Displacement-Based Power Series Solution Of Circular Membrane Problem

At present,the membrane structures has been widely used in many branches of engineering technology,such as architecture,instrumentation,electronics,aerospace.The deflection of the membrane is usually much larger than its thickness when it subjected loads,therefore,the deformation of thin membrane usually exhibits geometric nonlinearity,which makes the analytical study of membrane problems very difficult.But precisely because of this,many scholars has always been looking for the analytical methods with high accuracy and wide range of applications.The problem of axisymmetric deformation of circular membrane fixed at the outer edge under the action of transverse uniform load,namely Hencky problem for short,it's a classic problem among the membrane problems.An investigation into the detailed process of solving the problem by Hencky and Chien W.Z,we may draw the conclusion that the method used by this two scholars can be considered as stress-based power series solution method,that is to say,both of them take the stress component as the basic unknown quantity,and use the power series solution method to solve the equation.However,the displacement-based power series solution method has not been adopted to solve Hencky problem in the existing literature,this paper has carried out relevant research on the issue.This paper is based on the large-deflection theory of thin plates,by ignoring the term containing bending stiffness in the von Kármán equation,such a plate problem is transformed into a membrane,the basic equations of the axisymmetric deformation of circular membrane in the polar coordinate system were further derived through coordinate transformation,moreover,the power series solution of deflection stress and strain was presented by using the displacement-based solution method.The result shows that the displacement-based solution method is feasible for the Hencky problem,but in comparison with the stress-based solution method,the method used in this note can make the solution process more concise.This paper is composed of six chapters:The first chapter briefly states the research significance of the subject,reviews the research status of thin plate and membrane problems,and introduces the main research contents and innovations of this paper.The second chapter mainly introduces the von Kármán large-deflection theory of thin plates,if the bending stiffness term in von Kármán equation is neglected,such a plate problem is transformed into a membrane problem.The third chapter carries an investigation into the detailed process of solving circular membrane problem by Hencky and Chien W.Z.The fourth chapter presents the detailed process of solving the circular membrane problem by using the displacement-based solution method,and the power series solutions of deflection and stress and strwas obtained here.Moreover,some typical figures of the relative parameters of the common poisson was also given,and some related problems were discussed.The fifth chapter carry out a comparative analysis through a specific example,the general process of analyzing the static problem of circular membrane with ABAQUS was given out.Moreover,the analytical results and ABAQUS numerical results were compared and analyzed,thus,the validity of the analytical solution was verified here.The sixth chapter summarizes the main contain of this paper,and point out the work maybe conducted with respect to membrane problems in the future.The work of this paper not only enriches the solution method of the circular membrane problem,but also has some theoretical reference value for the study of similar mechanical problems.