1 0.Q318 0.0254 0.C206 0.0169 3.2 0.02 86 0.0227 0.0182 0.0149 3.3 0.0257 0.0202 0.0162 0.0131 3.4 0.0231 0. 01 80 0.0143 0.0115 3.5 0 .0208 0.0161 0.C127 C. O l d 3.6 0.0187 0.0144 0.0113 0.0089 3.7 0.0168 0. 012 8 0.0100 0.0079 3.8 0.0151 0.0115 C. C088 0.0069 3. 9 0.0136 0.0102 0.G078 0.0061 4.0 0.0123 0.0091 0.C070 0.0054 4. 1 0.0110 0.0082 0.C062 0.0048 4.2 0.0099 0.00 73 0.C055 0. 0042 4.3 0.0089 0.0065 0.0049 0.0037 4.4 0.0081 0.0058 0.0043 0.003 3 4.5 0.00 73 0.0052 O.C038 0.0029 4.6 0.0065 0.0047 0.0034 0.0026 4.7 0.00 59 G.0042 0.0030 0.0023 4.8 0.0053 0.0037 0.C027 0.0C20 4. 9 0.0048 0.00 33 0.0024 0.0018 5.0 0.0043 0.0030 0.0021 0.0016 A-26 (°VL) = O . I O G PROBABILITY OF FAILURE * s / L = 4 . 2 5 4 . 5 0 4 . 7 5 5 . 0 0 1.0 0 . 2 2 0 0 0 . 2 0 9 2 0 . 1 9 9 4 C . 1 9 0 5 1 . 1 0 . 1 9 4 6 0 . 1 8 3 9 0 . 1 7 4 3 0 . 1 6 5 6 1 . 2 0 . 1 7 1 8 C . 1 6 1 4 0 . 1 5 2 1 0 . 1 4 3 7 1 . 3 0 . 1 5 1 4 0 . 1 4 1 4 0 . 1 3 2 5 0 . 1 2 4 5 1 . 4 0 . 1 3 3 3 0 . 1 2 3 7 0 . 1 1 5 2 0 . 1 0 7 7 1 . 5 0 . 1 1 7 2 0 . 1 0 8 1 0 . 1 0 0 1 0 . C 9 3 1 1 . 6 0 . 1 0 2 9 0 . 0 9 4 4 0 . C 8 6 9 0 . 0 8 0 4 1 . 7 0 . C 9 C 3 0 . 0 8 2 3 0 . C 7 5 4 0 . 0 6 9 3 1 . 8 0 . 0 7 9 2 0 . 0 7 1 8 0 . 0 6 5 4 0 . 0 5 9 8 1 . 9 0 . 0 6 9 4 0 . 0 6 2 5 0 . G 5 6 6 0 . 0 5 1 5 2.0 0 . 0 6 0 8 0 . 0 5 4 5 0 . C 4 9 0 0 . 0 4 4 4 2 . 1 0 . 0 5 3 3 0 . 0 4 7 4 0 . 0 4 2 5 0 . 0 3 3 2 2 . 2 0 . 0 4 6 7 0 . 0 4 1 3 0 . 0 3 6 7 0 . 0 3 2 9 2 . 3 0 . 0 4 0 8 0 . 0 3 5 9 C . C 3 1 8 0 . 0 2 8 3 2 . 4 0 . 0 3 5 7 0 . 0 3 1 2 0 . 0 2 7 5 0 . 0 2 4 4 2 . 5 C . 0 3 1 3 0 . 0 2 7 2 0 . 0 2 3 8 0 . 0 2 1 0 2 . 6 0 . 0 2 7 4 0 . 0 2 3 6 0 . C 2 0 6 C . 0 1 8 0 2 . 7 0 . 0 2 3 9 0 . 0 2 0 6 0 . 0 1 7 8 0 . 0 1 5 5 2 . 8 0 . G 2 C 9 0 . 0 1 7 9 0 . 0 1 5 4 0 . 0 1 3 4 2 . 9 0 . C 1 8 3 C . 0 1 5 6 0 . C 1 3 3 0 . 0 1 1 5 3.0 0 . 0 1 6 0 0 . 0 1 3 5 0 . 0 1 1 5 0 . 0 0 9 9 3 . 1 0 . 0 1 4 0 0 . 0 1 1 8 0 . 0 1 0 0 0 . 0 0 8 5 3 . 2 0 . 0 1 2 3 0 . 0 1 0 2 0 . 0 0 8 6 0 . 0 0 7 3 3 . 3 0 . 0 1 0 7 0 . 0 0 8 9 0 . C 0 7 5 C . 0 0 6 3 3 . 4 0 . 0 0 9 4 0 . 0 0 7 8 0 . 0 0 6 5 0 . 0 0 5 4 3 . 5 0 . 0 0 8 2 C . 0 0 6 8 0 . 0 0 5 6 0 . 0 0 4 7 3 . 6 0 . 0 0 7 2 0 . 0 0 5 9 0 . C 0 4 9 C . 0 C 4 0 3 . 7 0 . 0 0 6 3 0 . 0 0 5 1 0 . 0 0 4 2 0 . 0 0 3 5 3 . 8 0 . 0 0 5 5 0 . 0 0 4 5 0 . 0 0 3 6 0 . 0 0 3 0 3 . 9 0 . 0 0 4 8 C . 0 0 3 9 0 . C 0 3 2 0 . 0 0 2 6 4 . 0 0 . 0 0 4 2 0 . 0 0 3 4 0 . 0 0 2 7 C . 0 Q 2 ? 4 . 