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of the 3-parameter Weibull distribution. The procedure for the 3-parameter Weibull distribution is more complex, as you would expect, due to the distribution being modelled by three rather than two parameters. Essentially it requires the determina- tion of the expected minimum value, xo, a location parameter on the x-axis. As shown in Appendix X, the linear recti®cation equations are a function of lnx i ÿ xo where xo < x min , the minimum variable value on the data. We don't know the value of xo initially, but by searching for a value of xo such that when lnx i ÿ xo is plotted against ln ln1=1 ÿ F i , the correlation coecient is its highest value, will give a reasonably accurate answer. The process can be easily translated to computer code to speed up the process. Determining the parameters for the common distributions can also be done by hand using suitably scaled probability plotting paper, a straight line through the data points being determined `by eye', as described earlier. See Lewis (1996) for examples of probability plotting graph paper for some of the statistical distributions mentioned. Further improvement in the selection of the best linear recti®cation model can be performed by comparing the uncertainty in model ®t as determined from the standard error of the data (Nelson, 1982). Also, the use of con®dence limits in determining the uncertainty in the estimates from linear regression is useful for assessing the nature of the data, particularly when small samples are taken and/or when outlying data points control the gradient of the regression line. Con®dence limits are generally wider than some inexperienced data analysts expect, so they help avoid thinking that estimates are closer to the true value than they really are. A discussion of their application in data analysis can be found in Ayyub and McCuen (1997), Comer and Kjerengtroen (1996), Nelson (1982), and Rice (1997). Example ± ®tting a Normal distribution to a set of existing data We will next demonstrate the use of the linear recti®cation method described above by ®tting a Normal distribution to a set of experimental data. The data to be analysed is in the form of a histogram given in Figure 4.9. It shows the distribution of yield strength for a cold drawn carbon steel (SAE 1018). The data is taken from ASM (1997a), a reference that provides data in the form of histograms for several important mechanical properties of steels. Data collated in this manner has been chosen for analysis because a designer may have to resort to the use of data from existing sources. Also, the analysis of this case raises some interesting questions which may not necessarily be met when analysing data collated in practice and displayed using the methods described in Appendix I. After a visual inspection, it is evident that the SAE 1018 yield strength data has a distribution approaching the Normal type, although there is an abnormally high fre- quency value around the mid-range of the data. Further analysis of the data as shown in Table 4.2 using the cumulative frequency modelling approach yields Figure 4.10. Note that the mean rank equation is used to determine the plotting positions on the y-axis, F i , the x-axis plotting positions being the mid-class values for the yield strength in MPa. The class width w 13:8 MPa. The values determined graphically for the mean and standard deviation are also shown. The estimated cumulative frequency ®ts the data well, where in fact the curve is modelled with a ®fth order polynomial using commercial curve ®tting software. Statistical methods for probabilistic design 145 (Commercial software such as MS Excel is useful in this connection being widely available.) Omissions in the ranked values of F i in Table 4.2 re¯ect the omissions of the data in the original histogram for several classes. As can be judged from Figure 4.10, inclusion of the cumulative probabilities for these classes would not follow the natural pattern of the distribution and are therefore omitted. However, when a very low number of classes exist their inclusion can be justi®ed. Linear recti®cation of the cumulative frequency, F i , is performed by converting to the Standard Normal variate, z. The linear plot together with the straight line