THÔNG TIN TÀI LIỆU
more so when process shift is taken into account, as shown by the term H . Anticipation of the shift or drift of the tolerances set on a design is therefore an important factor when predicting reliability. It can be argued that the dimensional variability estimates from CA apply to the early part of the product's life-cycle, and may be overconservative when applied to the useful life of the product. However, variability driven failure occurs throughout all life-cycle phases as discussed in Chapter 1. From the above arguments, it can be seen that anticipation of the process capability levels set on the important design characteristics is an important factor when predict- ing reliability (Murty and Naikan, 1997). If design tolerances are assigned which have large dimensional variations, the eect on the reliability predicted must be assessed. Sensitivity analysis is useful in this respect. Stress concentration factors and dimensional variability Geometric discontinuities increase the stress level beyond the nominal stresses (Shigley and Mischke, 1996). The ratio of this increased stress to the nominal stress in the component is termed the stress concentration factor, Kt. Due to the nature of manufacturing processes, geometric dimensions and therefore stress concentra- tions vary randomly (Haugen, 1980). The stress concentration factor values, however, are typically based on nominal dimensional values in tables and handbooks. For example, a comprehensive discussion and source of reference for the various stress concentration factors (both theoretical and empirical) for various component con®gurations and loading conditions can be found in Pilkey (1997). Underestimating the eects of the component tolerances in conditions of high dimensional variation could be catastrophic. Stress concentration factors based on nominal dimensions are not sucient, and should, in addition, include estimates based on the dimensional variation. This is demonstrated by Haugen (1980) for a notched round bar in tension in Figure 4.20. The plot shows Æ3 con®dence limits on the stress concentration factor, Kt, as a function of the notch radius, generated using a Monte Carlo simulation. At low notch radii, the stress concentration factor predicted could be as much as 10% in error from the results. The main cause of mechanical failure is by fatigue with up to 90% failures being attributable. Stress concentrations are primarily responsible for this, as they are among the most dramatic modi®ers of local stress magnitudes encountered in design (Haugen, 1980). Stress concentration factors are valid only in dynamic cases, such as fatigue, or when the material is brittle. In ductile materials subject to static loading, the eects of stress concentration are of little or no importance. When the region of stress concentration is small compared to the section resisting the static load, localized yielding in ductile materials limits the peak stress to the approximate level of the yield strength. The load is carried without gross plastic distortion. The stress concentration does no damage (and in fact strain hardening occurs) and so it can be ignored, and no Kt is applied to the stress function. However, stress raisers when combined with factors such as low temperatures, impact and materials with marginal ductility could be very signi®cant with the possibility of brittle fracture (Juvinall, 1967). For very low ductility or brittle materials, the full Kt is applied unless information to the contrary is available, as governed by the sensitivity index of the material, q s . For example, cast irons have internal discontinuities as severe as the stress raiser Variables in probabilistic design 165 itself, q s approaches zero and the full value of Kt is rarely applied under static loading conditions. The following equation is used (Edwards and McKee, 1991; Green, 1992, Juvinall, 1967; Shigley and Mischke, 1996): K H 1 q s Kt ÿ 14:22 where: K H actual stress concentration factor for static loading q s index of sensitivity of the material (for static loading ÿ 0.15 for hardened steels, 0.25 for quenched but untempered steel, 0.2 for cast iron; for impact loading ÿ 0.4 to 0.6 for ductile materials, 0.5 for cast iron, 1 for brittle materials): In the probabilistic design calculations, the value of Kt would be determined from the empirical models related to the nominal part dimensions, including the dimensional variation estimates from equations 4.19 or 4.20. Norton (1996) models Kt using power laws for many standard cases. Young (1989) uses fourth order polynomials. In either case, it is a relatively straightforward task to include Kt in the probabilistic model by determining the standard deviation through the variance equation. The distributional parameters for Kt in the form of the Normal distribution can then be used as a random variable product with the loading stress to determine the ®nal stress acting due to the stress concentration. Equations 4.23 and 4.24 show Figure 4.20 Values of stress concentration factor, Kt, as a function of radius, r, with Æ3 limits for a circumferentially notched round bar in tension [d $ N(0.5, 0.00266) inches, r 0:00333 inches] (adapted from Haugen, 1980) 166 Designing reliable products that the mean and standard deviation of the ®nal stress acting (Haugen, 1980): L Kt L Á Kt 4:23 L Kt 2 L Á 2 Kt 2 Kt Á 2 L 2 L Á 2 Kt 0:5 4:24 By replacing Kt with K H in the above equations, the stress for notch sensitive materials can be modelled if information is known about the variables involved. 