THE APPLICATION OF WEINSTEIN-CHIEN’S METHOD——THE UPPER AND LOWER LIMITS OF FUNDAMENTAL FREQUENCY OF RECTANGULAR PLATES WITH EDGES ARE THE MIXTURE OF SIMPLY SUPPORTED POR-TIONS AND CLAMPED PORTIONS

In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, os that it is enabled to evaluate the upper limit of fundamental frequency by Ritz’ method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the socalled exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.     

ON THE METHOD OF RECIPROCAL THEOREM TO FIND SOLUTIONS OF THE PLANE PROBLEMS OF ELASTICITY

In this paper the method of reciprocal theorem is extended to find solutions of planeproblems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem of the rectangular plate with fouredges built-in as the basic system and then find displacement expressions of the actualsystem by using the reciprocal theorem between the basic system and actual system withvarious edge conditions.When only displacement edge conditions exist,obtaining displacement expressions bymeans of the method of reciprocal theorem is actual.But in other conditions,when staticforce edge conditions or mixed ones exist,the obtained displacements are admissible.Inorder to find actual displacement,the minimum potential energy theorem must be applied.Calculations show that the method of reciprocal theorem is a simple,convenient andgeneral one for the solution of plane problems of elasticity of the rectangular plates withvarious edge conditions.Evidently,it is a new method.     

ANALYSIS OF SYMMETRIC LAMINATED RECTANGULAR PLATES IN PLANE STRESS

Symmetric laminated plates used usually are anisotropic plates. Based on the fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution is established accurately by method of stress function. Therefore the general formula of stress and displacement in plane is given. The integral constants in general formula can be determined by boundary conditions. This general solution is composed of solutions made by trigonometric function and hyperbolic function, which can satisfy the problem of arbitrary boundary conditions along four edges, and the algebraic polynomial solutions which can satisfy the problem of boundary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load, tangential load and nonuniform normal load on four edges is calculated and analyzed.     

The general solution for axial symmetrical bending of nonhomogeneous circular plates resting on an elastic foundation

In this paper,a new method,the exact analytic method,is presented on the basis of stepreduction method.By this method,the general solution for the bending of nonhomogenouscircular plates and circular plates with a circular hole at the center resting,on an elastfcfoundation is obtained under arbitrary axial symmetrical loads and boundary conditions.The uniform convergence of the solution is proved.This general solution can also be applieddirectly to the bending of circular plates without elastic foundation.Finally,it is onlynecessary to solve a set of binary linear algebraic equation.Numerical examples are givenat the end of this paper which indicate satisfactory results of stress resultants anddisplacements can be obtained by the present method.     

THE APPLICATION OF GENERALIZED VARIATIONAL PRINCIPLE IN FINITE ELEMENT-SEMIANALYTICAL METHOD

The method developed in this paper is inspired by the siewpoint in ref. [1] that sufficient attention has not been paid to the value of the generalized variational principle in dealing with the houndary conditions in the finite element method. This method applies the generalized variational principle and chooses the series constituted by spline function multiplied by sinusoidal function and added by polynomial as the approximate deflection of plates and shells. By taking the deflection problem of thin plate, it shows that this method can solve the coupling problem in the finite element-semianalytical method. compared with the finite element method and finite stripe method, this method has much fewer unknown variables and higher precision. Hence, it proposer an effective way to solve this kind of engineering problems by minicomputer.     

THE METHOD OF TRANSFORMATION OF THE BOUNDARIES FOR STRUCTURE THE ADMISSIBLE DISPLACEMENTS

In this paper the method of transformation of the boundaries for structure theadmissible displacements with various boundary conditions is presented What is called themethod of transformation of the boundaries is that. first we transform the actual systeminto the basic system and additional boundary forces and displacements on the basis of thesuperposition principle, then apply variational principles to the basic system,finally find thepermissible displacement of the actual system by means of the method of transformation ofthe series.In this paper, we also give mixed energy prinapies under Variation of boundaryconditions. The mixed energy principles as the potential and complementary energyprinciples under variation of boundary conditions are all the chief theoretical fundamentalof the method of transformation of the boundaries.Applying the method of transformation of the boundaries. we form the permissibledisplacements of rectangular plates of plane stress and bending problems with various edgeconditio     

