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loading stress is given by: FLexp ÿexp ÿ x ÿ L  L 4:50 and the strength, f S, is given by a Normal distribution, for example by: f S 1 S 2 p exp ÿ x ÿ S 2 2 S 2 4:51 the reliability is given by equation 4.33 by substituting in these terms: R n I 0 exp ÿexp ÿ x ÿ L  L ! Á 1 S 2 p exp ÿ x ÿ S 2 2 S 2 ! dx 4:52 which can be solved numerically using Simpson's Rule as shown in Appendix XII. The numerical solution of equation 4.35 is sucient in most cases to provide a reasonable answer for reliability with multiple load applications for any combination of loading stress and strength distribution (Freudenthal et al., 1966). 4.4.4 Reliability determination when the stress is a maximum value and strength is variable The assumption that a unique maximum loading stress (i.e. variability) is assigned as being representative in the probabilistic model when variability exists in strength sometimes applies, and this is treated as a special case here. The problem is shown in Figure 4.32. We can refer to this maximum stress, L max , from the beginning of application until it is removed. Several time dependent loading patterns may be treated as maximum loading cases, for example the torque applied to a bolt or pressure applied to a rivet. If the applied load is short enough in duration not to cause weakening of the strength due to fatigue, then it may be represented by a maximum load. The resulting reliability does not depend on time and is simply the Figure 4.32 Maximum static loading stress and variable strength Stress±Strength Interference (SSI) analysis 185 probability that the system survives the application of the load. However, in reality loads are subject to variability and if the component does not have the strength to sustain any one of these, it will fail (Lewis, 1996). Therefore, the reliability R L max is given by: R L max 1 ÿP 1 ÿ L max 0 f SdS 4:53 where: f Sstrength distribution: Equation 4.53 can be solved by integrating the f S using Simpson's Rule or by using the CDF for the strength directly when in closed form, i.e. R 1 ÿ FL max . In the case of the Normal and Lognormal distributions, the use of SND theory makes the calculation straightforward. The above formulation suggests that all strength values less than the maximum loading stress will fail irrespective of any actual variation on the loading conditions which may occur in practice. 4.4.5 Example ± calculation of reliability using different loading cases Consider the situation where the loading stress on a component is given as L $ N350; 40MPa relating to a Normal distribution with a mean of L 350 MPa and standard deviation L 40 MPa. The strength distribution of the component is S $ N500; 50MPa. It is required to ®nd the reliability for these conditions using each approach above, given that the load will be applied 1000 times during a de®ned duty cycle. Maximum static loading stress, L max , with variable strength If we assume that the maximum stress applied is 3 from the mean stress, where this loading stress value covers 99.87% those applied in service: L max 350 340470 MPa Because the strength distribution is Normal, we can determine the Standard Normal variate, z, as: z x ÿ 470 ÿ 500 50 ÿ0:6 From Table 1 in Appendix I, the probability of failure P 0:274253. The reliability, R, is given by: R 1 ÿ P 1 ÿ 0:274253 R L max 0:725747 The reliability, R L max , as a function of the maximum stress value used is shown in Figure 4.33. The reliability rapidly falls o at higher values of stress chosen, such 186 Designing reliable products as 4. A particular diculty in this approach is, then, the choice of maximum loading stress that re¯ects the true stress of the problem. Single application of a variable static loading stress with variable strength Substituting the given parameters for stress and strength in the coupling equation (equation 4.38) gives: z ÿ 500 ÿ 350 50 2 40 2 p ÿ2:34 From Table 1 in Appendix I, the probability of failure P 0:009642. The reliability is then: R 1 ÿ P 1 ÿ 0:009642 R 1 0:990358 This value is much more optimistic, as you would assume, because the reliability is the probability of both stress and strength being interfering, not just the strength being equal to a maximum value. Variable static loading stress with a de®ned duty cycle of `n' load applications with variable