Dynamic stability of thin-walled structures: a semi-analytical and experimental approach

Abstract

Buckling refers to a sudden large increase in the deformation of a structure due to a small increase of some external load. If this external load has a dynamic nature, (e.g. a harmonic load, shock load, a step load and/or a random load), such a sudden increase in deformations is denoted as dynamic buckling. Thinwalled structures are often met in engineering practice due to their favourable mass-to-stiffness ratio. Such structures are very susceptible to buckling and are often subjected to dynamic loading. However, fast (pre-) design tools for obtaining detailed insight in the dynamic response and the stability of thinwalled structures subjected to dynamic loading are still lacking. One of the research objectives of this thesis is, therefore, to develop (fast) modelling and analysis tools which give insight in the behaviour of dynamically loaded thinwalled structures. To illustrate and to test the abilities of the developed tools, a number of case studies are examined. The tools are developed for structures with a relatively simple geometry. The geometric simplicity of the structures allows to derive models with a relative low number of degrees of freedom which are, therefore, very suitable for extensive parameter studies (as essential during the design process of thin-walled structures). These models are symbolically derived using a Ritz method in combination with assumptions regarding geometric nonlinear (strain-displacement) relations and the effects of (in-plane) inertia. The resulting models, obtained from energy expressions, are sets of coupled ordinary differential equations which include stiffness nonlinearities and (sometimes) inertia and damping nonlinearities. The modelling approach is implemented in a generic manner in a symbolic manipulation software package, so that model variations can be easily performed. Furthermore, a set of designated numerical tools is combined (e.g. continuation tools for equilibria, periodic solutions and bifurcations, and numerical integration routines) to solve the analytically derived models in a computationally efficient manner. Using this semi-analytical (i.e. analytical-numerical) approach four case studies are performed which include the dynamic buckling of an arch type of structure due to shock loading, snap-through behaviour of a transversally, harmonically excited pre-buckled beam, and the dynamic buckling of a beam and a cylindrical shell structure, both with top mass, which are harmonically loaded in axial direction at their base. For all cases, the effects of several parameter variations are illustrated, including the effect of small deviations from the nominal geometry (i.e. geometric imperfections). For validation, the semi-analytical results are compared with results obtained using the computationally much more demanding finite element modelling technique. However, more important, for two cases (i.e. the axially excited beam and cylindrical shell structures carrying a top mass), the semi-analytical results are also compared with experimentally obtained results. For this purpose, a dedicated experimental set-up has been realized. For the beam structure, the experimental results are in good agreement with the semianalytical results whereas for the cylindrical shell structure, a qualitative match is obtained. It has been illustrated that the differences between the experimental results and the semi-analytical results for the cylindrical shell may be due to the strong dependency of the results with respect to the geometrical imperfections present in the shell. Next to the specific new insights obtained for each case considered, the major result of the thesis is the illustrated power of the semi-analytical approach to obtain practical relevant insights in the phenomena of dynamic buckling of thin-walled structures. In conclusion it can be stated that the semi-analytical approach is a valuable tool in the (pre-) design process of thin-walled structures under dynamic loading.