OPTIMUM PRESSURE VESSEL ESICN BASED OM FRACTURE MECHANICS AND RELIABILITY CRITERIA

Ewald Heer, et a1

California Institute of Technology Pasadena, California

February 1978

DISTRIBUTED BY:

National Technical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road, Springfield Va. 22151 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Technical Memorandum 33- 470

Optimum Presswe Vessel Design Based on Fracture Mechanics and Reliability Criteria

Ewald Heer Jann-Nan Yang

/-

LOION LAB CADIFORNsA INSTITUTE OF TECHNOLOGY

PASADENA, CALlFORNlA

February 1, 1970 Prepared Under Contract No. NAS 7-100 National Aeronautics and Space Administration PREFACE

The work described in this report was performed by the Engineering Mechanics Division of the Jet Propulsion Laboratory.

JpL Technical Memorandum 33-470 iii ACKNOWLEDGMENT

This study was partly supported by the National Research Council, under whose sponsorship Dr. Jann-Nan Yang held a Resident Research Associateship at the Jet Propulsion Laboratory. The authors most gratefully acknowledge the valuable comments by Professor M. Shinozuka and Mr. J. C. Lewis during the course of this investigation.

iv JPL Technical Memorandum 33-470 CONTENTS

I. Introduction ...... 1 11. Fracture Me chanic s C oncepts ...... 4 UI. Statistical Aspects of Fracture...... 7 IV. Time Effects...... 11 V. Probability of Failure ...... 13

VI. Optimization...... 15 VII. Numerical Example ...... 19 VIII.- Discussion and Conclusion...... 24 References ...... 27

Table 1. Optimum design of pressure vessel...... 22

Figures 1. Typical load and strength distributions...... 29 2. Elliptically shaped flaws ...... 29

3. Flaw shape parameter curves for surface and internal flaws ...... 30

4. Stress intensity magnification factors for deep surface flaws ...... 30 5. Fracture toughness of annealed 6A1-4V titanium plate at various test temperatures ...... 31 6. Fracture toughness variation with material thickness...... 31 7. Applied stress vs flaw size, for plane strain ...... 32 8. Operational life curves, lower bound, cyclic load ...... 32 9. Operational life curves, lower bound, sustained load ...... 33 10. Typical schematic design case ...... 33

11. Relative expected cost as a function of proof-load level for varying values of p ...... 34

12. Relative expected cost as a function of optimum proof-load level for varying values of p ...... 34

JPL Technical Memorandum 33-470 V ABSTRACT

Spacecra€t structural systems and subsystems are subjected to a number of qualification tests in which the proof loads are chosen at some level above the simulated loads expected during the space mission. Assuming fracture as prime failure mechanism, and alluwing for time effects due to cyclic and sustained loadings, this paper treats an optimization method in which the statistical variability of loads and material properties are taken into account, and in which the proof load level is used as an additional design variable. In the optimization process, the structural weight is the objective function while the total expected cost due to coupon testing for material characterization, due to failure during proof testing, and due to mission degradation is a constraint. Numerical results indicate that for a given expected cost constraint, substantial weight savings and improvements of reliability can be realiz-ed by proof testing.

vi JPL Technical Memorandum 33-470 I. INTRODUCTION

Structural -design is the sizing and synthesiz- desire to set the probability of failure as low as ing of structural elements in space for -certain possible, i. e., to set the structural strength as intended purposes. Although the purposeful geo- high as possible, is unfortunately countered by the metrical arrangement of the structural elements necessity of reducing the structural weight to a can be accomplished with adequate, better, or minimum and/or by the necessity of keeping costs optimum results, in structural optimization it is within tolerable limits. It follows that an increase usually assumed that the geometrical configura- in structural reliability (probability of structural tion of the structure is given as constant. It is survival) must usually be paid for by an increase also assumed that the design and optimization are in weight, or cost, or both. performed with the sizes of the individual struc- Once the structural is designed and built, the tural elements being used as generalized coordi- question then arises concerning the reliability of nates. These coordinates are varied to deter- the particular structure. The answer to this ques- mine, within specified constraints, those values tion is not simple. The most careful analysis and that will yield an extremism of a given objective design before building the structure, taking into function. account the statistical variabilities of material The objective function to be extremized may properties, loads, and other measurable param- be structural weight, cost, reliability, etc. In eters, become meaningless when, during the many cases, it is necessary to determine how manufacturing process, an error is committed weak the structural elements can be designed with- that was not allowed for in the design. In a sta- out resulting in too many failures. Indetermjning Yistical sense, such errors could be allowed for in this weakness, one should recognize that struc- the design procedure if estimates of their proba- tural loads and structural strength properties are bility of occurrences and severity are included in satistical variables and, therefore, it is not pos- the reliability evaluation of the structure. How- sible to eliminate &l structural failures. There ever, for structural systems involving new proc- will always be a finite probability of structural esses and techniques, such estimates are by failure, and the most one can hope for is to limit necessity very vague because of the lack of infor- this probability to a tolerable value. Thus, the mation on which to base them.

JPL Technical Memorandum 33-470 1 Therefore, one is left with two essential con- that the choice of the proof-load test level consid- tributions to the statistical strength variability of erably influences the reliability of the structure. the structure: (1) the statistical variability of In the associated optimization problem, with measurable quantities, such as material proper- weight as the objective function and total expected ties, dimensions, etc., and (2) the statistical var- cost as a constraint, Ref. 2 indicates that an opti- iability of chance events, such as gross errors mum proof-load test level can usually be obtained during design, gross errors during fabrication, with an increase of reliability and a decrease in etc. In this report, only the statistical strength weight as compared to conventional optimum de- variability due to measurable quantities is con- signs in which the proof load is not considered in sidered. It can be shown later that as far as the reliability evaluation. In Ref. 2, the expected structural reliability is concerned this considera- cost consisted of the expected cost of mission tion is on the safe side. Additional information, failure due to structural failure plus the expected which may become available regarding chance cost of failed subsystems (or components) during events, can be incorporated in the structural proof - load testing. strength distributions without causing a change in Frequently the optimum proof-load test level the basic approach as set forth in this report. is considerably different from the mean struc- Spacecraft structural systems and subsystems tural strength. Since the probability density func- are subjected to a number of qualification tests in tion of the structural strength is usually only known which the structural components must withstand with a certain degree of confidence close to the specified environments (proof loads) that are in- mean strength, it would be necessary, in such tended to simulate the various types of induced cases, to establish, through extensive specimen stresses at some- level above those expected dur- testing, strength density function at points far ing the space mission. The purpose of these apart from its mean. This effort can add sub- proof-load tests is to eliminate those structures or stantial additional costs to the overall evaluation structural components that are too weak. process if the strength density function is not available. It is clear that after the structural system or subsystem has passed the proof-load tests, its In Ref. 3, this additional cost item has been statistical strength characteristics are radically allowed for by the fact that a linear increase of changed. For instance, assuming negligible time testing cost was assumed with an increase of the effects during proof-stress application, the sta- difference between mean strength and proof- load tistical strength distribution, at the particular test level. The effect of this additional cost is an point under consideration in the structure after the increasing tendency to reduce the difference be- test, will be truncated at the lower end up to the tween mean strength and proof-load test level as proof-stress level. A similar statement is true the additional testing cost is increased. for the proof load applied to a structure and the In this report, which includes the essential resulting truncated overall structural strength aspects of Ref. 4, an approach similar to that in distribution. This truncation eliminates, through Ref. 3 is taken in the optimization process. This the proof test, preciselythat portion of the strength approach also includes in the expected cost con- distribution that is least known and that has the straint, the cost of establishing the truncated greatest interaction with the applied load distribu- strength distribution function by specimen testing. tion (Fig. 1). Reference 1 contains some impli- Fracture mechanics dealing with essentially brit- cations of screening out weak elements a ::‘I tle fracture has been chosen here as an area of proof test when the proof- stress distributions are application, since fracture mechanics design, as similar, dissimilar, and identical to the applied- compared to other structural design approaches load induced stresses; Ref. 1 also includes some in which considerable bulk yield occurs before simple aspects of optimum designs. failure, tends to be more susceptible to statistical Reference 2 introduces the proof-load test . strength variations. The wide scatter of the re- level as a design parameter. It is shown in Fig. 1 sults of tests leading to failure is assumed to be

