Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 13 doc

3.2 Theoretical Overview of Reliability and Performance in Engineering Design 103 α 0 = the total number of items put on test at time t = 0, α S = the number of items surviving at time t or at t + Δt. Substituting Eq. (3.64) into Eq. (3.63) leads to R i (t)=1− t  0 f i (t)dt . (3.65) A more common notation for the ith component reliability is expressed in terms of the mathematicalconstant e. Themathematical constant e is the uniquereal number, such that the value of the derivative of f(x)=e x at the point x = 0 is exactly 1. The function so d efined is called the exponential function. Thus, the alternative, commonly used expression for R i (t) is R i (t)=e −  t 0 λ i (t) dt , (3.66) where λ i (t) is the ith component hazard rate or instantaneous failure rate. In this case, component failure time can follow any statistical distribution func- tion of which the hazard r ate is known. The expression R i (t) is r e duced to R i (t)=1−F i (t) R i (t)=e − λ i t . (3.67) A redundant configuration or single component MTBF is defined by MTBF = ∞  0 R(t)dt . (3.68) Thus, substituting Eq. (3.67) into Eq. (3. 66), and in tegr ating the results in the series gives the model for MTBF, which in effect is the sum of the inverse values of the component hazard rates, or instantaneous failure rates o f all the components in the series MTBF =  n ∑ i=1 λ i  −1 (3.69) MTBF = sum of inverse values of component hazard rates = instantaneous failure rates of all the components. b) Parallel Network This type of redundancy can be used to improve system and equipment reliabil- ity. The redundant system or equipment will fail only if all of its components fail. To develop this mathematical model for application in reliability evaluation,itis 104 3 Reliability and Performance in Engineering Design assumed that all units of the system are active and load sharing, and units are sta- tistically independent. The unreliability, F P (t), at time t of a parallel structu re with non-identical components is F P (t)= k ∏ i=1 F i (t) (3.70) F i (t)=ith component unreliability (failure probability). Since R P (t)+F P (t)=1, utilising Eq. (3.70) the parallel structure reliability, R P (t), becomes R P (t)=1− k ∏ i=1 F i (t) . (3.71) Similarly,as was donefor the series network componentswith constant failure rates, substituting for F i (t) in Eq. (3.71) we get R P (t)=1− k ∏ i=1  1− e − λ i t  . (3.72) In order to obtain the series MTBF, substitute Eq. (3.69) for identical components and integrate as follows MTBF = ∞  0  1− k ∑ j= 0 (n j )(−1) j e − λ j t  dt MTBF = 1 λ + 1 2 λ + 1 3 λ + + 1 k λ (3.73) λ = the component hazard or instantaneous failure rate. c) A k-out-of-m Unit Network This type of redundancy is used when a certain number k of components in an ac- tive parallel redundant system orassembly must work forthe system’s or assembly’s success. The binomial distribution, system or assembly reliability of the indepen- dent and identical components at time t is R k/m (t),whereR(t) is the component reliability R k/m (t)= m ∑ i=k t (m)[R(t)] i [1−R(t)] k−i (3.74) m = the total number of system/assembly components k = the number of components required for system/assembly success at time t. 3.2 Theoretical Overview of Reliability and Performance in Engineering Design 105 Special cases of the k-out-of-m unit system are: k = 1: = parallel network k = m: = series network. For exponentially distributed failure times (constant failure rate) o f a component, substituting in Eq. (3.74) for k = 2andm = 4, the equation becomes R 2/4 (t)=3e −4 λ t −8e −3 λ t + 6e −2 λ t . (3.75) d) Standby Redundant Systems R S (t)= K ∑ i=0 ⎡ ⎣ t  0 λ (t)dt ⎤ ⎦ i e −  t 0 λ (t) dt (i!) −1 . (3.76) In this case (Eq. 3.76), one component is functioning, and K components are on standby, or are not active. To develop a system/assembly reliability model, the com- ponents must be identical and independent, and the standby components as new. The general components hazard rate, λ , is assumed. 3.2.3.5 Reliability Evaluation of Three-State Device Networks A three-state device (component) has one operational and two failure states. De- vices such as a fluid flow valve and an electronic diode are examples of a three- state device. These devices have failure modes that can be described as failure in the closed or open states. Such a device can have the following functional states (Dhillon 1983): State 1 = Operational State 2 = Failed in the closed state State 3 = Failed in the o pen state a) Parallel Networks A parallel network composed of active independent three-state componentswill fail only if all the components fail in the open mode, or at least one of the devices must fail in the closed m ode. The network (with non-identical devices) time-dependent reliability, R P (t),is R P (t)= k ∏ i=1 [1−F C i (t)] − k ∏ i=1 F O i (t) , (3.77) 106 3 Reliability and Performance in Engineering Design where: t = time k = the number of three-state devices in parallel F C i (t)= the closed mode probability of device i at time t F O i (t)= the open mode