Blending hierarchical economic decision matrices (EDM) with FE and stochastic modeling I Methodology and refining EDM

FINITE ELEMENTS I N ANALYSIS AND DESIGN

ELSEVIER

Finite Elements in Analysis and Design 28 (1997) 1-17

Blending hierarchical economic decision matrices (EDM) with FE and stochastic modeling I Methodology and refining EDM J.N. Majerus a'*, J.A. J a n n o n e a, S.P. L a n p h e a r b, D.A. Tenney c aMechanical Engineering Department, Villanova University, Villanova, Pennsalvania, 19085, USA bBoeing Defense and Space Group: Helicopter Division, P.O. Box 16858, Mail Stop P34-44, Philadelphia, Pennsalvania, 19142-0858, USA ¢C & D Charter Power Systems, Washington & Cherry St., Conshocken, Pennsalvania, 19428, USA

Abstract

This paper introduces the concept of a three-level hierarchy of economic-decision matrices (EDM) and discusses how finite element (FE) and stochastic modeling can be coupled in decisions on quality control. A disadvantage of the deterministic-engineering approach is that quality control cannot be properly accessed by a corresponding EDM. A four-step design strategy is outlined and various suggestions associated with these steps are presented in the paper. This design strategy is then quantified by the consideration of a hypothetical reverse-engineering problem involving a U-shaped bracket with cutouts.and holes. The paper presents the results of numerical experiments (sensitivity study) involving six preliminary variates (yield strength, Young's modulus, thickness, magnitude and phase of the forces, and displacement BC). The final primary variates are assumed to be governed by a Gaussian distribution, and the methodology of applying: the classical Z-statistic method to the FE results are presented. © 1997 Elsevier Science B.V. Keywords: Finite-element analysis; Design; Low-cost design; Loss-matrix; Quality control; Variability; Stochastic design; Nondeterministic design; z-Statistic; Uncertain forces; Uncertain properties; Statistical variables; Stress-strength interference

1. Introduction

Business decisions are reached via "what-if games" utilizing economic decision matrices and the weighing of various decisions to either minimize the cost (loss) or maximize the payoff (gain). These decisions are based upon three levels-of-knowledge(information): (a) "certainty", i.e., deterministic, (b) "at risk" (stochastic with known statistics), (c) "with uncertainty" (stochastic with unknown statistics). This information involves both the "states-of-nature (SON) and the "levels-of-severity" (LOS) of each state. Note that states-of-nature vary from engineering conditions, e.g., chemical, electrical, magnetic, ~Ihermal and loading environment, all the way to economic conditions such as S0168-874X/97/$17.00 ~) 1997 Elsevier Science B.V. All rights reserved PII S0 1 6 8 - 8 7 4 X ( 9 " 1 ) 0 0 0 2 3 - 1

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J.N. Majerus et al./Finite Elements in Analysis and Design 28 (1997) 1-17

overhead rates for a particular cost-center (a good overview is given by Dieter [1]). The manager/designer selects which states-of-nature (SON) she/he wants to include in that particular analysis, and this selection is highly influenced by the ISO 9000 Standard being implemented. In engineering, the LOS are specific values associated with the product design specifications (PDS), and the specific values depend upon the desired level of reliability (the so-called "three-sigma", or "six-sigma", etc.). Whereas the product can vary from simple parts to complex factories, the current paper will concentrate on design as applied to a single part. Pugh [2,3] believes that the usage of the common "decision matrices" do not lead to a good "competitive" design because the approach is too subjective, fraught with numerous uncertainties, and decision makers tend to treat the quantitative numbers as "Gospel". Heirarchical EDM attempts to eliminate these objections and utilize the best ideas of Pugh's approach by using three levels of decision matrices: screening, refining and detailing. This usage of different levels is similar to Pugh's controlled convergence since each level introduces a new set of creative ideas which must be processed. Furthermore, these decision matrices are termed "economic" since their purpose is to determine the most "competitive" design, i.e., economic decision matrices (EDM) are concerned with all aspects of the product. The interested reader should consult Prasad [4], who has an excellent overview of these various aspects. Note that the hierarchy of EDM and the concepts of statistical control and feedback are readily associated with each of Taguchi's three phases of design: system, parametric and tolerance (see Ross in Ref. [5]). The screenin9 EDM related to the system design which utilizes technology improvements for new ideas of product improvement. The refining EDM is related to parametric design which aims at improving uniformity at no cost, or even at a cost savings, via selecting "design" parameters so that "performance" is less sensitive to causes of variation. The detailin9 EDM is related to tolerance design where quality is improved (at a minimal cost) by tightening tolerances on product, or process, parameters to reduce performance variation. Obviously, FE analysis could be utilized in each of these phases of design. The most creative level of EDM is the first screenin9 level, which involves the same critical thinking process as Pugh's evaluation matrix [3]. The screenin9 EDM uses the same two key items as the evaluation matrix: (a) the "Concepts" and (b) the "Criteria" used to judge the concepts. Specifically, the screenin9 matrix involves the selection of "alternative course-of-action". These alternatives involve each aspect of the engineering design-triplet (Geometry Material, Processing the GMP). Each triple of alternatives forms an "outcome", or, a design candidate, and this screening produces a large set of design candidates. However, FE analyses and manufacturing simulations are usually not involved in these preliminary decisions and hence the screenin9 EDM will not be discussed further. Using this large set of design candidates, a refinement EDM attempts to cull a smaller number of viable alternatives (design candidates) associated with more specific design requirements. Finite element (FE) modeling enters into both the cost decisions via manufacturability models (simulations) and the reliability decisions via thermal/stress/deflection analyses coupled with an appropriate Analytical Criteria for Failure (ACF). The common Von Mises yield criteria is used in this paper, and the more complicated ACF of fatigue will be considered in the companion paper. Usually, manufacturability simulations are not used in the refinement EDM since many details of the design still need to be ascertained. Consequently, only "Expert Systems" or "Rules of Thumb" (see Trucks in Ref. I-6] and Bralla in Ref. [7] are concurrently used in the refinement EDM. The -

