Blending hierarchical economic decision matrices (EDM) with FE and stochastic modeling. II. Detailing EDM

Finite Elements in Analysis and Design 30 (1998) 219—234

Blending hierarchical economic decision matrices (EDM) with FE and stochastic modeling. II. Detailing EDM J.N. Majerus!,*, M.L. Mimnagh", J.A. Jannone!, D.A. Tenney#, S.P. Lamphear$ ! Mechanical Engineering Department, Villanova University, Villanova, PA 19085, USA " Advanced Engineering Research Association, 33 S. Delaware Av., Suite 106 Yardly, PA 19067, USA # C & D Charter Power Systems, Washington & Cherry St., Conshocken, PA 19428, USA $ Boeing Defense and Space Group; Helicopter Division, P.O. Box 16858, Mail Stop P34-44, Philadelphia, PA 19142-0858, USA

Abstract The purpose of the detailing EDM is to “fine tune” a previously ameliorated design. First, in order to determine the best fillet radius for formability, forging simulations are conducted using commercial software. Once the fillet dimension is ascertained, the 3-D model is generated and maximum stresses determined for a trial force. Two different commercial programs were used to determine the three-dimensional stresses. The only statistical quantities involve the loading (Gaussian distribution), the material “strength” in the analytical criteria for failure (ACF), and possibly, the boundary conditions in the 3-D models. This paper considers the ACF to be the resistance to fatigue fracture under complete reversal of loads at 5]108 cycles. The paper overviews three different methods of combining stochastic behavior with FE analysis, and presents a methodology for using the interference method with non-symmetrical distributions. The EDM are then presented for the three product criteria of forging formability, prime cost and 3-D reliability with respect to the selected ACF. ( 1998 Elsevier Science B.V. All rights reserved.

1. Introduction Whereas the first paper [1] dealt with combining the classical Z-statistic and FE methods, the current paper deals with the more complicated problem of non-Gaussian distributions of variates. Using statistical variations in thickness, radial force, displacement BC, and yield strengths, the first paper determined the 2-D reliability (w.r.t. Von Mises yield criteria) of aluminum, steel and titanium alloy brackets, where the thickness of steel and titanium were governed by forging minimums rather than the mean radial load of 4000 N. Moreover, for realistic load levels (886 N or

* Corresponding author. E-mail: [email protected]. 0168-874X/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 1 6 8 - 8 7 4 X ( 9 8 ) 0 0 0 1 8 - 3

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200 lbf ) on the small shafts of this bracket, the non-aluminum brackets are considerably overdesigned. Hence, it is difficult to ascertain the influence of quality control (size of variance, and number of standard deviates used in tolerances) on the calculated reliability. Therefore, the current study considers the aluminum bracket subjected to in-phase radial forces with a mean value of 886 N. The method of manufacturing is that of precision, closed-die, (PCD) forging with secondary machining operations used for machining of fillets, and the drilling and taping of the threads. The missing detail from the previous 2-D design involves the fillet radius between the base and upright legs of the bracket. This radius is best considered from a forging viewpoint, since the formability of the bracket is greatly influenced by this radius. The formability is rated via ability to form a defect-free forging, the required tonnage, and type of forging (cold, warm or hot). The forging processing cost also depends upon the “tonnage” of the equipment that must be used, and the forging temperature. Consequently, an appropriate forging simulation model (FE) is required. DEFORMTM is a well-established, commercial code, capable of performing a 3-D analysis of a PCD forging. However, the authors only had access to the microprocessor version DEFORMPCTM, and hence had to restrict the modeling to a planar problem. Therefore, the forging involved a U-shaped bracket. The paper first addresses the problem of selecting the final configuration that offers the best formability and a defect-free forging. Several candidate processes (forging plus machining steps) are selected for evaluation in the EDM. The final configuration is then modeled and analyzed utilizing several different commercial codes (ALGORTM and MSC/NASTRANTM). The stress-interference method is then used to determine the reliability of the configuration. Step-by-step details of using non-normal distributions are given, and a parametric study is conducted to illustrate the influence of various statistical parameters. Lastly, the reliability results are used to both select the best candidate process via an Economic Decision Matrix, and reach some general conclusions.