4.3.3 Service loads One of the topical problems in the ®eld of reliability and fatigue analysis is the prediction of load ranges applied to the structural component during actual operating conditions (Nagode and Fajdiga, 1998). Service loads exhibit statistical variability and uncertainty that is hard to predict and this in¯uences the adequacy of the design (Bury, 1975; Carter, 1997; Mùrup, 1993; Rice, 1997). Mechanical loads may not be well characterized out of ignorance or sheer diculty (Cruse, 1997b). Empirical methods in determining load distribution are currently superior to statisti- cal-based methods (Carter, 1997) and this is a key problem in the development of reliability prediction methods. Probabilistic design then, rather than a deterministic approach, becomes more suitable when there are large variations in the anticipated loads (Welling and Lynch, 1985) and the loads should be considered as being random variables in the same way as the material strength (Bury, 1975). Loads can be both internal and external. They can be due to weight, mechanical forces (axial tension or compression, shear, bending or torsional), inertial forces, elec- trical forces, metallurgical forces, chemical or biological eects; due to temperature, environmental eects, dimensional changes or a combination of these (Carter, 1986; Ireson et al., 1996; Shigley and Mischke, 1989; Smith, 1976). In fact some environ- ments may impose greater stresses than those in normal operation, for example shock or vibration (Smith, 1976). These factors may well be as important as any load in conventional operation and can only be formulated with full knowledge of the intended use (Carter, 1986). Additionally, many mechanical systems have a duty cycle which requires eectively many applications of the load (Schatz et al., 1974), and this aspect of the loading in service is seldom re¯ected in the design calcu- lations (Bury, 1975). Failures resulting from design de®ciencies are relatively common occurrences in industry and sometimes components fail on the ®rst application of the load because of poor design (Nicholson et al., 1993). The underlying assumption of static design is that failure is governed by the occurrence of these occasional large loads, the design failing when a single loading stress exceeds the strength (Bury, 1975). The overload mechanism of mechanical failure (distortion, instability, fracture, etc.) is a common occurrence, accounting for between 11 and 18% of all failures. Design errors leading to overstressing are a major problem and account for over 30% of the cause (Davies, 1985; Larsson et al., 1971). The designer has great responsibility to ensure that they adequately account for the loads anticipated in service, the service life of a product being dependent on the number of times the product is used or operated, the length of operating time and how it is used (Cruse, 1997a). Variables in probabilistic design 167 Some of the important considerations surrounding static loading conditions and static design are discussed next. Initially focusing on static design will aid the development of the more complex dynamic analysis of components in service, for example fatigue design. Fatigue design, although of great practical application, will not be considered here. The concepts of static design The most signi®cant factor in mechanical failure analysis is the character of loading, whether static or dynamic. Static loads are applied slowly and remain essentially constant with time, whereas dynamic loads are either suddenly applied (impact loads) or repeatedly varied with time (fatigue loads), or both. The degree of impact is related to the rapidity of loading and the natural frequency of the structure. If the time for loading is three times the fundamental natural frequency, static loading may be assumed (Juvinall, 1967). Impact loading requires the structure to absorb a given amount of energy; static loading requires that it resist given loads (Juvinall, 1967) and this is a fundamental dierence when selecting the theory of failure to be used. An analysis guide with respect to the load classi®cation is presented in Table 4.8. A static load, in terms of a deterministic approach, is a stationary force or moment acting on a member. To be stationary it must be unchanging in magnitude, point or points of application and direction (Shigley and Mischke, 1989). It is a unique value representing what the designer regards as the maximum load in practice that the product will be subjected to in service. Kirkpatrick (1970) de®nes a limit of strain rate for static loading as less than 10 ÿ1 but greater than 10 ÿ5 strain rate/second. This ®ts in with the strain rate at which tensile properties of materials are tested for static