2 JPL Technical Memorandum 33-470 an inherent property of the material and is treated In spacecraft structural design, fracture me- as such for the purposes of this report, which chanics concepts can be most readily and meaning- concerns the investigation of trends of behavior fully applied to pressure vessel subsystems. This rather than absolutes. A detailed discussion of report is mainly concerned with pressure vessel various probability models for the characteriza- design optimization, although the concepts put forth tion of material properties and their implications are equally applicable in other areas of design. As with respect to specimen size is given in Ref. 5. will become apparent from this investigation, this This report also considers volume effects and time approach tends to become more meaningful as the effects, such as fatigue due to cyclic loading and systems, subsystems, and components become less flaw growth due to sustained loading. expensive relative to the overall project cost.

JPL Technical Memorandum 33-470 3 II. FRACTURE MECHANICS CONCEPTS

0 Fracture mechanics treats failures that occur the authors of Refs. 9 and 10, have extendedthese because of the presence of existing flaws in the investigations to include (1)the effects of the prox- material. The stress intensifications that are imity of flaws to the material surface, (2) the induced at the edges of a flaw correspondingly in- effects of sustained loading, and (3) the effects of crease locally the stored energy. If this stored cyclic loading. Mainly as a result of these works, energy just ahead of the flaw tip becomes larger fracture mechanics can be put on a quantitative than the energy needed to create the new surfaces basis by the following equation: resulting from an extension of the flaw tip, then the flaw increases until energy balance has been restored (Ref. 6). If the applied stress is high enough, or if additional energy in some other form is applied, the energy balance at the flaw tip will not be restored and catastrophic failure will occur. Hence, for a given flaw size, an applied critical where a is the minor semiaxis of an elliptically stress exists that causes rapid propagation of the shaped flaw at onset of rapid fracture as shown in flaw without further increase of the applied stress. Fig. 2. Figure 3 shows that Q is a flaw shape In ductile materials that generally have small ini- parameter that depends on the ratio of applied tial flaw sizes, this critical stress approaches the stress to yield stress. The subscript cr denotes material yield strength, while in brittle materials the critical state at the outset of rapid fracture. the critical stresses are usually less thanthe yield On the right side of Eq. Il), c is ~ numericalcor- strength of the material. rection factor that varies with the location of the The original work in fracture mechanics is flaw with respect to the free surface. For surface associated with the name of Griffith, who solved flaws, c = 0.83 and for internal (completely em- the problem of a flaw in an elastic material (Ref. 7). bedded) flaws, c = 1. 00 (Fig. 2). The factor Mk Griffith's theory has been generalized by Irwin indicates the effect of flaw size compared to that (Ref. 8) to include plastic deformations at the of material thickness. Figure 4 shows typical crack tip. Recently, a number of authors, e. g., curves for Mk and experimentalvalues as functions

4 JPL Technical Memorandum 33-470 of the ratio of flaw size to material thickness h. thereof. These are time-dependent cases in The fracture toughness KI~indicates the capability which not only the level of applied stress is of of a material to resist fracture in the presence of importance but also the duration of stress or the a flaw under an applied tensile stress R (resisting number of applied stress cycles. Such cases in- strength) perpendicular to the flaw plane; it is volve subcritical flaw growth; i. e., the slow sometimes called the critical stress intensity fac- growth of a flaw until it reaches its critical value tor. The subscript I refers to the opening mode and catastrophic fracture occurs. Subcritical of fracture for plane strain conditions. The value flaw growth is conveniently indicated by curvns of of Klc generally depends upon such factors as heat KI/KI~ versus time or cycles of loading to failure treatment, temperature, etc. Figure 5 gives a that are determined from tests on preflawed spe- typical curve and some experimental values of KIc cimens by a change in both the initial flaw size and as a function of temperature. The fracture tough- the applied stress. Figure 8 shows typical curves ness is higher for the plane stress fracture mode for cyclic loading, and Fig. 9 gives typical curves than for the plane strain fracture mode, with some for sustained loading. transition values between these two modes. Fig- The curves for sustained stress applications ure 6 shows a typical variation of the fracture have characteristic trends showing that for a given toughness with thickness h, where Kc is the frac- material and environmental condition practically ture toughness referring to the opening mode of no flaw growth occurs until a certain threshold fracture for plane stress conditions. ratio of K/KIc is reached. In Fig. 9, this thresh- If the applied stress S perpendicular to the old is approximately 0.9 for titanium and 0.75 for plane of a flaw is less than the critical fracture aluminum in a liquid nitrogen environment. stress Eq. has the form R, (i) The effects of corrosive and oxidizing envi- ronments vary considerably for different mate- rials, but they usually lower the threshold stress 2 a intensity values. Temperature also affects the breakdown process in a number of ways. For in- stance, as shown in Fig. 5, KI~depends on tem- perature, and corrosion and oxidation are usually where K is any applied noncritical plane strain accelerated with increasing temperature. I intensity factor. Figure 7 shows schematically A typical schematic design case is shown in the applied stress as a function of the flaw size Fig. 10. It is assumed that the loading history is parameter a/Q along constant lines of K and KI. IC as shown in Fig. loa, in which a proof load is Equations (1) and (2) are the basic equations initially applied at time TA, then a cyclic load is describing the relationship between :rack size and applied between times Tg and Tc, and then a sus- applied stress for the simplest case; i. e. , that of tained load between times TC and TD. Also, it is a single, short-time stress application at a con- assumed that the maximum initial embedded flaw stant temperature in an inert environment. The size parameter is (ai/Qi). The proof load, being design of structures requires additional consider- of short duration, has a negligible effect on sub- ations beyond these. Environmental effects such critical flaw growth. The cyclic and sustained as temperature, cyclic loading, and corrosionhave loads, however, determine the maximum allowable a pronounced influence on the fracture character- operating load as indicated by the arrows in Fig. istics of structures. Many materials fracture not lob. During cyclic loading, the flaw increases only when a certain critical stress is reached but from point @to point @ (Fig. lob), and under also at relatively low stresses after being subjected sustained loading it increases from point @ to for a certain time to sustained loading, or after a point @ when the critical flaw size for the applied certain number of load cycles, or any combination operating load is reached and fracture takes place.

JPL Technical Memorandum 33-470 5 This design process requires knowledge of the The preceding discussion indicates that to largest initial embedded flaw size, which is usually fully characterize the material properties, all the not available. By proof loading the structure, it is possible environmental and loading conditions established that the largest initial flaw is not must be taken into account and extensive experi- larger than that indicated by point @ in Fig. 10; mental data must be generated. This is not only however, no knowledge is provided as to how much time-consuming; it can also become intolerably smaller it might be. expensive.

