Power Series Solutions Of Circular Membrane Problems And Disproval On Extended Hencky Transformation Method

The membrane has wide applications in many advanced fields. The membrane thatis under the loads, relative to its deflection, usually presents much larger deflectionvalue, so its transformation problems show much stronger non-linear, which makesanalytic study complex, difficulty, and its solutions are not usually fundamentalfunctions. The difference of analytic study, is the charm of membrane problems, andmotivates scholars’ interests of research.The problem of axisymmetric deformation of the circular membrane fixed at itsperimeter under the action of uniformly-distributed loads, namely well-known Henckyproblem, is a classic plate and shell problem of elastic mechanics. The problem’sanalytic solution is expressed by the pattern of power series, and includes the studyingworks of three famous scientists: Hencky, Chien and Alekseev, and is usually short forwell-known Hencky solution. The solution is the first solution of circular membraneproblems and is widely cited.However, seeing from existing literature search results, the solve process ofHencky problem, is not entirely presented in the pattern of power series method ofdifferential equation. Especially, in the solve process of Professor Chien, he uses apattern that seeming a general mathematical method to express the process, and themethod is called after Hencky transformation method by later scholars. The method isregarded as a general mathematical method,namely extended Hencky transformationmethod, and is extended to use in the solutions of general membrane problems.To verify the effectiveness of so-called Hencky transformation method and itsextension, the thesis strictly uses the pattern of power series method of differentialequation to resolve the membrane equation of well-known Hencky problem. The thesisconsists of seven chapters:①Introduction: Briefly stating the issue’s research background, researchsignificance and main research contents;②Basic theory: Briefly clarifying the basic of elastic theory and Von Karman’sthin plate large deflection theory, which is needed by studying circle membraneproblem;③Henckyproblem: Introducing Hencky’s and Chien’s work about the problem of axisymmetric deformation of the circular membrane fixed at its perimeter under theaction of uniformly-distributed loads;④P ower series method of differential equation of Hencky problem: Strictly usingthe pattern of power series method of differential equation to resolve the membraneequation of well-known Hencky problem, and giving the typical parameter figures;⑤D isproof onthe effectiveness of extended application of Hencky transformationmethod: Adopting mathematics theory and specific examples, proves that so-calledHencky transformation method is only applied to the second order differential equationof specific form, but it can’t be regarded as a general mathematical method of solvingsecond order differential equations, and be extended to apply;⑥Finite element calculation and experiment: Using experiment and finite elementanalysis to verify the theoretical work’s effectiveness;⑦Conclusion and prospect: Summarizing and concluding conclusions of researchwork, and prospecting related problems.The thesis firstly gives the detailed process of power series method of Henckyproblem, and points out the mistake of so-called extended application of Henckytransformation method in mathematics theory, which embodies novelty and theoreticalvalue of the thesis’ research work.