conditions (10 ÿ3 ). With regard to probabilistic design, a load can also be considered static when some variation is expected (Shigley and Mischke, 1989). Static loading also requires that there are less than 1000 repetitions of the load during its designed service (Edwards and McKee, 1991), which introduces the concept of the duty cycle appropriate to reliability engineering. Static loads induce reactions in components and equilibrium usually develops. Where the total static loading on a mechanical system arises from more than one independent source, a statistical model combining the loads may be written for the resultant loading statistics using the algebra of random variables. The correlation of the loading variables is important in this respect. That is the load on the component may be the function of another applied load or associated somehow (Haugen, 1980). For static design to be valid in practice, we must assume situations where there is no deterioration of the material strength within the time period being considered for the loading history of the product. With a large number of cyclic loads the material will eventually fatigue. With an assumed static analysis, stress rupture is the mechanism of failure to be considered, not fatigue. The number of stress cycles in a problem could Table 4.8 Loading condition and analysis used (Norton, 1996) Load type M/c element Constant Time-varying Stationary Static analysis Dynamic analysis Moving Dynamic analysis Dynamic analysis 168 Designing reliable products be ignored if the number is small, to create a steady stress or quasi-static condition, but one where a variation in loading still exists (Welling and Lynch, 1985). Although, if the loading is treated as a random variable, then this could imply a dynamic analysis (Freudenthal et al., 1966). While static or quasi-static loading is often the basis of engineering design practice, it is often important to address the implications of repeated or ¯uctuating loads. Such loading conditions can result in fatigue failure, as mentioned above (Collins, 1981). In a typical load history of a machine element, most of the applied loads are relatively small and their cumulative material damage eects are negligible. When the applied loading stresses are above the material's equivalent endurance limit, the resulting accu- mulation of damage implies that the component fails by fatigue at some ®nite number of load repetitions. Relatively large loads occur only occasionally suggesting their cumulative damage eects are negligible (Bury, 1975). Evidently, failure will occur as soon as a single load exceeds the value of the applicable strength criterion. Some typical load histories for mechanical components and systems are shown in Figure 4.21. The load (dead load, pressure, bending moment, etc.) is subject to variation in all cases, rather than a unique value, the likely shape of the ®nal Figure 4.21 Typical load histories for engineering components/systems Variables in probabilistic design 169 distribution shown schematically to the right of each load history. Even permanent dead loads that should maintain constant magnitude show a relatively small and slow random variation in practice. At the other end of the loading type spectrum are transient loads which occur infrequently and last for a short period of time ± an impulse, for example extreme wind and earthquake loads (Shinozuka and Tan, 1984). The above would assume that the load distributions for static designs are often highly unsymmetrical, indicating that there is a small proportion of loads that are relatively large (Bury, 1975). During the conditions of use, environmental and service variations give rise to temporary overloads or transients causing failures (Klit et al., 1993). Data collected from mechanical equipment in service has shown that these transient loads developed during operation may be several times the nominal load as shown in Figure 4.22. There is very little information on the variational nature of loads commonly encountered in mechanical engineering. Several references provide guidance in terms of the coecient of variation, C v , for some common loading types as shown in Table 4.9 (Bury, 1975; Ellingwood and Galambos, 1984; Faires, 1965; Lincoln Figure 4.22 Transient loads may be many times the static load in operation (Nicholson et al., 1993) Table 4.9 Typical coecient of variation, C v , for various loading conditions Loading condition C v Aerodynamic loads in aircraft 0.012 to 0.04 Spring force 0.02 Bolt pre-load using powered screwdrivers 0.03 Powered wrench torque 0.09 Dead load 0.1 Hand wrench torque 0.1 Live load 0.25 Snow load 0.26 Human arm strength 0.3 to 0.4 Wind loads 0.37 Mechanical devices in service 0.5 170 Designing reliable products et al., 1984; Shigley and Mischke, 1996; Smith, 1995; Woodson et al., 1992). These values are representative of the variation experienced during typical duty cycles. These values can only act as a guide in mechanical design and should be treated with caution and understanding, but they do indicate that loads vary quite consider- ably, when you observe that the ultimate tensile strength, Su, of