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

3

detailing EDM involves the usage of concurrent manufacturability models (simulations), and will be presented in an accompanying paper. The next section discusses the general approach needed to consider the effects of statistical variation in an FE analysis.

2. Consideration for stochastic effects

One fringe benefit of the ISO 9000-9004 type of standards is that the importance of minimizing "variations" becomes more apparent to the design community. When the design parameters and input are considere,d as deterministic, improved product-reliability normally means a larger factor-of-safety (FOS = critical value of ACF/design value of ACF) or margin of safety (MOS = FOS - 1). Because of expediency, many designers obtain these larger factors via usage of so-called "stronger" material. Stronger refers to increasing the material characteristic of whatever is being considered for the design criteria, e.g., stiffness implies Young's modulus, fatigue-fracture resistance implies fatigue strength or endurance limit, resistance to flaw-propagation implies plane strain fracture toughness. Hence, only the material aspect of the design GMP triplet is considered. Conversely, deterministic concurrent-engineering utilizes the other two GMP aspects involving improved part configuration and improved manufacturing processes. A great disadvantage of the deterministic approach is that the influence of quality control cannot be economically assessed except for changes that influence the mean values. Actually, quality control enters into any FE analyses via control of the variation (uncertainty) in material, geometry, surface condition, environmental conditions, and the modeling boundary conditions. The material variation stems from variation in the parent material (so-called billet or feed-stock), process:ing method, and secondary processes, e.g., heat treatment. The geometric variation is due to uncertainty in lengths, thicknesses, fillets, and locations of holes, etc., due to specified dimensional tolerances and wear of the processing equipment. The uncertainty in environmental conditions involve uncertainties in loading, thermal, and hostile fluids as a function of time. The loading variation involves uncertainty in magnitude, direction, and the location and type of loading distrJLbution. The thermal variations involve uncertainty in magnitude, location and distribution of both the boundary and far-field temperatures, and the heat fluxes and internal heat-generation rates. The boundary-condition variations are caused by uncertainty in the location and fixity (displacement vs rotational) of displacement constraints, and the uncertainties in heat-transfer coefficients and emissivities. The surface-condition variation is due to uncertainty in smoothness of the surface, localized surface hardness, the effectiveness of protective finishes, and damage from handling and shipping. While most of the above listed variations involve continuous variables, some variables can only have discrete (off/on, yes/no) values, such as BC or damage. Furthermore, for FE analysis, continuous variable,s can always be replaced by a set of discrete values associated with the histogram representation of the statistical frequency distribution. The authors believe that the key to utilization within an EDM involves the selection of a proper design strategy. Four key-steps in a design-strategy are: (a) selection of the appropriate stochastic model, (b) preliminary selection of pertinent statistical variates, (c) estimation of sensitivity factors, (d) proper utilization of generic physical properties and linear analysis.

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J.N. Majerus et al./Finite Elements in Analysis and Design 28 (1997) 1-17

2.1. Selection o f the appropriate stochastic model

Numerous authors have explicitly considered combining stochastic analysis with finite-element analysis, and a brief overview of their various approaches is given in the second paper. This section merely reviews the basic concept of reliability viewed from the method known as the "interference method" (see Ref. [8]). The first step is to realize that the normalized probability of failure F of a part is simply the area under the overlapping statistical distributions of "stress" and "strength". Physically, what stress and strength corresponds to depends upon the ACF. Since the statistical distributions are normalized, reliability R is simply 1 - F. This is represented by either of the following two integral equations: R = | f(strength) f(stress) dxstress dXstrength ,J LL(strength) I ,) LL(stress)

(1)

or

R =

IUL (stress) f ~UL(strength) f(stress) ~ | f ( s t r e n g t h ) dXstrength( dxstress. J LL(stress) I..](stress) )

(2)