6 JPL Technical Memorandum 33-470 111. STATISTICAL ASPECTS OF FRACTURE

It is recognized that the strength of material is a random variable. In fact, the statistical properties of material strength were investigated extensively (Refs, 11-15) under one-dimensional in which R. and S. are, respectively, the resisting stress-field applications. In fracture mechanics, 1 J the major reason for the statistical variation of the stress and applied stress normal to the plane of the flaw of size parameter a. = (ala). contained material strength is attributed to the statistical J J in the jth volume element V variation of the embedded flaw sizes that appear j' to be inherent characteristics of the material. It Since a. has by far the largest statistical J is mainly due to this fact that the strength of the dispersion as compared to the statistical disper- material is far below the theoretical value com- sion of the other parameters in Eqs. (3) and (4), puted from atomic bond considerations. it is assumed that Ac and A are deterministic constants Without essential loss of generality, a two- . dimensional stress field (plane stress), as com- Under the weakest-link hypothesis, which monly associated with the stresses in thin-walled states that failure of the structure occurs when pressure vessels and other thin-walled structures, any one of the material elements is subjected to is assumed here for the derivation of the statisti- its critical stress, the statistical distribution of cal distribution of the resisting strength of the the resisting strength of the structure can be de- structure. rived from that of Rj, whereas the statistical dis- tribution of Rj can be determined from that of aj Let the structure be divided into small mate- by a transformation of Eq. The present state rial volume elements, each containing one flaw. (3). Equations (1) and (2) are then written as of technology does not allow, in most cases, the direct measurement of the distribution of a. J' Therefore, the distribution of Rj is determined KIc -112 - 112 from results of uniaxial tensile tests of specimens, R. = (+) - a. =Aa (3) J Mk cj henceforth referred to as coupon tests.

JPL Technical Memorandum 33-470 7 Let S and R be, respectively, the applied load For uniaxial tensile tests (coupon tests) in to and the resisting strength of the structure, and which +. = 1 and +.2 = 0, Eq. (8) yields 11 1 let S+. and S+ be the analyzed principal stresses 11 j2 at V. due to the application of S. Here, + and 1 jl +j2 are functions of the spatial coordinates and stiffness properties defining the structure. If 8. J is the angle between the plane of the flaw contained in V. and theprincipal stress S+.l, the applied where Fk (x) is the distribution function of the 1 3 ju stress normal to the plane of the flaw is then uniaxial tensile strength for the jth volume ele- ment, and

S. = S+ cos2 e. t S+ sin2 e (5) J jl I j2 j R. R =+ j‘ cos It is reasonable to assume that the a 9 j’ j=1,2,3, ***, m are statistically independent and identically distributed and that the angles e. From experience, e.g., Refs. 5 and 11-15, J’ the distribution function FR (x) of the uniaxial j=1,2,3, *.* , m are also statistically independent and uniformly distributed between 0 and R/2. Thus, tensile strength Rc of the coupon specimen with volume Vc can be represented by the Weibull distribution 2 a fg (x) = fe(x) = a’ 05x5- (6) - 2 j - =o other wise in which f (x) is the probability density function e in which x xo and k are parameters characteriz- of e. EL’ ing the particular material and u is the unit

The probability of failure of the entire struc- volume e ture due to a deterministic applied load S is The distribution function of the uniaxial fen- sile strength R of the jth volume element fol- ju lows from Eq. (11) and Refs. 1,5, and 10-12,

where S. is the applied stress due to S normal to 3 the plane of a. and P[E] is the probability of J occurrence of the event E. Equation (7) simply With the aid of Eqs. (9), (lo), and (12), Eq. states that the survival of the structure implies (8) yields the survival of each volume element.

The unconditional probability of the event [~jI sj]for given e. = P[R~ I sjlej = SI, J 4, follows from Eq. (5),

The unconditional probability can be obtained 5 S+. cos 2 9 t S+. sin2 (8) 11 12 $1 from Eqs. (6) and (13) as follows:

2 2 (+jl t +j2 tan + 2 PR.- . =1-- S(+jl t ‘pj2 tan S) 2 x (14) [J<’I] xO EL

8 JPL Technical Memorandum 33-470 The distribution function cf the structural strength F (S) follows then from Eqs. (7) and (14) as R

FR(S) = P[R 5 S] 1 -

(15)

The integration with respect to in Eq. (15), in general, cannot be carried out analytically. If it is assumed that the mean value 3 = ~r/4as an approximation for the orientation 6 Eq. (15) yields j j’

with the approximation

The last approximation leading to Eq. (16) is equivalent to the criterion that fracture occurs when- ever the sum of the two principal stresses exceeds some critical constant value C; i. e. ,

s+,+ s+2 = c (17)

For a spherical pressure vessel in which approximately = +2 = 4, = constant, the distribution function of the vessel strength becomes

in which V is the total material volume of the pressure vessel.

It follows from Eqs. (16) and (18) that the distribution function of the strength of the entire structure under two-dimensional stress-field applications is also a Weibull distribution in which the parameters x xo and k can be estimated from coupon tests. The estimation of x xo and k can be obtained from P ’ P’ the results of coupon tests by the method of moments; i.e. , the population mean ml, the variance m 2 and the third central moment m3 are, respectively, equated to the unbiased sample mean El, the vari- ance E2and the third central moment Ei3, where

JPL Technical Memorandum 33-470 9 and Here n

- l/k j=1 X" = xo(>)

n - 1 2 - (19b) n is the number of coupon tests, Y. is the observa- m2 = El) 1 j=1 tion of the jth test result, and r (. ) is the gamma function. Equating Eq. (19a) to Eq. (19b), one n obtains three equations for the determination of 3 - x xo and k. Other methods of estimation can be m3 = (n - l)(n - 2)C(Yj - P' j=1 found, e. g., in Ref. 10.

10 JPL Technical Memorandum 33-470 IV. TIME EFFECTS - In Section 11, it was mentioned that during From Eqs. (3), (4), (20) and (21), one obtains cyclic loading, and during sustained loading above a certain threshold value, subcritical flaw growth is expected. In this report, it is assumed that the time relationships of these subcritical flaw growths are representable by deterministic relations that are determined experimentally from tests on pre- flawed coupons by varying both the flaw size and SJ the applied stress. As can be seen from such in which S and S are, respectively, the stresses typical data as shown in Figs. 8 and 9, the changes cj sj due to cyclic and sustained loading. of the ratios KI/KI~for the cyclic and the sustained The terms R . and Rsj are the corresponding strengths of the case, respectively, can be written in the general CJ form jth element before the application of Scj and Ssj. The function U(t) usually has a characteristic shape that can be approximated by an exponential (20) decay to the threshold value. Thus

U(t) = b t re-t/T (24)

KI where b and T represent the threshold value and a (21) KIc = u(t) characteristic time, respectively, and t denotes the time to fracture in log scale. The parameters b,

T and r are material properties depending on the in which n indicates the number of cycles to fail- environmental conditions such as temperature, etc. ure and t is the time to failure. The functions It should be noted that the above equations are W(n) and U(t) are monotonically decreasing func- valid for a particular volume element; i. e., for a tions of their arguments. given stress distribution they are also valid for

JPL Technical Memorandum 33-470 11 the weakest volume element. However, since the with amplitude Sc, can be derived from Eqs. (3), structural strength R can be assumed to be equal (4), and (22) in the following form: to the strength of the weakest material volume ele- ment, the strength deterioration of the structure is sC R(n) = equivalent to the strength deterioration of the weak- est volume element.