equation through the data and the correlation coecient, r, is shown in Figure 4.11. From Figure 4.11, it is evident that the mean is 530 MPa because the regression line crosses the Standard Normal variate, z, at 0 representing the 50 percentile or median in the non-linearized domain. The mean and standard deviation can also be found from the relationships given in Appendix X. For the Normal distribution Figure 4.9 Yield strength histogram for SAE 1018 cold drawn carbon steel bar (ASM, 1997a) Table 4.2 Analysis of histogram data for SAE 1018 to obtain the Normal distribution plotting positions Mid-class (MPa) Frequency ( f ) Cumulative freq. (i) F i i N 1 z È ÿ1 SND F i (x-axis) (N 52) ( y-axis) ( y-axis) 431.0 1 1 0.01887 ÿ2.08 444.8 0 1 458.6 2 3 0.05660 ÿ1.59 472.4 3 6 0.11321 ÿ1.21 486.2 2 8 0.15094 ÿ1.03 500.0 3 11 0.20755 ÿ0.82 513.8 4 15 0.28302 ÿ0.57 527.6 5 20 0.37736 ÿ0.31 541.4 13 33 0.62264 0.31 555.2 5 38 0.71698 0.58 569.0 5 43 0.81132 0.88 582.8 5 48 0.99566 1.32 596.6 3 51 0.96226 1.78 610.4 0 51 624.2 0 51 638.0 1 52 0.98113 2.08 146 Designing reliable products we can calculate the mean and standard deviation from: ÿ A0 A1 ÿ ÿ11:663 0:022 530:14 MPa 1 ÿ A0 A1 A0 A1 1 11:663 0:022 ÿ11:663 0:022 45:45 MPa The conclusion is that the Normal distribution is an adequate ®t to the SAE 1018 data. A summary of the Normal distribution parameters calculated from Figures 4.10 and 4.11 and other values for the mean and standard deviation from various sources (including commercial software and a package developed at Hull University called FastFitter à ) are given in Table 4.3. The frequency distributions derived from the Normal distribution parameters from source are shown graphically overlaying the original histogram in Figure 4.12 for comparison. It can be seen from Table 4.3 that there is no positive or foolproof way of determin- ing the distributional parameters useful in probabilistic design, although the linear recti®cation method is an ecient approach (Siddal, 1983). The choice of ranking equation can also aect the accuracy of the calculated distribution parameters using the methods described. Reference should be made to the guidance notes given in this respect. The above process above could also be performed for the 3-parameter Weibull distribution to compare the correlation coecients and determine the better ®tting distributional model. Computer-based techniques have been devised as part of the approach to support businesses attempting to determine the characterizing distributions Figure 4.10 Cumulative frequency distribution for SAE 1018 yield strength data à The FastFitter software is available from the authors on request. Statistical methods for probabilistic design 147 from sample data. As shown in Figure 4.13, the users screen from the software, called FastFitter, is the selection of the best ®tting PDF and its parameters representing the sample data, here for the yield strength data for SAE 1018. The software selects the best distribution from the seven common types: Normal, Lognormal, 2-parameter Weibull, 3-parameter Weibull, Maximum Extreme Value Type I, Minimum Extreme Value Type I and the Exponential distribution. Using the FastFitter software, it is found that the 3-parameter Weibull distribution gives the highest correlation coecient of all the models, at r 0:995, compared to r 0:991 for the Normal distribution. The mean and standard deviation in Table 4.3 for FastFitter are calculated from the Weibull parameters, the relevant information is provided in Appendix IX. 4.2.3 The algebra of random variables Typically, if the stress or strength has not been taken directly from the measured distribution, it is likely to be a combination of random variables. For example, a Table 4.3 Normal distribution parameters for SAE 1018 from various sources Source of Normal distribution parameters Mean, Standard deviation, Reference (Mischke, 1992) 541 41 FastFitter 532 44 Moment calculations 537 41 Normal linear recti®cation 530 45 Cumulative. freq. graph 534 40 Commercial software 545 38 Average 537 42 Figure 4.11 Normal distribution linear recti®cation for SAE 1018 yield strength data 148 Designing reliable products failure governing stress is a function of the applied load variation and maybe two- or three-dimensional variables bounding