Reliability represents all possible events (area under the curve) to the right of all possible stresses. Some authors [9] refer to this as the "stress-strength interference approach to reliability". As illustrated by Rao [10], there are three general methods of evaluating the integral equations: (a) assumed Gaussian (Normal) distributions and the Z-statistic, (b) assumed non-Gaussian distributions and the interference method, (c) unknown distributions and a Monte-Carlo type of method. The predominant method is to assume normal distributions for the different variates and use the so-called z-statistic (actually a special case of the interference method). This method is easy to implement and will be demonstrated in the next section. However, it is one (JNM) of the author's opinion that this technique is limited to reliabilities on the order of 0.999-0.9999 because of asymmetries in the tails of actual distributions as compared to the assumed Gaussian. Hence, for the higher levels of reliability, one must assume more realistic, non-normal distributions, and use the interference method. This method will be demonstrated in the second paper. 2.2 Preliminary selection o f the statistical variates

The most important step involves narrowing the field of possible variates. This involves engineering judgement, the range of variate values, the sensitivity of the reliability to a particular variate, and obviously, CPU, personnel and time constraints. Any variate that exhibits a large range is a prime candidate for a statistical variate. The range of variate-value (referred to as the "design range") involves the upper and lower limits of each variate of the design. Sometimes these ranges are specified in the product design specification, e.g., "acceleration levels not to exceed 7-g". Other times, these limits are obtained by adding/subtracting tolerances from a specified deterministic value, e.g., "the length is 145 + 0.12 mm". If not specified, the tolerances associated with any processing method can be found in various manufacturing handbook. So-called " + tolerances" are usually taken to imply __+3SD (standard deviations). Sometimes, there are in-house or vendor

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

5

related constraints or specifications, e.g., "minimum tensile strength is 85 Mpa". Normally, "minimal" physical properties refer to either the lowest observed value in a large number of tests, or a value less than the., lowest observed. Other prime candidates for a statistical variable are those that are highly influenced by another variables within the design range. This implies the influence of temperature and hostile fluids on the so-called material faiilure property associated with the ACF. Note that, temperature variations can cause so-called deterministic values to behave like a statistical variate, e.g., Young's modulus may have a very small variation ( + a few percent) at any temperature, but a large temperature variation may imply a large wtriation in Young's modulus. Hence, the FE stresses may show a variation due to the modulus variation (see next section).

2.3. Estimation of sensitivity factors As discussed above, numerous statistical (stochastic) factors enter into any FE analysis o r FE simulation. After the preliminary selection of the pertinent factors (variates), the sensitivity of the "stress" (the value of ACF as determined by an FE analysis) must be ascertained. Only those variates which exhibit "stress-sensitivity" should be considered as a statistical variate in the final reliability analysis. The statistical sensitivity involves the change in reliability vs. a variate, and the classical method will be discussed in a companion paper. However, in the current study, the authors take a simplier view of "sensitivity" involving the change in "stress" vs. change in variate. This is readily implemented via the so-called 2-level factorial approach (see Dieter in Ref. [1]) combined with "numerical experiments" using FE analysis. The 2-level approach uses a high/low level of the variate (other variates fixed) to determine if the variate has any "significance" on the behavior. Obviously, the definition of significance depends upon both the desired levels of accuracy and reliability. The current Refining EDM uses the following definitions: insignificant ~ <~ 1%, minor =~ ~- 10%, and significant =~ >~ 50% change in stress.

2.4. Proper utilization of generic physical properties and linear analysis An important a priori observation concerning FE analysis is that, at a fixed temperature, the modulus and Poisson's ratio are somewhat constant for all alloys of a generic material. The major influence of changirLg alloys is changing of the value associated with the "strength" of the generic material. Consequently, for a specific geometry and BC, Only one FE analysis need be made for each type of generic material, e.g., one for "steel", one for "aluminum", etc. Another observation is that for linear analysis and a fixed location, spatial distribution and direction or applied force, the results can be scaled[ by the peak magnitude of the force. Consequently, only one FE analysis needs to be conducted for a statistical distribution of force magnitudes. In the next section, a sample problem is considered using the above design strategy. First, the problem is defined, and this is followed by a preliminary consideration of the statistical variables (variates). Then, a sensitivity study is conducted, and the primary variates are determined. Assuming that these variates are normally distributed, i.e., the frequency distribution function is a Gaussian function, the reliabilities of several candidate designs are then evaluated.

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J.N. Majerus et al./Finite Elements in Analysis and Design 28 (1997) 1-17

3. Application using Von-Mises's yield criteria for the ACF The hypothetical project involved a pre-existing aluminum part (see Fig. 1) which is used by a "competitor" within precision machinery driven by an electric stepper motor and utilized in

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J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