In this report it is assumed that subcritical flaw growth, i. e. , the time-dependent deteriora- =o otherwise (25) tion of structural strength is negligible under the short duration proof-load test. This assumption where Ro is the structural strength after proof- seems justified on the basis of the following two load application. Similarly, by the use of Eqs. (3), consider ations. (4), (23), and (24), the deteriorated strength after application of the sustained load with amplitude Ss First, during proof loading the embedded for a time period T following applicationof nload- flaws in the structure may undergo some growth.^ ing cycles with amplitude S is If at the instant of unloading from the proof-load level the structure has not failed, then the maxi- mum flaw in the structure at that moment was less S S than the critical flaw associated with the proof- load level. During unloading, additional flaw growth may occur. However, based on available test results for rates of unloading commonly employed, e. 8,during pressure vessel proof- load testing, this additional flaw growth is either not detectable or it is negligibly small,

Second, during proof loading, some rearrange- ment of the statistical flaw size distribution ma,y R(n) take place. To this time, it has not been possible to assess the effect of such a change in flaw size distribution. Although in some cases (depending =o otherwise on the degree of ductility of the material) it can be argued that proof loading has a beneficial effect on the subsequent strength behavior of the structure 1 1 In Eqs. (25) and (26), W- (x)and U- (y) are the because of the "shakedown" phenomena, the r&- inverse functions of W(x) and U(y) in Eqs. (22)and tribution effect on ultimate strength is probably (23), which represent the aumber of cycles-to- negligible compared to other effects. failure associated with Sc/RO = x and the time-to- Based on the above assumptions and consider- failure associated with S /R(n) = y, respectively. ing a loading history as shown in Fig. 10, the Expressions similar to those in Eqs. (25) and (26) deteriorated strength of the structure, after appli- can be derived if the loading sequence of Sc and S, cation of the proof load So and n loading cycles is interchanged.

12 JPL Technical Memorandum 33-470 V. PROBABILITY OF FAILURE

Those structures- that have passed the proof-load test are to be used in actual mission applications. The original strength distribution for these vessels has been changed by the proof-load test. This change must be accounted for in the reliability evaluation. If Ro is the structural strength after the application of the proof load So, then the structural strength distribution is given by the conditional probability

Use of Eq. (16b) gives

for XM1 + +2) So(+, -!- 4,2) x (28) 1 1 P

Equation (28) valid when ~(4,~9,) is proportional to S (4, If this proportionality does not hold, is t 01+ 4,2). then the strength distribution, after application of So, is given by

for ~(4,~+ 9,) 1 so($,+62) 2 x (29) I*

To simplify the algebra, Eq. (28) is used in what follows without essential loss of generality.

JPL Technical Memorandum 33-470 13 For a spherical pressure vessel in which of Sc and S,, and FR~(*)i,s given by Eqs. (28) approximately = +2 = + = constant, or (30). The conditional distribution function

for x I So (30) of R(n) given R(n) > 0 can be obtained fromEq. (25) as follows:

The probability of structural failure po due to the pdof load So follows then from Eq. (16b) as po=F (S)= RO

Y fS (Y) dY = joFRo(W[w-I(<) t q) c

and for the spherical pressure vessel this becomes (35)

Substitution of Eq. (35) into Eq. (34) yields

Let pc be the probability of structural failure pcs = due to n cycles of the cyclic load Sc after So has been applied, and let pcs be the probability of structural failure due to the sustained load Ss for a period of time T, given that the structure has survived Sc. Then, it follows from Eqs. (22) and (23) that

in which it should be realized that W-’(x) = 0 for x I 1 and FR~(x)= 0 for x < S 0’

Pm The probability of structural failure due to the loFRO($iijFSc(X)dx (33) application of sustained loading S, for a period of time T after passing the proof load test, i. e., and without applying the cyclic loading Sc, is

rm

The probability of failure psc due to n cycles of Sc, given that the structure has survived Ss In Eqs. (33) and (34), fsc(x) and fs (x) are, for a period T after SO, can be obtained in a simi- S respectively, the probability density functions lar fashion as the probability of failure pcs.

14 JPL Technical Memorandum 33-470 VI. OPTIMIZATION

Obtaininfthe best possible performance, or be substantial. In particular, if this information the least possible cost, or the least possible is required at the tail end of the.distributions, the weight, etc., is an integral part of every struc- number of coupon tests and the associated cost tural design. The optimization task is to find the soon become intolerably high. Thus, in the over- values of the controllable parameters, subject to all cost picture, the required expenditures for the various constraints, that make a desired objec- material characterization should be taken into tive function an extremum. In this report, the account. objective function to be optimized is the structural As indicated by Eq. (28), the truncated struc- weight or the statistically expected cost: i. e., the tural strength distribution FR (x), after the appli- mean cost due to coupon testing, proof testing and 0 cation of the Droof load, is zero for strength.. mission degradation. The structural weight is values less than So. Therefore, the lower tail of expressible in terms of the physical parameters the original strength distribution FQ(x)does not - - A. such as density and structural dimensions, and give any contribution to .the probability of failure. the cost items are expressible in terms of the The uDDer tail of the truncated streneth distribu- AI - proof-load test levels, as well as the physical tion contributes to the probabilityof failure depend- parameters. It is the objective of the presentopti- ing on its relative interaction with the upper tailof mization process to determine those proof-load the load distribution. Since the intera6tion between test levels and physical parameters which yield load and strength distribution is of a general form, minimum expe cted cost, or which yield minimum as shown in Fig. 1, the upper tail contribution weight subject to an expected cost constraint. FR(x)to the probability of failure diminishes very Coupon testing has as its prime purpose the quickly with increasing distance from the mean characterizations of the statistical strength prop- strength. Consequently, the greatest contribution erties of the structural material. The efforts and of the strength density function to the probability costs that must be expended to establish, with suf- of failure during service stems from the region ficient confidence, the material strength distribu- close to the proof-load test level (Fig. 1). Since tions for one or more environmental conditions can the determination of the probability of failure with

JPL Technical Memorandum 33-470 15 certain confidence requires knowledge of the dis- subsystems is the sum of the statistically expected tribution functions of the load and of the truncated costs ECi for each ith structural subsystem, which strength Ro, it can be inferred that the required can be written as cost of coupon tests to enable the determination of the probability of failure with certain confidence n is strongly dependent on the proof-load level. This cost will be called coupon testing cost or i=l material characterization cost. If different struc- tural subsystems are used with differing materials, the total coupon testing cost is the sum of the characterization costs for each material. where the three terms on the right side of Eq. The expected cost due to proof testing is the (38b) represent coupon testing cost, proof testing statistically expected cost of structural testing in cost, and mission degradation cost, respectively, which one structure after another is tested at a of the ith subsystem. The proof test level E i for certain proof-load level, until a structure is ob- the ith strGctura1 subsystem is the ratio of the tained that passes the applied proof load. This proof load Soi to the mean structural strength Ei; cost includes the cost of the structures after their C. is the coupon testing cost for the ith structural completion plus the actual cost of proof testing. subsystem; qi is the expected number of the ith As Eq. (31) clearly indicates, this structural structural subsystem failing before the one sur- qualification cost is also strongly dependent on the viving the proof load is obtained; Coi is the cost of proof-load test level. It should be recalled that losing one of the ith structural subsystem during the term structure refers here to structural sub- proof load; C is the actual cost of mission degra- f systems such as sfruts, pressure vessels, etc., dation; pfi is the probability of structural failure and that the structural system, such as a space- of the ith structural subsystem during the mission. craft structure, may consist of more than one sub- The approximation of the summation sign for the system. If a number of structural subsystems mission degradation cost in Eq. (38) is on the require qualification, then the total proof testing conservative side (Refs. 2 and 3). It follows from cost is the sum of the statistically expected costs the developments in the previous section, that pfi due to proof testing for each subsystem. is not only a function of ei but also of the central a safety factor, vi = Bi/gi, which is the ratio of From structural utilization point of view, - it is not only important to consider the costs of mean strength wi to mean load Si. or of some coupon testing and proof testing, but also the cost other central measure of location. It should be that will be incurred if the structure fails during noted that v. is numerically different from the the time of its use. In space applications, this conventional safety factor, which is usually based cost may range from cost of total mission loss to on percentiles of Ri and Si, but plays, in principle, negligibly small cost, depending on whether struc- the same role.

