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The purpose of this paper is to study the regions of parameter space of engineering design in which performance is sensitive to design parameters. Some of these parameters (for example, the dimensions and compositions of components) constitute the design, but others are intrinsic properties of materials or Nature. The paper is concerned with narrow regions of parameter space, “cliffs”, in which performance (some measure of the final state of a system, such as ignition or nonignition of a flammable gas, or failure or nonfailure of a ductile material subject to tension) is a sensitive function of the parameters. In these regions, performance is also sensitive to uncertainties in the parameters. This is particularly important for intrinsically indeterminate systems, those whose performance is not predictable from measured initial conditions and is not reproducible.

We develop models of ignition of a flammable mixture and of failure in plastic flow under tension. We identify and quantify cliffs in performance as functions of the design parameters. These cliffs are characterized by large partial derivatives of performance parameters with respect to the design parameters and with respect to the uncertainties in the model. We calculate and quantify the consequences of small random variations in the parameters of indeterminate systems.

We find two qualitatively different classes of performance cliffs. In one class, performance is a sensitive function of the parameters in a narrow range that separates wider ranges in which it is insensitive. In the other class, the final state is not defined for parameter values outside some range, and performance is a sensitive function of the parameters as they approach their limiting values. We find that sensitivity of performance to control (design) parameters implies that it is also sensitive to other parameters, some of which may not be known, and to uncertainties of the initial state that are not under the control of the designer. Near or on a cliff performance is degraded. It is also less predictable and less reproducible.

Frequently, design optimization or cost minimization leads to choices of engineering design parameters near cliffs. The sensitivity of performance to uncertainty that we find in those regimes implies that caution and extensive empirical experience are required to assure reliable functioning. Because cliffs are defined as behavior on the threshold of failure, this is a reflection of the trade-off between optimization and margin of safety, and implies the importance of ensuring that margins and uncertainties are quantified. The implications extend far beyond the model systems we consider to engineering systems in general.

Many of these considerations have been part of the informal culture of engineering design, but they were not formalized until the methodology of “Quantification of Margins and Uncertainty” was developed in recent years. Although this methodology has been widely used and discussed, it has only been published in a small number of reports (cited here), and never in a journal article or book. This paper may be its first formal publication, and also its first quantitative application to and illustration with explicit model problems.

HAVING discussed in the standard longhand notation the main ideas and methods for the calculation of redundant structures on the basis of forces as unknowns we now turn our attention to the matrix formulation of the analysis. Consider a system consisting of s structural elements with a total number n of redundancies which may be forces (stresses), moments or any generalized forces. We select a basic system by ‘cutting’ a number r of redundancies where r<n. Thus, the simple idea of a statically determinate basic system (r=n) is but a particular case of our investigations.

The series of illustrations to the elimination procedures is concluded with two examples in which the various techniques are again examined critically. The first application considers the important problem of a major cut‐out in a fuselage, extending over a number of panels and bays, and including the attached ring segments. Here the dismembering process appears in its straightforward simplicity and generality a particularly effective choice. The final example investigates the removal of a major substructure from a system in the form of an ‘open’ or ‘closed chain’ of substructures. The discussion on cut‐outs is rounded off with a synopsis of the techniques used and a description of the automatic programme for checking an elimination scheme for any hidden singularities.

The general theory of the cut‐out and modification analysis is reviewed and extended for a structure involving primary, secondary and tertiary redundancies. Some important points of practical application are illustrated on simple examples and the influence of the form chosen for the unassembled flexibility matrix is discussed. The question of the selection and number of actual cuts which will simulate a given major cut‐out is treated in general and illustrated on a simple type of structure.

THE general theorems given in Sections 4 and 6 include, from the fundamental point of view, all that is required for the analysis of redundant structures. However, to facilitate practical calculations it is helpful to develop more explicit methods and formulae. To find these is the purpose of this Section.

A series of examples illustrates the application and results of the method previously given for improving the conditioning of the primary redundancies. The transformation matrix giving locally orthogonal redundancies for the parent station is used also to generate the primary self‐equilibrating stress systems for sections with different (but related) geometry and varying ring stiffness. A double‐cell cross‐section is also treated and a brief investigation shows that the general methods developed for establishing primary redundancies in the fuselage can be usefully and economically applied to certain kinds of wing structure. Preparatory to a detailed discussion of cut‐outs and modifications in the fuselage structure, the necessary general theory is collected and summarized.

A DSIR Sponsored Research Programme on the Development and Application of the Matrix Force Method and the Digital Computer. The present issue gives a summary of the basic theory of the matrix force method together with some necessary extensions for the fuselage problem. The equilibrium conditions for the idealized structure are then examined in detail and the relevant equations of equilibrium established in matrix form.

The first Principia Mathematica (1686) by Sir Isaac Newton with reference to natural philosophy and his system of the world has largely contributed to the first revolution in scientific thinking in modern times. It has created the conceptual basis of modern science in the classical tradition by providing the tools of analysis and the technique of reasoning in terms of stability—from—within or, as we would say today, the model of stable equilibrium conditions.

We test the determinacy properties of the standard and financial-sector-augmented Taylor rules in a new Keynesian model with a presence of banking activities. We extend the basic fully rational environment to the setting with heterogeneous expectations. We observe that the benefits from extra financial targeting are limited. Financial targeting, if well designed, can compensate for the improper output-gap targeting through the financial-production channel. The analysis demonstrates however possible threats resulting from the misspecification of the augmented rule. A determinate mix of output-gap and inflation weights can turn indeterminate if compensated by too extreme financial targeting. The results are robust to the presence of heterogeneous expectations.

IT is natural in reviewing the developments of Sections 3 and 4 to inquire if it is possible to enlarge upon the conception of complementary work and strain energy in a similar way as accomplished for work and strain energy by the introduction of virtual displacements.