steel generally has a C v 0:05. The smaller the variance in design parameters, the greater will be the reliability of the design to deal with unforeseen events (Suh, 1990), and this would certainly apply to the loads too due to the diculty in determining the nature of overloading and abuse in service at the design stage. Example ± determining the stress distribution using the coef®cient of variation When dimensional variation is large, its eects must be included in the analysis of the stress distribution for a given situation. However, in some cases the eects of dimen- sional variation on stress are negligible. A simpli®ed approach to determine the likely stress distribution then becomes available. Given that the mean load applied to the component/assembly is known for a particular situation, the loading stress can be estimated by using the coecient of variation, C v , of the load and the mean value for the stress determined from the stress equation for the failure mode of concern. For example, suppose we are interested in knowing the distribution of stress associated with the tensile static loading on a rectangular bar (see Figure 4.23). It is known from a statistical analysis of the load data that the load, F, has a coecient of variation C v 0:1. The stress, L, in the bar is given by: L F ab Figure 4.23 Tensile loading on a rectangular bar Variables in probabilistic design 171 where: F load a and b mean sectional dimensions of the bar: Assuming the variation encountered in dimensions `a'and`b' is negligible, then the mean stress, L , on the bar is: L F a  b 100 000 0:03  0:05 66:67 MPa The load coecient of variation C v 0:1. Rearranging the equation for C v to give the standard deviation of the loading stress yields: L C v  L 0:1  66:67 6:67 MPa The stress, L, can therefore be approximated by a Normal distribution with par- ameters: L $ N66:67; 6:67MPa Probabilistic considerations For many years in mechanical design, load variations have been masked by using factors of safety in a deterministic approach, as shown below (Ullman, 1992): . 1 to 1.1 ± Load well de®ned as static or ¯uctuating, if there are no anticipated over- loads or shock loads and if an accurate method of analysing the stress has been used. . 1.2 to 1.3 ± Nature of load is de®ned in an average manner with overloads of 20 to 50% and the stress analysis method may result in errors less than 50%. . 1.4 to 1.7 ± Load not well known and/or stress analysis method of doubtful accuracy. These factors could be in addition to the factors of safety typically employed as discussed at the beginning of this section. Because of the diculty in ®nding the exact distributional nature of the load, this approach to design was considered adequate and economical. It has been argued that it is far too time consuming and costly to measure the load distributions comprehensively (Carter, 1997), but this should not prevent a representation being devised, as the alternative is even more reprehensible (Carter, 1986). Practically speaking, deciding on the service loads is frequently challenging, but some sort of estimates are essential and it is often found that the results are close to the true loads (Faires, 1965). Since static failures are caused by infrequent large loads, as discussed above, it is important that the distributions that model these extreme events at the right-hand side of the tail are an accurate representation of actual load frequencies. Control of the load tail would appear to be the most eective method of controlling reliability, because the tail of the load distribution dictates the shape of the distribution and hence failure rate (Carter, 1986). The Normal distribution is usually inappropriate, although some applied loads do follow a Normal distribution, for example rocket motor thrust (Haugen, 1965) or the gas pressure in the cylinder heads of reciprocating engines (Lipson et al., 1967). Other loads may be skewed or possess very little scatter 172 Designing reliable products and in fact load spectrums tend to be highly skewed to the left with high loads occurring only occasionally as shown in Figure 4.24 (Bury, 1974). Civil engineering structures are usually subjected to a Lognormal type of load, but in general the Normal and 3-parameter Weibull distributions are commonly employed for mechanical components, and the Exponential for electrical components (Murty and Naikan, 1997). It is useful in the ®rst instance to describe the load in terms of a Normal distribution because of the necessity to transform it into the loading stress parameters through the variance equation and dimensional variables. Although the accuracy of the variance equation is dependent on using to near-Normal distributions and variables of low coecient of variation (C v < 0:2), it is still useful when the anticipated loads have high coecients of variation. Even if the distribution accounting for the static load is Normal, the stress model is usually Lognormal (Haugen, 1980) due to the complex nature of the variables that make up the stress function. Experimental load analysis Information on load distributions was virtually non-existent until quite recently