| This paper focuses on general elastic buckling problem based on3-D solidstructure and the solving by boundary face method (BFM). In this paper, a generalpartial differential equation (PDE) for buckling analysis is presented. According tothe partial differential equation, we derive a boundary integral equation and establishthe process of BFM for buckling analysis. Then we code a program for generalbuckling analysis by C++language. Several numerical examples given in this articleillustrated the validity of the elastic buckling theory based on3-D solid structure andits good applicability for different structures.Traditional buckling analysis, linear or nonlinear, presents different forms ofPDE for different structures such as beam, plate or shell and so on. All these equationsare based on the deformation hypothesizes of thin and slender structures for theirdeformation characteristics. Though problems are simplified for applying thesehypothesizes, some contradiction in theory appears, for example, discordances inconstitutive relations and boundary conditions between traditional buckling theoryand factual. Additionally, lots of experiments by predecessors also confirmed thatthere is great deviation between theoretical and experimental results.For bridging the gap between traditional buckling analysis and actual results, wepropose a new way of thinking that is giving up all deformation hypothesizes of beam,plate or shell structures and deriving the buckling equation based on3-D elasticitytheory directly. To study the equilibrium path of structure, we find that bucklingproblem essentially is due to the presence of multiple equilibrium paths, which meansmultiple solutions for a differential equation in mathematic. That implies the propertyof nonlinear in buckling problem. The mutating of structure’s topological form inbuckling phenomenon indicates that the geometric relations and equilibrium equationscan not be constructed based on the small deformation assumption but on thenonlinear geometric relation theory. In Chapter2, applying the nonlinear geometricrelations, that is the Green strain tensor, and the energy criterion for stability judging,we established the general elastic buckling equation based on3-D solid structure inTotal Lagrange formulation (T.L.). In order to simplify the problem, we introduce twoassumptions: one is assuming that the constitutive relation is linear and satisfies thegeneralized Hooke’s law; another is assuming that the pre-buckling state to meet the assumption of small deformation which makes it is possible to get initial stress vialinear elastic analysis. These two assumptions simplified the general elastic bucklingequation with complex form into a linear eigenvalue buckling equation.Boundary face method, proposed by Professor Zhang Jianming, which developedfrom boundary element method (BEM), inherited all the advantages of the BEM.Compared to BEM, BFM with higher accuracy for all boundary integration andgeometric data calculating is based on surface parametric space, which avoids thegeometric error on boundary surface.In Chapter3, the process of applying the BFM for solving buckling analysis isestablished. Using the Kelvin fundamental solution, the boundary integral equation ofbuckling problem based on3-D solid structure is derived. Coupling with dualreciprocity method (DRM), the domain integration is well disposed. And eventually,the discrete format of buckling problem is established and the eigenvalue matrix isassembled. Then we write a program of BFM for general buckling analysis by C++language. Some numerical examples are given in Chapter4, by comparing with finiteelement method (FEM), the proxmity of the both results shows the validity of theboundary surface method for buckling analysis. Meanwhile, compared with the resultsof finite element method for general elastic buckling problem, BFM requires lessnodes and degrees of freedom can get the similarly accuracy results. This indicatesthe advantage of dimensionality reduction of BFM. |
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