the geometry of the problem. The mathematical manipulation of the failure governing equations and distributional parameters of the random variables used to determine the loading stress in particular are complex, and require that we introduce a new algebra called the algebra of random variables. We need this special algebra to operate on the engineering equations as part of probabilistic design, for example the bending stress equation, because the parameters are random variables of a distributional nature rather than unique values. When these random variables are mathematically manipulated, the result of the operation is another random variable. The algebra has been almost entirely developed with the application of the Normal distribution, because numerous functions of random variables are normally distributed or are approximately normally distributed in engineering (Haugen, 1980). Engineering variables are found to be either statistically independent or correlated in some way. In engineering problems, the variables are usually found to be unrelated, for instance a dimensional variable is not statistically related to a material strength (Haugen, 1980). Table 4.4 shows some common algebraic functions, typically with one variable, x, or two statistically independent random variables, x and y. The mean and standard deviation of the functions are given in terms of the algebra of Figure 4.12 Normal distributions from various sources for SAE 1018 yield strength data Statistical methods for probabilistic design 149 random variables. Where the variables x and y are correlated in some way, with correlation coecient, r, several common functions have also been included. When a function, , is a combination of two or more statistically independent variables, x i , then equation 4.5 can be eectively used to determine their combined variance, V (Mischke, 1980). V % n i 1 @ @x i 2 Á 2 x i 1 2 n i 1 @ 2 @x 2 i 23 2 Á 4 x i 4:5 To determine the mean value, , of the function : % x 1 ; x 2 ; FFF; x n ÿÁ 1 2 n i 1 @ 2 @x 2 i Á 2 x i 4:6 Equation 4.5 is exact for linear functions, but should only be applied to non-linear functions if the random variables have a coecient of variation, C v < 0:2 (Furman, 1981; Morrison, 2000). If this is the case, then the approximation using just the ®rst term only diers insigni®cantly from using higher order terms (Furman, 1981). However, for a function whose ®rst derivative is very small, the higher terms cannot be ignored (Bowker and Lieberman, 1959). Approximate solu- tions for the mean and standard deviation, , are provided by omitting the higher order terms, for example equation 4.5 is often written as: % n i 1 @ @x i 2 Á 2 x i 23 0:5 4:7 Data Correlation coefficient Distribution parameters Figure 4.13 FastFitter analysis of SAE 1018 yield strength data 150 Designing reliable products and % x 1 ; x 2 ; FFF; x n ÿÁ 4:8 Equation 4.7 is referred to as the variance equation and is commonly used in error analysis (Fraser and Milne, 1990), variational design (Morrison, 1998), reliability Table 4.4 Mean and standard deviation of statistically independent and correlated random variables x and y for some common functions Function () Mean ( ) Standard deviation ( ) x x x x 2 2 x 2 x 2 x Á x Á 1 0:25 x x 2 ! x 3 3 x 3 2 x Á x 3 x Á 2 x Á 1 x x 2 ! x 4 4 x 6 2 x Á 2 x 4 x Á 3 x Á 1 9 4 x x 2 ! x n n x Á 1 0:5nn ÿ 1 x x 2 ! n Á x Á n ÿ1 x Á 1 0:25n ÿ 1 2 x x 2 ! x 0:5 0:5 x 1 ÿ 1 8 x x 2 ! x Á 0:5 x 2 x 1 1 16 x x 2 ! 1 x 1 x 1 x x 2 ! x 2 x 1 x x 2 ! 1 x 2 1 2 x 1 3 x x 2 ! 2 x 3 x 1 9 4 x x 2 ! 1 x 3 1 3 x 1 6 x x 2 ! 3 x 4 x 1 4 x x 2 ! x Æy x Æ y 2 x 2 y 0:5 2 x 2 y Æ 2r Á x Á y 0:5 x Á y x Á y x Á y r Á x Á y 2 x Á 2 y 2 y Á 2 x 2 x Á 2 y 0:5 2 x Á 2 y 2 y Á 2 x 2 x Á 2 y 1 r 2 0:5 x y x y 2 y Á x 3 y 1 y 2 x Á 2 y 2 y Á 2 x 2 y 2 x 0:5 x y y Á x 2 y y y ÿ r Á x x x y Á 2 x 2 x 2 y 2 y ÿ 2r Á x Á y x Á y 0:5 Statistical methods for probabilistic design 151 analysis (Haugen, 1980) and sensitivity analysis (Parry-Jones, 1999). Most impor- tantly in probabilistic design, through the use of the variance equation we have a means of relating geometric decisions to reliability goals by including the dimensional and load random variables in failure governing stress equations to determine the stress random variable for any given problem. The