ON EXACT SOLUTION OF KÁRMÁN’S EQUATIONS OF RIGID CLAMPED CIRCULAR PLATE AND SHALLOW SPHERICAL SHELL UNDER A CONCENTRATED LOAD

It is extremely difficult to obtain an exact solution of von Karman’s equations because the equations are nonlinear and coupled. So far many approximate methods have been used to solve the large deflection problems except that only a few exact solutions have been investigated but no strict proof on convergence is presented yet. In this paper, first of all, we reduce the von KÁrmÁn’s equations to equivalent integral equations which are nonlinear, coupled and singular. Secondly the sequences of continuous function with general form are constructed using iterative technique. Based on the sequences to be uniformly convergent, we obtain analytical formula of exact solutions to von Karman’s equations related to large deflection problems of circular plate and shallow spherical shell with clamped boundary subjected to a concentrated load at the centre.     

THE SCHR(?)DINGER EQUATION IN THEORY OF PLATES AND SHELLS WITH ORTHORHOMBIC ANISOTROPY

This work is the continuation of the discussion of Refs.[1-5].In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection fororthorhombic anisotropic thin shells or orthorhombic anisotropic thin plates on Winkler’sbase are classified as several of the same solutions of Schr?dinger equation.and we canobtain the general solutions for the two above-mentioned problems by the method in Refs.[1]and[3-5].[B]The von Kármán-Vlasov equations of large deflection problem for shallow shellswith orthorhombic anisotropy(their special cases are the von Kármán equations of largedeflection problem for thin plates with orthorhombic anisotropy)are classified as thesolutions of AKNS equation or Dirac equation,and we can obtain the exact solutions forthe two abovementioned problems by the inverse scattering method in Refs.[4-5].The general solution of small deflection problem or the exact solution of largedeflection problem for the corrugated or rib-reinforced plates and shells as special c     

The Schrödinger equation in theory of plates and shells with orthorhombic anisotropy

This work is the continuation of the discussion of Refs. [1-5]. In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection for orthorhombic misotropic thin shells or orthorhombic anisotropic thin plates on Winkler’s base are classified as several of the same solutions of Schrodmger equation, and we can obtain the general solutions for the two above-mentioned problems by the method in Refs. [1] and [3-5].[B] The. von Karman-Vlasov equations of large deflection problem for shallow shells with orthorhombic anisotropy (their special cases are the von Harmon equations of large deflection problem for thin plates with orthorhombic anisotropy) are classified as the solutions of AKNS equation or Dirac equation, and we can obtain the exact solutions for the two abovementioned problems by the inverse scattering method in Refs. [4-5].The general solution of small deflection problem or the exact solution of large deflection problem for the corrugated or rib-reinforced plates and shells as special cases is included in this paper.     

A PERTURBATION-VARIATIONAL SOLUTION OF THE LARGE DEFLECTION OF RECTANGULAR PLATES UNDER UNIFORM LOAD

In this paper,von Kármàn’s set of nonlinear equation for large deflection ofrectangular plates is at first converted into several sets of linear equations by taking centraldimensionless deflection as perturbation parameter,and then,the sets of linear equationsfor plates with various ratios of length to width are solved with application of variationalmethod.The analytical expressions for displacements and stresses as well as formulas fornumerical calculation are worked out.The figures of maximum deflection-load andmaximum stress with ratio λ of length to width as a parameter are given in this paper.Through comparison,it is found that the results of this paper are quite in accord withexperiments.     

FURTHER STUDY OF THE RELATION OF VON KF8B6RMN EQUATION FOR ELASTIC LARGE DEFLECTION PROBLEM AND SCHRDINGER EQUATION FOR QUANTUM EIGENVALUES PROBLEM

This work is the continuation and improvement of the discussion of Ref.[1]. We alsoimprove the discussion of Refs.[2-3] on the elastic large deflection problem by results ofthis paper.We again simplify the von Kármán equation for elastic large deflection problem,and finally turn it into the nonlinear Schr(?)dinger equation in this paper.Secondly,weexpand the AKNS equation to still more symmetrical degree under many dimensionalconditions in this paper.Owing to connection between the nonlinear Schr(?)dinger equationand the integrability condition for the AKNS equation or the Dirac equation,we can obtainthe exact solution for elastic large deflection problem by inverse scattering method.In otherwords,the elastic large deflection problem wholly becomes a quantum eigenvalues problem.The large deflection problem with orthorhombic anisotropy is also deduced in thispaper.     

ON THE GENERAL EQUATIONS OF AXISYMMETRIC PROBLEMS OF IDEAL PLASTICITY

In this paper, introducing a velocity potential, we reduce the fundamental equations of axisymmetric problems of ideal plasticity to two nonlinear partiCal differential equations. From these equations we discuss compatibilitg of Harr-Kármán hypothesis with von Mises yield criterion and the associated flow law.     

