On Stochastic Load Combination
This series includes technical reports prepared by faculty, students and staff who are associated with the John A. Blume Earthquake Engineering Center at Stanford University. While the primary focus of Blume Center is earthquake engineering, many of the reports in this series encompass broader topics in structural engineering and materials, computational mechanics, geomechanics, structural health monitoring, and engineering life-cycle risk assessment. Each report includes acknowledgments of the specific sponsors for the report and underlying research. In addition to providing research support, the Blume Center provides administrative support for maintaining and disseminating the technical reports. For more information about the Blume Center and its activities, see https://blume.stanford.edu.
In the Introduction, we first address the questions why (Section 1.1) and how (Section 1.2) stochastic load combination problems for structural elements should be studied. Previous work in the field is reviewed. A case is made for second moment analysis of safety using-results from parametric stochastic load models and first-order second moment procedures applied to all other random variables in the interaction equation for a specific failure mode of a structural element. A classification of stochastic load combination problems according to the formal structure of load terms is proposed. The resulting safety index (the logarithmic Rosenblueth β) is understood as an approximately ordinal scale of measurement for the (inaccessible) probability of failure. The question for the size of ordinal classes (how large is the class of problems whose β can be compared to each other) is asked. From Section 1.2.4 on, attention is focused on technical aspects of stochastic load combination. Simulation procedures--which are otherwise not used in this report--are outlined as an unavoidable last resort for all but the simplest cases. A simple step function load process with Poisson load occurrences in time and discrete or mixed amplitude distributions is defined in Section 1.3. Its interpretation as a special equilibrium Markov pure jump process is given, together with statistical estimators for the process parameters from observed mean times in discrete states. Chapter 2 discusses the basic case of superposition with constant coefficients for a process (Section 1.3) and a process of random Poisson events of short duration. Elementary combinatorial methods are used. A special case with deterministic amplitudes and randomness in time only is carried out in detail. Numerical results and apowerseries expansion for the probability of coincidence ,are given, together with a discussion of the accuracy of certain approximations. Section 2.2 gives a combinatorial method for the general case with random amplitudes. Section 2.3 generalizes a model due to Borges and Castanheta to cases with randomness of events in time. Though closely related to Section 2.2, this model stands out from the context by allowing for general renewal processes in time and combination of more than two load effects, under certain well defined circumstances. Chapter 3 discusses the superposition of two processes of the family (Section 1.3), with constant coefficients. Finite Markov chain methods are used throughout. Again; the special case of deterministic amplitudes (randomness only in time) is considered first (Section 3.1); numerical results and approximations are given. The general case (Section 3.2) is not a straightforward generalization, but requires a more elaborate structure of the chain, together with a discrete (or discretized) amplitude density function for one of the two processes.
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