strength using approaches by Bury (1974), Carter (1997) and Freudenthal et al. (1966) Using Carter's approach ®rst, from equation 4.47 we can calculate LR to be: LR 40 50 2 40 2 p 0:62 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 Figure 4.33 Reliability as a function of maximum stress, L max Stress±Strength Interference (SSI) analysis 187 p % 0:0009 from Figure 4.31. Using equation 4.48, and given that n 1000, gives the reliability as: R n 1 ÿ p n 1 ÿ 0:0009 1000 R 1000 0:406405 Next using Bury's approach, from Table 4.11 the extremal parameters, and Â, from an initial Normal loading stress distribution are determined from: 2lnnÿ0:5lnlnnÿ1:2655 2lnn p 45 350 40 2ln1000ÿ0:5lnln1000ÿ1:2655 2ln1000 p 45 474:66 MPa and  2lnn p 40 2ln1000 p 10:76 MPa Substituting in the parameters for both stress and strength into equation 4.52 and solving using Simpson's Rule (integrating between the limits of 1 and 1000, for example) gives that the reliability is: R 1000 0:645 This value is more optimistic than that determined by Carter's approach. Next, solving equation 4.35 directly using Simpson's Rule for R n as described by Freudenthal et al. (1966): R n I 0 FL n Á f SdS A 3-parameter Weibull approximates to a Normal distribution when 3:44, and so we can convert the Normal stress to Weibull parameters by using: xo L % L ÿ 3:1394473 L L % L 0:3530184 L L 3:44 Therefore, the loading stress CDF can be represented by a 3-parameter Weibull dis- tribution: FL1 ÿ exp ÿ x ÿ xo L L ÿ xo L L and the strength is represented by a Normal distribution, the PDF being as follows: f S 1 S 2 p exp ÿ x ÿ S 2 2 2 S 188 Designing reliable products Therefore, substituting these into equation 4.35 gives the reliability, R n , as: R n I S xo L 1 ÿ exp ÿ x ÿ xo L L ÿ xo L L ! n Á 1 S 2 p exp ÿ x ÿ S 2 2 2 S ! dx From the solution of this equation numerically for n 1000, the reliability is found to be: R 1000 0:690 Figure 4.34 shows the reliability as a function of the number of load applications, R n , using the three approaches described to determine R n . There is a large discrepancy between the reliability values calculated for n 1000. Repeating the exercise for the same loading stress, L $ N350; 40MPa, but with a strength distribution of S $ N500; 20MPa increases the LR value to 0.89 and SM 3.35. Figure 4.35 shows that at higher LR values, the results are in better agreement, up to approxi- mately 1000 load applications, which is the limit for static design. The above exercise suggests that if we had used equation 4.32 to determine the reliability of a component when it is known that the load may be applied many times during its life, an overoptimistic value would have been obtained. This means that the component could experience more failures than that anticipated at the design stage. This is common practice and a fundamentally incorrect approach (Bury, 1975). A high con®dence in the reliability estimates is accepted for the situation where a single application of the load is experienced, R 1 . However, the con®dence is Figure 4.34 Reliability as a function of number of load applications using different approaches for LR 0:62 (medium loading roughness) and SM 2:34 Stress±Strength Interference (SSI) analysis 189 lower when determining the reliability as a function of the number of load applica- tions, R n , when n ) 1, using the various approaches outlined at low LR values. At higher LR values, the three approaches to determine the reliability for n ) 1do give similar results up to n 1000, this being the limit of the number of load applica- tions valid for static design. It can also be seen that a very high initial reliability is required from the design at R 1 to be able to survive many load applications and still maintain a high reliability at R n . 