An iteration algorithm for solving postbuckling equilibrium path of simply-supported rectangular plates under biaxial compression

In this paper,on the basis of von Kárman large deflection equations and itsdouble trigonometric series solution,we present a simple,fast and effective iterationalgorithm for solving simply-supported rectangular plate subjected to biaxial compression.     

Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis

Kármán-type nonlinear large deflection equations are derived occnrding to the Reddy’s higher-order shear deformation plate theory and used in the thermal postbuckling analysis The effects of initial geometric imperfections of the plate areincluded in the present study which also includes th thermal effects.Simply supported,symmetric cross-ply laminated plates subjected to uniform or nomuniform parabolictemperature distribution are considered. The analysis uses a mixed GalerkinGolerkinperlurbation technique to determine thermal buckling louds and postbucklingequilibrium paths.The effects played by transverse shear deformation plate aspeclraio, total number of plies thermal load ratio and initial geometric imperfections arealso studied.     

A WAVELET METHOD FOR BENDING OF CIRCULAR PLATE WITH LARGE DEFLECTION

A wavelet method for solving strongly nonlinear boundary value problems is described, which has been demonstrated early to have a convergence rate of order 4, almost independent of the nonlinear intensity of the equations. By using such a method, we study the bending problem of a circular plate with arbitrary large deflection. As the deflection increases,the bending behavior usually exhibits a so-called plate-to-membrane transition. Capturing such a transition has ever frustrated researchers for decades. However, without introducing any additional treatment, we show in this study that the proposed wavelet solutions can naturally cover the plate-membrane transition region as the plate deflection increases. In addition, the high accuracy and efficiency of the wavelet method in solving strongly nonlinear problems is numerically confirmed, and applicable scopes for the linear, the membrane and the von Karman plate theories are identified with respect to the large deformation bending of circular plates.     

NONLINEAR DYNAMIC ANALYSIS OF A LAMINATED HYBRID COMPOSITE PLATE SUBJECTED TO TIME-DEPENDENT EXTERNAL PULSES

Nonlinear dynamic responses of a laminated hybrid composite plate subjected to time-dependent pulses are investigated. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von Kármán large deflection theory of thin plates. Approximate solutions for a clamped plate are assumed for the space domain. The single term approximation functions are selected by considering the nonlinear static deformation of plate obtained using the finite element method. The Galerkin Method is used to obtain the nonlinear differential equations in the time domain and a MATLAB software code is written to solve nonlinear coupled equations by using the Newmark Method. The results of approximate-numerical analysis are obtained and compared with the finite element results. Transient loading conditions considered include blast, sine, rectangular, and triangular pulses. A parametric study is conducted considering the effects of peak pressure, aspect ratio, fiber orientation and thicknesses.     

THE SCHRDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of[50]and[51].In this paper:(A)The Love-Kirchhoff equation of small deflection problem for elastic thin shellwith constant curvature are classified as the same several solutions of Schr(?)dingerequation,and we show clearly that its form in axisymmetric problem;(B)For example for the small deflection problem,we extract the general solution ofthe vibration problem of thin spherical shell with equal thickness by the force in centralsurface and axisymmetric external field,that this is distinct from ref.[50]in variable.Today the variable is a space-place,and is not time;(C)The von Kármán-Vlasov equation of large deflection problem for shallow shellare classified as the solutions of AKNS equations and in it the one-dimensional problem isclassified as the solution of simple Schr(?)dinger equation for eigenvalues problem,and wetransform the large deflection of shallow shell from nonlinear problem into soluble linearproblem.     

THE PERTURBATION-ITERATIVE METHOD APPLIED TO THE PROBLEMS OF THE LARGE DEFLECTION OF THE ELASTIC CIRCULAR THIN PLATES

In this paper, we present a perturbation-iterative method for solving certain boundary value problems encountered in the nonlinear theory of elastic circular thin plates. At the same time, with this method, we strictly prove the convergence of the solutions for the large deflection equations of circular plates subjected to certain distributed loads.     

Variational principles for hydrodynamic impact problems

We first establish the rigorous field equations of the two continuous stages before and after entering water. Then correspondently, we obtain the specific variational principles, bounded theorems, and boundary integral equations of the second stage problems. The existence of solutions are proved and the scheme of solving the solutions are provided. Finally, as a numerical example, the ship's wave resistence problem is used to demonstrate the specific application of the second stage problems and its accuracy. Then we provide a rigorous and sound theoretical basis of variational finite element method and boundary element method for calculating the accurately fundamental equations.     

