Biaxial isotropic stochastic visco-elastic creep

This is second part of a paper dealing with the formulation of constitutive equations for statistically isotropic multi-axial visco-elastic stochastic creep in terms of a second moment white noise field model. Herein the biaxial linearized model is studied. After an extensive retrospective introduction recapitulating the basic concepts, the paper presents the solution to the spatial covariance structure of a stress field history which is homogeneous in the mean. Results for the corresponding strain field are also presented. It turns out to be necessary to let the spatial second moment while noise character of the strain tensor field history for given deterministic stress tensor history be approached through a sequence of genuine covariance functions corresponding to isotropic random fields. In the limit the variances of both stress and strain become infinite. Interesting asymptotic results show up in this connection.The appendices give some useful mathematical results concerning Fourier transforms related to the Laplace operator and covariance functions of isotropic random fields.     

Slepian Approach to a Gaussian Excited Elasto-Plastic Frame Including Geometric Nonlinearity

Taking geometric non-linearity into account, an oscillator inthe form of a portal frame with a rigid traverse and withideal-elastic-ideal-plastic clamped-in columns behaves under horizontalexcitation as an ideal-elastic-hardening/softening-plastic oscillatorgiven that the columns carry a tension/compression axial force. Assumingthat the horizontal excitation of the traverse is Gaussian white noise,statistics related to the plastic displacement response are determinedby use of simulation based on the Slepian model process method combinedwith envelope excursion properties. Besides giving a physical insight,the method gives good approximations to results obtained by slow directsimulation of the total response. Moreover, the influence of a randomlyvarying axial column force is investigated by direct responsesimulation. This case corresponds to parametric excitation as generatedby the vertical acceleration component in an earthquake.     

Generalized Second Moment Reliability Index

ABSTRACT

A crucial property of any measure of structural reliability should be comparativeness. With this point in mind this paper discusses some well-known versions of so-called reliability indices. Such reliability indices, defined by use of second moment information, have been used for the last decade, specifically within safety code committee work. This paper defines a generalized second moment reliability index that satisfies some few fundamental canonical rules and principles of simplicity. The reliability index is specifically defined to be used when no high quality information is available to the engineer other than the limit state surface and a second moment representation for the set of basic variables of the structural problem. In a parallel paper the author demonstrates the practical operability of this reliability index even for multimode failure systems.     

Narrow Reliability Bounds for Structural Systems

ABSTRACT

For structural systems that may fail in any one of several possible modes, reliability analysis is greatly simplified by use of upper and lower bound techniques. General bounds based on all the single mode failure probabilities and all the pairwise mode intersection failure probabilities are established. For systems where the single mode limit state surfaces are hyperplanes in the space of basic variables, a simple geometrical interpretation of the correlation between mode safety margins combined with a well-known geometrical interpretation of the single mode reliability index makes the practical calculation of the system reliability bounds easy. This is particularly true when the set of basic variables is jointly normally distributed. Examples show very narrow bounds which, in the practically important domain of high reliability, are almost coincident.     

Anomalous jumping in a double-well potential

Noise-induced jumping between metastable states in a potential depends on the structure of the noise. For an alpha-stable noise, jumping triggered by single extreme events contributes to the transition probability. This is also called Levy flights and might be of importance in triggering sudden changes in geophysical flow and perhaps even climatic changes. The steady-state statistics is also influenced by the noise structure leading to a non-Gibbs distribution for an alpha-stable noise.     

Climate transitions on long timescales

The climate has changed through the history of the Earth as evidenced in the geological records. Today we might be experiencing a climate change of the same magnitude as the transition into an ice age caused by very rapid burning and emission to the atmosphere of a substantial part of the fossilised carbon. Whether this leads to a gradual warming or if we will experience a transition into a different climatic state is presently unknown. The present day state-of-the-art numerical climate models are capable of producing fair representations of the current climate and are as such trusted to also predict the climate changes due to increasing atmospheric concentrations of greenhouse gases. However, the models are not presently capable of reproducing the rapid transitions from one climatic state, such as a glacial climate, into another, such as the present climate. The reason for this is unknown. The transitions are inherently ‘non-linear’ and thus not accessible through linear response theory. The term ‘non-linear’ is in this context defined as the phenomenon that the response of the system to a change in the forcing of the system is not linearly proportional to the forcing. This would happen if a threshold is reached such that the state of the system becomes unstable and the system bifurcates into a different state. There are strong indications in the geological records of this kind of behaviour for the climate. These dynamics can be understood in the context of fairly simple models of the climate.     

Reliability Against Defect Generated Fracture

ABSTRACT

The subject of this study is the reliability determination for structures that may fail due to fracture generated from some material defect among a random number of small random defects distributed in some way throughout the body of the structure. The considered failure mechanism may be characterized as a weakest link mechanism with a random number of links. The calculation technique is based on some previously published general narrow probability bounds for unions of small probability events. In the case of high reliability structures, as are usually of interest in structural engineering, some very useful conclusions may be drawn from the study. Given only the defect fracture probability corresponding to the actual stress field and the mean number of defects per unit volume both as functions of the space coordinates, a sufficiently accurate assessment of the total reliability may be calculated. With rare exceptions, no dependency information is needed and no information about the defect point process is needed except its mean function. The reliability is insensitive to very large variations of these properties. If there is doubt about the dependency level being too high to neglect it, conditioning followed by fast probability integration may be useful. This is illustrated in the last section of the paper.