THE SOLUTION OF LARGE DEFLECTION PROBLEM OF THIN CIRCULAR PLATE BY THE METHOD OF COMPOSITE EXPANSION

In this paper, the method of composite expansion in perturbation theory is used for the solution of large deflection problem of thin circular plate. In this method. the outer field solution and the inner boundary layer solution are combined together to satisfy all the boundary conditions. In this paper, Hencky's membrane solution is used for the first approximation in outer field solution, and then the second approximate solution is obtained. The inner boundary layer solution is found on the bases of boundary layer coondinate. In this paper, the reciprocal ratio of maximum deflection and thickness of the plate is used as the small parameter. The results of this paper improves quite a bit in comparison with the results obtained in 1948 by Chien Wei-zang.     

THE GENERAL DERIVATION OF RITZ METHOD AND TREFFTZ METHOD IN ELASTOMECHANICS

This paper derives the Ritz method and Trefftz method in li-near elastomechanis with the help of general mathematicalexpressions.Thus it is proved that Ritz method gives theupper bound of the corresponding functional extremum,whileTrefftz method gives its lower bound.At the same time ithas been found that the eigenvalue problem(e.g.the naturalfrequency problem)concerning the functional variational me-thod in Trefftz method is in concord with the lower boundmethod of the loosened boundary condition which seeks forthe eigenvalue.of course,the results of this derivationare also applicable to the sort of functional variationalmethod of which Euler’s equation is linear positive defi-nite.     

Green quasifunction method for vibration of simply-supported thin polygonic plates on Pasternak foundation

A new numerical method—Green quasifunction is proposed.The idea of Green quasifunction method is clarified in detail by considering a vibration problem of simply-supported thin polygonic plates on Pasternak foundation.A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem.The mode shape differential equation of the vibration problem of simply-supported thin plates on Pasternak foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula.There are multiple choices for the normalized boundary equation.Based on a chosen normalized boundary equation,a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome.Finally,natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations.Numerical results show high accuracy of the Green quasifunction method.     

THEOREMS ON THE LIMIT ANALYSIS OF FINITE DEFORMATION

A generalized variational principle (theorem 1) which is equivalent mathematically to the whole set of equations and conditions and must be satisfied by the limit analysis of finite deformation is proposed in this paper. It is also proved that the limit load deduced from theorem 1 will lie between the lower and upper bounds given by the bound theorems of finite deformation.     

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.     

THE CONVERGENT CONDITION AND UNITED FORMULA OF STEP REDUCTION METHOD

The step reduction method was first suggested by Prof. Yeh Kai-yuan. This method has more advantages than other numerical methods. By this method, the analytic expression of solution can he obtained for solving nonuniform elastic mechanics. At the same time. its ealculuting time is very short and convergent speed very fast. In this paper, the convergent condition and nited formula of step reduction method are given by mathemutical method. It is proved that the solution of displacement and stress resultants obtained by this method can eonverge to exact solution uniformly, when the convergent condition is sutisfied. By united formula, the analytic solution solution can be expressed as matrix form, and therefore the former complicated expression can be avoMed. Two numerical examples are given at the end of this paper which indicate that. by the theory in this paper, a right model can be obtained for step reduction method.     

Nonlinear Bending of Circular Sandwich Plates

In this paper,fundamental equations and boundary conditions of non-linear axisymmetrical bending theory for the circular sandwich plates with asoft core are derived by means of the method of calculus of variations.Especial-ly in the case of very thin faces,the preceding fundamental epuations andboundary conditions simplity considerably.For example,a circular sandwichplate with edge clamped but free to slip under the action of uniform lateralload is considered.A more accurate solution of this problem has been ob-tained by means of the modified iteration method.     