7

a aircraft. The part is to be reversed-engineered and, if possible, ameliorated w.r.t, three product criteria: (a) weight, (b) billet-material cost, and (c) 2-D reliability. Due to the company ownership of extensive forging and machining equipment, the acceptable generic process is internally constrained to be forging, followed by secondary machining processes. Based upon strength/weight ratios, forgability, and availability of materials, a previous screening EDM determined that the three acceptable generic materials are alloy-steels, alloy-titanium and alloy-aluminums. Since metals lose ductility and fracture toughness at low temperatures, and strength at high temperatures, the environmental temperature is quite important in reliability. A typical range might be from - 50°C to + 150°C, and for this study the temperature was assumed constant at 150°C. In addition to the constant thermal environment, two other states of nature (SON) were considered for the analysis: (a) distributed radial forces acting at each interface of the two shaft holes, and (b) displacement boundary conditions (DBC). Since the Refinement Level of EDM involves a large number of possible designs, 2-D FE analysis are utilized. Consequently, the tapped holes and fillets are neglected in the geometry, and the 2-D FE model consists of three interconnected plates, with the center plate containing two holes and the radial cut out. The bronze inserts were ignored in the modeling representation. The surface of the two bearing holes are assumed to be loaded via radial shaft forces whose resultant vector F is inclined 30 ° w.r.t, a axis passing through the center of both bearings. The mean magnitude of the radial load is 4000 N, with a coefficient of variation (standard deviation/mean value) of 10%, and the "phase" of the loads is uncertain. The load variation is assumed to have a Gaussian (normal) distribution. 3.1. Preliminary selection of the statistical variates Since the ACF is the Von Mises yield criteria, one primary statistical variate is the yield strength of the different "alloys" associated with the three generic materials. Based upon forging processability and strength/density ratios, the selected alloys were 4340 HT steel, 2025-T6 aluminum, and Ti6V-4N. Note that, if the product was being designed for room temperature (RT), then the 7075-T6 alloy would be selected since it is stronger at RT. The following properties (ASM Metals Reference Book, 2nd ed.) were used in this study: (1430 MPa, 7.84 Mg/m3), (248 Mpa, 2.8 Mg/m3) and (1100 Mpa, 4.43 Mg/m3) respectively for Sy and 7 of steel, aluminum and titanium. Using these strengths and densities and a fixed deterministic factor of safety (FOS), the normalized weight ratios would be 2.1, 1.0 and 0.75, respectively, for aluminum, steel, and titanium. Since the magnitude of the load varies, all the analysis will be conducted using a trial radial load of 100 N applied at each of the two bearing locations. The stresses can then be scaled up or down according to a specific force value. Obviously, because of the linearity of stress with load magnitude, the load is a primary statistical variate. Another possible statistical variate is the loading "phase". The phase of the load means whether or not the resultant 30 ° forces point towards or away from each other. This might be a important variable since the d:irection of the forces determines the type of maximum stress in the bracket, and the symmetry of the problem. When the loads point towards each other (out of phase), the significant stress is a localized bearing type of stress, little bending stress is induced in the vertical legs, and a horizontal plane of symmetry exists in the problem. Hence, the direction of loading could be a significant statistical variate, and for illustrative purposes, this paper will consider both "phases" of loading.

8

J.N. Majerus et al./Finite Elements in Analysis and Design 28 (1997) 1-17

Another preliminary variable is that of Young's modulus (E) and Poisson's ratio (n) associated with the three generic materials. For this study the following set of properties were used: {207 Gpa, 0.3}, {70.3 Gpa, 0.33} and {112 Gpa, 0.34}, for steel, aluminum and titanium, respectively. For this reverse-engineering problem, the only geometric unknowns on the configuration are the thicknesses of the various "sides" of the configurations, and the size of the fillet. The size of the fillet does not enter into the problem involving the intersection of 2-D plates. The base thickness will be designated as "t 1", and the side-thicknesses will be denoted as "t2". If bending loadings are applied, t2 may need to be thicker than tl, and the original bracket used t2 = 10% thicker than tl. The recommended forging tolerance value of _+ 0.76 mm [-6] was taken to be the _+ 3 standarddeviation ( _+ 3s) value. Both tl and t2 will be considered as preliminary variates. Since the manufacturing method is dictated as forging, the minimal forging thickness are quite important. For the longitudinal dimensions of the bracket, the minimal steel thickness is 4.3 mm for steel and 5.0 mm for titanium [6]. The risk of the displacement B.C. (DBC) is associated with the uncertainty of supports associated with the given configuration. A clamped edge vs. four pinned corners vs. resting on one of the flat surfaces, may yield considerably different stresses and hence different solutions. Hence, another preliminary variate is that of the DBC. For the current sensitivity study, three different levels of severity (LOS) of DBC are associated with the two taped holes in each side of the bracket. The first set of DBC represents the highest constraint conditions, whereas the second is the "most likely" based upon good engineering judgement, and the third offers the least constraint. Hence, BC2 will be assigned a probability of 70%, whereas the other two DBC (LOS = BC1, BC3) are assigned probabilities of 15 % each. Note that the sum of probabilities assigned to the LOS of a variate must sum to unity.