1 0 . 0 0 3 7 0 . 0 0 3 0 0 . 0 0 2 4 0 . 0 0 1 9 4 . 2 0 . 0 0 3 3 0 . 0 0 2 6 0 . 0 0 2 1 0 . 0 G 1 7 4 . 3 0 . 0 0 2 9 0 . 0 0 2 3 0 . C 0 1 8 0 . 0 0 1 5 4 . 4 0 . 0 0 2 5 0 . 0 0 2 0 0 . 0 0 1 6 0 . 0 0 1 3 4 . 5 0 . 0 0 2 2 0 . 0 0 1 7 0 . 0 0 1 4 0 . 0 0 1 1 4 . 6 0 . 0 0 1 9 0 . 0 0 1 5 0 . C 0 1 2 0 . 0 0 0 9 4 . 7 0 . 0 0 1 7 0 . 0 0 1 3 0 . 0 0 1 0 0 . 0 0 0 8 4 . 8 0 . 0 0 1 5 0 . 0 0 1 2 0 . 0 0 0 9 0 . 0 0 0 7 4 . 9 0 . 0 0 1 3 C . O O I O 0 . C 0 0 8 0 . 0 0 0 6 5 . 0 0 . 0 0 1 2 0 . 0 0 0 9 0 . 0 0 0 7 0 . 0 G 0 5 APPENDIX VI Values of r vs. A-28 10.0 9 . 0 ao 7 0 o ^ 5 - 0 4 . 0 3 0 2 0 1.0 1 2 Figure VI-1 r(i+ 4/» [ rci+|) - r2<i+jb] 5 6 B E T A 8 9 10 Relation between the Parameter £ of the Material Strength D i s t r ibut ion and the Ratio Jt of the Mean Strength and the Standard Deviation APPENDIX VII TABLES FOR MEAN VALUE OF S.F. V A-30 P °<s/z= 1 . 2 5 I. 50 1 . 7 5 2 . 0 0 1. 00 1 . 2 5 0 5 1 .5 0 0 4 1 . 7 003 2 . 0 0 0 9 1. 1C 1 . 2 0 7 7 1 . 4 4 9 3 I . 6 9 0 7 1 . 9 3 2 4 1 . 2 0 1. . 164 9 1 . 3 9 33 1 . 6 3 1 0 1 . 0 6 4 -1 . 30 1 .1 5 2 a 1 . 382 7 .1 . 6.L 39 1 . 0 4 4 3 1. AO 1 . 1 4 0 8 1 . 3 6 7 2 1 .5 968 1 . 8 2 4 6 1 . 5 0 1 . 1 0 3 7 1 . 3 2 1 4 1 .54 44 1 . 762 7 1 . 6 0 1 . 0 6 6 3 1 . 2 7 5 7 1 . 4 9 1 9 1 . 7 0 0 0 1. 70 I . 0 5 5 7 1.2 0 4 5 1 .4 76 9 ] . 6 3 70 1 . 6 0 1 . 0 4 4 6 1 . 2 5 3 3 1 . 4 6 1 9 1 . 6 70 7 1 . 9 0 1 . J 4 5 1 1.2 542 1 . 4 6 3? 1 . 6 7 3 0 2. 00 1 . 0 4 5 7 1 . 2 5 5 2 1.4 64 6 1 . 6 7 3 0 2 . 10 1 . 0 4 6 0 1 . 2 5 8 1 I . 4 6 0 1 1 . 6 7 5 0 2 . 2 0 1.0 464 1 . 2 6 1 1 1 . '+717 1 .6 760 2 . 30 1 . 0 4 0 ) 1 . 2 6 2 2 1. 4 74 1 1. < 0:. 2 . 40 1 . 0 0 1. 3 1 . 2 6 3 3 1 , 4 7 7 0 1 . 6 0 5 1 2 . 5 G 1 .05 3 3 1 . 2 6 6 3 1 . 4 7 9 ? 1 . 6 9 0 9 2 . 6 0 1 . 0 5 6'-t 1 . 2 6 9 4 1 .-^8 03 1 . 6 9 6 0 2. 70 1 . 0 5 87 1 . 2 7 1 5 1 .48 35 1 . 6 9 79 2 . 8 0 1 . 0 6 1 1 1 . 2 73 7 I . 4 0 6 3 1 . 6 9 0 9 2 . 9 0 1 . 0 6 3 3 1 . 2 7 6 4 1 . 4 8 9 0 1 . 7 0 2 1 3. 00 1 . 0 6 5 5 1 . 2 7 9 2 1 . 4 9 1 f 1. 7 0 54 3 . 1 0 1 . 0 6 7 6 1 . 2 8 1 2 1.4 948 1 . 7 0 0 4 3 . 2 0 1 . 0 6 0 6 1 . 2 8 3 1 1 .4 9 7 9 1 . 7 1 1 4 3 . 3 0 1 . 0 6 7 3 1 . 2 8 0 0 1 . 4 9 5.1 1 . 7 0 79 3 . 4 0 1 . 0 6 5 0 1.2 7 7 0 1 . 4 9 2 4 1 . 7 0 44 3 . 5 0 L .0 66 7 1 . 2 7 9 3 L.4 9 5 5 1 . 7 0 8 1 3 . 6 0 1 . 0 5 8 5 1 . 2 8 1 7 I . 4 9 8 6 1 . 7 1 2 0 3. 70 I . 0 6 9 3 1 . 2 8 3 3 1 . 4 9 9 5 \ 1 . 7 1 3 0 3 . 8 0 1.0 712 1.2 8 5 9 1 . 