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On Postbuckling Analysis of Thin-walled Structures Numerical and Statistical Approaches

Doctoral thesis, 1999

Thin-walled structural elements, such as plates, shells and beams are important load carrying components in civil-, mechanical-, automotive- aeronautical- and marine engineering structures. In order to reduce the weight thin-walled structures are given an optimum design making them sensitive to unforeseen deviations in geometry, boundary conditions, material properties and applied loads. Therefore to assure safe, reliable structures accurate deterministic numerical methods for nonlinear analysis along with probabilistic methods are needed.
In part one of the thesis an analytical probabilistic model is proposed for predicting the scatter of the buckling load of axially compressed thin linear elastic cylindrical shells with geometrical imperfections. The probability density function of the buckling load is deduced using an asymptotic relation according to Koiter between an assumed geometric imperfection and the buckling load. The scatter in the load-carrying capacity predicted by the proposed model, agrees well with experimental results from the buckling of cylindrical shells of a certain manufacturing tolerance class.
In general, for complex structures, it is not possible to deduce closed form probability density functions for failure load predictions. Then a computer method using the Monte-Carlo simulation technique can be applied. However, the success of adopting a Monte Carlo technique strongly depends on the ability of the deterministic numerical method used to simulate the non-linear structural behavior.
As an important step in developing a probabilistic computer simulation method geometrically non-linear thin structures with conservative deterministic one parameter loading has been studied in part two of the thesis and a computer code has been developed.
The finite deformation theory for a linear elastic continuum is expressed in terms of displacement gradients. In the finite element approximation the displacement field is expressed only in translational variables. Standard linear shape functions are used for truss elements and p-hierarchical basis functions based on integrals of the Legendre polynomials are adopted for two-dimensional plane stress elements and for three-dimensional solid elements. The structural deformations are measured with respect to the initial undeformed reference configuration in a total Lagrange formulation. In order to follow the non-linear equilibrium path an incremental algorithm driven by the dominant displacement component is implemented. Numerical tests on a number of geometrically non-linear space trusses, a plane frame and a three dimensional beam show excellent agreement with analytical solutions as well as with numerical results presented in literature. The adopted p-hierarchical solid element formulation is demonstrated to work excellent for linear analysis of thin beams and shells.

postbuckling

hierarchical elements

imperfection sensitivity

displacement gradients

total Lagrange formulation

geometrically non-linear structures