4.4.6 Extensions to SSI theory The question arises from the above as to the amount of error in reliability calculations due to the assumption of normally distributed strength and particularly the loading stress, when in fact one or both could be Lognormal or Weibull. Distributions with small coecients of variations (C v 0:1 for the Lognormal distribution) tend to be symmetrical and have a general shape similar to that of the Normal type with approximately the same mean and standard deviation. Dierences do occur at the tail probabilities (upper tail for stress, lower tail for strength) and signi®cant errors could occur from substituting the symmetrical form of the Normal distribution for a skewed distribution. If the form of the distributional model is only approximately correct, then the tails may dier substantially from the tails of the actual distribution. This is because the model parameters, related to low order moments, are determined from typical rather than rare events. In this case, design decisions will be satisfactory for bulk Figure 4.35 Reliability as a function of number of load applications using different approaches for LR 0:89 (rough loading) and SM 3:35 190 Designing reliable products occurrences, but may be less than optimal for rare events. It is the rare events, catastrophes, for example, which are often of greatest concern to the designer. The suboptimality may be manifested as either an overconservative or an unsafe design (Ben-Haim, 1994). Many authors have noted that the details of PDFs are often dicult to verify or justify with concrete data at the tails of the distribution. If one fails to model the tail behaviour of the basic variables involved correctly, then the resulting reliability level is questionable as noted (Maes and Breitung, 1994). It may be advantageous, therefore, to use statistics with the aim of determining certain tail characteristics of the random variables. Tail approximation techniques have developed speci®cally to achieve this (Kjerengtroen and Comer, 1996). Con®dence levels on the reliability estimates from the SSI model can be determined and are useful when the PDFs for stress and strength are based on only small amounts of data or where critical reliability projects are undertaken. However, approaches to determine these con®dence levels only strictly apply when stress and strength are characterized by the Normal distribution. Detailed examples can be found in Kececioglu (1972) and Sundararajan and Witt (1995). Another consideration when using the approach is the assumption that stress and strength are statistically independent; however, in practical applications it is to be expected that this is usually the case (Disney et al., 1968). The random variables in the design are assumed to be independent, linear and near-Normal to be used eec- tively in the variance equation. A high correlation of the random variables in some way, or the use of non-Normal distributions in the stress governing function are often sources of non-linearity and transformations methods should be considered. These are generally called Second Order Reliability Methods, where the use of independent, near-Normal variables in reliability prediction generally come under the title First Order Reliability Methods (Kjerengtroen and Comer, 1996). For economy and speed in the calculation, however, the use of First Order Reliability Methods still dominates presently. 4.5 Elements of stress analysis and failure theory The calculated loading stress, L, on a component is not only a function of applied load, but also the stress analysis technique used to ®nd the stress, the geometry, and the failure theory used (Ullman, 1992). Using the variance equation, the par- ameters for the dimensional variation estimates and the applied load distribution, a statistical failure theory can then be formulated to determine the stress distribution, f L. This is then used in the SSI analysis to determine the probability of failure together with material strength distribution f S. Use of the classical stress analysis theories to predict failure involves ®rstly identi- fying the maximum or eective stress, L, at the critical location in the part and then comparing that stress condition with the strength, S, of the part at that location (Shigley and Mischke, 1996). Among such maximum stress determining factors are: stress concentration factors; load factors (static, dynamic, impact) applied to axial, bending and torsional loads; temperature stress factors; forming or manufacturing stress factors (residual stresses, surface and heat treatment factors); and assembly stress factors (shrink ®ts and press ®ts) (Haugen, 1968). The most signi®cant factor Elements of stress analysis and failure theory 191 in failure theory is the character of loading, whether static or dynamic. In the discus- sion that follows, we constrain the argument to failure by static loading only. Often in stress analysis