5 0 0 5 1 . 7 1 5 1 3 . 9 0 1 .0 743 1 . 2 8 9 5 I . 50 ?•-) 1 . 7 1 0 1 4 . 00 1 . 0 7 70 1.2 933 1 • 50 5 3 1 . 7 2 1 2 4 . 1 0 1.0 734 1 . 2 9 4 3 I . 50 74 1 . 7 2 0 > 4 . 2 0 1 . 0 7 9 5 1 . 2 9 6 5 1.0 095 1 . 72 66 4 . 3 0 1 . 0 6 01 1 . 2 9 7 8 I . 5 1 1 2 1. 7209 4. 40 1 . 0 80 f 1 .2 991 I . 5 ! ? 9 1 . 7 3 1 3 4 . 5 0 i . ;o 34 .1. 3000 I.5166 . 1 . 70 3 4 . 60 1 . 0 8 6 L 1 . 3 0 1 0 1 . 5 204 . 1. 735 4 4 . 70 I . 0 8 3 4 1.. 302 7 1. . 0 04 i 1 . 7 39 0 4 . 8 0 1. 0 9 0 9 L . 3 044 \ . 5 0 79 1. . 74 3 2 4 . 9 0 l . 0 y ? i 1 . 3 1 1 6 1 . 0 3 30 1. 7 4 0 ? 5. 00 1 . 0 3 4 1 . 3 1 0 0 I . 0 3 0 5 1. 7 0 30 A-31 0 . 1 0 0 °<s/c = 2 . 2 5 2 . 5 0 2 . 7 5 3 . 0 0 1 . CO 2 . 2 4 1 9 2 . 4 8 8 6 2 . 7 3 9 0 2 . 9 8 5 6 1 . 10 2 . 1 2 6 9 2 . ' 3 6 1 0 .?. 5 9 8 ? 2 . 8 3 2 5 1 . 2 0 . 2 . 0 1 I 9 2 . 2 3 3 4 2 . 4 5 7 4 2 . 6 7 9 5 1. 3 0 1 . 9 / 3 6 2 . 1 9 4 ? ? . 4 1 ? 0 2 . 6 3 1 0 1 . 4 0 1 . 9 3 9 4 2 . 1 5 5 1 2 . 3 6 8 4 2 . 5 8 4 1 1 . 5 0 1 . 9 2 3 4 2 . 1 3 6 4 2 . 3 4 0 0 2 . 5 6 0 9 1 . 6 0 1 . 9 0 7 4 2 . 1 1 7 7 2 . 3 2 7 7 2 . 5 3 7 0 1 . 7 C 1 . 8 9 7 6 2 . 1 0 8 6 2 . 3 1 8 7 2 . 5 2 01 I . 8 0 1 . S 8 7 9 2 . 0 9 9 6 0 . 3 0 9 9 2 . 5 1 8 0 1 . 9 0 1 . 0 8 7 4 2 . 0 9 7 9 2 . 3 0 6 3 2 . 5 1 5 7 2 . CO 1 . 0 6 70 2 . 0 9 6 3 2 . 3 0 3 0 2 . 5 1 3 1 2 . 10 1 . 0 0 3 9 2 . 0 9 5 9 2 . 3 0 6 0 2 . 51 4 9 2 . 2 0 1 . 8 8 4 9 2 . 0 9 5 6 2 . 3 0 0 2 2 . 5 1 6 C 2 . 3 0 1 . 3 8 7 1 2 . 0 9 7 5 2 . 3 0 0 0 2 . 5 1 3 1. 2 . 4 C 1 . 8 0 9 3 2 . Q 9 9 4 2 . 3 0 9 4 2 . 5 1 9 4 2 . 5 0 1 . 8 9 2 7 2 . 1 0 3 5 2 . 3 1 3 ? 2 . 5 2 4 0 2 . 60 1 . 3 9 6 2 2 . 1 0 7 7 2 . 3 1 7 0 2 . 5 2 8 6 2 . 7 0 1 . 8 9 9 4 2 . 1 1 1 7 2 . 3 2 1 7 2 . 5 3 2 0 2 . 8 0 1 . 9 0 2 6 2 . 1 1 5 7 2 . 3 2 6 4 2 . 5 3 7 ! 2 . 90 1 . 9 0 6 7 2 . 1 1 9 3 2 . 3 3 0 0 2 . 5 4 2 2 3 . C O 1 . 9 1 0 0 2 . 1 2 3 0 2 . 3 3 5 1 2 . 5 4 7 0 3 . 10 1 . 9 1 4 5 2 . 1 2 9 0 2 . 3 3 9 0 2 . 5 5 1 9 3 . 2 0 1 . 9 1 8 3 2 . 1 3 5 0 2 . 3 4 3 0 2 . 5 5 6 5 3 . 3 0 1 . 9 2 1 7 2 . 1 3 6 2 2 . 3 4 7 6 1 . 2 7 5 0 3 . 4 C 1 . 9 2 5 1 2 . 1 3 7 5 2 . 3 5 2 3 0 . 0 0 0 1 3 . 5 0 1 . 9 2 01 2 . 1 4 1 1 2 . 3 5 5 ? 1 . 2 6 7 2 3 . 6 0 1 . 9 3 1 1 2 . 1 4 4 7 2 . 3 5 8 ? 2 . 5 7 4 4 3 . 7 0 1 . 9 3 3 7 2 . 1 4 7 8 2 . 3 6 1 9 2 . 5 7 7 0 3 . 8 0 1 . 9 3 6 3 2 . 1 5 1 0 2 . 3 6 5 7 2 . 5 0 0 0 3 . 9 0 1 . ^ 3 8 2 . 1 5 3 7 0 . 3 6 0 9 2 . 5 8 4 1 • 4 . CO 1 . 9 4 0 6 2 . 1 5 6 4 ? . 3 7 2 1 2 . 5 0 7 9 4 . 