although much of it is very rudimentary, probably because collecting data is very expensive, the measuring transducers being dicult to install on the test product or prototype (Carter, 1997). It has been cited that at least one prototype is required to make a reliability evaluation (Fajdiga et al., 1996), and this must surely be to under- stand the loads that could be experienced in service as close as possible. In experimental load studies, the measurable variables are often surface strain, acceleration, weight, pressure or temperature (Haugen, 1980). A discussion of the techniques on how to measure the dierent types of load parameters can be found in Figliola and Beasley (1995). The measurement of stress directly would be advantageous, you would assume, for use in subsequent calculations to predict reliability. However, no translation of the dimensional variability of the part could then be accounted for in the probabilistic model to give the stress distribution. A better test would be to output the load directly as shown and then use the appropriate probabilistic model to determine the stress distribution. A key problem in experimental load analysis is the translation of the data yielded from the measurement system (as represented by the load histories in Figure 4.21) into an Figure 4.24 A loading stress distribution with extreme events Variables in probabilistic design 173 appropriate distributional form for use in the probabilistic calculations. Measuring the distribution of the load peak amplitudes is a useful model for both static and dynamic loads, as peaks are meaningful values in the load history (Haugen, 1980). A much sim- pler method, described below, analyses the continuous data from the load history. Assuming that the statistical characteristics of a load function, xt, are not changing with time, then we can use the load±time plot, as shown in Figure 4.25, to determine the PDF for xt. The ®gure shows a sample history for a random process with the times for which x xt x dx, identi®ed by the shaded strips. During the time interval, T, xt lies in the band of values x to x dx for a total time of dt 1 dt 2 dt 3 dt 4 . We can say that if T is long enough (in®nite), the PDF or f x is given by: f xdx fraction of total elapsed time for which xt lies in the x to x dx band dt 1 dt 2 dt 3 FFF T n i 1 dt i T 4:25 The fraction of time elapsed for each increment of x can be expressed as a percentage rounded to the nearest whole number for use in the plotting procedure to ®nd the characterizing distributional model. Estimation of the distribution parameters and the correlation coecient, r, for several distribution types is then performed by using linear recti®cation and the least squares technique. The distributional model with a correlation coecient closest to unity would then be chosen as the most appropriate PDF representing the load history. The above can be easily translated into a computer code providing an eective link between the prototype and load model. See the case study later employing this technique for statistically modelling a load history. For discrete data resulting from many individual load tests, for example spring force for a given de¯ection as shown in Figure 4.26, a histogram is best constructed. The optimum number of classes can be determined from the rules in Appendix I. Again, the best distribution characterizing the sample data can be selected using the approach in Section 4.2. Figure 4.25 Determination of the PDF for a random process (adapted from Newland, 1975) 174 Designing reliable products [...]... stress, L, and which has a strength, S It is assumed that both L and S are random variables with known PDFs, represented by f
S and f
L (Disney et al., 1968) The probability of failure, and hence the reliability, can then be estimated as the area of interference between these stress and strength functions (Murty and Naikan, 19 97) SSI is recommended for situations where a considerable potential for... this loading stress value covers 99. 87% those applied in service: Lmax 350 3
40 470 MPa Because the strength distribution is Normal, we can determine the Standard Normal variate, z, as: x ÿ 470 ÿ 500 z ÿ0:6 50 From Table 1 in Appendix I, the probability of failure P 0: 274 253 The reliability, R, is given by: R 1 ÿ P 1 ÿ 0: 274 253 RLmax 0 :72 574 7 The reliability, RLmax , as a function... approaches by Bury (1 974 ), Carter (19 97) and Freudenthal et al (1966) Using Carter's approach ®rst, from equation 4. 47 we can calculate LR to be: 40 LR p 0:62 2 50 402 We already know SM 2.34 because it is the positive value of the Standard Normal variate, z, calculated above The probability of failure per application of load 1 87 188 Designing reliable products p % 0:0009... readily available and presented in terms of the mean and standard deviation for the property of interest (Pheasant, 19 87; Woodson et al., 1992) Many studies have been undertaken to collate this data, particularly in the US armed forces and space programmes Anthropometric data is also provided for population weights For example, Cv 0:18 for the weight of males aged between 18 and 70 (Woodson et al.,...