we may be required to make simpli®ed assumptions, and as a result, uncertainties or loss of accuracy are introduced (Bury, 1975). The accuracy of calculation decreases as the complexity increases from the simple case, but ultimately the component part will still break at its weakest section. Theoretical failure formulae are devised under assumptions of ideal material homogeneity and isotropic behaviour. Homogeneous means that the materials properties are uniform throughout; isotropic means that the material properties are independent of orientation or direction. Only in the simplest of cases can they furnish us with the complete solution of the stress distribution problem. In the majority of cases, engineers have to use approximate solutions and any of the real situations that arise are so complicated that they cannot be fully represented by a single mathematical model (Gordon, 1991). The failure determining stresses are also often located in local regions of the component and are not easily represented by standard stress analysis methods (Schatz et al., 1974). Loads in two or more axes generally provide the greatest stresses, and should be resolved into principal stresses (Ireson et al., 1996). In static failure theory, the error can be represented by a coecient of variation, and has been proposed as C v 0:02. This margin of error increases with dynamic models and for static ®nite element analysis, the coecient of variation is cited as C v 0:05 (Smith, 1995; Ullman, 1992). Understanding the potential failure mechanisms of a product is also necessary to develop reliable products. Failure mechanisms can be broadly grouped into overstress (for example, brittle fracture, ductile fracture, yield, buckling) and wear-out (wear, corrosion, creep) mechanisms (Dasgupta and Pecht, 1991). Gordon (1991) argues that the number of failure modes observed probably increases with complexity of the system, therefore eective failure analysis is an essential part of reliability work (Burns, 1994). The failure governing stress must be determined for the failure mode in question and the use of FMEA in determining possible failure modes is crucial in this respect. 4.5.1 Simple stress systems In postulating a statistical model for a static stress variable, it is important to distinguish between brittle and ductile materials (Bury, 1975). For simple stress systems, i.e. uniaxial or pure torsion, where only one type of stress acts on the component, the following equations determine the failure criterion for ductile and brittle types to predict the reliability (Haugen, 1980): For ductile materials in uniaxial tension, the reliability is the probabilistic require- ment to avoid yield: R PSy > L4:54 For brittle materials in tension, the reliability is given by the probabilistic requirement to avoid tensile fracture: R PSu > L4:55 192 Designing reliable products For ductile materials subjected to pure shear, the reliability is the probabilistic requirement to avoid shear yielding: R P y 0:577Sy > L4:56 where: Sy yield strength Su ultimate tensile strength L loading stress y shear yield strength: The formulations for the failure governing stress for most stress systems can be found in Young (1989). Using the variance equation and the parameters for the dimensional variation estimates and applied load, a statistical failure theory can be formulated for a probabilistic analysis of stress rupture. 4.5.2 Complex stress systems Predicting failure and establishing geometry that will avert failure is a relatively simple matter if the machine is subjected to uniaxial stress or pure torsion. It is far more dicult if biaxial or triaxial states of stress are encountered. It is therefore desirable to predict failure utilizing a theory that relates failure in the multiaxial state of stress by the same mode in a simple tension test through a chosen modulus, for example stress, strain or energy. In order to determine suitable allowable stresses for the complicated stress conditions that occur in practical design, various theories have been developed. Their purpose is to predict failure (yield or rupture) under combined stresses assuming that the behaviour in a tension or compression test is known. In general, ductile materials in static tensile loading are limited by their shear strengths while brittle materials are limited by their tensile