10 1 . 9 4 3 0 2 . 1 6 0 2 2 . 3 7 4 8 2 . 5 9 1 0 4 . 2 0 1 . 9 4 7 1 2 . 1 6 4 0 2 . 3 7 7 6 2 . 5 94 5 4 . 3 0 1 . 9 4 0 3 2 . 1 . 65 8 2. 3 0 1 0 2 . 5 9 72 4 . 4 0 1 . .^4 9 5 2 . 1 6 7 6 2 . 3 0 5 6 2 . 5 9 9 9 4 . 5 0 1 . 9 5<+? 2 . 1 7 3 0 2 . 3 ' H O ? . 6 0 6 0 4 . 6 0 1 . 9 6 :c 2 . 1 7 0 5 2 . 39 7 6 2 . 6 1. ? 4 4 . 7 0 1 . 9 6 72 2 . 1 3 4 0 2 . 4 02 9 2 . 6 2 2 4 4 . 0 0 1 . 9 7 4 5 2 . 1 8 9 4 2 . 4 0 0 3 .? . 6 3 ? 0 4 . 9 0 1 . 9 7 4 3 2 . 1 9 1 5 2 . 4 1 3 0 2 . 6 3 5 3 5 . CO 1 . 9 7 4 1 2 . 1 9 3 7 2 . 4 1 9 3 2 . 6 3 9 ] A-32 = 3.2 :> 3 .5 0 3 .7 5 4 .00 1.00 3.2358 3.404 4 3. 7 3 2.3 .5 .9307 1.10 3.07 Ob 3 .3055 3.5^06 0.7 766 1.2 0 2.9054 3.1267 3 . 348Q 3. 5 72 6 1.30 2 . 3 5 2 6 3.0 704 3.2 0 93 3 . 50 8 9 I .40 2 . 7998 3.0141 }. .22.97 3.4452 1. 50 2 . 7 73*3 2.9832 3.00 12 3.4 1 54 1. 6G 2. 74 RO 2.962 5 3.172 6 3.385 7 1 .70 2.7 399 2.9514 5. 10 93 .3 . 0 7 2 2 1.80 2.7313 2.9 40 3 3. 14 71 3.3500 1. 90 2 . 72 8 9 2.9360 3.1440 3 . 3 5 4 5 2. CC 2.7261 2.9317 3.1410 3 . .3 5 0 .3 2.10 2.7208 2 .9 3 39 .3.14 10 3. 3 0.2 7 2. 20 2 . 72 7 5 2 .936 1 3. 14 2 7 3.3 5 52 ,2. 3G 2.7 30 6 2.9399 .3. 1402 3 .357 "3 2 .40 2.7 33 3 2 .9438 3 . 1. 0 38 3.3095 2. 50 2.73 6 9 .?. . 9466 3. 159 7 .3. .30 71 2.60 2.740 0 2 .94 93 3 . 10 5 0 3.3 74 0 2 .70 2.74 51 2.9551 '3. 16 06 3.3 786 2 . 80 2 .75 92 2 .9609 3 . 1716 3. 3 2 4 2. 90 2 . 75 4 8 2.966 3 3.17 77 0 . .3 0 91 3 . G 0 2.7 594 2 . 9 7 16 3. 1 0 3 8 3 . 3 • i 5 9 3.10 2.7535 2 .9 764 3. 18 00 3.40 00 3. 20 2.76 77 2 . 9 H1 2 3.19/2 3.4030 3. 3 0 2.7736 2.9366 3.19 95 3.4124 3 .40 2.779 6 2 . ) 9 1 9 0 . 2 0 0 8 3.4191 3. 50 2 . 78.3 7 2 .9966 3.21 ? .2 3. 4 0 0 1 3. 60 2.7879 3.0014 3.2176 3. 4 31 1 3 . 70 2.792H 3.00 69 3.2223 3.4 3 64 3.80 2.7 977 3.0 124 3.2271 3.4^10 3. 90 2 .80 0 7 3.0159 3 .2.3 11. 3.4464 4 . C C ' 2.8037 3. 0 1 9 5 3 . 2 3 50 3 . 4 0 I 0 4. 10 2 . 0 0 7 5 3.0 2 39 3. >'••:> 3.4 540 4. 20 2.3114 3.0233 i. -'4 51 3 . 4 0 0 7 4. 3 0 2 . 8 1 4 /• .3 . 0 321 3 . 2 '+ 7 7 .3. 4 o 5 0 4 .40 2.8130 3 .0060 .3. ••• 3.46 0 0 4. 50 2.0270 3.044 9 3.2619 3 . 4 7 0 3 4. 60 2 . 3 3 0 J 3.0530 ' . 0 7 0 4 3 .4000 4 .70 2 . 841 6 3.0 60 1 . 2 . 2 7 0 3.4'-.'6 0 4.80 2 .3473 3 .0 6 65 1 . 00 00 .3 . 0 0- 5 4. 9C 2. 3506 i.0701 3.0 097 0.0120 5. CC 2.850 0 3.0 736 3 . 0 '5 0 ;3 .0 . 5 1 89 A-33 = 0 . LOO = 4.25 4.50 4.75 5.00 1 .00 •'t .22 07 4.4769 4.7 2 47 4.972 7 1. IC 4. 01 1 3 4.2476 4 . 