tural failure occurs at the beginning of a mission The equations in Section V are valid for any or after the mission objectives have been fulfilled. ith structural subsystem. Since Soi = the E.R1 i' This cost would also depend on whether structural probability of failure poi of the ith structural sub- failure causes complete destruction or only some system due to the proof load Soi given by Eq. (31) mission degradition. The statistically expected can be expressed in terms of E.. It can be shown value of this cost, which will be called mission that degradation cost, is the product of the actual cost of mission degradation and the probability of occurrence of this degradation, which is the prob- (39) ability of structural failure.

The preceding discussion indicates that the which gives the functional dependence of qi

total statistically expected cost EC for n structural on E i'

16 JPL Technical Memorandum 33-470 The coupon testing cost requires some expla- If E. < 1, the significant part of the truncated nation. Note that it is the subsystem that is sub- strength distribution for the evaluation of the jected to proof testing. For this reason, the origi- probability of failure is located between the proof- nal subsystem strength Ri is truncated into Rei. load level'and the central portion of the strength Since the distribution of Ri and hence the distribu- distribution, whereas, if E. > 1, the significant tion of Roi are derived from the distribution of the part lies beyond the central portion of the distribu- coupon strength Rci for the ith subsystem (i. e., Eq. tion. To establish the strength distribution with (1 l), and since only the strength distribution func- certain confidence, this suggests that a larger tion associated with the strength value Ri 2 Soi sample of coupons is required if ci > 1 as com-

(i. e., Roi), is needed in evaluating the probability pared to ei < 1 for the same value /E 11. i - of failure, it is necessary to establish the coupon When one considers the preceding remarks strength distribution, with certain statistical con- and divides Eq. (38) by Cf, the total relative fidence, only for those coupon strength values expected cost EC' = EC/C and the relative f RCi > SCi with SCi being the value associated with expected cost for the ith subsystemECr = ECi/Cf, the truncation point Soi (or proof load) of the sub- become system strength. Let Ti be the ratio of SCi to the mean coupon strength xci for the ith subsystem; - i. e., ~i = Sci/Eci. The functional dependence of Ci onTi should be such that Ci is increasing with in- creasing absolute difference between Ti and 1, i.e., with ITi - 11 (see Ref. 3 for a detailed discussion). Thus Ti should be expressible as a function of ~i so that Ci in E;. (38) can be written as a function of ~i.In this report, the following assumptions are made: (1) in the Weibull distribution, the param- eter xo is equal to zero, and (2) in Eq. (16b), for where a. = Ai/Cf, pi = Bi/Cf and yi = Coi/Cf. each ith structural subsystem the expression Note that p. and yi indicate the relative importance (4 ) is independent of the space coordinates. 12+ Q of the ith subsystem with respecttothe actual cost While the first assumption is not particularly of mission degradation if the ith subsystem fails restrictive, the second assumption implies a and, as will be shown later, these values are the homogeneous stress field within each structural important parameters in the optimization process. subsystem. With these two assumptions, and The optimization problem can now be stated using Eqs. (11) and (16b), it can be easily shown as either one of minimizing the structural weight thatTi = ci. subject to a constraint on the relative expected It is now assumed that the coupQn testing cost cost given in Eq. (41),or one of minimizing the for the ith structural subsystem can be approxi- relative expected cost EC9' subject to a constraint mated by an expression of the following form on the structural weight. Both approaches are (Ref. 3): essentially the same. In this report, the optimiz- ation problem is stated as follows: m. Ci(ci) Ai t biBi)Ei - 11 ' (40) Minimize the total structural weight G sub- ject to the maximum expected cost constraint EC" where A. is the minimum cost of coupon tests Eta.

necessary for the determination of the mean value The objective function of coupon strength with certain confidence. The

terms Bi and m. are constants, and 6. is a constant n that may take diifferent values and'6; for ti > 1 6' 6' i G= Gi and ei < 1, respectively. izl

JPL Technical Memorandum 33-470 17 with G. being the ith subsystem weight, can be equality sign should hold in the constraint. Note written as a linear function of the design variables that the central safety factor vi in Eq. (41) can be hi(i=1,2, **e , n), where h. may, e. g., represent expressed as a function of h j=1,2, a, n; i.e., j' the cross-sectional area of the ith subsystem strut V. = vi (hl, hZ, a, hn). Using the method of or the thickness of the ith subsystem pressure Lagrangian multipliers, one can show that at opti- vessel; thus mum the following equations hold:

n G = xgihi i=1 where g. represents functions of physical and geo- metrical parameters of the ith subsystem.

It is emphasized that if the proof-load test is Equation (43) states that for an optimum structural not performed or is not considered (i.e., if all weight, the proof-load level E. to be applied to the qi(c.) = 0), and if the material properties are well ith structural subsystem should also be optimum known to engineers so that coupon tests are not in the sense that corresponding to a given safety needed (i.e., if all = p = 0), then the maximum factor v. the relative expected cost should be a.ii constraint EC* becomes the maximum constraint minimum at that level. a of the probability of failure, and the problem As for the optimization technique, depending reduces the optimum design based on a reliability on whether the structural system is statically constraint criteri_a that is discussed in Refs. 16-21. determinate or statically indeterminate, the itera- Since Eq. (41) is the only constraint, and tive procedure or the gradient move method can since the objective function is linear in h. the be employed, respectively. This subject is dis- 1' constraint is always active; i. e., at optimum, the cussed in detail in Ref. 2.

JPL Technical Memorandum 33-470 VII. NUMERICAL EXAMPLE

A 20-in. -Tiam spherical pressure vessel is to be designed to sustain an internal pressure Ss after proof testing for 360 h. The weight of the . vessel is to be minimized for an appropriate choice of proof-load level €(orSo) and vessel wall thick- and ness h in such a way that the total relative expected cost EC" does not exceed a certain assigned value (45) EC: (constraint). The sustained pressure is assumed to be normally distributed with mean value 5 = 1000 psi and 2% coefficient of varia- tion. (Since only one subsystem is considered, where r~ is the coefficient of variation of the the subscript i is dropped.) It is assumed that the coupon st:ength equal to 0.1, and Rc is the mean material is titanium Ti-61A-4V and that the coupon coupon strength equal to 160,000 psi. Hence, the strength has a Weibull distribution with a mean parameter k can be evaluated from Eq. (44), and value of 8, 160,000 psi and coefficient of varia- then xo can be computed from Eq. (45). tion of 1070, with a coupon size of 8 in. long, 1/2 In accordance with the discussion in the pre- in. wide and 1/4 in. thick. Based on the discus- vious sections, sion in Section VI, it is further assumed that in

Eq. (40), 6- = 1 for E < 1 and 6' = 2 for E > 1, and m = 1. The vessel is to be designed for room temperature at which the parametric values in Eq. (24) for U(t) are b = 0.5, r = 0.5 and

T = 0.713. The stress field is assumed to be 4, =-d such that = 4,z = + = constant. 4h (47) The Weibull distribution for the coupon strength is given in Eq. (ll), from which, it fol- v =-w (48) lows that SS

JPL Technical Memorandum 33-470 19 in which the daimeter of the vessel d = 20 in., the vessel material volume V = ad'h, and the proof load

level E = So/R with So being the proof load.