probability of device i at time t b) Series Networks A series network is the reverse of the parallel network. A series system will fail only if all of its independent elements fail in a closed mode or any one of the components fails in open mode. Th us, because of duality, the time-dependen t reliability of the series network with non-identical and independent devices is the difference of the summations of the respective values for the open mode probability,[1−F O i (t)], and the closed mode probability,[F C i (t)], of device i at time t. The series network with non-identical and independent devices time-dependent reliability, R S (t),is R S (t)= k ∏ i=1 [1−F O i (t)] − k ∏ i=1 F C i (t) , (3.78) where: t = time k = the number of devices in the series configuration F C i (t)= the closed mode probability of device i at time t F O i (t)= the open mode probability of device i at time t Closing comments to theoretical overview It was stated earlier, and must be iterated here, that these techniques do not represent the total spectrum of reliability calculations, and have been considered as the most applicable for their application in determining the integrity of engineering design during the conceptual, preliminary and detail design phases of the engineering de- sign process, based on an extensive study of the available literature. Furthermore, the techniques h ave been grouped according to significant differences in the approaches to the determination of reliability of systems, compared to that of assemblies or of components. This supports the premise that: • predictions of the reliability of systems are based on prognosis of systems perfor- mance under conditions subject to failure modes (reliability prediction); • assessments of the reliability of equipment are based upon inferences of failure according to various statistical failure distributions (reliability assessment); and • evaluations of the reliability of components are based upon known values of fail- ure rates (reliability evaluation). 3.3 Analytic Development of Reliability and Performance in Engineering Design 107 3.3 Analytic Development of Reliability and Performance in Engineering Design Some of the techniques identified fo r reliability prediction, assessment and evalua- tion, in the conceptual, preliminary and detail design phases respectively, have been considered for further analytic development.This has been done on the basis of their transformational capabilities in developing intelligent computer automated method- ology. The techniques should be suitable for application in artificial intelligence- based modelling, i.e. AIB modelling in which knowledge-based expert systems within a blackboardmodel can be applied in determiningthe integrityof engineering design. The AIB model should be suited to applied concurrent engineering design in an online and integrated collaborative engineering d esign environment in which au- tomated continual design reviews are conducted throughout the engineering design process by remotely located design groups communicating via the internet. Engineering designs are usually composed of highly integrated, tightly coupled systems with complex interactions, essential to the functional performance of the design. Therefore, concurrent, rather than sequential considerations of specific re- quirements are essential, such as meeting the design criteria together with design integrity constraints. The traditional approach in industry for designing engineered installations has been the implementation of a sequential consideration of require- ments for process, thermal, power, manufacturing,installation and/or structural con- straints. In recent years, concurrent engineering design has become a widely ac- cepted concept, particularly as a pre ferred altern ative to the sequential engineerin g design process. Concurrent engineering design in the context of design integrity is a systematic ap proach to integratin g the various continual design reviews within the engineering design process, such as reliability prediction, assessment, and evalua- tion throughout the preliminary, schematic, and detail design phases respectively. The objective of concurrent engineering design with respect to design integrity is to assure a reliable design throughout the engineering design process. Parallelism is the prime concept in concurrent engineering design, and design integrity (i.e. de- signing for reliability) becomes the central issue. Integrated collaborative engineer- ing design implies information sharing and decision coordinationfor conductingthe continual design reviews. 