2. Final configuration via forging simulations The ameliorated bracket involves an unknown fillet radius between the base and vertical sides. The formability and integrity of a precision, closed-die, forging, depends upon both the billet and cavity configuration of the die (the bracket shape), in addition to the forging variables of die speed, lubricant, temperature, etc. Some of the critical cavity dimensions are the size of the fillet and the wall (vertical sides) thicknesses. The wall thickness is greater than the minimum recommended [2,3], and the fillet radius is probably the critical-cavity parameter. Hence, assuming no constraints due to interference with parts mounted between the vertical sides, the fillet radius should be determined via forging simulations. Consequently, a series of 2-D planar analyses are done to estimate the forgability or formability. Formability is a somewhat nebulous term that involves the amount of time to form the object, the required tonnage of the equipment, the required temperature, and the number of steps (intermittent shaped blocker-dies or annealings) that must be done. Good formability is also taken to imply higher values of mechanical properties and a low probability of internal defects. Hence, this section will discuss some numerical experiments (simulations) done to study formability (forgeability) and the selection of a “formability index” or algorithm for usage in the EDM.

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Some important items that can be obtained from the simulated forging are: internal flow of material (associated with enhanced ultimate and fatigue strengths), contours of equivalent (Von Mises) strains and strain rates, and total forces on the moving and constraining dies. The internal flow is observed using a “flow net” that utilizes an initially orthogonal grid (not the FE mesh) whose deformed position is obtained by interpolation from the deformed FE mesh. The flow net is a qualitatively indication of metallurgical flow and provides insight towards improving the surface properties and eliminating forging folds. The strain and strain-rate contours are utilized to minimize the possibility of internal defects. There are numerous methodologies for determining defect generation, ranging from the simple to the complex, and some of these can be found in the publication edited by Dwivedi et al. [4]. The authors selected a simple approach which utilizes forging information provided in the Atlas of Formability [5]. The Atlas’s experimental “formability index” involves a parametric map of the ultimate uniaxial strain as a function of temperature and strain rate. Consequently, once the maximum value of either uniaxial strain or strain rate was reached, the forging step was terminated. Using the above approach for internal defects, initial simulations varied the initial billet shape for a fixed die-cavity configuration. These simulations showed that a multiple step process was required using the same die configuration combined with intermittent annealing. Since annealing removes the internal stresses and strains, this corresponds to “restarting” the forging process using the deformed billet as the initial billet configuration with the internal strains and stresses removed. Since EDM involves the selection from possible “candidates”, “course-of-action” or “events”, four possible forging processes were considered involving the parameters of top-die speed, forging temperature, and fillet radius. Two rectangular billets were used in these processes (68.6 mm long, 25.4 mm wide and either 13.1 or 13.6 mm thick) and Table 1 lists the processing conditions and some computed results. The data show that decreasing the radius by a factor of two, caused the forging force to increase by 60%. This modeling condition required 10 more modeling steps and two more remeshings (done automatically) of the FE grid. Decreasing the billet temperature by 129°C increased the required die force by 145%, and the number of analytical steps increased by a factor of four. As expected, increasing the die speed by a factor of 25 had the largest influence (325%) on the forging force.

Table 1 Input conditions and forging-simulation results for four different cases (alternative course of action) Case

Forging radius (mm)

Billet temp. (°C)

Top die speed (mm/s)

Forging time! (s)

Max. die. force (MN)

Steps"

Formability

1 2 3 4

9.52 4.76 9.52 9.52

500 500 371 500

0.509 0.509 0.509 12.7

9.97 9.52 10.2 0.64

0.15 0.24 0.37 0.66

3 3 3 4

4.69 4.99 5.24 7.62

!Associated with each forging step to reach either maximum strain or strain rate. "Number of annealings followed by reforging using same die cavity.

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The formability index was selected as a simple linear sum involving N (billet temp/932°C &03'*/'4 * #forging time/10.2 s#forging force/650 kN), with each processing variable normalized to the range of zero to unity. In this format, low formability indices are desirable. Using this linearized index, the vastly shorted forging time for the faster die (Cast 4) is offset by the increase in the number of annealings and reforging steps. A more realistic formability index would utilize the details of the forging/annealing facility. Note that the larger radius will be machined to the dimension of the smaller radius, and the bracket has a fixed configuration. Hence, the FE model can now be generated and the elastic stresses determined for a test loading.