strengths (though there are exceptions to these rules when ductile materials behave as if they were brittle) (Norton, 1996). This observation required the development of dierent failure theories for the two main static failure modes, ductile and brittle fracture. Ductile fracture Of all the theories dealing with the prediction of yielding in complex stress systems, the Distortion Energy Theory (also called the von Mises Failure Theory) agrees best with experimental results for ductile materials, for example mild steel and aluminium (Collins, 1993; Edwards and McKee, 1991; Norton, 1996; Shigley and Mischke, 1996). Its formulation is given in equation 4.57. The right-hand side of the equation is the eective stress, L, for the stress system. 2Sy 2 s 1 ÿ s 2 2 s 2 ÿ s 3 2 s 3 ÿ s 1 2 4:57 where s 1 , s 2 , s 3 are the principal stresses. Elements of stress analysis and failure theory 193 Therefore, for ductile materials under complex stresses, the reliability is the prob- abilistic requirement to avoid yield as given by: R P Sy > s 1 ÿ s 2 2 s 2 ÿ s 3 2 s 3 ÿ s 1 2 2 s 4:58 Again, using the variance equation, the parameters for the dimensional variation estimates and applied load to determine the principal stress variables, a statistical failure theory can be determined. The same applies for brittle material failure theories described next. In summary, the Distortion Energy Theory is an acceptable failure theory for ductile, isotropic and homogeneous materials in static loading, where the tensile and compressive strengths are of the same magnitude. Most wrought engineering metals and some polymers are in this category (Norton, 1996). Brittle fracture The arbitrary division between brittle and ductile behaviour is when the elongation at fracture is less than 5% (Shigley and Mischke, 1989) or when Sy is greater than E/1034.2 (in Pa) (Haugen, 1980). Most ductile materials have elongation at fracture greater than 10% (Norton, 1996). However, brittle failure may be experienced in ductile materials operating below their transition temperature (as described in Section 4.3). Brittle failure can also occur in ductile materials at sharp notches in the component's geometry, termed triaxiality of stress (Edwards and McKee, 1991). Strain rate as well as defects and notches in the materials also induce ductile-to-brittle behaviour in a material. Such defects can considerably reduce the strength under static loading (Ruiz and Koenigsberger, 1970). Brittle fracture is the expected mode of failure for materials like cast iron, glass, concrete and ceramics and often occurs suddenly and without warning, and is associated with a release of a substantial amount of energy. Brittle materials, there- fore, are less suited for impulsive loading than ductile materials (Faires, 1965). In summary, the primary factors promoting brittle fracture are then (Juvinall, 1967): . Low temperature ± increases the resistance of the material to slip, but not cleavage . Rapid loading ± shear stresses set up in impact may be accompanied by high normal stresses which exceed the cleavage strength of the material . Triaxial stress states ± high tensile stresses in comparison to shear stresses . Size eect on thick sections ± may have lower ductility than sample tests. By de®nition, a brittle material does not fail in shear; failure occurs when the largest principal stress reaches the ultimate tensile strength, Su. Where the ultimate compres- sive strength, Su c , and Su of brittle material are approximately the same, the Maximum Normal Stress Theory applies (Edwards and McKee, 1991; Norton, 1996). The probabilistic failure criterion is essentially the same as equation 4.55. Materials such as cast-brittle metals and composites do not exhibit these uniform properties and require more complex failure theories. Where the properties Su c and Su of a brittle material vary greatly (approximately 4 X 1 ratio), the Modi®ed Mohr Theory is preferred and is good predictor of failure under static loading conditions (Norton, 1996; Shigley and Mischke, 1989). 194 Designing reliable products [...]... to determine the eective or maximum stress are shown below and involve all three principal stresses (Dowling, 19 93): ! 1 Suc ÿ 2Su C 1 j s1 ÿ s2 j s1 s2 4:59 2 Suc ! 1 Su ÿ 2Su s2 s3 4:60 C 2 j s2 ÿ s3 j c 2 Suc ! 