According to Eq. (37), the probability of vessel failure due to Ss, after the vessel passed the proof test, can be written as

where u = 0.02 is the coefficient of variation of S and where ps = pf. The introduction- of the transformation y = x/E into Eq. (51) yields

In this particular example, since there is only one subsystem, Eq. (41) can be written as

EC* = a t 6pI~- II t q(E)y tpf(E,v) (53)

in which the subscript i has been dropped and it can be shown that the optimum values of Y and E can be determined from the following two equations:

Q :: EC = ECa (54b)

The relative expected cost EC'. in Eq. (53) is plotted as a function of E for different values of y and p

with q(c) and p (E, v) being given by Eqs. (50) and (52). These plots are shown in Fig. f 11 for a particu- lar value v = 2. 1. 'i'ue constant value a in Eq. (53) has been disregarded in these figures, since it has no effect on the optimization process. Including a nonzero value for a would give to the plots a corre- sponding shift parallel to the EC* axis.

Those values of E for which EC"; becomes minimum for a given v are denoted by E". The solution space of the optimum design, Eq. (54), can then be constructed by plotting the locuses for different

values of Y as shown in Fig. 12. This figure is the extended version of Fig. 11, and is referred to as the optimum design space.

20 JPL Technical Memorandum 33-470 The optimum design procedure can now be expected cost design can be obtained either by summarized as follows: plotting ECQ as a function of E for given P,y, and constraint v such as Fig. 11, to find and Construct the optimum design space, =:. E* minimum EC''., or by reading E-'' and minimum e. g., Fig. 12. EC" directly from the optimum design space con- Read v and E* from the optimum design structed previously, for a specified constraint space constructed in step (1) for speci- v and given value of y. Numerical results for a fied constraint EC: and given values of three specific cases are given in Table 1. y and p. It should be noted that if the proof-load test With the safety factor v obtained in step is not performed or not considered (i. e., if (2), the optimum (minimum) thickness h q(E) = 0), and if the material property is well of the vessel or the minimum weight G known so that the coupon tests are not needed can be determined from Eq. (49). (i.e., if (Y = p = 0), then the maximum constraint If the relative expected cost is to be on the relative expected cost becomes the EC* EC"a minimized, subject to the constraint on the vessel maximum constraint on the probability of failure. wei.ght (or safety factor v ), the relationship of The problem reduces then to the optimum design a Eq. (54a) is still valid but Eq. (54b) should be based on the reliability criteria. This design is replaced by v = v Hence, the minimumrelative termed "Standard Optimum Design" in Table 1. a'

JPL Technical Memorandum 33-470 21 Table 1. Optimum design of pressure vessel

7 6 4 Standard Optimum y = 10- y = 10- y. = 10-5 y = 10- Designa

p = 0.0, EC$ =

0.1869 0.2086 0.2472 0.300 0.6311

1.11634 1.00845 0. 8568 0.70928 0.0

2130 2128 2113 2089 0.0

1.9082 2.1104 2.4668 2.9450 5. 8300

0.7934x IO-^ 0.4787 X 0.4223 X 0.706 x IO-^

p = 0.0, EC: = IO-^

0.1746 0.1852 0.2065 0.2447 0.5134

1.179 1.114 1.0091 0. 850 0.0

2113 2108 2110 2088 0.0

1 a- 7925 1. 8926 2.091 2.444 4. 8238 6 0.174 x 0.177 x IO-^ 0.472 X 0.72 X 10- IO-^

p = 0.0, ECE =

0.1661 0.1727 0.1833 0.2042 0.4176

1.2172 1.179 1.116 1.008 0.0

2090 2094 2091 2087 0.0

1.713 1. 775 1. 874 2. 07 3.99

0.374 X 0.2115 X 0.1242 X 0.5731 X

p = EC~= IO-^

0.1765 0.1870 0.2066 0.2484 -

1.169 1.096 1.0086 0. 8510

2117 2093 2110 2089

1. 81 1.91 2. 092 2.477

0.1042 X 0.8236 X 0.4639 X 0.6941 X

22 JPL Technical Memorandum 33-470 Table 1 (contd)

Standard Optimum 7 -6 4 Paramete r Y = 10- Y = 10 Y = 10- 5 Y = 10- Design a

Thickness 0.1665 0.1728 .O. 1834 0.2042 h, in.

€* 1.2170 1.178 1.115 1.008

So, psi 2088 2092 2090 2087

V 1.716 1.776 1.876 2. 07

x X X Pf 0.34 0.1642 0.1207 0.573

4 p = EC: = 0.5 x io-

~~ Thickness 0.1734 0.1807 0.1920 0.2158 h, in.

€* 1.1865 1.138 1.073 0.9594

So, psi 2114 2104 2098 2090

V 1. 781 1. 850 1.960 2.178

x x X Pf 0.176 0.349 IO-^ 0.358 0.253 x

p = io -4, EC: =

Thickness 0.1671 0.1744 0.185 0.2042 h, in.

a* 1.216 1.170 1.107 1.008

SO, psi 2090 2094 2093 2087

V 1.7218 1.791 1. 891 2.070

x X x Pf 0.1 0.1277 0.93 0.574 x IO-^

a The Standard Optimum Design is possible only when p = 0.

JPL Technical Memorandum 33-470 23 VIII. DISCUSSION AND CONCLUSION

Using Eq. (53), one obtains curves of the rel- suchasO.9Se Sl.Oand~>l.l;~>0.97, EC"

ative expected cost EC" versus proof load level E is also sensitive to y. Similar statements hold for different p, and for different safety factors v. also for safety factors different from v = 2. 1. Figure 11 shows representative curves for four y In Fig. 12, the relative expected cost EC" is values, three p values, and a particular safety plotted as a function of the optimum proof-load factor v = 2.1. In Fig. 11, the relative expected level E" for the same three values of p as in Fig. cost EC" changes very little for E less thanapprox- 11. Figure 12 is an extension of Fig. 11 in that imately 0.85. For other safety factors, EC" be- the lines for v = 2.1 give the optimum points haves similarly, indicating that there is no advan- indicated in Fig. 11, while the lines for the other tage in conducting proof testing below a certain -5 values of v reflect the optimum points of similar value oft. In all three cases, = 0, = 10 and p p curves as those in Fig. 11. /3 = the optimum proof-load levels are in the vicinity of E = 1. 0 with a slight and expected ten- The first set of curves for p = 0 in Fig. 12

dency of the optimum proof-load levels E" toward shows that when the coupon test is not needed, the unity with increasing p. It is due to this fact that relative expected cost EC* can be made as small

under reasonable relative expected cost con- as desired simply by decreasing E" and increasing straints, the optimum proof-load level E * will fall the safety factor v, or the weight which, in this with great 1ikeliPood within the range of two stan- case, is proportional to v. This result is a con- dard deviations around the mean value x. From sequence of the fact that EC" is not, in this case, the designer's point of view, this is desirable, a function of the cost of coupon test. If the cost of since in general, a considerably greater number coupon testing is considered (i.e., if f3 # 0 as for of coupon tests is required for characterizing the the second and third set of curves in Fig. 12), then truncated strength distribution with a certain level the relative expected cost EC" has a lower limit.