3. FE modeling of bracket In this study, the geometric configuration is defined as deterministic, and hence the stresses are stochastic variates only because the applied radial shaft forces are variable. A test loading was selected to be 4.448 N (1 1bf) so that stress variation due to the load variation (mean value of 886 N) can be readily scaled. The radial shaft forces were inclined at 30° w.r.t. the longitudinal X-axis, out-of-phase (both point in the !X, #Z direction), and distributed in a sinusoidal fashion over half-a-hole circumference. The displacement boundary conditions (DBC) were selected to represent the bolting of the bracket to a supporting member. In the ALGORTM model (Model A), 4244 brick (eight-node, hexahedral) elements were used and one row of the outer most top nodes of the vertical sides were subject to º " º " 0. The MSC/NASTRAN model (Model x y B) utilized 7980 brick elements with º "º "0 constraints on two rows of nodes of the x y outermost elements (see Fig. 1). Poisson’s effects were permitted by pinning either the corner, or a line of nodes, in the Z-direction. The MSC/NASTRAN model was also used with the DBC involving all top nodes on the bracket ends (Model C), which restrains the rotation of the vertical brackets and is similar to BC1 in the 2-D study (see Fig. 2). Models A, B and C represent uncertainty due to the DBC. Since Model A was constrained in the Z-direction at only two nodes (but a significant force points in that direction), it should demonstrate large “concentrated type” of stresses at those nodes. Conversely, Models B and C used about twice the elements of Model A and more nodes are used to support the Z-direction forces. Figs. 3 and 4 illustrate some stress results associated with Model B, and the maximum deflection of the model was 0.035 mm. As in the 2-D bracket, there are three critical regions of stress: (a) near the support corners, (b) on the surface of the second shaft-hole at the start of the applied load, and (c) in the fillet region between the base and vertical side. The stresses predicted by both models and various measures of error are shown in Table 2 for selected nodes in these three critical regions. As expected, the highest values are at single line of the pinned supports, and decreases as more nodes are used to support the loads. Note that the fully constrained model C exhibits very low stresses in Region A. Also, the measure of error is extremely high at these constraint nodes (Region A) and indicate poor estimated stresses. If the load fully reverses its direction, the line or two lines of nodes are fully unloaded and the opposite edge carries the load. Consequently, the ACF for these locations would be a yield criteria and the elemental values are on the order of one-half the yield strength. The corresponding (factor of safety) (FOS) is about 2,

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Fig. 1. Three-dimensional “Model B” of bracket with four nodal rows subjected to DBC.

Fig. 2. Displaced “Model C” of bracket showing the restrained nodes.

and the reliability is on the order of 0.9999 (see Ref. [1]). Consequently, Region A is not used in the reliability analysis. Models A and B predicted similar stresses in Region B, and the measures of error are about 5%. This region is identical to that predicted by the 2-D model, and adds credence to the results. Since the mean applied load is only 22% of that used in the 2-D analysis, the stress magnitudes are also of the right order as compared to the 2-D out-of-phase results. If the load reverses its direction, the stresses at this location remain tensile. Consequently, excluding the case of shutting the motor off so that the stresses go to zero, there is only a small alternating component of stress. Therefore,

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Fig. 3. Isostress contours for “Model B” showing the three critical regions of Von Mises effective stress.

Fig. 4. Sectional view of “Model B” showing region B at the loaded surface of the bearing hole and Region C at the start of the fillet.

fatigue fracture is not a problem at this location, and the Yield-FOS is over 4.0. Hence, Region B is not used in the reliability analysis either. Inspection of Region C under reversal of force direction reveals that it changes from tension to compression. This immediately implies a large alternating stress component. The magnitudes are not identical because of the twisting action changes its direction, but the worst fatigue scenario is complete reversal in magnitude which implies zero mean value. Hence, for illustrative purposes, zero mean value will be assumed in this study. The alternating component will be taken as 4.67 Mpa (average of the two FE analysis) per 100 N per radial shaft. Note that uncertainty due to modeling DBC could readily be considered as a statistical variate.

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Table 2 Comparison of numerical results from finite element models for trial load of 100 N per shaft Parameter

Nodal values of Model A

Region A: max. Von Mises stress Measure of error % Region B: max. Von Mises stress Measure of error % Region C: max. Von Mises stress Measure of error %

37.57 Mpa!

27.7 Mpa!

15.3 Mpa

25.8 6.90 Mpa

32.6 6.20 Mpa

14.2 4.31 Mpa

31.8 4.27 Mpa

3.4 5.07 Mpa

6.8 4.28 Mpa

1.9 NA

39.8 NA

(1.0 over fillet cross section 0.031 mm

NA

NA

Max. deflection

Max of 1.0 over fillet cross section 0.035 mm

Nodal values of Model B

Element values of Model B

NA

Nodal values of Model C 4.30 Mpa!