1 Suc ÿ 2Su s3 s1 4: 61 C 3 j s3 ÿ s1 j 2 Suc where: L max C1 ; C2 ; C3 ; s1 ; s2 ; s3 4: 62 The eective stress is then compared to the materials ultimate... materials, for example mild steel and aluminium (Collins, 19 93; Edwards and McKee, 19 91; Norton, 19 96; Shigley and Mischke, 19 96) Its formulation is given in equation 4.57 The right-hand side of the equation is the eective stress, L, for the stress system 2Sy2 s1 ÿ s2 2 s2 ÿ s3 2 s3 ÿ s1 2 where s1 , s2 , s3 are the principal stresses 4:57 19 3 19 4 Designing reliable products Therefore, for ductile... treatment factors); and assembly stress factors (shrink ®ts and press ®ts) (Haugen, 19 68) The most signi®cant factor 19 1 1 92 Designing reliable products in failure theory is the character of loading, whether static or dynamic In the discussion that follows, we constrain the argument to failure by static loading only Often in stress analysis we may be required to make simpli®ed assumptions, and as a result,... reliability and number of components in series (adapted from Michaels and Woods, 19 89) 20 0 Designing reliable products (Amster and Hooper, 19 86) Other formulations exist for components in parallel with equal reliability values, as shown in equation 4.69, and for combinations of series, parallel and redundant components in a system (Smith, 19 97) The complexity of the equations to ®nd the system reliability... mechanical systems is determined by their weakest link (Broadbent, 19 93; Carter, 19 86; Furman, 19 81; Roysid, 19 92) Carter (19 86) illustrates this rule using Figure 4.38 relating the loading roughness and the number of components in the system Failure to understand it can lead to errors of judgement and wrong decisions which could prove expensive and/ or " Figure 4.38 Overall system reliability as a function... al., 19 96) In static failure theory, the error can be represented by a coecient of variation, and has been proposed as Cv 0: 02 This margin of error increases with dynamic models and for static ®nite element analysis, the coecient of variation is cited as Cv 0:05 (Smith, 19 95; Ullman, 19 92) Understanding the potential failure mechanisms of a product is also necessary to develop reliable products. .. isotropic and homogeneous materials in static loading, where the tensile and compressive strengths are of the same magnitude Most wrought engineering metals and some polymers are in this category (Norton, 19 96) Brittle fracture The arbitrary division between brittle and ductile behaviour is when the elongation at fracture is less than 5% (Shigley and Mischke, 19 89) or when Sy is greater than E /10 34 .2 (in... fracture, yield, buckling) and wear-out (wear, corrosion, creep) mechanisms (Dasgupta and Pecht, 19 91) Gordon (19 91) argues that the number of failure modes observed probably increases with complexity of the system, therefore eective failure analysis is an essential part of reliability work (Burns, 19 94) The failure governing stress must be determined for the failure mode in question and the use of FMEA... when stress and strength are characterized by the Normal distribution Detailed examples can be found in Kececioglu (19 72) and Sundararajan and Witt (19 95) Another consideration when using the approach is the assumption that stress and strength are statistically independent; however, in practical applications it is to be expected that this is usually the case (Disney et al., 19 68) The random variables... components in the system and the number of parallel paths (Burns, 19 94): Ri 1 ÿ 1 ÿ Rsys 1= m 4:69 where: m number of components in parallel: In very complex systems, grave consequences can result from the failure of a single component (Kapur and Lamberson, 19 77), therefore if the weakest item can endure the most severe duty without failing, it will be completely reliable (BompasSmith, 19 73) It follows . stress are shown below and involve all three principal stresses (Dowling, 19 93): C 1 1 2 s 1 ÿ s 2 jj Su c ÿ 2Su Su c s 1 s 2 ! 4:59 C 2 1 2 s 2 ÿ s 3 jj Su c ÿ 2Su Su c s 2 s 3 ! 4:60 C 3 1 2 s 3 ÿ. the stress system. 2Sy 2 s 1 ÿ s 2 2 s 2 ÿ s 3 2 s 3 ÿ s 1 2 4:57 where s 1 , s 2 , s 3 are the principal stresses. Elements of stress analysis and failure theory 19 3 Therefore, for. s 3 jj Su c ÿ 2Su Su c s 2 s 3 ! 4:60 C 3 1 2 s 3 ÿ s 1 jj Su c ÿ 2Su Su c s 3 s 1 ! 4: 61 where: L maxC 1 ; C 2 ; C 3 ; s 1 ; s 2 ; s 3 4: 62 The eective stress is then compared