3.2. Results of sensitivity study and selection of primary variates The easiest case for the sensitivity study involves the out-of-phase radial loads since only half the physical problem needs to be modeled. Therefore, the first set of "FE experiments" were for these loads, and the runs were done using either steel or aluminum. A constant mesh (763 3-D ALGOR TM plate-elements) was used, and the ALGOR TM precision for stress (0.5{max. nodal v a l u e - min. nodal value}/global max) measured at the maximum stress (at location C). This measure never exceeded a few percent. The modeling precision was also studied using a finer mesh (1148 elements) and the maximum stress only changed by 0.37%. Fig. 2 shows the influence of both DBC and thicknesses (tl and t2) upon the maximum Von Mises stress. Note that two sets (thin vs thick) of aluminum were run to ensure low stresses (high reliabilities or FOS) using aluminum. Fig. 2 illustrates several trends. First, either increasing t2 by 10% to 20% (average --- 12.5%), or, the additional rotational constraint of BC1, has a minor influence on the maximum stress (Smax) at location C. Secondly, BC3 has a significant influence on Smax-A third trend is that the data follow a quadratic influence (the correlation coefficient of a least squares curve fit R z = Unity implies a perfect correlation) indicating that bending is significant. While the curve-fit equations have very similar coefficients for BC1 and BC2, the coefficients are approximately one-half for the BC3 condition. This indicates a much smaller influence of bending for this DBC. Actually, the vertical sides carry no stress for BC3, and the problem involves contact stress and bending about the Z-axis of the base plate. For a fixed radial force, increasing the thickness is equivalent to a linear reduction

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

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in the applied "nominal bearing pressure (p)". The classical stress concentration KT at a hole is given by smax(C)/P, and changing thickness give essentially the same values (1.687 - 1.705). Note that this trend was also exhibited by the aluminum material. Obviously, the selected DBC must be considered as a principal statistical variate (discrete)• Fig. 3 shows a plot of all the data for all the DBC, and an interesting trend is observed. The values of Smaxfor aluminum and steel appear to overlap together for all three DBC. Consequently, in the second set of FE experiments for the out of phase radial forces, the material was randomly mixed among the various thicknesses. The thickness ratio of t2/tl was also slightly varied to confirm the earlier observation on its minor importance on stresses at location C. The model for this loading direction involved 981 3-D plate elements• Fig. 4 shows the some typical results showing the influence of both location and boundary conditions. Contrary to the results for the symmetrical case, significant stresses can occur at several locations. For BCL location E near the lower L.H.S. boundary constraint exhibited either the

10

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

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largest stress, with location C at circumference of hole at start of loading exhibiting the second largest value. Location B near the joint between the base and side plate, and location D on the circumference of the cutout were about equal in magnitude and about 50% of the highest stress. For BC2, location C was the highest stress, with locations B and D being about 50% of the highest. Note that the precision measure at these ifferent locations ranged from about 1% up to 12%, with most values being around 5%. Fig. 5 shows the influence of material and relative thicknesses upon the value of Smaxat location C. Similar to that of the symmetry case, the material has no significant influence and the relative thicknesses has a minor influence on the stress at position C for all DBC. However, a 10-20% increase in t2 thickness did cause a 25-35% decrease in the maximum BC1 stress in the vertical legs. The additional rotational constraint of BC1 induced Smax to Occur at constraints A and E. However, for BC2 and BC3, the largest stress occurs at location C, which is the same as that of the symmetrical case. Note that the location Smaxis important if the ACF involves fatigue, which will be considered in a companion paper. Comparison of in and out of phase forces shows a significant increase in Smaxat location C for the out of phase forces. Consequently, the phase of the force is a primary statistical variate. The ratio of Smax(non-steel)/smax(Steel) was found to vary from 0.931 to 1.032 (average = 1.003) for all five stress locations and all DBC with the largest influence being at location B. Hence it was concluded that, for single material parts that are subjected to zero-displacement constraints and applied forces, within a range of three in Young's modulus, the material is a insignificant statistical variate in determining the level of stress. Note that for ACF that involve the maximum displacement, such as snap-fits, interacting machine parts, etc., then both the modulus and thickness will be significant statistical variates. However, in this problem, the displacement of the supporting hole was not a design constraint. Also note that, for multiple material parts, such as including a bronze fitting to support the shaft, then the stress will depend upon the modulus of the bracket material.

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

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10.0

Fig. 4. Influence of DBC and location upon the maximum Von Mises stress for in-phase trial loads of 100 N.

Summarizing, the primary statistical variates for the stress are base thickness, with t2 fixed at about ten percent thicker than tl, loading magnitude, phase of the load, and DBC. Since the largest stresses occurred for the out of phase force, its probability of occurrence was taken as a parameter in the reliability study. The variate for the selected ACF is the tensile yield strength Sy. The next section utilizes these variates for the evaluation of the reliabilities.

3.3 Evaluation of tk~e reliabilities The continuous loads, strengths and thickness variation are considered as governed by a Gaussian (normal)distribution. If we also assume that the resultant stresses are normally distributed, then the so-called Z-statistic approach can be utilized. Basically, the strength Sy and stress sc

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

12

¢b

25 BC2 = 61.499

- 12.395x

+ 0.69675x'2

R'2 = 0.999

,,=, fv k-

t2=

20 0

<> BC1,t'itanium r~ BC1,locationA,steel a BC1, loo A, titanium

u~ I 5

+

u~ m TI0 o 0 :>

tl

BCI, s(eel

+ BC,3

B

+ BC2

o

o

I-

R*2 - 1.000 <

0

....

= . . . .

4.0

= ....

5.0

i

6.0

....

i . . . .

7.0

i

~].0

....