0.014 mm

!See text for discussion of the reasons for these differences in the stress.

4. Statistical reliability In this section, a brief overview of various methods of combining stochastic behavior with FE is presented. Then, some comments concerning the ACF for fatigue are presented, followed by an application of the stress-interference method to the selected ACF. 4.1. A brief overview of three methods of combining stochastic behavior with finite elements Reviews of the standard methods of combining statistical or stochastic methods with design or analysis can be found in the four texts by Thoft-Christensen and Baker [6], Melchers [7], Rao [8], and Vanmarcke [9]. The first three texts consider the engineering views that either the geometry or the physical properties are statistical variates, and consider the reliability of either single parts, or multiple parts making up a structural product. The text by Vanmarcke is not concerned with reliability per se, but the representation and evaluation of the interactions between randomly varying fields (one would be the design object and another would be the environmental conditions). Two other important references are by Nakagiri and Hisada [10,11] who extended the concept of randomly varied fields combined with FE. Overall, there are three methods contained within these six references for determining “statistical results”: (a) the reliability index (b), (b) the interference between “loads” and “strength”, and (c) stochastic finite elements (SFEM). In SFEM, the geometric domain is represented by a set of finite elements, whose average value and standard deviation are represented by correlated local statistics, whereas the global properties depend upon the global averages, etc. This general methodology can yield the local approximation associated with the linearized reliability index (method a). Representative of SFEM are the recent work of Zhu and Wu [12] who studied the influence of stochastic Young’s modulus and density

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upon the coefficient of variation (COV"standard deviation/average value) of eigenvalues, and Vanmarcke [13] who reviewed the methodology and its application to uncertainty in experimental measurements. Vanmarcke applied the first-order, second-moment, approximation to a seepage problem and showed the importance of using Bayesian probability methods to correctly update a state of knowledge. More recently, Chakraborty and Dey [14] used SFEM by considering the stiffness matrix and load vector per se as the statistical variates. The average deflection and COV of deflection are computed directly by splitting the variates into a sum of random and average parts, and the resulting stiffness matrix is approximately inverted by Neumann expansion. The random parts are modeled via zero mean, homogeneous, isotropic Gaussian fields. After obtaining the statistical mean and COV for assumed variations via SFEM, these statistical values must be utilized in methods (a) or (b) in order to determine the “reliability”. Conversely, the reliability-index b (method a) deals directly with reliability and the design parameters associated with a set of parts or a structure. This method attempts to determine the mean (average) variates associated with the optimal design located near the most likely failure “point” (in variate space). It has been widely used in the space industry, for example, see the NASA Memorandums of Shiao and Chamis [15], and Abumeri et al. [16]. Recent overviews of the reliability index can be found in the papers by Reddy et al. [17], and Shen and Shen [18]. Shen and Shen mention ongoing work at NASA Lewis Research Center to develop computer programs that integrate finite-element structural analysis with probabilistic algorithms, such as nonlinear evaluation of stochastic structures under stress (NESSUS). Shen and Shen extend this method to the problem of mixed mode (Modes I and II) crack propagation to be used for the ACF. Reddy, Grandhi and Hopkins attempt to reduce the computational cost by the safety-index optimization approach which combines an interpolation method with the advanced mean-value method and the fast probability integration technique developed by Cruse and associates [19—21]. Recently, Gopalakrishna and Donaldson [22] reformulated the minimization of b2, subject to a failuresurface constraint G(Z )"0, to an optimization problem that could be solved using standard FE i optimization programs. Method b, the so-called “interference method”, uses two of the key steps associated with the above reliability-index method. These steps are the normalization of the statistical variates using the “standard deviate Z”, and the estimation of “design sensitivity” to changes in each variate. The design sensitivity utilized by the MSC/NASTRAN method was reviewed recently by Riha et al. [23], and Ahammed and Melchers [24] illustrated that parametrics via incrementing the COV of each variate can readily determine whether or not the reliability is sensitive to the variability of a specific variate. The interference method also uses the same definition of reliability as the reliability-index method, i.e., the joint probability that the critical design variables (EDMN analytical criteria for failure MACFN) are each higher that the minimum acceptable or constraint value. However, the interference method is usually applied to reliability of single-parts since the number of variates is small and the joint-probability integrals can be directly integrated. The reliability index method relies on normal distributions for final evaluation of reliability from the calculated Z-value. However, the interference method can easily consider non-Gaussian distributions, e.g., see the application by Boehm and Lewis to the reliability of ceramic parts [25]. The authors also believe that the interference method is more comprehensible to designers and easy to adapt for quantitative evaluation of quality control via EDM. Consequently, the interference method is used in this study.