NEW FORMULATIONS OF THE SECOND FUNDAMENTAL PROBLEM IN PLANE ELASTICITY

Some general formulations of the second fundamental problem in plane elasticity areproposed here when the displacements given on the closed boundary contours of a multi-connected elastic region are relative to certain rigid motions which are different to eachother for different boundary contours.In such case, it is proved that,for the uniqueexistence of solution,there must be given in addition to the principal vector and the principalmoment of the external forces on each boundary contour. A method of solution is given alsotogether with some illustrative examples.     

A GENERAL SOLUTION OF RECTANGULAR THIN PLATES IN BENDING

In this paper a general solution of rectangular plates in bending is given.The integralconstants are determined by means of boundary conditions.This method is simpler andeasier than the method of superposition.     

PERTURBATION ANALYSES FOR THE POSTBUCKLING OF SIMPLY SUPPORTED RECTANGULAR PLATES UNDER UNIAXIAL COMPRESSION

In this paper, applying perturbution method to yon Krmn nonlinear large deflection equations of plates by taking deflection as perturbation parameter, the pa(?)buckloing behavior of simply supporied rectungular plates under umaxial compression is (?)igated. Two types of in-plane boundary conditions are now considered and the efferis of initial imperfections are also studied. It is found that the theoretical results are in good agreement with experiments. The method suggested in this paper whieh has not been found in previons papers is rather simple and easy for the postbuckling analysis of rectangular plates.     

THE BENDING, STABILITY AND VIBRATIONS OF RECTANGULAR PLATES WITH FREE EDGES ON ELASTIC FOUNDATIONS

This paper discusses the problems of the bending, stabilio, and vibrations of the rectangular plates with free boundaries on elastic foundations, In the present paper we select a flexural function, which satisfies not only all the boundary Conditions of free edges but also the conditions at free corner points, and consequently we obtain a better approximate solution. The energy method is used in this paper.     

A direct method for deriving fundamental solution of half-plane problem

A fundamental solution for half-plane problems which will play a key role in calculation of the stress concentration around a hole embedded in half-plane is derived by a method combining images with direct integrations. It is wore intuitive than the Fourier transform method used by Gladwell[6]. In addition, the principle and procedure of boundary element method to solve the half-plane problems are also presented by means of Betti’s reciprocal theorem in this paper.It is shown that the. computing procedure for half-plane problems is much more convenient using the fundamental solution presented here than the one adopted by C.A.     

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅰ)—RECTANGULAR PLATES WITH FOUR CLAMPED EDGES AND WITH THREE CLAMPED EDGES

In this paper the method of the reciprocal theorem(MRT)is extended to solve thesteady state responses of rectangular plates under harmonic disturbing forces.A series ofthe closed solutions of rectangular plates with various boundary conditions are given andthe tables and figures which have practical value are provided.MRT is a simple,convement and general method for solving the steady state responsesof rectangular plates under vartous harmonic disturbing forces.The paper contains three parts:(Ⅰ)rectangular plates with four clamped edges andwith three clamped edges:(Ⅱ)rectangular.plates with two adjacent clamped edges:(Ⅲ)cantilever plates.We are going to publish them one after another.     

THE GENERALIZED GALERKIN’S EQUATION OF THE FINITE ELEMENT,THE BOUNDARY VARIATIONAL EQUATIONS AND THE BOUNDARY INTEGRAL EQUATIONS

Based on[1 ],we have further applied the variational princi-ple of the variable boundary to investigate the discretizationanalysis of the solid system and derived the generalized Ga-lerkin’s equations of the finite element.the boundary varia-tional equations and the boundary integral equations.These e-quations indicate that the unknown functions of the solid sys-tem must satisfy the conditions in the element S_a or on the boun-dariesГ_a.These equations are applied to establishing the discreti-zation equations in order to obtain the numerical solution ofthe unknown functions.At a time these equations can be usedas the basis for the simplified calculation in various corres-ponding cases.In this paper,the results of boundary integral equationsshow that the calculation of integration is not accurate alongthe surfaceГ_a of interior element S_a by J-integral suggestedby Rice[2].