Robust Reliability of Structures

Yakov Ben-Haim    Faculty of Mechanical Engineering Technion—Israel Institute of Technology Haifa, Israel

I Introduction

Though the Twentieth century began with the promulgation of the profoundest deterministic theory since the Seventeenth century, Einstein remained an anomaly; scientific thought for the past hundred years has been mainly influenced by concepts of uncertainty. Also, just as the early years of the century witnessed profound revision of the philosophical understanding of mathematical thought, so too has our understanding and treatment of uncertainty broadened and changed enormously. The technological applications of the mechanical sciences have followed suit and uncertainty thinking has won a place in the engineer’s practice. The major application is reliability: its assessment and attainment in mechanical design.

In accord with the diversification of uncertainty models that have emerged in recent decades, are a variety of reliability theories. In this paper we concentrate on one approach: robust reliability based on convex models of uncertainty. The comparison of alternative possibilities is avoided because this tends to be polemical rather than technological. The choice of an uncertainty model is, to some extent, a matter of taste. Also, some comparison is to be found elsewhere [5,6,15,16]. Finally, it is too early for a far-reaching analysis of the merits of alternative theories of reliability because the monopoly of classical probabilistic reliability has only recently been challenged. Some eligible theories, such as reliability based on a nonstandard probability logic like fuzzy theory, have yet to be thoroughly developed for applied mechanics.

To rely on something means to have confidence based on experience. This is a plain English word that has a classical etymology and it has carried this meaning long before engineers started thinking scientifically or scientists starting thinking probabilistically. Reliability rests on two more primitive concepts: performance and uncertainty. Still speaking lexically and not technically, we can rely on something when, despite uncertainties, its performance is acceptable.

We now ask for a quantitative theory that reflects the intuitive idea of reliability. The current standard theory of reliability is based on probability: the reliability of a system is measured by the probability of no failure. This approach is exceedingly useful and has been developed in recent decades by many able authors.

In this paper we describe a different formulation. We measure the reliability of a system by the amount of uncertainty consistent with no failure. A reliable system will preform satisfactorily in the presence of great uncertainty. Such a system is robust with respect to uncertainty, and hence the name robust reliability. On the other hand, a system has low reliability when small fluctuations can lead to failure. Such a system is fragile with respect to uncertainty.

The relevance of noise-robustness to the quality of technological systems is widely recognized. Consider for example the comment by Taguchi et al. [18, p. 3]:

In robust reliability, a system has high reliability when it is robust with respect to uncertainties. It has low reliability when even small amounts of uncertainty entail the possibility of failure.

A similar idea is found in robust adaptive control, where one seeks a control strategy which will perform acceptably in the presence of considerable uncertainty. Robust control has been motivated in large measure by severely limited information about the uncertainties, usually system-model imperfections, which characterize complex-controlled systems such as aerospace vehicles, power, or chemical plants.1 Robustness-to-uncertainty is a useful guideline in formulating control laws in the absence of detailed probabilistic information [1,13].

Finally, self-learning systems are notable for their ability to adapt themselves to uncertain environments. This is important in on-line learning precisely because this adaptability enhances the reliability of the system despite the uncertainties accompanying its operation.

Robustness-to-uncertainty is a natural starting point for a theory of reliability. It is not the only possibility starting point, but it is a fruitful one, well suited considering the limited information available about the uncertainties of mechanical structures and devices.

We will identify three primary components in our analysis of the robust reliability of mechanical systems: (1) a model of the mechanics, (2) a model of the uncertainties, and (3) a criterion of failure. For items (1) and (3) we will employ standard mechanical and physical theories. For item (2) we will use convex models of uncertainty. The relation between convexity and uncertainty, and the theoretical and practical aspects of convex models, are developed extensively elsewhere [3,9], but a brief description is included here in Section II for convenience.

Following that discussion, we provide a sequence of examples of robust reliability analysis, beginning in Section III with a heuristic example of the reliability of a truss supporting a platform bearing uncertain static loads. In Section IV we analyze the reliability of a cylindrical shell with uncertain geometrical imperfections that is subject to a static axial load. The third example in Section V deals with a dynamic system: a lifting device subject to time-varying and spatially uncertain loadings. These three case studies demonstrate the method of robust reliability analysis. In Section VI we discuss the topic from a more general point of view by developing the idea of modal reliability: the assessment of the relative reliability of dynamic degrees of freedom of a structure. Up to this point, all the examples have dealt with linear, or linearized, mechanical models. In Section VII we analyze the robust reliability of an inherently non-linear phenomenon: structural degradation and failure by material fatigue resulting from uncertain, time-varying loads far below the yield limit of the material. Finally, in Section VIII we modify our subject a bit, and discuss the reliability of mathematical models of structures, rather than of the structures themselves.

II Convexity and Uncertainty

Uncertainty: events occur, one after the other, accumulating into clouds, clusters, and patterns. Fragmentary information about the uncertain phenomena can be used to characterize the clustering of uncertain events. Without specifying anything about the probabilistic frequency-of-recurrence of events, a convex model is a set whose structure is derived from the underlying properties of the event clusters.

A convex model is a set of functions or vectors. Each element of the set represents a possible realization of the uncertain phenomenon. The usual engineering approach to formulating a convex model is to start with what we know about the phenomenon and to define the set of all functions consistent with that information.

An an example, consider transient excursions in the vector f(t) of loads applied to a structure. Suppose that the nominal load history is known, ¯t, and that the cumulative energy of deviation from this nominal load is bounded. The set of all functions consistent with this information is a cumulative energy-bound convex model, defined as:

α=ft:∫0∞ft−f¯tTft−f¯tdt≤α2

We may not know the value of α, and in robust reliability analysis, we do not need to know it. α is the uncertainty parameter whose value determines the “size” of the set, the latitude of variation of the uncertain phenomenon, or the information gap between the anticipated nominal load and the loads which may occur in practice. The convex model defines a set of functions consistent with particular prior information. It defines a cluster of events.

Because we do not know the value of α, we will often think of a convex model as a family of sets: (α) for α ≥ 0. Thinking in this way, we realize that every L2-integrable function f(t) belongs to the family for sufficiently large values of α. The convex model is not a black-and-white division of events into “completely allowed” and “strictly forbidden.” Rather, the family of convex models (α), α ≥ 0, arranges the space of all events in a particular order. It is not an order in terms of frequency of occurrence as in probability, but an order in terms of clustering. In the convex model of eq. (1) the events cluster around the nominal load profile, ¯t, and the variation is ordered by the cumulative energy of deviation.

We will often define convex models in such a way as...