variance equation can be solved directly by using the Calculus of Partial Derivatives, or for more complex cases, using the Finite Dierence Method. Another valuable method for `solving' the variance equation is Monte Carlo Simulation. However, rather than solve the variance equation directly, it allows us to simulate the output of the variance for a given function of many random variables. Appendix XI explains in detail each of the methods to solve the variance equation and provides worked examples. The variance for any set of data can be calculated without reference to the prior distribution as discussed in Appendix I. It follows that the variance equation is also independent of a prior distribution. Here it is assumed that in all the cases the output function is adequately represented by the Normal distribution when the random variables involved are all represented by the Normal distribution. The assumption that the output function is robustly Normal in all cases does not strictly apply, particularly when variables are in certain combination or when the Lognormal distribution is used. See Haugen (1980), Shigley and Mischke (1996) and Siddal (1983) for guidance on using the variance equation. The variance equation provides a valuable tool with which to draw sensitivity inferences to give the contribution of each variable to the overall variability of the problem. Through its use, probabilistic methods provide a more eective way to determine key design parameters for an optimal solution (Comer and Kjerengtroen, 1996). From this and other information in Pareto Chart form, the designer can quickly focus on the dominant variables. See Appendix XI for a worked example of sensitivity analysis in determining the variance contribution of each of the design variables in a stress analysis problem. 4.3 Variables in probabilistic design Design models must account for variability in the most important design variables (Cruse, 1997b). If an adequate characterization of these important variables is performed, this will give a cost-eective and a fairly accurate solution for most engineering problems. The main engineering random variables that must be ade- quately described using the probabilistic approach are shown in Figure 4.14. The variables conveniently divide into two types: design dependent, which the designer has the greatest control over, and service dependent, which the design has `limited' control over. Typically, the most important design dependent variables are material strength and dimensional variability. Material strength can be statistically modelled from sample data for the property required, as previously demonstrated; however, diculties exist in the collation of information about the properties of interest. Dimensional variability and its eects on the stress acting on a component can be great, but information is typically lacking about its statistical nature and its impact on geometric stress concentration values is rarely assessed. 152 Designing reliable products Important service dependent variables are related to the loading of the component and stresses resulting from environmental eects. These are generally dicult to determine at the design stage because of the cost of performing experimental data collection, the nature of overloading and abuse in service, and the lack of data about service loads in general. Also, the eect that service conditions have on the material properties is important, the most important considerations arising from extremes in temperature, as there is a tendency towards brittle fracture at low temperatures, and creep rupture at high temperatures. To this end, it has been cited that the quality control of the environment is much more important than quality control of the manufacturing processes in achieving high reliability (Carter, 1986). Among the most dramatic modi®ers encountered in design of those mentioned above are due to thermal eects on strength and stress concentration eects on local stress magnitudes in general (Haugen, 1980). As seen from Figure 4.14, there are several important design dependent variables that terms of an engineering analysis are then: . Material strength (with temperature and residual processing eects included) . Dimensional variability . Geometric stress concentrations . Service loads. 