of statistical confidence if E" falls outside this This implies that in such cases a relative expected region. It is noteworthy that for v = 2. 1, EC" is cost constraint less than this lower limit yields no

very sensitive to changes of E in certain regions, feasible solution. It is evident from Fig. 12 that

24 JPL Technical Memorandum 33-470 for p # 0 the absolute optimum proof-load level is often incompatible with prevalent thinking during in the vicinity of E* = 1. 0 for p = 10-5withv = 2.2 project applications, particularly if the subsys- -4 and for p = 10 with v = 2.1. Additional details tems are pressure vessels. It is often expected regarding.the influence of the cost of coupon tests that no pressure vessel will be destroyed during on the optimum design are given in Ref. 3. proof testing and that pressure vessels are de- signed to fulfill that expectation. This implies Table 1 gives specific numerical results for that pressure vessels are designed for the proof different values of p and for specified relative load rather than the expected mission environment expected cost constraints EC:. It is particularly and..are proof tested at a level E that corresponds instructive to compare the results of the Standard to the nearly horizontal portion of the curves for Optimum Design E* = 0 with those of the optimum EC" in Fig. 11. As stated above, proof loads at design considering the proof-load test; i. e., for 7 such levels have no advantage in terms of expected y = 10- to Not only is considerable weight cost. saving realized (weight is proportional to h), but also a great reduction of the probability of failure In the development of this report, various pf is obtained if the proof-load level is considered simplifying assumptions were made that could be as a design variable. The percentage of weight relaxed in a more extensive study. Nevertheless, saving of the optimum design with proof-load tests it is believed that the results of this report are as compared to the design without proof-load tests representative and would not undergo major quali- is much higher in this case than in the examples tative changes if these assumptions were relaxed, given in Refs. 2 and 3. This is because in the although quantitative changes would be expected. present case, the coefficient of variation of the These major assumptions and some of their impli- strength R of the vessel (uR = 100J0)is higher than cations are discussed below. the coefficient of variation of loading 2%) so (us = The first specific assumptions were that (1) that the probability of failure comes mainly from the statistical variation of the material strength is the lower portion of the strength distribution, only due to the flaw size parameter a = (a/Q).in which is truncated by the proof-load. In Refs. 2 j 1 Eq. (3),and (2) it is sufficiently accurate to use the and 3, low-dispersion material = 5%) is used (uR mean value of 45" for the flaw orientation so that The fact for high-dispersion loading (us = 20%). the computational effort involved becomes tractable. that the proof-load test improves the statistical The first assumption is not important in the pre- confidence of the reliability estimate is discussed sent development because the results derived here in Refs. 2 and 3. are based on experimentally determined statistical As in Refs. 2 and 3, the conclusion can be distributions of material strength by coupon tests drawn here that the weight saving of the optimum rather than a determination of the strength based design depends to a large degree on the parameter on the statistical distribution of flaw size. The value y. For low values of y. one c'an afford to second assumption, which implies that failure lose more vessels during proof-load testing; i. e., occurs when the sum of the two principal stresses higher values of E can be applied and these result exceeds a critical value (Eq. 16), is considered a in higher strength vessels and the saving of reasonable first approximation for cases in which structural weight. This follows from Fig. 12 and both principal stresses are tensile stresses, as in Table 1. thin pressure vessels. It is believed, however,

A general conclusion that can be 6rawn from that this assumption warrants additional extensive the preceding discussion is that in proof testing investigations based on Eq. (14) with the objective structural subsystems it is to be expected that of determining, for various combinations of prin- the effect of the distribution of some of these subsystems will be lost. In fact, in cipal stresses, flaw orientation on the strength distribution FR(S). many cases where E = 1.0, it should be expected that approximately half of these subsystems will Another assumption is that strength deterio- be destroyed during proof testing for the achieve- ration due to time effects, i.e., due to cyclic ment of minimum expected cost EC". This is loading and sustained loading, can be represented

JPL Technical Memorandum 33-470 25 by equations with deterministic parameters as than a certain value, it is necessary to also con- shown in Section IV. This assumption can only be sider the statistical distribution of the cost. This accepted in a qualitative fashion, since it is known aspect of the problem has not yet been treated in from experience that the time-to-failure of a spec- the literature. imen has a considerable statistical variation even A further assumption in this report (not ex- if the initial flaw size is the same from test to test. plicitly stated) is that the mission load statistical Considerable additional investigations, both exper- distribution is independent of time, although the imental and analytical, are required befare this most important problems deal with dynamic loads problem can be adequately understood; perhaps it and wide time changes of the environments, such must be assumed that the parameters involved in as temperature fluctuations, radiation, etc. It is Eqs. (20) and (21) are statistical variables. expected that the investigation in this report is In Section VI, the assumption that xtL = 0 in only quantitatively influenced by these environ- the Weibull distribution is equivalent to saying that mental changes while, qualitatively, the results there is no stress threshold value below which no are still valid. failures occur. It is expected that this assumption The objective function in this report is the has little effect on the results of the present weight of the structural system. Other objective investigations , since the proof load eliminates the functions can be chosen, of course; for instance, lower end of the strength distribution. the expected cost, or the reliability, can be used Equation (38) represents the statistically as an objective function that is to be minimized or expected cost that is used in the optimization pro- maximized. For electronic systems, in which the cess as a constraint. In this process, it is implied subsystems are the electronic components or the that the cost has-a statistical distribution, which integrated circuits that are proof tested before is not considered in this report. If the cost con- use, weight is usually not the critical quantity to straint is stated so that the probability of exceed- be minimized. In such cases, cost or reliability ing a given cost level is required to remain less would be nore appropriately extremized.

26 JPL Technical Memorandum 33-470 REFERENCES

1. Barnett, R. L., and Hermann, P. C., Edited by H. Liebowitz. Academic Press "Proof Testing in Design with Brittle Mate- Inc., New York, 1968. rials, '' J. Spacecraft Rockets, Vol. 2, No, 6, 6. Irwin, G. R., "Fracture, It Encyclopedia of pp. 956-961, Dec. 1965. Physics, Vol. 6, pp. 551-509, 1958. 2. Shinozuka, M., and Yang, J. -N., "Optimum 7. Griffith, A. A., "The Phenomena of Rupture Structural Design Based on Reliability and and Flow in Solids, Phil. Trans., Vol. 221, Proof-Load Test, It in Annals of Assurance pp. 163-198, 1920. Science, Proceedings of Eighth Reliability and Maintainability Conference, Vol. 8, 8. Irwin, G. R., "Crack-Extension Force for a pp. 375-391, 1969. Part-Through Crack in a Plate, I' J. Appl. Me.ch. Vol. 84E, No, 4, pp. 651-654, Dec. 1962. 3. Shinozuka, M., Yang, J. -N., and Heer, E., "Optimum Structural Design Based on Reli- 9. Tiffany, C. F., Masters, J. N., and Pall, ability Analysis, in Proceedings of the F. A., "Some Fracture Considerations in the Eighth International Symposium on Space Design and Analysis of Spacecraft Pressure

Technology and Sciences, pp. 245-258, Vessels, It paper given at 1966 National Metal Tokyo, Japan, Aug. 1969. Congress, Chicago, Ill., Oct. 1966. Pub- lished through American Society for Metals 4. Heer, E., and Yang, 3. -N., "Optimum Report System as ASM Technical Report Pressure Vessel Design Based on Fracture C6-2.3. Mechanics and Reliability Criteria, It in Pro- ceedings of ASCE-EMD Specialty Conference 10. Brown, W. F., and Srawley, J. E., "Plane on Probabilistic Concepts and Methods in Strain Crack Toughness Testing of High Engineering, pp. 102-106, Purdue University, Strength Metallic Materials, ASTM Special Nov. 1969. Technical Publication No. 41 0, American 5. Foreudenthal, A. M. , "Statistical Approach Society for Testing and Materials, to Brittle Fracture, in Fracture: Volume 11. Philadelphia, Pa., 1967.