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4.2. Comments on the fatigue criteria for ACF Since the loads can reverse direction in this design problem, product failure can occur by the generation and growth of fatigue cracks. A good overview of fatigue failure and various criteria can be found in the texts of Shigley and Mischke [26], or Rao [8]. A significant statistical fact is that the distribution of fatigue strength is highly skewed, and the log-normal and Weibull distributions have been widely used to represent the data. Also, the COV (range of 0.2—0.5) is an order-ofmagnitude higher than that observed for mechanical properties, e.g., see the work of Schijve [27] or Lasserre and Froustey [28]. As discussed by Shigley and Mischke, a variety of stress criteria have been proposed for analysis to prevent fatigue fracture, and they take the form of

A B A B

p B p A . # K ! "1, &S S % 6-5

(1)

where constants A and B are 1.0 for the Modified Goodman, or 2.0 for the so-called non-linear criteria, S is the ultimate tensile strength, K is the fatigue—stress concentration factor, S is the 6-5 & % fatigue life (at some specified number of cycles), p is the mean (average per time cycle) and p is the . ! alternating amplitude. Fatigue fracture is taken to occur when the equation is satisfied, and a deterministic design is “safe” when the LHS of Eq. (1) is less than unity. Examples of current research concerning an appropriate “stress measure” for the non-uniaxial case are: Du Quesnay et al. [29] promoting the effective stress range, Berkovits and Fang [30] utilizing a non-dimensional parameter similar to Eq. (1), and Froustey and Lasserre [28] demonstrating that a distorsionalenergy-based criteria matches the experimental observation (for steels) concerning (p )2#3(q )2"constant at fracture. Of particular interest for our investigation are the works of ! ! Berkovitz and Fang [30], and Du Quesnay et al. [29], who studied aluminum alloys. The current paper considers the ACF to be the resistance to fatigue fracture under complete reversal of loads at 5]108 cycles. The selected criteria is that of Modified Goodman, and the stress measure will be that of the Von Mises equivalent stress (p ). Hence, p corresponding % . 0.5(p #p ) and p corresponding 0.5(p !p ) at some critical nodal location and %.!9 %.*/ ! %.!9 %.*/ evaluated over one reversal of load cycle. Note that the FE analysis determines the product K p where K is the geometric T T stress—concentration factor. This factor must be reduced by the so-called notch-sensitivity factor (q) which, for aluminum, depends mainly upon the “stress state” and the radius of curvature. The stress state refers to axial, bending or torsional stress, with torsion showing a measurable difference in notch sensitivity. Obviously, for this bracket problem, the stress-state involves all three of these “components”, and an effective notch sensitivity is written as q " q #(q !q )*%Torsion. % 1 2 1 Using the fillet radius of 4.76 mm, the values of q and q are 0.85 and 0.95, respectively. Analysis of 1 2 the stress field at the critical fillet node reveals about 15% axial, 5% shear (considered as torsion) and 80% bending. Therefore, the reliability analysis will use these values as %axial, %torsion and %bending, respectively. Consequently, q "0.855 is used as the average notch-sensitivity factor, % which has a COV of 10% and is assumed to be a Gaussian variable. All values of the equivalent stress will be multiplied by this factor. Modifications to the handbook fatigue strength (124 Mpa @ 150 C°) are considered via the non-dimensional Marin factors (k ) as modified by Miskchke [31] for stochastic considerations, i

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i.e., 4 S " < k S (¹), % i 6-5 i/1