9.0

10.0

BASE T H I C K N E S S ( t i ) m m

©

14.0

Q.

£

u~ u3

BCI, LOCATION C 12.0 steel

!

•

I0.0

a'kcaii~m s~eel, t2=

p. u~ u~

1.15 tl

ti(ankam,

~2 = 1.15 tl

8.0

U,J

¢~ E

6.0

>o

4.0 o

X

< T-

4,

2.0

•

4.0

,

•

i

•

6.0

,

•

i

•

8.0

•

•

i

•

I0.0

•

•

i

•

12.0

BASE THICKNESS ( t l )

,

•

i

•

14.0

,

,

16.0

mm

Fig. 5. Influence of material, BCD and relative thicknesses upon the maximum Von Mises stress for in-phase loads of 100N.

(Von Mises "equivalent" stress) variates are replaced by another variate M = Sy - se. Since the sum or difference of normal distributions yields another normal distribution, the reliability is given by another Gaussian function with m e a n value ( M ) = ( S y ) - (s~) and standard deviation sM - [(Ssy)2 + (ss~)2] 1/2, where Ssy and Ss~ are the standard deviations associated with the tensile yield strength and the Von Mises stress respectively. The M-variable is sometimes referred to as the stress or strength margin. Obviously, failure occurs whenever M < 0. In order to readily utilize functions and tables associated with the Gaussian distribution, the M < 0. In order to readily utilize functions and tables associated with the Gaussian distribution, the M-variable is normalized into the so-called "standard variate" (Z - statistic) by the following transformation z = ( M - ( M ) ) / s u . The corresponding reliability integrals of Eqs. (1) and (2) take the following form: R=A

exp d Z~....

-\

Z 2 dz--1-~(Zlower)

(3)

J.N. Majerus et al./Finite Elements in Analysis and Design 28 (1997) 1-17

13

where Z l o w e r = {(M := 0 ) - (M)}/SM, A = lx/(2r 0, and F(z) is the standard integral (see next section). Note that the value of the integral in Eq. (3) can be readily determined from tables or estimating functions. Hence, the reliability can be evaluated once the limit Zlow~ris known. The mean-values of Sy were given earlier, and the corresponding standard deviations [9] were selected as follows: 2024 aluminum = 3.0%, 4340 steel = 5.0 or 9.0%, Ti-6AL-4V = 7.0%. Two different standard deviations were selected for steel since explicit data was not found for the specific heat-treated 4340 steel. The 5.0% corresponds to high carbon, hot-rolled, steel, whereas the 9.0% corresponds to high-strength Type 301 stainless steel. The remaining unknown is the average and standard deviation of the stresses associated with some selected average base thickness. This can be obtained by using the statistical definitions of average and standard deviation assuming that the four primary variates are statistically independent. Independence appears to be a good assumption since the FE "factorial" tests produced approximately parallel curves for stresses vs. thickness vs. DBC vs. phase of loading. Hence, the mean and standard deviations are obtained from Eqs. (4) and (5), respectively,

3 (s~)=

IUL(/) IUL(t) 2

~

~ IdF

dbc=l

,JLL(/) ,JLL(t) p=l

3

IUL(/) IUL(t) 2

dbc=l

,./LLb!) ,JLL(t) p=l

a,°= •

2Ildr

(4)

(5)

where I =f(bcd)f(L)f(t)f(p)se, 11 =f(bcd)f(L)f(t)f(p) {se - (s~)} 2 and t h e f ' s are the frequencydistribution functions and dG represents the product of the incremental values. The summations correspond to the discrete variates and hence fdx corresponds to the assigned probabilities associated with either the DBC. or the phase of the loading. The lower and upper limits correspond to the "tolerances" associated with the desired ISO level of quality. If one data point per 100 is permitted outside t:he tolerance, then the limits correspond to about 2.6 standard deviations. Conversely, if only ene data-point per 104 is permitted, then the limit would be about 3.7 standard deviations. The program methodology for evaluating the bracket reliabilities are shown in Fig. 6. 4. Conclusions

Table 1 shows the results of the reliability calculations associated with the bracket problem. In addition to the three candidate brackets, the tolerance level and percentage of in-phase force were considered as parameters. Several important trends can be observed. The high stresses caused by the in-phase forces cause severe reductions in the part's reliability. Also, including broader tolerance limits reduces the calculated reliability, i.e., the actual part reliability is reduced by accounting for a broader band of uncertainty in the variates. The steel part, at the higher stress levels associated with 75% in-phase forces, show that high variance in material strength can easily reduce the reliability by one order, e.g., 0.9999 becomes 0.999. Surprisingly, even with an FOS greater than 2.0, the high variance steel can exhibit several failures per 104. This is also seen by comparing the reliability of the high-variance steel that of with the titanium alloy. Although, the titanium alloy has a lower FOS, its lower variance produces about half the failures per 105 parts than that of the steel. Similarly, the aluminum parts demonstrate that, when the FOS is on the

14

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

sigma(L,t,D.g.C.,p)

sigma(t,DBC,p)* Load/(test-valuc

=>

of

100

I

I

N)