S f
L dL F
L A2 P
L < S1 ÿI Figure 4. 27 SSI theory applied to the case where loading stress does not exceed strength
4:29 177 178 Designing reliable products The probability of these two events occurring simultaneously is the product of equations 4.28 and 4.29 which gives the elemental reliability, dR, as:
S dR f
S1 dS1 Á f
L dL
4:30 ÿI For all possible values of strength, the... q
4:38 2 2 L S where: z Standard Normal variate S mean material strength S material strength standard deviation L mean applied stress L applied stress standard deviation: 2 s3 2 1 CL ln S 2 L 1 CS z ÿ q  à 2 2 ln
1 CL
1 CS
4:39 179 180 Designing reliable products Figure 4.28 Derivation of the coupling... the mean, 65 kg, and the standard deviation, 11 :7 kg Therefore, at 3, you would expect 1 in 1350 males in this age range to be over 100 kg in weight, as determined from SND theory In the US, this probability applies to males of 118 kg in weight The designer should use this type of data whenever human interaction with the product is anticipated 176 Designing reliable products Table 4.10 Human... structural reliability, and is the most commonly used method because of its simplicity, ease and economy (Murty and Naikan, 19 97; Sundararajan and Witt, 1995) It is a practical engineering tool used for quantitatively predicting the reliability of mechanical components subjected to mechanical loading (Sadlon, 1993) and has been described as a simulative model of failure (Dasgupta and Pecht, 1991) The... sucient empirical data (Kapur and Lamberson, 1 977 ; Verma and Murty, 1989) 4.4.3 Reliability determination with multiple load application The approach taken by Carter (1986, 19 97) to determine the reliability when multiple load applications are experienced (equation 4.34) is ®rst to present a Safety Margin, SM, a non-dimensional quantity to indicate the separation of the stress and strength distributions... stress and strength interference conditions and where `n' is much greater than 1 The ®nal reliability is given by (O'Connor, 1995): Rn
1 ÿ pn where: p probability of failure per application of load `n':
4:48 Stress±Strength Interference (SSI) analysis Figure 4.30 Relative shape of loading stress and strength distributions for various loading roughnesses and arbitrary safety margin Bury (1 974 ; 1 975 ; . 4.25 Determination of the PDF for a random process (adapted from Newland, 1 975 ) 174 Designing reliable products Given the lack of available standards and the degree of diversity and types of load application,. 0.1 Hand wrench torque 0.1 Live load 0.25 Snow load 0.26 Human arm strength 0.3 to 0.4 Wind loads 0. 37 Mechanical devices in service 0.5 170 Designing reliable products et al., 1984; Shigley and. heads of reciprocating engines (Lipson et al., 19 67) . Other loads may be skewed or possess very little scatter 172 Designing reliable products and in fact load spectrums tend to be highly skewed
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