(2)

where the k are the statistical variates and S (¹) corresponds to a deterministic ultimate strength i 6-5 at the design temperature. The indices refer to the surface finish, size effects, stress state, and loading-proportionality factor, respectively. Although these Marin factors were developed for steel materials, aluminum fatigue data (the Military Standards-HDBK-5D, and the Atlas of Fatigue Curves [32]) shows similar type of influences. Hence, for demonstration purposes, the steel relationships of Eq. (2) and COVs will be utilized. Therefore, k "aS" where Ma,bN "M39.9,!0.995N and Ma,bN "M2.7, 1 6-5 &03'%$ .!#)*/%$ !0.2653N are considered as deterministic, whereas k is stochastic with a COV"0.08 for forged 1 and 0.06 for machined. Because of the localized nature of the stress at location C, the size-effect factor k can be taken as deterministic and equal to unity. The equivalent stress-state factor (mean 2 factor and COV) is written in terms of the percentages of “stress states” as, k

"%torsion* 0.774#%axial* 0.583#%bending, 3%2

COV "%torsion* 0.16#%axial* 0.12. 3

(3) (4)

Note that the bending value of k is considered deterministic with a COV of zero. Similarly, the 3%2 loading-proportionality factor is scaled from the aluminum fatigue data under axial and reversed bending as k

"M%axial*0.18#%torsion*0.25#%bending*0.32NS (¹), 4%2 6-5

COV "%axial*0.27#%torsion*0.31#%bending*0.15 4

(5) (6)

and the COVs quoted by Roe [8] for steel are assumed for the aluminum. Now that the basic ACF equations are known, the next section illustrates how to determine the reliability using the interference definition and the FE stress-analysis results. Various problems associated with using the modified Goodman/Marin factors with FE analysis are also discussed. 4.3. Application of the stress-interference method to the modified Goodman criteria There are essentially three ways of using Eq. (1) in a reliability analysis involving the interference between “stress” and “strength”. If the material properties are considered as deterministic, then Eq. (1) can be rewritten in the form Stochastic stress (X )"Constant strength S, 453%44

(7)

where X "Ap !Bp , with A"1#S /S , and B"S /S !1, and S"2S . Con453%44 %.!9 %.*/ 6-5 % 6-5 % 6-5 versely, if the “stress” is considered as deterministic, Eq. (1) takes the form Constant stress (,p )"Stochastic strength (,S [1!p /S ]). ! % . 6-5

(8)

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If both stress and material properties are considered as stochastic, then Stochastic stress(,p /S )"Stochastic strength(,1!p /S ). (9) ! % . 6-5 Hence, for our situation of zero mean, the stochastic stress is p and the stochastic strength is S . ! % Due to the large uncertainties and skewness in the fatigue strengths, the stochastic strength will be represented by a Beta-distribution function, f (x)"(x!LL)p~1(UL!x)q~1/MB(p, q) (UL!LL)p`q~1N,

(10)

where B( p.q)"C( p)C(q)/C( p#q), LL"lowest possible value of x, UL"largest possible value of x. After some algebra, the p and q parameters can be related to the known statistical values of COV and average value SxT as p"M(SxT!LL)#(UL!SxT)*[(1!LL/SxT)/COV]2N/(UL!LL),

(11)

q"pMp[(UL!LL)/(SxT!LL)]!1N*COV2/(1!LL/SxT)2.

(12)

The beauty of this distribution is that it can have a non-negative lower bound, a finite upper bound, and can represent highly skewed experimental behavior. The normal or Gaussian distribution is assumed for the loading and notch-sensitivity factor. Now that the distribution functions are selected, the reliability of the part is found from the evaluation of the following integral equation:

P

P

UL(4*'.!)

AP

B

UL(453%/'5)) f (strength) dx dx , (13) 453%/'5) 4*'.! LL(4*'.!) 4*'.! where sigma"q p (FE) load/8.86, and strength"k k k S . The easiest way to get the distri% %! 1 3%2 4%2 6-5 bution of strength is to directly use Eqs. (10)—(12) with the definition that SstrengthT" Sk TSk TSk TSS T and the propagation-of-error law for products and sums of independent 1 3%2 4%2 6-5 variates that states COV "[(COV )2#(COV )2#(COV )2]1@2. 453%/'5) 1 3%2 4%2 Conversely, the integral over sigma corresponds to a double integral over the upper and lower ranges of the Gaussian variables of radial shaft loads and the notch-sensitivity factor. Numerical examples indicate that each Gaussian integral is best integrated over Z via the following -*.*5 approximation: R"

f (sigma)

