Study On Thin Circular Membrane Problems Without Small-rotation-angle Assumption

Membrane structures and structural components have found increasing applications in many fields.The membrane that is under the loads,relative to its thickness,often exhibit large deflection value,so its deformation problem usually show much stronger nonlinear,and the governing differential equation of membrane usually also are non-linear differential equations.These nonlinear differential equations generally present serious analytical difficulties when applied to boundary-value problems.Due to the somewhat intractable non-linear equations,analytical solutions are available in a few cases,but usually a variety of methods,such as shooting method,iterative method and so on,are utilized to obtain numerical solutions for displacements,strains,and stresses.In practice,however,analytical solutions are often found to be necessary.In this paper,the large deflection problem of axisymmetric deformation of uniformly loaded circular membranes,was resolved,where the small-rotation-angle assumption usually adopted in membrane problems was given up,and a better understanding of the non-linear behavior of the considered problem could thus be reached.The presented closed-form solution has a higher accuracy than well-known Hencky solution,which is suitable for situations where the load is large.When the rotation-angle is small,the validity of the analytical solution given by this paper is proved by comparing the results obtained by the analytical solution given in this paper and the results obtained by Hencky solution and the finite element calculation results.Then when the rotation-angle is relatively large,showing the error of the small-rotation-angle assumption by comparing the results obtained by the analytical solution given in this paper and the results obtained by Hencky solution.The results show that with the increase of the applied transverse loads,the calculation error of the small-rotation-angle also increases,and the famous Hencky solution will no longer apply when the applied transverse loads is relatively large.Through the above comparative analysis,it is also found that an important integral constant in the analytical solution given in this paper is changed with the applied transverse loads,but in well-known Hencky solution it becomes a constant independent of the load due to the adopted small-rotation-angle assumption,which is why the small-rotation-assumption leads to the error.The thesis consists of seven chapters:The first chapter present the main research content in brief,research background,research significance;The second chapter introduces basic theories of circle membrane problem for elastic theory,and the power series solution of a differential equation;The third chapter mainly introduces the Hencky problem and the analytical solution of Hencky problem with the small-rotation-assumption.The fourth chapter established the governing differential equations of circle membrane problem under uniformly distributed loads,where the small-rotation-assumption was given up,which resolved by exactly using power series method,getting the analytic expression of membrane deflection,radial stress and circumferential stress.Then some corresponding discussions are made.The fifth chapter mainly gives the steps and the results of finite element calculation of the large deflection problem of axisymmetric deformation of uniformly loaded circular membranes;The sixth chapter verifies the validity of the theoretical work by comparing the finite element calculation results with the analytical results,and analyzes the errors caused by the small-rotation-angle assumption;The seventh chapter presents the main conclusions of this thesis and prospects of membrane problem.In this paper,the large deflection problem of axisymmetric deformation of uniformly loaded circular membranes,was resolved,where the small-rotation-angle assumption usually adopted in membrane problems was given up.By comparing the analytical solution with the Hencky solution,this paper shows the effect of the small-rotation-angle assumption of the membrane problem.The work of this paper has some theoretical significance for the analysis and design of flexible membrane structures and structural components.