4 P ? 1 4.71 F9 1.2 0 3.79 0 9 4.0183 4. 2 3 94 4.465 I 1 .30 i. 72 61 3.9462 4.1645 4.3851 1 .40 3 .65 8 5 3 .0741 4 .00'")6 4 . 30 5 1. 1. 50 3. 62 71 3.8421 4.0549 4 .2 60], 1 .60 3.5958 3.8102 4.02 03 4.2 3 32 1 . 70 3.5624 3.7938 4.0 0 56 4.2162 1. 80 3. 56 0 9 3 . 7 775 8.990 9 4.190J 1 . 90 3 . 566 1 3.7750 3. 9 86:3 4.195 2 2 .CO 3.5630 3 .7726 3.9318 4.1911 2. 10 3 . 563 0 3.7735 3.9024 4.192 4 2. 20 3.56 30 3.7 745 3 .9031 4.19 3 7 2 .30 3.5 66 6 3.7 7 70 3.9 862 4 . 1 9 R 7 2 .40 3 .56 95 3 . 7795 3.0 895 4 . .2 0 3 8 2. 50 3.5755 3.7 863 3.9959 4 .20 89 2.60 3.5017 3.7932 4.0r?5 4.2139 2. 70 3.5910 3 .7996 4.00^7 4.2215 2. 80 3.6004 3.8062 4 . 0 16*> . 4.2 299 2.90 3.6o4 3 3.8133 4. 0274 4.2 37 3 3 .00 3. 60 82 3.3204 4.9 3 79 4. 2 44 7 3. 10 3.6150 3.8279 4 .0 42.2 4. 255 2 3.2 0 3.6218 3.8354 4.04 05 4.2650 3 .30 3. 62 7 9 3 . 8421 4.0 5 38 4.2 703 3. 40 3.03 4 0 3 .8 489 4 . 0 6 13 4.2 76 1 3. 50 3.6400 3.8548 4.0 6 78 4.2833 3.60 3.647 3 3.8608 4.0 744 4.2 900 3 .70 3.651 8 3 .8659 4.08 14 4 . 2 5 0 1. 3. 80 3.6 5 64 8.8711 4.0 8 85 4 . 0 "I 8 3 • 9 0 3.6616 3.8 768 4.0 98 4- 4.3100 4 .00 3 . 6 u 6 8 3.8 82 6 4.0003 4 . 3 14 1 4.10 3.6712 3 . 8 8 7 5 4.10 34 4. 3 2 0,? 4.20 3.6 75 6 3.852 5 4. 1094 4 . 3 2 6 3 4. 30 3.6 791 3.8966 4.114 1 4 . 32 9 7 4.40 3.6827 3 .9003 4.1188 4 . "3 0 3 ! 4. 50 3 . fc 9 6 5 3 . 9.1 2 9 4 . 1 3 1 0 4.34 7 0 4.60 3.71 0 4 8.92 5 0 4. 1448 4 . 3 6 ?. 6 4.70 3.7174 .3 .9 346 4.1040 0.3 700 4. 80 3. 72 4 5 3 .9443 4 . 1 0 <4 4. 3 7 0 1 4. 9 0 3.731 6 3.94 8 8 4.16 82 4 . 3 0 5 5 5 .00 3. 7388 3. 9 5 34 4. 1731. 4.3930 APPENDIX VIII Numerical Example In th i s appendix, the three design models developed are i l l u s -trated with the help of numerical examples. For the design of a c r i t i c a l component, the maximum load p.d.f. i s given by E . -V £ / ^ A y model and the minimum material strength d i s t r i bu t i on is considered as E -V . m . As the loads are assumed to be purely a x i a l , for a tens i l e f a i l u r e by separation the relevant material property w i l l be the ultimate ten s i l e strength. However, in the case of a d i f fe rent nature of the load or the f a i l u r e c r i t e r i o n , some other appropriate material property may be con-sidered. I f the e f fec t i ve load acting on the component i s a combination of several loads such as ax ia l thrust, tors ion and bending loads, e t c . , the same procedure can be used by obtaining the p.d.f. of the e f fec t i ve load and the values of i t s parameter.