Numerical solution of static and dynamic problems of imprecisely defined structural systems

Abstract

Static and dynamic problems with deterministic structural parameters are well studied. In this regard, good number of investigations have been done by many authors. Usually,
structural analysis depends upon the system parameters such as mass, geometry, material properties, external loads and boundary conditions which are defined exactly or considered as deterministic. But, rather than the deterministic or exact values we may have only the vague, imprecise and incomplete informations about the variables and parameters being a result of errors in measurements, observations, experiments, applying different operating conditions or it may be due to maintenance induced errors, etc. which are uncertain in nature. Hence, it is an important issue to model these types of uncertainties. Basically these may be modelled through a probabilistic, interval or fuzzy approach. Unfortunately, probabilistic methods may not be able to deliver reliable results at the required precision without sufficient experimental data. It may be due to the
probability density functions involved in it. As such, in recent decades, interval analysis and fuzzy theory are becoming powerful tools. In these approaches, the uncertain
variables and parameters are represented by interval and fuzzy numbers, vectors or matrices.In general, structural problems for uncertain static analysis with interval or fuzzy parameters simplify to interval or fuzzy system of linear equations whereas interval or fuzzy eigenvalue problem may be obtained for the dynamic analysis. Accordingly, this thesis develops new methods for finding the solution of fuzzy and interval system of linear equations and eigenvalue problems. Various methods based on fuzzy centre, radius, addition, subtraction, linear programming approach and double parametric form of fuzzy numbers have been proposed for the solution of system of linear equations with fuzzy parameters. An algorithm based on fuzzy centre has been proposed for solving the
generalized fuzzy eigenvalue problem. Moreover, a fuzzy based iterative scheme with Taylor series expansion has been developed for the identification of structural parameters from uncertain dynamic data. Also, dynamic responses of fractionally damped discrete and continuous structural systems with crisp and fuzzy initial conditions have been obtained using homotopy perturbation method based on the proposed double parametric form of fuzzy numbers.
Numerical examples and application problems are solved to demonstrate the efficiency and capabilities of the developed methods. In this regard, imprecisely defined
structures such as bar, beam, truss, simplified bridge, rectangular sheet with fuzzy/interval material and geometric properties along with uncertain external forces have been considered for the static analysis. Fuzzy and interval finite element method have been applied to obtain the uncertain static responses. Structural problems viz. multistorey shear building, spring mass mechanical system and stepped beam structures with uncertain structural parameters have been considered for dynamic analysis. In the
identification problem, column stiffnesses of a multistorey frame structure have been identified using uncertain dynamic data based on the proposed algorithm. In order to get the dynamic responses, a single degree of freedom fractionally damped spring-mass mechanical system and fractionally damped viscoelastic continuous beam with crisp and fuzzy initial conditions are also investigated.Obtained results are compared in special cases for the validation of proposed methods.