4.3.1 Material strength The largest design dependent strength variable is material strength, either ultimate tensile strength (Su), uniaxial yield strength (Sy), shear yield strength ( y ) or some Figure 4.14 Key variables in a probabilistic design approach Variables in probabilistic design 153 other failure resisting property. For de¯ection and instability problems, the Modulus of Elasticity (E) is usually of interest. Shear yield strength, typically used in torsion calculations, is a linear function of the uniaxial yield strength and is likely to have the same distribution type (Haugen, 1980). With mass produced products, extensive testing can be carried out to characterize the property of interest. When production is small, material testing may be limited to simple tension tests or perhaps none at all (Ayyub and McCuen, 1997). Material properties are often not available with a sucient number of test repetitions to provide statistical relevance, and remain one of the challenges of greater application of statistical methods, for example in aircraft design (Smith, 1995). Another problem is how close laboratory test results are to that of the material provided to the customer (Welling and Lynch, 1985), because material properties tend to vary from lot to lot and manufacturer to manufacturer (Ireson et al., 1996). However, this can all be regarded as making the case for a probabilistic approach. Ideally, information on material properties should come from test specimens that closely resemble the design con®guration and size, and tested under conditions that duplicate the expected service conditions as closely as possible (Bury, 1975). The more information we have about a situation before the trial takes place and the data collected, the more con®dence there will be in the ®nal result (Leitch, 1995). One of the major reasons why design should be based on statistics is that material properties vary so widely, and any general theory of reliability must take this into account (Haugen and Wirsching, 1975). Material properties exhibit variability because of anisotropy and inhomogeneity, imperfection, impurities and defects (Bury, 1975). All materials are, of course, processed in some way so that they are in some useful fabrication condition. The level of variability in material properties associated with the level of processing can also be a major contribution. There are three main kinds of randomness in material properties that are observed (Bolotin, 1994): . Within specimen ± inherent within the microstructure and caused by imperfec- tions, ¯aws, etc. . Specimen to specimen ± caused by the instabilities and imperfections of the manufacturing processes with the batch. . Batch to batch ± natural variations due to processing, such as material quality, equipment, operator, method, set-up and the environment. Other uncertainties associated with material properties are due to humidity and ambient chemicals and the eects of time and corrosion (Farag, 1997; Haugen, 1982b). Brittle materials are aected additionally by the presence of imperfections, cracks and internal ¯aws, which create stress raisers. For example, cast materials such as grey cast iron are brittle due to the graphite ¯akes in the material causing internal stress raisers. Their low tensile strength is due to these ¯aws which act as nuclei for crack formation when in tensile loading (Norton, 1996). Subsequently, brittle materials tend to have a large variation in strength, sometimes many times that of ductile materials. Strain rate also aects tensile properties at test. An increasing strain rate tends to increase tensile properties such as Su and Sy. However, a high loading rate tends to promote brittle fracture (Juvinall, 1967). The average strain rate used in obtaining a 154 Designing reliable products [...]