C u r v e fit t h e FE r e s u l t s f o r m a x i m u m s t r e s s v e r s u s b a s e t h i c k n e s s f o r e a c h d i s p l a c e m e n t boundary

condition

a n d l o a d i n g p h a s e => s i g m a ( t , D B C , p h a s e )

Normalize each continuous lower and upper integration

G a u s s i a n v a r i a b l e x i v i a z i = (x i -t~i ) / c~i limits become standard

T r a n s f o r m t h e z i v a r i a b l e s to G a u s s i a n Q u a d r a t u r e

Approximate

each Integral

using a large integer-number

a n d w e i g h t s w i, w i t h e a c h d i s t r i b u t i o n

normal variate

:l: Z L L

v a r i a b l e s via z i =

zi/ZLL

of Gaussian Quadrature

frequency-function

,

r o o t s ~li

becoming

A w i Z L L exp{- 1/2 ( Z L L q i ) 2 }

S e l e c t t h e t o l e r a n c e level f o r t h e a s s o c i a t e d Z L L , e.g. 0.99 a n d 0 . 9 9 9 c o r r e s p o n d s 3.15 r e s p e c t i v e l y , a n d t h e f r e q u e n c y o f t h e i n - p h a s e f o r c e ( l o u t - o f - p h a s e

to 2.66 a n d

= I - fin-phase )

I

N u m e r i c a l l y e v a l u a t e d E q s (4) a n d (5) u s i n g 4 D o - L o o p s a n d a L o g i c a l R e p e a t

e v a l u a t e Zlowe r a n d F(z) u s i n g t h e f o l l o w i n g a p p r o x i m a t i o n

]

(Rao,1992)

F(z) ~* f(z){ 0 . 3 1 9 3 8 1 5 3 t - 0 . 3 5 6 5 6 3 7 8 2 t 2 + 1 . 7 8 1 4 7 7 9 3 7 t 3 - 1 . 8 2 1 2 5 5 9 7 8 t 4 + 1 . 3 3 0 2 7 4 4 2 9 t 5 }, w i t h f(z) = A * e x p ( - z 2 / 2 ) , t = 1/(1 + 0 . 2 3 1 6 4 1 9 z), a n d e r r o r less t h a n 7.5 x 10 -8

Fig. 6. Methodology used to evaluate reliabilities using FE experiments associated with the four primary stress variates and assumed Gaussian distribution for all the variates

order of 1.5, the reliability is drastically reduced by small increases in the stress. Although the aluminum part shows only 4 failures per 104 at an F O S = 1.66, an F O S of 1.40 corresponds to a b o u t 3 failures per 100 ! Also, at these lower factor of safety levels, increasing the breadth of the tolerance limits (increasing U L and LL in standard deviations) adds several more failures per 100. N o t e that the advantage of considerin9 broader tolerance limits is that the part can be correctly

designed to eliminate these additional failures. The economic decision matrix can now be evaluated based upon the desired quality-control level, and the percentage of in-phase forces. Table 2 shows a typical type of results assuming 75% in-phase and ___3.15 sigma in the tolerance levels. The final weighted utility depends u p o n the

15

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

Table 1 Influence of out-of-phase force and the upper/lower limits of tolerances on the reliabilities of the three candidate brackets Material

25% in-phase

2024_T6 aaluminum

Reliability FOS = Reliability FOS = Reliability FOS = Reliability FOS =

4340 steel b low sigma 4340 steel high sigma Ti-6AL-4V ~

= = = =

Material

75% in-phase

2024_T6 aluminum

Reliability FOS = Reliability FOS = Reliability FOS = Reliability FOS =

4340 steel low sigma 4340 steel high sigma Ti-6AL-4V

= = = =

___ZLL ~ 0.99

0.999601 1.66 0.9999999 3.73 0.9999999 3.73 0.9999999 3.47 _ ZLL ~ 0.99

0.827584 1.21 0.9999912 2.33 0.9999093 2.33 0.9999999 2.23

_+ ZLL ~ 0.999

0.999416 1.63 0.9999999 3.67 0.9999999 3.67 0.9999999 3.41 _+ ZLL ~ 0.999

0.806215 1.19 0.9999869 2.29 0.9998795 2.29 0.9999188 2.19

_+ ZLL ~ 0.9999

0.999404 1.63 0.9999999 3.67 0.9999999 3.67 0.9999999 3.41 +_ ZLL ~ 0.9999

0.8046447 1.19 0.9999865 2.29 0.9998769 2.29 0.9999168 2.18

"Average thicknesses: t2 = 9.30 mm, tl = 8.30 mm. bAverage thicknesses: t2 = tl = 4.30 mm. c Average thicknesses: t2 = tl = 5.00 mm.