- In the example given here, values of various parameters of the maximum load ( L , ) and the minimum material strength ( o<s^ p ) are assumed to be ava i lab le to the designer, as a resu l t of previous s t a t i s t i c a l analys is . The values of various parameters of the decision theory are also assumed to be ava i lab le to the designer as inputs to the design problem. A-35 Conventional Approach P robab i l i s t i c Approach Model 1 Given: ( i ) Load L c = 45,000 lbs. Given: ( i ) Maximum Load E . V . r ^ a x ( t > L ) ( i i ) Strength of material M L = 45,000 l b s . , °<i_= 4,500 lbs . = 30,000 p . s . i . ( i i ) Minimum Material Strength Design: Select a value of the o(s = 30,000 p . s . i . , p = 4.0 safety factor , ( i i i ) Desired r e l i a b i l i t y say = 4.0. R = 0.995 Then, S 0 = i ; 0 L 0 = 180,000 lbs . Des i gn: The area of cross section A w i l l be: A = 180,000 30,000 Probab i l i ty of f a i l u r e Rf = 1-0.995 = 0.005 = 6.0 sq. inches = 0.10 This design with area of cross - P = 4.0 section of 6,0 sq. inches is con- refer r ing to the tables in sidered to have the r e a d a b i l i t y of Appendix V and p lo t t i ng the curve 1.0, i e . the f a i l u r e is impossible. as shown in f i g . (V I I I -1) + ( ° V t ) * - 4.05 for Ff =0.005 Using eq., (2.4), A.30,000 = 4 0 5 = ( « s / ~ < * 45,000 4 > U 3 S. s / Q Hence, A = 6.075 sq. inch. Thus a cross section area of 6.075 sq. inch for the design w i l l provide a r e l i a b i l i t y of 99.5 p.c. A-36 for the given load d i s t r i bu t i on and the material strength d i s t r i bu t i on over a time period for which the value of L is given. Model 2 Let there be another material For the second mater ia l , l e t with given strength = 27,000 p . s . i . <Ks = 27,000 p . s . i . for the same load and safety factor £ = 5.0 value. The cross section area Thenfor R = 0.995, using i s given as, table 5 in Appendix V and f i g . A., = 180,000 = 6.667 sq. in 27,000 and A < C A ± Let the density of the f i r s t (VIII.1), + C* S/L)*= 3.125 and A , = 3.125 x 45.0 27.0 material f = 8 Ibs./sq. f t . / unit length and for the second material l e t the density f \ =7.5 lbs./sq. f t . unit length = 5.208 sq. inch and so, A^ <C A Comparing the mean strength values of the two materials ava i l ab le , If the design