Reliability and Risk Analysis in Engineering and Medicine

This chapter deals with discussion of important probability density functions used in practice.

In a probabilistic analysis, since the outcome is uncertain, these variables are called as random variable (RV). There are two types of random variables. One is called discrete random variable and the other is called continuous random variable. In a discrete random variable, the outcomes are whole numbers. Famous examples of discrete random variables are tossing of a coin or a dice. In a continuous random variable, the probability of the outcome can use fractions. Typical examples of continuous variables in civil engineering are cylinder strength of concrete (f_c) and yield stress of steel (f_y). There are many discrete distributions such as binomial, and poisson distribution. Similarly, commonly used probability distributions are normal distribution, log normal distribution, uniform distribution, exponential distribution and Weibull distribution. Each distribution is identified with certain parameters. In order to make use of probability theory for a set of experimental data, data analysis is done to start with. It is described below. Illustrative examples are solved for each of the probability density function.

Reliability and Risk Analysis, while related are distinct. It is very important to first study the fundamentals before one goes into actual analysis. This chapter deals with the relation between Reliability and Risk Analysis. Reliability Analysis basically uses the basic principles of probability and statistics and calculates Reliability of an element or the structure as a whole. So, from that point of view Reliability Analysis can be considered as a sophisticated term for probabilistic Analysis. In this book, previous chapters dealt with the basic concepts of random variables, their density functions and cumulative density functions or distribution functions. The concepts of regression analysis were applied to develop a relation between dependent and independent variables and then obtain distribution function of desired. Another term so often used in the general context of Reliability Analysis and that is Risk (the related field is Risk Analysis). Illustrative examples are solved to determine reliability and risk of few selected structures.

A system is composed of several components. The system Reliability depends on the number of components, reliability of components and also as to how they are connected. Any structure can be considered as a system of various components put together to serve the purpose for which the structure is built to start with. To determine system reliability, one has to first obtain the component reliability using basic methods. The system reliability depends upon how the various components in the system. The two extremes are – series system and parallel system. In series system, all the components in the system have to function for the system to function successfully. In parallel system, functioning of one component is enough for the system to work. A system can also have combined series-parallel system components. In addition, there is another class of system referred to as k-out-of-n redundancy. In this case, if k components out of the total n components work, the system is considered successful. To start finding, the system reliability, one has to draw RBD (Reliability Block Diagram) depicting the way the components are connected in the actual physical system. Each of the cases is discussed below in detail. Mathematical expressions for system reliability are first derived for each case. This is followed by actual generic examples. Finally, reliability examples in various specific disciplines are discussed. It is to be noted that the extreme cases are series system and parallel system. Most of the real systems are mixed systems.

Regression analysis is a very important concept which is used not only in Engineering but other disciplines as well. Engineering problems of the present scenario are quite challenging to solve mainly due to their analytical complexities that requires numerical modeling approach. This chapter deals on the use of surrogate models for handling the solution of large-scale and high-dimensional computation intensive real-life problems. In large-scale high-dimensional computational problems, to evaluate the system response/behavior the majority of computation is involved in the repetitive function calls. To adhere to the quality of solution which depends on the system response estimation, the high-fidelity models are used to get accuracy in results. Their use of incorporates the conventional techniques a.k.a., the evolutionary algorithms require a great number of such high-fidelity function calls. To this end, a much-appreciated low-cost surrogate models or metamodels, which approximates the original model mathematically by reducing the computation cost for a desired accuracy level in an optimization solution is well accepted. During training, the surrogate model requires a minimum number of evaluations of the original model at support points and an efficient design of experiment can be performed for that. This chapter discusses in detail various regression models like linear, non-linear, multiple linear and Equivalent linear. The Equivalent Linear Regression model is extremely useful in real life as it has lot of practical applications.

Complex problems are difficult to solve by exact methods. Simulation methods are used in such cases. Probabilistic simulation of systems has received much attention particularly in the research as of reliability, and risk analysis, sensitivity analysis, optimization under uncertainty, to mention a few. In the context of uncertainty- based simulation, one of the major problem is computational demand of the numerical (finite element) model that is used to analyze the large scale engineering systems under consideration. However, probabilistic simulation is the only alternative for those cases in reliability analysis for those cases where the limit state function is not available in explicit form. However to address computational issue, efficient simulation techniques or design of experiments (DoE) are carried out i.e. determining the design points (in the input space), where the original (high fidelity) computational model needs to be evaluated. The accuracy level of reliability analysis depends on the region of simulation in the limit state function depends on the DoE over the input design space. This chapter will introduce state-of-the-art probabilistic simulation methods in uncertainty quantification of engineered systems with varying input dimensionality and computational complexity.

Decision Theory used in lot of engineering problems. This is because at every stage in an engineering project, there are lot of choices with lot of uncertainty associated with it and associated probabilities. In any industry, there are sequence of decisions the management has to make for the success of the company. It is true in the life of a person as well. All these decisions don’t have results with certainty. In other words, they are uncertain events. Hence, they are probabilistic. The decision theory is very important and has lot of usefulness associated with it. A typical decision tree has a root node, leaf nodes and branches. Irrespective of the type of the decision tree, it starts with a decision which is depicted in a box called the root node. Root and leaf nodes contain questions or some kind of criteria. Generally, these nodes appear as squares or circles. Squares depict decisions while circles represent uncertain outcomes. As stated above, since the decisions involved in a decision theory are uncertain, these have certain probabilities associated with them. A decision tree is very similar to a flow chart except the events listed in a flow chart have no uncertainty associated with it. Since, the final outcome of an engineering is not clear to start with, drawing a decision tree helps making a considered and optimal decision.

As is well known, the areas of Reliability Risk Analysis are interdisciplinary in nature. Hence these concepts can easily be applied to various disciplines including Engineering and Medicine. This chapter discusses dome applications to Medical area. Concepts of Reliability/Risk Analysis play lot of role in arriving at decisions in complicated medical cases. The principles of reliability have also been applied to the field of medical science. The occurrence of Subdural Hematoma disease was predicted, and the application was based on the premises that the factors influencing the occurrence of subdural hematoma remain unchanged in space and/or time domain. The disease reflects the accumulation of blood outside the brain between the dura and the next layer-the arachnoid, which eventually results in stroke with consequential increased pressure on brain. The additional pressure on the brain develops subdural hematoma’s symptoms that reflect in terms of the loss of consciousness and immediate collapse of patient into coma state depending on the rate of bleeding. Subdural hematoma can also be caused by a head injury or stroke. This is because stroke itself can be a hemorrhage or blood clotting in cerebral vessels of all sizes, therefore, the occurrence of subdural hematoma was predicted by reverse approach that works on consequence- the stroke. Since most of the systems including medical systems are probabilistic in nature, it is prudent to apply the principles of Reliability and Risk Analysis in medical field also. That is what this chapter does. It should be noted that while few applications of Reliability/Risk analysis are discussed , there are many more situations in which these concepts can be applied.

This chapter deals with Although the book focuses on the reliability aspects of statistical domain, but other statistical techniques also find significant utility in the studies of medical science problems. Clinical interventions, uses of drugs, quantity of doses, frequency of dose intake, duration for taking up dose and treatment protocols are usually specified with the help of statistical analysis and consequential inferences. This chapter analyses the effect of clinical intervention (multimodal group therapy) on the post-traumatic stress (PTS) index determined for general population. Correlations between PTS index and psychological/ social manifestations such as substance abuse, self-medication, divorce and suicide are investigated using the correlation and regression analyses.