assigned weights for each of these p r o d u c t design criteria. T a b l e 2 uses values of 4 0 % , 5 0 % a n d 10% for weight, reliability a n d cost, respectively. This is a "loss" m a t r i x since high weight a n d cost are c o n s i d e r e d non-beneficial, a n d hence the first choice b r a c k e t has the lowest weighted utility. T h e l o g a r i t h m o f (1 - R) is utilized since high reliability is desirable a n d h e n c e a negative "loss". N o t e t h a t the table indicates t h a t the steel material, with low variance, has twice the utility o f the steel with a high variance. H e n c e , an e x a m p l e of the e c o n o m i c justification o f utilizing q u a l i t y c o n t r o l to r e d u c e the v a r i a n c e of the material's yield strength. N o t e that, if the m i n i m u m forging thickness is t a k e n to be the l o w e r limit thickness o f the steel, t h e n the a v e r a g e steel-thickness m u s t a b o u t 5.2 mm. In this case, the increase of reliability does n o t offset the increase in weight, a n d the a l u m i n u m b r a c k e t a n d steel h a v e a b o u t the same weighted utility. In conclusion, this s t u d y has illustrated a design m e t h o d o l o g y for utilizing F E analysis with e c o n o m i c decision m a t r i c e s t h a t can a c c o u n t for the statistical v a r i a t i o n s in the g e o m e t r y , m a n u f a c t u r i n g process a n d material. A sensitivity s t u d y o n a sample p r o b l e m i n d i c a t e d that, using a V o n Mises yield criteria for a single-material p a r t subjected to zero d i s p l a c e m e n t b o u n d a r y conditions, the p r i m a r y variates for d e t e r m i n i n g reliability are: thicknesses, l o a d i n g m a g n i t u d e a n d direction, l o c a t i o n a n d degree o f b o u n d a r y constraints, a n d the yield strength of the material.

16

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

Table 2 Typical e c o n o m i c decision m a t r i x utilizing reliabilities a n d o t h e r p r o d u c t design criteria Bracket

Estimated weight (gr)

M a t e r i a l cost ~

Normalized reliability a

Normalized weight b

Normalized cost c

Weighted utility +

2024-T6 A l u m t l = 8.3 m m t2 = 9.3 m m

71.3

57.4

- 0.146

1.148

2.944

68.1

4340 Steel t l = t2 = 4.30 m Low variance

92.6

19.5

- 1.000

1.490

1.000

19.6

4340 Steel t l = t2 = 4.30 m m High variance

92.6

19.5

- 0.758

1.490

1.000

31.7

62.1

2170 + +

- 0.839

1.000

Ti-6A1-4V t l = t2 = 5.00 m m

111

113

a I n ( u n i t y - R e l i a b i l i t y ) / m a g n i t u d e o f L a r g e s t V a l u e o f ln(1 - R). b E s t i m a t e d w e i g h t / l o w e s t e s t i m a t e d weight. c M a t e r i a l c o s t / l o w e s t m a t e r i a l cost. d {Sum o f the w e i g h t e d n o r m a l i z e d c o l u m n s (4)-(6)} • 10. e Bulk purchase.

Notation

ACF DBC BCD,BCD1,BCD2 EDM GMP FE FOS LOS M MOS PDS tl t2 R Sy

(Sy) Zorz ZLL

analytical criteria for failure displacement boundary conditions specific DBC used in model economic decision matrix set of geometry, material and process finite element factor of safety level of severity variate of stress or strength margin margin of safety product design specification thickness of bracket base thickness of bracket sides Reliability tensile yield strength mean values of tensile yield strength standard variate or Z-statistic normalized lower limit in terms of the number of standard deviations from the mean

J.N. Majerus et al./ Finite Elements in Analysis and Design 28 (1997) 1-17

m Se

(so5 SSy Sse SM

17

mean value of Z-statistic Von Mises equivalent stress mean value of equivalent stress standard deviation of yield strength standard deviation of equivalent stress standard deviation of strength margin.

References [1] G.E. Dieter, Engineering Design: A Materials And Processing Approach, 2nd. ed. McGraw-Hill, New York, 1991. [2] S. Pugh, The application of CAD in relation to dynamic/static product concepts, Proceedings of ICED 83, Vol. 2, Copenhagen, 1983, pp. 564-71. 1-3] S. Pugh, Total Design: Integrated Methods For Successful Product Engineering, Addison-Wesley, Readings MA, 1990. 1-4] B. Prasad, Concurrent Engineering Fundamentals, Vol. 1, Prentice-Hall, Englewood Cliffs, NJ, 1996. [5] P.J. Ross, Taguchi Techniques For Quality Engineering, McGraw-Hill, New York, 1988. 1-6] H.E. Trucks, Designing For Economic Production, 2nd ed., Society of Manufacturing Engineers, Dearborn, MI, 1987. [7] J.G. Bralla, (as editor-in-chief), Handbook Of Product Design For Manufacturing: A Practical Guide To Low-Cost Production, McGraw-Hill, New York, 1986. 1.8] H.P. Bloch, F.K. Geitner, An Introduction To Machinery Reliability Assessment, Van Nostrand Reinhold, 1990. [9] F. Boehm, E.E. Lewis, A stress-strength interference approach to reliability analysis of ceramics: Part I-fast fracture, and Part II-delayed fracture, Probab. Eng. Mech, 7, 92, 1-8, 9-14. [10] Rao, S. Singiresu, Reliability-Based Design, McGraw-Hill, New York, 1992.