c r i t e r i on is minimum M ( S ) = 30,000 r (1.25) weight, = 27,192 p . s i . \N = 8.0 x 6.0 = 0.334 lbs ./ unit length, M<(S) = 27,000 T (1.20) = 24,800 p . s i . W± = 7.5 x 6.667 = 0.5 lbs ./ 144 unit length, Thus, i t can be seen that though as K|< (A/^ > the f i r s t material is selected for the design with the mean value of the minimum strength has decreased in the second case, the cross section area = 6.0 sq. inch. value of the cross section area, for A-37 the same load d i s t r i bu t i on and the same r e l i a b i l i t y of 99.5 p . c , has reduced from 6.075 sq. inch to 5,208 sq. inch. The reason is that the standard deviation is reduced even more s i g n i f i c an t l y from 7629 p . s . i . for the f i r s t material to 5670 p . s . i . for the second mater ia l . Assuming the same values of the density of two mater ia l s , as given in the conventional approach, W = 8.Q x 6.075 = 0.3375 lbs./ 144 unit length, W 1 = 7.5 x 5.208 = 0.2712 lbs ./ 144 unit length, and the design c r i t e r i o n being minimum weight, the second material with a cross section area of 5.208 sq. inch is selected for the design. + Using tables given in Appendix V, for f ixed values of parameters °Vt and £ , a set of probab i l i ty of f a i l u r e values i s ava i lable corresponding to d i f fe rent values of the parameter (see the values underlined in Appendix V for O^wt) = 0.10 and p = 4.0). When these values of % are plotted against the values of Oxs/t) , curve as shown in f i g . (VIII-1) is obtained. Using this curve, then,for a par t i cu la r speci f ied value of Pf , a required value of the th i rd para-meter is obtained as C^s/t)*". A-38 Model 3: If a se lect ion between these two given materials i s to be made on the basis of the economic criterion of minimum cost, the addit ional information required consits of the values of various decision parameters. These values are obtained with the help of actual data and the appl icat ion of the decision theory. In this numerical example, these values are assumed to be ava i lab le for the purpose of i l l u s t r a t i n g the design process. Let Vj_ = $1000.00 = 4.0 = 5 $/lb. km-\* = 3.0 CQ_ = 7 $/lb. k. m x = 7.0 For the f i r s t mater ia l , A = 6.075 sq. inch 1 = 3.565 (Appendix VI) f = 8.0 lbs./sq. f t .Vun i t length V = 3.50 (Appendix VII and f i g . (VIII.2)) Using eq. (3.15), the tota l expected cost of the design = 