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## Über dieses Buch

During the last two decades more and more universities offer courses on modern structural reliability theory. A course on structural reliability theory is now a natural part of the curri­ culum for mechanical and structural engineering students. As a result of this, a number of textbooks have been published in this decade. In PlOst of these books it is shown how the reliability of single structural members can be evaluated in a rational way. The methods used are usually so-called level 2 methods, i. e. methods involving certain approximate iter­ ative calculations to obtain an approximate value of the probability of failure of the struc­ tural members. In these methods the joint probability distribution of relevant variables (re­ sistance variables, loads, etc. ) is simplified and the failure criteria are idealized in such a way that the reliability calculations can be performed without an unreasonable amount of work. In spite of the approximations and idealizations made it is believed that a rational treatment of uncertainties in structural engineering can be obtained by level 2 methods. Usually, in­ sufficient data are at hand to make a more advanced estimate of the reliability of a struc­ tural member. It has been recognized for many years that a fully satisfactory estimate of the reliability of a structure must be based on a systems approach. In some situations it is sufficient to estimate the reliability of the individual structural members of a structural system.

## Inhaltsverzeichnis

### Chapter 1. Fundamentals of Structural Reliability Theory

Abstract
During the last decade structural reliability theory has been treated in a large number of research reports, conference papers, textbooks, etc. Structural reliability theory is now taught extensively at universities all over the world and several organizations offer elementary and advanced courses on this subject. Therefore, from being a subject only well-known by a relatively small number of researchers it is now an accepted engineering discipline.

### Chapter 2. Modelling of Structural Systems

Abstract
A real structural system is so complex that direct exact calculation of the probability of failure is completely impossible. The number of possible different failure modes is so large that they cannot all be taken into account, and even if they could all be included in the analysis exact probabilities of failure cannot be calculated. It is therefore necessary to idealize the structure so that the estimate of the reliability becomes manageable. Not only the structure itself but also the loading must be idealized. Because of these idealizations it is important to bear in mind that the estimates of e.g. probabilities of failure are related to the idealized system (the model) and not directly to the structural system. The main objective of a structural reliability analysis is to be able to design a structure so that the probability of failure is minimized in some sense. Therefore, the model must be chosen carefully so that the most important failure modes for the real structures are reflected in the model.

### Chapter 3. Reliability of Series Systems

Abstract
In section 2.1, page 34, a simple example is used to illustrate the need for estimating the reliâbility of series systems. On page 34 a structural system with two potential failure modes defined by the safety margins M1 = f1(X1, X2) and M2 = f2 (X1, X2) is considered. If Fi = {Mi ≤ 0}, i = 1, 2 the probability of failure Pf of the structural system is
$${P_f}=P\left({{F_1}\cup{F_2}}\right)$$
(3.1)
corresponding to evaluating the probability of failure of a series system with two elements. An approximation of Pf can be obtained by assuming that the safety margins M1 and M2 are linearized in their respective design points Al and A2 (see figure 2.1)
$${M_1}={a_1}{X_1}+{a_2}{X_2}+{\beta_1}$$
(3.2)
$${M_2}={b_1}{X_1}+{b_2}{X_2}+{\beta_2}$$
(3.3)
where β 1 and β 2 are the corresponding reliability indices when $$\bar a = \left( {a_1 ,a_2 } \right)\;and\;\bar b = \left( {b_1 ,b_2 } \right)$$ are chosen as unit vectors.

### Chapter 4. Reliability of Parallel Systems

Abstract
In section 2.4, page 43, it is suggested to model the reliability of a structural system by a series system of parallel systems. Each parallel system corresponds to a failure mode and this modelling is called systems modelling at level N, N = 1, 2, ... if all parallel systems have the same number N of failure elements. In chapter 6 it is shown how the most significant failure modes (parallel systems) can be identified by the β-unzipping method. After identification of significant (critical) failure modes (parallel systems) the next step is an estimate of the probability of failure $$P_{f_P }$$ for each parallel system and the correlation between the parallel systems. The final step is the estimate of the probability of failure Pf of the series system of parallel systems by the methods discussed in chapter 3.

### Chapter 5. Automatic Generation of Safety Margins

Abstract
There are many modes of failure in structural systems, depending on the configuration of the systems shapes and materials of the members, the loading conditions, etc. In order to perform the reliability assessment of the systems, those failure modes and their safety margins are to be given. For a simple type of structure the safety margins can be obtained by hand calculation. In the field of conventional systems reliability analysis, potential collapse mechanisms are specified and their safety margins are derived by using the principle of virtual work [5.1]. However, it is difficult in practice for a large structure with a high degree of redundancy to determine a priori which failure modes are probabilistically significant.

### Chapter 6. Reliability Analysis of Structural Systems by the β — Unzipping Method

Abstract
The β-unzipping method is a method by which the reliability of structures can be estimated at a number of different levels. The aim has been to develop a method which is at the same time simple to use and reasonably accurate. The method was first suggested by Thoft-Christensen [6.1] and is further developed by Thoft-Christensen & Sorensen [6.2], [6.3]. The β-un-zipping method is quite general in the sense that it can be used for two-dimensional and three-dimensional framed and trussed structures, for structures with ductile or brittle elements and also in relation to a number of different failure mode definitions.

### Chapter 7. The Branch-and-Bound Method

Abstract
There are so many failure modes in large structures with a high degree of redundancy that it is impossible to identify all of them a priori for estimating structural systems reliability or probability of failure. For the purpose, the β-unzipping method is discussed in chapter 6. An alternative and more rigorous method [7.1–9] is given in this chapter. First, the basic concepts are explained through a simple structure in sections 7.2 and 7.3. Second, the generalization is given in section 7.4 to identify the stochastically dominant failure paths. Third, some procedures are discussed in section 7.5 for evaluating the systems reliability by using the generated failure paths. Fourth, section 7.6 is concerned with application to a jacket-type offshore platform. Finally, heuristic operations which are essential for reliability assessment of large-scale structures, and numerical examples are presented in section 7.7.