UL

1 j0& */5%37!-4 2+ + Interval width exp[!(Z )2/2], j #%/5%3 0& j5) */5%37!J2p j/1 LL

(14)

where the Z-range of 2*Z is divided into a number of equal intervals. Fig. 5 outlines a scheme -*.*5 for the numerical evaluation of Eq. (13) which uses the approximation of Eq. (14)

5. Results and summary Table 3 lists some typical results obtained for the forged bracket. The average fatigue strength for the bracket is 80.1 vs. 102.4 Mpa for forged vs. machined radius. However, the reliability varies greatly depending on the assumed upper and lower limits of this strength. For the forged finish and

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Fig. 5. Methodology used to evaluate reliability from Eq. (13). Table 3 Influence of stress, surface finish and the upper (UL) and lower (LL) limits of fatigue strength on the predicted reliability Reliability for forged! surface finish

Reliability for machined" surface finish

Alternating Von Mises stress# Mpa

LL/UL of fatigue strength

0.5778 0.6579 0.6936 0.8600 0.9537 0.9703 0.9963 0.99960 0.99980 NA NA NA

NA NA 0.7542 0.8877 0.9628 0.9771 0.9972 0.99964 0.99985 0.99969 0.99980 0.99981

44.9 41.4 44.9 41.4 37.9 41.4 41.4 41.4 41.4 44.9 41.4 41.1

35/95 35/95 40/120 40/120 40/120 45/120 50/120 55/120 60/120 60/120 60/140 80/140

!SS T"80.1 Mpa, standard deviation"15.4 Mpa. % "SS T"102.4 Mpa, standard deviation"19.0 Mpa. % #Mean-value for the average-load of 886 N per shaft, Z

"4 for the stress integrals. -*.*5

a symmetric distribution (40/120 corresponds to $2.7 SD about mean), the predicted reliability is only 0.860. Conversely, for the machined finish and a symmetric distribution (60/140 corresponds to $2.22 SD about mean), the reliability (0.9998) is in the desired reliability range. Fixing the upper limit at 120 (handbook-strength without loading or surface finish effects), and slowly

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dropping the lower limit (LL) shows small changes in reliability until LL is about 45 Mpa. This LL corresponds to about the mean stress plus one SD. At the 40/120 limit (!3 SD#1 SD) assumed for the strength range, the machined reliability drops to only 0.8877. The corresponding distribution is highly skewed to the low side and hence the large change in reliability. Note that the average and SD are the same in both of these machinedradius reliabilities. Similarly, for a forging whose strength is highly skewed to the low side (35/95N!3 SD#1 SD), the reliability drops to only 0.6579. Conversely, highly skewing the distribution to the high side (60/120N!1.30 SD#2.60 SD) drastically increases the forging reliability to 0.99980. Obviously, the tails of the distributions are critical in the calculated reliability. One last observation involves the influence of changing stress for a fixed strength (LL/UL and surface finish). At high reliabilities, a 10% increase in stress causes a small decrease in the reliability. However, at the lower reliability, a 10% increase in stress can cause a large reduction in reliability since the mean stresses now approach the lower limit of the strengths. Now that the reliabilities are known, the detailing EDM can be evaluated. For illustrative purposes, the product criteria were selected to be: (a) formability, (b) Prime cost and (c) 3-D reliability w.r.t fatigue fracture. The prime cost was determining using a typical cost model for the manufacturing cost plus the material cost based upon bulk purchase. The manufacturing consisting of a three-step forging/annealing process followed by two secondary fabrication processes: (a) drilling of small holes (4.0! @ shallow and 12! @ deep) and tapping of threads (4.0! @), (b) an end-milling operation (0.2! per cubic mm) for machining of the fillet radius and “clearance”. The assumed cost per annealing is a model based upon weight, e.g., 0.15* billet cost. The forging processing cost depends upon the “tonnage” (assuming a ton"104 N) of the equipment that must be used and the in-house formula is (machine#operator#dies)!"3.0!#0.1! *(Tonnage N]10~4)0.7. Note that the cost increases with tonnage, and this is another reason why tonnage was a parameter in the previous “formability index”. The U-shaped forged bracket will have to be end-milled to produce the semi-circular notch. However, in an actual 3-D forging, this notch will be formed via the forging process. Consequently, the projected bracket cost does not include this expensive operation. The corresponding prime cost estimations are given in Table 4. Note that the forging cost excludes the pro-rating of the die costs. Now that the product criteria are known for each of the alternative, the ameliorated bracket can be obtained. This involves selecting the bracket with the lowest weighted utility. Table 5 shows typical results of this approach. The different cases refers to different forging or machining steps. The higher reliabilities of Cases 1, 3 and 4 far outweigh the lower cost of Case 1, and yield considerably lower utilities. Recall that the shaft forces are only 22% of the forces used for refining EDM with the ACF of Von Mises Yield [1]. However, some of the reliabilities are considerably lower w.r.t. an ACF of fatigue fracture. In summary, the detailing EDM involves the usage of concurrent manufacturability models (FE simulations) with numerical “experiments” to obtain the lowest cost that is compatible with the desired reliability of the product. In addition to ascertaining an ameliorated bulk material, material processing, and secondary processing steps, FE simulations also allow one to study the possible development of internal defects in forging/casting/extrusion processes. This permits fine tuning of the material alloy, the part’s geometry and the manufacturing process to prevent these “flaw developments”. Properly used, this greatly increases the reliability of the finished product. In the