1000 4.0 + 2 x 3.565 x 6.075 x 8 - 1000 x 3.50 1.5 4 144 1.5 = 333.34 + 0.60 = 334.04 do l la r s In the case of the second mater ia l , \ = 5.208 sq. inch ^1= 4.366 P i = 7.5 lbs/ sq. f t . / u n i t length VL = 2.7375 (from Appedix VII and f i g . (VIII.2)) A-39 The tota l expected cost of design = 1000 x 4.0 + 2 x 4.366 x 5.208 x 7.5 - 1000 x 2.7375 1.5 4 144 1.5 = 845.00 + 0.592 = 845.59 do l l a r s . As the to ta l expected cost of the design i s higher in the case of the second mater ia l , the f i r s t material with cross section area of 6.075 sq. inch is selected for the design to minimise the cost of the design. By comparing model 2 and model 3 for the p robab i l i s t i c approach, i t may be seen that while the second material is the optimum choice in model 2, on the basis of the weight c r i t e r i o n , the f i r s t material should be selected i f the design c r i t e r i o n has to be minimum cost, using model 3. These examples i l l u s t r a t e that the chosen decision c r i t e r i a as well as the input data for the load and the material strength d i s t r ibut ions influence the optimum decision for the design. Thus, not only does the p robab i l i s t i c approach to design quantify the r i s k of the design, but i t also allows to optimize the decision e x p l i c i t l y w. r . t . a relevant decision c r i t e r i o n . This is in contrast to the conventional approach to design where the r i s k and relevant decision c r i t e r i a are ignored by designing to a f ixed value of the safety factor . ** Using the tables given in Appendix VII, for f ixed values of parameters <°<L./tL) and p> , set of mean strength values is ava i lab le corresponding to d i f fe rent values of the parameterC^s/t) (see the values underlined in Appendix VII for C^u/t) = 0.10 and £ = 4.0). When these values of U are plotted against the values of parameter Q^s/t) , a curve as shown in f i g . VI11-2 i s obtained for a par t i cu la r set of values of (^L/t) and . Using this curve, then, for a par t i cu la r value of the th i rd parameter C^S/L) ( i e . C ^ s / t ) * ) , a corresponding value of 7J can be obtained. •04i 0 I ' I , I U J 2.5 aO 3-5 4.0 425 C.S.F VQ Figure VIII-1 - P robab i l i t y of Fa i lure F% vs. Central Safety Factor l)Q = O^S/L.) Figure VII1-2 Mean Strength Value V vs. Central Safety Factor )J0 — 

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