JPL Technical Memorandum 33-470 11. Weibull, W., "A Statistical Theory of the 17. Kalaba, R., "Design of Minimal-Weight

Strength of Material, It Ing. Vetenskaps Akad. Structures for Given Reliability and Cost, Handl., 151, Stockholm, 1939. J. Aerosp. Sci., pp. 355-356, Mar. 1962.

12. Weibull, W., "The Phenomenon of Rupture 18. Switzky, H. , "Minimum Weight Design with in Solids, 'I Ing. Vetenskaps Akad. Handl. Structural Reliability, If in AIM 5th Annual 153, Stockholm, 1939. Structures and Materials Conference, pp. 316 13. Weibull, W., "A Statistical Representation of 322, New York, 1964.

Fatigue Failures of Solids, I' Kungl. Tekniska 19. Broding, W. C., Diederich, F. W., and Hogskolans Handl., 27, 1949. Parker, P. S. , "Structural Optimization 14. Weibull, W., "A Statistical Distribution and Design Based on a Reliability Design Function of Wide Applicability, J. Appl. Criterion, J. Spacecraft Rockets, Vol. 1, Mech., pp. 293-297, Sept. 1951. No. 1, pp. 56-61, Jan. 1964.

15. Gumbel, E. J. , Statistical Theory of Extreme 20. Murthy, P. N., and Subramanian, G., values and Some Practical Applications, "Minimum Weight Analysis Based on Struc-

National Bureau of Standards, Applied tural Reliability, I' AIM J., Vol. 6, Xo. 10, Mathematics Series 33, Feb. 1954. pp. 2037-2038, Oct. 1968.

16. Hilton, J., and Feigen, M., "Minimum 21. Moses, F., and Kimser, D. E., "Optimum Weight Analysis Based on Structural Reli- Structural Design and Failure Probability ability," J. Aerosp. Sci., Vol. 27, pp. 641- Constraints," AIM J., Vol. 5, No. 6, 652, 1960. pp. 1152-1158, Jan. 1967.

28 JPL Technical Memorandum 33-470 STRUCTURAL LOAD DENS ITY FUNCTION f S TRU CTURAL STRENGTH DENSITY FU NCTI 0 N

LOAD S +MEAN LOAD 3 4 STRENGTH R --.) -PROOF LOAD TEST LEVEL So- p-MEAN STRUCTURALSTRENGTH R-

Fig. 1, Typical load and strength distributions

P”_I

Fig. 2. Elliptically shaped flaws: (a) surface flaw; (b) embedded flaw

JPL Technical Memorandum 33-470 29 0.50

0.40

= NEGLIGIBLE 0.30

0 %

0.20 $ = COMPLETE ELLIPTICAL INTEGRAL OF THE SECOND KIND c = GROSS STRESS CT = 0.2% OFFSET TENSILE YIELD STRESS YS 0.10

0 I I 0 1 .o 1.5 2.0 2.5 FLAW SHAPE PARAMETER Q

Fig. 3. Flaw shape parameter curves for surface and internal flaws

1.8

- KOBAYASHI'S SOLUTION FOR SMALL a/2c VALUES -

-

1.4

NUMBER NEXT TO DATA POINT INDICATES a/2c VALUE 1.2 SMITH'S SOLUTION

1 .o

a/h

Fig. 4. Stress intensity magnification factors for deep surface flaws

30 JPL Technical Memorandum 33-470 -500 -400 -300 -200 -1 00 0 100 200 300

TEST TEMPERATURE OF

Fig. 5. Fracture toughness of annealed 6A1-4V titanium plate at various test temperatures

v, v, Y sI 0, w p:

2U d LL

4m THICKNESS h Fig. 6. Fracture toughness variation with material thickness

JPL Technical Memorandum 33-470 31 YIELD FAILURE LINE /-

CONSTANT FRACTURE TOUGHNESS

\\ 7CONSTANT STRESS INTENSITY

FLAW SIZE (a/Q)

Fig. 7. Applied stress vs flaw size, for plane strain

1 .o

0.8

0.6

0.4

0.2

0 1 10 100 1,000 10,000

CYCLES TO FAILURE, N Fig. 8. Operational life curves, lower bound, cyclic load

32 JPL Technical Memorandum 33-470 1 .o

0.9

0.8 2219-T87 ALUMINUM -0 0.7 - y. '- - y 0.6 0.5

0.4 -

0.3

0.01 0.1 1 .o 10 100

TIME TO FAILURE, h

Fig. 9. . Operational life curves, lower bound, sustained load

1PROrF LOAD - 1000 cycles 100 hours 2 2 I TA TB TC TC TD TW

(a). TYPICAL LOAD HISTORY FOR PRESSURE VESSELS

I I I I I I c LOAD I 100 hours

TIME

(b). STRUCTURAL LIFE PREDICT1 ON AND OPERATING LOAD DETERMI NATl ON

Fig. 10. Typical schematic design case: (a) load history for pressure vessels; structural life prediction and operating load determination

JPL Technical Memorandum 33-470 33 * z1

W 2

4W

Lv.2.1 0.7 0.8 0.9 1.0 1.1 1.2 1. 3 0.7 0.8 0.9.1.0 1.1 1.2 1.3 PROOF STRESS LEVEL c

Fig. 11. Relative expected cost as a function of proof-load level for varying values of p

__ 0.7 0.8 0.9 1.0 1.1 1.2~ 1.3 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.7 0.8 0.9 1.0 1.1 1.2 1.3 I OPTIMUM PROOF STRESS LEVEL E*

Fig. 12. Relative expected cost as a function of optimum proof-load level for varying values of p

.34 JPL Technical Memorandum 33-470 NASA - IPL - Cod., LA.. Calif. TECHNICAL REPORT STANDARD TITLE PAGE

9. Performing Organization Name and Address I IO. Work Unit No. JET PROPULSION LABOMTORY California Institute of Technology 71. Contract or Grant No. 4800 Oak Grove Drive Pasadena, California 91107 13. Type of Report and Period Covered Technical Memorandm 12. Sponsoring Agency Name and Address NATIONAL AERONAUTICS AND SPACE ADMINISTRATIQN 14. Sponsoring Agency Code Washington, D.C. 20546

, 15. Supplementary Notes

16. Abstract Spacecraft structural systems and subsystems are subjected to a number of qualification tests in which the proof loads are chosen at some level above the simulated loads expected during the space mission. Assuming fracture as a prime failure mechanism, and allowing for time effects due to cyclic and sustained loadings, this paper treats an optimization method in which the statistical. variability of loads and material properties are taken into account, and in which the proof load level is used as an additional design variable, In the optimization process, the structural weight is the objec- tive furaction while the total expected cost due to coupon testing for material characterization, due to failure during proof testing, and due to mission degradation is a constraint, Hmerical results indicate that for a given expected cost constraj.nt, substantial weight savings and im- provements of reliabiliw can be realized by proof testing,

17. Key Words (Selected by Author(s)) 18. Distribution Statement

Chi1 Engineering Unclassified -0 Ulalimited Interplanetary Spacecraft, Advanced Planetary Spacecrdt, Advanced (Stractwal Engineering

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