3.3.1 Analytic Development of Reliability and Performance Prediction in Conceptual Design Techniques for reliability and performance prediction in determining the integrity of engineering design during the conceptual design phase include system reliability modelling based on: i. System performance measures ii. Determination of the most reliable design 108 3 Reliability and Performance in Engineering Design iii. Conceptual design optimisation and iv. Comparison of conceptual designs v. Labelled interval calculus and vi. Labelled interval calculus in designin g for reliability 3.3.1.1 System Performance Measures For each process system, there is a set o f performance measures that require particu- lar attention in design—for example, temperature range, pressure rating, output and flow rate. Some measures such as pressure and temperature rating may be common for different items of equipment inherent to each process system. Some measures may apply only to one system. The performance measures of each system can be described in matrix form in a parameter profile matrix (Thompson et al. 1998), as showninFig.3.22where: i = number of performance measure parameters j = number of process systems x = a data point that measures the performance of a system with respect to a par ticular parameter. It is not meaningful to use actual performance—for example, an operating temperature—as the value of x ij . Rather, it is the proximity of the actual perfor- mance to the limit of process capability of the system that is useful. In engineering design review, the proximity of performance to a limit closely relates to a measure of the safety margin. In the case of process enhancement, the proximity to a limit may even indicate an inhibitor to proposed changes. For a pro- cess system, a non-dimensionalnumerical value of x ij may be obtained by determin- ing the limits of capability ,suchasC max and C min , with respect to each performance parameter, and specifying the nominal point or range at which the system’s perfor- mance parameter is required to operate. The limits may be r epresented diagrammatically as shown in Figs. 3.23, 3.24 and 3.25, where an example of two performance limits, of one upper performance limit, and of one lower performance limit is given respectively (Thompson et al. 1998). The data point x ij that is entered into the performance of systems with two p er- formance limits is the lower value of A and B (0 < score < 10), which is the closest Process systems Performance x 11 x 12 x 13 x 14 x 1i parameters x 21 x 22 x 23 x 24 x 2i x 31 x 32 x 33 x 34 x 3i x j1 x j2 x j3 x j4 x ji Fig. 3.22 Parameter profile matrix 3.3 Analytic Development of Reliability and Performance in Engineering Design 109 Fig. 3.23 Determination of a data point: two limits Fig. 3.24 Determination of a data point: one upper limit the n ominal design co ndition does approach a limit. The value of x ij always lies in the range 0–10. Ideally, when design condition is a single point at the mid-range, then the data point is 10. 110 3 Reliability and Performance in Engineering Design Fig. 3.25 Determination of a data point: one lower limit It is obvious that this process of data point determination can be generated quickly by computer modelling with inputs from process system performance mea- sures and ranges of capability. If there is one operating limit only, then the data point is obtained as shown in Figs. 3.24 and 3.25, where the upper or lower limits respectively are known. Therefore, a set of data points can be obtained for each system with respect to the performance parameters that are relevant to that system. Furthermore, a method can be adopted to allow designing for reliability to be quantified, which can lead to optimisation of design reliability. Figures 3.23, 3.24 and 3.2 5 illustrate how a data point can be generated to mea- sure performance with respect to the best and the worst limits of pe rformance. 3.3.1.2 Determination of the Most Reliable Design in the Conceptual Design Phase Reliability prediction through system reliability modelling based on system perfor- mance may be carried out by the following method (Thompson et al. 1999): a) Identify the criteria against which the process design is measured. b) Determine the maximum and minimum acceptable limits of performance for each criterion. c) Calculate a set of measurement data points of x ij for each criterion according to the algorithms indicated in Figs. 3.23, 3.24 and 3.25. 3.3 Analytic Development of Reliability and Performance in Engineering Design 111 d) A design proposal that has good reliability will exhibit uniformly high scores of the data points x ij . Any low data point represents system performance that is close to an unacceptable limit, indicating a low safety margin. e) The conceptual design may then be reviewed and revised in an iterative manner to improve low x ij scores. When a uniformly high set of scores has been obtained, then the design, or alter- native design that is most reliable, will conform to the equal strength principle,also referred to as unity, in which there are no ‘weak links’ (Pahl et al. 1996). 