Application of differential constraint method to exact solution of second-grade fluid

A differential constraint method is used to obtain analytical solutions of a second-grade fluid flow. By using the first-order differential constraint condition, exact solutions of Poiseuille flows, jet flows and Couette flows subjected to suction or blowing forces, and planar elongational flows are derived. In addition, two new classes of exact solutions for a second-grade fluid flow are found. The obtained exact solutions show that the non-Newtonian second-grade flow behavior depends not only on the material viscosity but also on the material elasticity. Finally, some boundary value problems are discussed.     

Nonlinear three-dimension analysis for axially symmetrical circular plates and multilayered plates

Analytic nonlinear three-dimension solutions are presented for axially symmetrical homogeneous isotropic circular plates and multilayered plates with rigidly clamped boundary conditions and under transverse load.The geometric nonlinearily from a moderately large deflection is considered.A developmental perturbation method is used to solve the complicated nonlinear three-dimension differential equations of equilibrium.The basic idea of this perturbation method is using the two-dimension solutions as a basic form of the corresponding three-dimension solutions,and then processing the perturbation procedure to obtain the three-dimension perturbation solutions.The nonlinear three-dimension results in analytic expressions and in numerical forms for ordinary plates and multilayered plates are presented.All of the plate stresses are shown in figures.The results show that this perturbation method used to analyse nonlinear three-dimension problems of plates is effective.     

DETERMINATION OF THE PERTURBATION PARAMETER IN LARGE DEFLECTION OF PLATES AND SHALLOW SHELLS BY MEANS OF THE LEAST SQUARE METHOD

In this paper the least square method of determination of the perturbation parameteris presented when the perturbation technique is used in the solution of large deflection ofaxisymmetrical plates and shallow shells.The examples of circular plates are calculatedand compared with the exact solution and other perturbation solutions. The results show thebest agreement with the exact solution among those perturbation solutions.     

Refined differential equations of deflections in axial symmetrical bending problems of spherical shell and their singular perturbation solutions

This paper deals with the research of accuracy of differential equations of deflections.The basic idea is as follows.Firstly,considering the boundary effect the meridianmidsurface displacement u=0,thus we derive the deflection differential equations;secondly we accurately prove that by use of the deflection differential equations or theoriginal differential equations the same inner forces solutions are obtained;finally,weaccurately prove that considering the boundary effect the meridian surface displacementu=0 is an exact solution.In this paper we give the singular perturbation solution of thedeflection differential equations.Finally we check the equilibrium condition and prove theinner forces solved by perturbation method and the outer load are fully equilibrated.Itshows that perturbation solution is accurate.On the other hand,it shows again that thedeflection differential equation is an exact equation.The features of the new differential equations are as follows:1.The accuracies of the new differentia     

EXISTENCE AND COMPARISON RESULTS OF SOLUTIONS FOR NONLINEAR RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS RELATIVE TO WEAK TOPOLOGY

In this paper,we prove several existence theorems of random solutions to nonlinearrandom Volterra integral equations under the weak topology of Banach spaces.Then,asapplications,we obtain the existence theorems of weak random solutions to randomdiffrential equations. Existence of extremal random solutions and a random comparisontheorem for these random equations are also obtained.Our theorems imaprove and extendthe corresponding resulls in [4,5,10,11,12].     

AN EXACT ELEMENT METHOD FOR THE BENDING OF NONHOMOGENEOUS REISSNER’S PLATE

In this paper,based on the step reduction method and exact analytic method,a new method,the exact element method for constructing finite element,is presented.Since the new method doesn’t need variational principle,it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficients.By this method,a triangle noncompatible element with15 degrees of freedom is derived to solve the bending of nonhomogenous Reissner’s plate.Because the displacement parameters at the nodal point only contain deflection and rotation angle.it is convenient to deal with arbitrary boundary conditions.In this paper,the convergence of displacement and stress resultants is proved.The element obtained by the present method can be used for thin and thick plates as well,Four numerical examples are given at the end of this paper,which indicates that we can obtain satisfactory results and have higher numerical precision.     

ON PROBLEMS OF U-SHAPED BELLOWS WITH NONLINEAR DEFORMATION OF LARGE AXISYMMETRICAL DEFLECTION(Ⅱ)——COUNTING VARIATION OF THICKNESS DISTRIBUTION

On the basis of paper[1],assuming the logarithm of thickness at arbitrary point on a U-shaped bellows meridian is linear with the logarithm of distance between that point and axis of symmetry,perturbation solutions of the corresponding problems of large axisymmetrical deflection are given.The effects of thickness distribution variation,which result from technology factors,on stiffness of bellows are discussed.