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Table 4 Estimated prime cost for the four cases (alternative course of action) Case

Billet cost (!)

Forging cost (!)

Annealing cost (!)

Drill/tap screw holes

Drill shaft holes

Endmilling cost

Total (prime) cost (!)

1 2 3 4

70.2 68.0 70.2 70.2

29.4 28.7 31.5 50.5

10.5 10.2 10.2 21.1

64.0 64.0 64.0 64.0

8.0 8.0 8.0 8.0

333.6 138.6 333.6 333.6

515.7 317.5 517.5 547.4

Table 5 Detailing EDM results associated with 3-D bracket Case

Prime cost (!)

Reliability

Formability

Weighted utility!

1 2 3 4

515.7 317.5 517.5 547.4

0.9998 0.8877 0.9998 0.9998

4.69 4.99 5.24 7.69

!55.0 4.00 !50.0 !46.0

!Sum of M0.15 Col. 2/Min. Col. 2#0.8 ln(1!Col. 3)/Min. Col. 3#0.05 Col. 4/Min. Col. 4N 100.

EDM approach, the trade-off between quality control and prime part cost can be studied via FE simulations of manufacturing coupled with TSD analyses. Another synergistic benefit of the coupled manufacturing/TSD analyses is that material “flow lines” and solidification profiles can be used to study the alignment of metallurgical grains which decrease the notch sensitivity of the part to the generation of surface fatigue cracks. Also, the cost effectiveness of either enhancing residual stresses (e.g., shot peening) or removing residual stresses (e.g., annealing) upon the overall product reliability can be readily accessed.

6. Software used in investigation ALGORTM ver. 11.08H, ALGOR Inc., 150 Beta Drive, Pittsburg, PA, 15238 MSC-NASTRANTM for Windows, MacNeal-Schwendler Corp., 815 Colorado Blvd, Los Angeles, CA 90041 DEFORM-PCTM ver. 2.0, Scientific Forming Technologies Corp., 700 Ackerman Rd., Suite 255, Columbus, OH 43202

Nomenclature ACF COV

analytical criteria for failure, standard deviation/average value,

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DBC EDM FE FOS K 46"4#3*15 LL LOS M MOS PDS SFEM Subscripts q 46"4#3*15 R S 6-5 S % t1 t2 TSD UL X 46"4#3*15 Z or z Z LL %Axial %Torsion S2T Greek letters b C k p % p ! p .

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displacement boundary conditions, economic decision matrix, finite element, factor of safety, nondimensional Marin-factors, lower limit of variable, level of severity, variate of stress or strength margin, margin of safety, product design specification, stochastic finite elements,

notch-sensitivity under specific stress state and radius of curvature, reliability, tensile ultimate strength, fatigue strength at 5]108 cycles, thickness of bracket base, thickness of bracket sides, temperature, stress, deformation, upper limit of variable, stochastic variable, standard variate or Z-statistic, normalized lower limit in terms of the number of standard deviations from the mean, percentage of axial stress, percentage of shear stress, mean value of inclosed variable,

reliability index, gamma function, mean value of Z-statistic, Von Mises equivalent stress, alternating component of Von Mises equivalent stress, mean component of Von Mises equivalent stress.

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