3.3.1.3 Comparison of Conceptual Designs If it is required to compare two or more conceptual designs, then an overall rating of reliability may be obtained to compare these designs. An overall reliability may be determined by calculating a systems performance index (SP) as follows SP = N  N ∑ i=1 1  d i  −1 (3.79) where N = the sum of the p erformances considered d i = the scores of the performances considered. The overall SP score lies in the range from 0 to 10. The inverse method of combina- tion o f scores readily identifies low safety margins, unlike normal averaging through addition where almost no safety margin with respect to one criterion may be com- pensated for by high safety margins elsewhere—which is unacceptable. Alternative designs can therefore be compared with respect to reliability, by comparing their SP scores; the highest score is the most reliable. In a proposed method for using this overall rating approach (Liu et al. 1996), caution is required because simply choosing the highest score may not be the best solution. This requires that each de- sign should always be reviewed to see whether weaknesses can be improved upon, which tends to defeat the purpose of the method. Although other factors such as costs may be the final selection criterion for conceptual or preliminary design pro- posals with similar overall scores (which oft is the case), the objective is to achieve a design solution that is the most reliable from the viewpoint of meeting the re- quired performance criteria. This shortcoming in the overall rating approach may be avoided by supplementing performance measures obtained from mathematical models in the form of mathematical algorithms of process design integrity for the values of x ij , rather than the ‘direct’ performance parameters such as temperature range, pressure rating, output or flow rate. The performance measures obtained fro m these mathematical models consider the prediction, assessment or evaluation of parameters particular to each specific stage of the design process, whether it is conceptual design, preliminary design or detail design respectively. 112 3 Reliability and Performance in Engineering Design The approach defines performancemeasures that, when met, achievean optimum design with regard to overall integrity. It seeks to maximise the integrity of design by ensuring that the criteria of reliability, availability, maintainability and safety are concurrently being met. The choice of limits of performance for such an approach is generally made with respect to the consequences and effects of failure, and reliabil- ity expectations based on the propagation of single maximum and minimum values of acceptable performance for each criterion. If the consequences and/or effects of failure are h igh, then limits of acceptable performance with high safety margins that are well clear of failur e criteria are chosen. Similarly, if failure criteria are imprecise, then high safety margins are adopted. These considerationshave been further expanded to represent sets of systems that function under sets of failures and performance intervals, applying labelled interval calculus (Boettner et al. 1992). The most significant advantage of this expanded method is that, besides not hav- ing to rely on the propagation of single estimated values of failure data, it also does not have to rely on the determination of single values of maximum and minimum acceptable limits of performance for each criterion. Instead, constraint propaga- tion of intervals about sets of performance values is applied. As these intervals are defined, it is possible to compute a multi-objective optimisation of performance val- ues, in order to d etermine optimal solution sets for different sets of performance intervals. 3.3.1.4 Conceptual Design Optimisation The process described attempts to improvereliability continually towards an optimal result (Thompson et al. 1999). If the design problem can be modelled so that it is possible to compute all the x ij scores, then it is possible to optimise mathematically in order to maximise the SP function, as a result of which the x ij scores will achieve a uniformly high score. Typically in engineering design, several conceptual design alternatives need to be optimised for different design criteria or constraints. To deal with multiple design alternatives, the parameter profile matrix,inwhich the scores for each system’s performance measure of x ij is calculated, needs to be modified. Instead of a one-variable matrix, in which the scores x ij are listed, the analysis is completed for each specific criterion y j . Thus, a two-variable matrix of c ij is constructed, as shown in Fig. 3.26 (Liu et al. 1996). Design alternatives y 1 y 2 y 3 y 4 y n Performance x 1 c 11 c 12 c 13 c 14 c 1n parameters x 2 c 21 c 22 c 23 c 24 c 2n x 3 c 31 c 32 c 33 c 34 c 3n x m c m1 c m2 c m3 c m4 c mn Fig. 3.26 Two-variable parameter profile matrix . be applied in determiningthe integrityof engineering design. The AIB model should be suited to applied concurrent engineering design in an online and integrated collaborative engineering d esign. concept in concurrent engineering design, and design integrity (i.e. de- signing for reliability) becomes the central issue. Integrated collaborative engineer- ing design implies information sharing. environment in which au- tomated continual design reviews are conducted throughout the engineering design process by remotely located design groups communicating via the internet. Engineering designs