... (Dieter, 19 86; Haugen, 19 80; Smith, 19 95 ; Welling and Lynch, 19 85), which 16 4 Designing reliable products Table 4.7 Empirical factors for determining standard deviation based on tolerance No of parts manufactured Factor 4 to 5 10 25 10 0 500 to 700 1 1.5 2 2.5 3 relates to a maximum Process Capability Index, Cp 1 This estimate does not take into account process shift, typically 1: 5 from the target... than static properties (Bury, 19 74) For example, Cv 0:7 has been cited for the creep time to fracture for copper (Yokobori, 19 65) Although little statistical data has been found on the properties highlighted, creep strength data and properties at high temperatures for various materials can be found in ASM ( 19 97 a), ASM ( 19 97 b), Furman ( 19 80) and Waterman and Ashby ( 19 91 ) Many mechanical components... reliability and long life of a product and the act of assigning tolerances in fact ®nalizes reliability (Dixon, 19 97 ; Rao, 19 92 ; Vinogradov, 19 91 ) Large tolerances and/ or large variance can result in signi®cant degradation of reliability because the failure probability is a function of the magnitude of dimensional variability and tolerance allocated, and aects load induced stress in a component (Haugen, 19 80;... hours, which ranges from 0. 01 to 1% deformation in 10 00 hours; and the nominal 700 600 Stress (MPa) 500 Modulus of Elasticity (GPa) 400 250 300 200 15 0 200 10 0 10 0 50 0 0 0 10 0 200 300 400 500 Temperature (°C) Figure 4 .15 Mechanical properties of a low carbon steel as a function of temperature (adapted from Waterman and Ashby, 19 91 ) Variables in probabilistic design 15 9 Figure 4 .16 Short-term tensile strength... comm., 19 98 ) Table 4.5 shows the coecient of variation, Cv , for various material properties at room temperature compiled from a number of sources (Bury, 19 75; Haugen, 19 80; Haugen and Wirsching, 19 75; Rao, 19 92 ; Shigley and Mischke, 19 96 ; Yokobori, 19 65) Further insight into the statistical strength properties of some commonly used metals is provided by a data sheet in Table 4.6 Again caution should... 4340 (BS 817 M40) Structural steel BS Grade 43C Stainless steel BS 316 S16 Aluminium alloy 7075-T6 Titanium alloy Ti-6Al-4V Su Sy Sy Cold drawn 517 27 447 36 Normalized 506 25 ± ± Cold drawn 604 40 540 41 Hot rolled 594 27 342 26 Cold drawn 812 49 658 45 Cold drawn annealed 1 10 mm Hot rolled t 16 mm Sheet annealed t 3 mm Sheet aged 803 9 ± ± Bar ± ± 324 16 5 79 20 ± ± 555 27.5 484 22 93 4 46 90 0 50 Finally,... 4. 21 which is based on the bilateral tolerance, t, and various empirical factors as shown in Table 4.7 (Dieter, 19 86; Haugen, 19 80; Smith, 19 95 ) The factors relate to the fact that the more parts produced, the more con®dence there will be in producing capable tolerances: t
4: 21 Factor i Historically, in probabilistic calculations, the standard deviation, , is expressed as t=3 (Dieter, 19 86;... v2 =v1 P 1 ÿ exp ÿ ÿ xo
4 :11 where: xo expected minimum value: A high shape factor in the 2-parameter model suggests less strength variability The Weibull model can also be used to model ductile materials at low temperatures which exhibit brittle failure (Faires, 19 65) (See Waterman and Ashby ( 19 91 ) for a detailed discussion on modelling brittle material strength.) 15 6 Designing reliable products. .. references such (ASM, 19 97 a; ASM, 19 97 b; Haugen, 19 80; Mischke, 19 92 ) is the best available to the designer who requires rapid solutions An example of such data was shown in Figure 4 .9 Although the property data strictly applies to US grade ferrous and non-ferrous materials, conversion tables are available which show equivalent material grades for UK, French, German, Swedish and Japanese grades However,... tooling and production errors (Evans, 19 75), and relies heavily on the tolerances being within Æ3 during inspection and process control Unless there is 10 0% inspection, however, there will be some dimensions that will always be out of tolerance (Bury, 19 74) In equations 4 . 19 and 4.20, improved estimates for the standard deviation are presented based on empirical observations This is shown in Figure 4 . 19 . ÿ0.82 513 .8 4 15 0.28302 ÿ0.57 527.6 5 20 0.37736 ÿ0. 31 5 41. 4 13 33 0.62264 0. 31 555.2 5 38 0. 716 98 0.58 5 69. 0 5 43 0. 811 32 0.88 582.8 5 48 0 .99 566 1. 32 596 .6 3 51 0 .96 226 1. 78 610 .4 0 51 624.2 0 51 638.0. F i i N 1 z È 1 SND F i (x-axis) (N 52) ( y-axis) ( y-axis) 4 31. 0 1 1 0. 018 87 ÿ2.08 444.8 0 1 458.6 2 3 0.05660 1. 59 472.4 3 6 0 .11 3 21 1. 21 486.2 2 8 0 .15 094 1. 03 500.0 3 11 0.20755. 51 638.0 1 52 0 .9 811 3 2.08 14 6 Designing reliable products we can calculate the mean and standard deviation from: ÿ A0 A1 ÿ 11 :663 0:022 530 :14 MPa 1 ÿ A0 A1 A0 A1 1
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