ON THE MODIFIED CASTIGLIANO"S THEOREM

This paper gives the modified Castigliano's theorem, which is more convenient and more extensive for applications than the classical Castigliano's theorem. The modifications to the classical Castigliano's theorem are in following two respects, the first respect is that the expression of the partial derivative with respect to the concentrated load p of the complementary energy density in the clasical Castigliano's theorem is replaced by the expressions with teh products of the external loads and influence functions, the modification brings us greatest simplicity and greatest convenience for calculations under various loads; the second respect is that the expressions with the products of the inhomogeneous boundary displacements and the influence functions are introduced into the classical Castigliano's theorem, the modification provides the theoretical fundamental for solving the problems of various boundary conditions. We show also the method of how to apply the modified Castigliano's theorem to solve th     

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅱ)—RECTANGULAR PLATES WITH TWO ADJACENT CLAMPED EDGES

In this paper,applying the method of reciprocal theorem,we give the distributions ofthe amplitude of bending moments along clamped edges and the amplitude of deflectionsalong free edges of rectangular plates with two adjacent clamped edges under harmonicdistributed and concentrated loads.     

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.     

AXISYMMETRY PROBLEMS OF RING SHELLS UNDER ARBITRARY DISTRIBUTED LOADS

Extending Novozhilov’s transformation,this paper simplifies successfully theaxisymmetry problems of ring shells under arbitrary distributed loads, and the equation ofthe problems in complex form which is similar to Novozhilov’s is obtained The particularsolution is obtained.Combining with Chien’s general solution of homogeneous equation,this paper gives the general solution for the general symmetry problems of ring shells.Various examples of useful loads and closed ring shells are discussed respectively.     

CALCULATION FOR CYLINDRICAL SHELL UNDER LOCAL VERTICAL LOADINGS

In this paper,we use the method of mixed-type series to derive the analytical solutionsof cylindrical shell,which is simply supported along the transverse edges and subjected tothe local vertical loads,and give the analytical expressions of the solutions for this kind ofshell under five types of local vertical loading.A numerical example for a cylindrical shellroof,which is simply supported along the transverse edges and is free along the longitudinaledges.is given in this paper and from the calculated results it may be seen that theconvergence of the solutions is considerably satisfactory.Using the solutions of this paper,we can deal with some practical problems of underground structure.     

Application of the reciprocal theorem for calculating the natural frequencies of rectangular elastic thin plates

This paper further extends the applications of the reciprocal theorem to calculating thenatural frequencies of rectangular elastic thin plates on the basis of [1]. Applying thepresented method. there is no need to solve governing differential equations. it is onlynecessary to solve a simple integral equation after using the reciprocal theorem between thebasic system and the actual system.Using the idea of the generalized edge simply supported and introducing the frequencymatrix. then all frequency equations of the rectangular plates with two opposite edgessimply supported and other two opposite edges variously suppored are obtained together.This is a simple convenient and general method for calculating the frequencyequations of the rectangular plates.     

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.     

Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method

The size-dependent effect on the biaxial and shear nonlinear buckling analysis of an isotropic and orthotropic micro-plate based on the surface stress,the modified couple stress theory(MCST),and the nonlocal elasticity theories using the differential quadrature method(DQM)is presented.Main advantages of the MCST over the classical theory(CT)are the inclusion of the asymmetric couple stress tensor and the consideration of only one material length scale parameter.Based on the nonlinear von K′arm′an assumption,the governing equations of equilibrium for the micro-classical plate considering midplane displacements are derived based on the minimum principle of potential energy.Using the DQM,the biaxial and shear critical buckling loads of the micro-plate for various boundary conditions are obtained.Accuracy of the obtained results is validated by comparing the solutions with those reported in the literature.A parametric study is conducted to show the effects of the aspect ratio,the side-to-thickness ratio,Eringen’s nonlocal parameter,the material length scale parameter,Young’s modulus of the surface layer,the surface residual stress,the polymer matrix coefficients,and various boundary conditions on the dimensionless uniaxial,biaxial,and shear critical buckling loads.The results indicate that the critical buckling loads are strongly sensitive to Eringen’s nonlocal parameter,the material length scale parameter,and the surface residual stress effects,while the effect of Young’s modulus of the surface layer on the critical buckling load is negligible.Also,considering the size dependent effect causes the increase in the stiffness of the orthotropic micro-plate.The results show that the critical biaxial buckling load increases with an increase in G12/E2and vice versa for E1/E2.It is shown that the nonlinear biaxial buckling ratio decreases as the aspect ratio increases and vice versa for the buckling amplitude.Because of the most lightweight micro-composite materials with high strength/weight and stiffness/weight ratios,it is anticipated that the results of the present work are useful in experimental characterization of the mechanical properties of micro-composite plates in the aircraft industry and other engineering applications.     

A numerical approach based on the meshless collocation method in elastodynamics

In this paper, a collocation technique with the modified equilibrium on line method （ELM） for imposition of Neumann （natural） boundary conditions is presented for solving the two-dimensional problems of linear elastic body vibrations. In the modified ELM, equilibrium over the lines on the natural boundary is satisfied as Neumann boundary condition equations. In other words, the natural boundary conditions are satisfied naturally by using the weak formulation. The performance of the modified version of the ELM is studied for collocation methods based on two different ways to construct meshless shape functions： moving least squares approximation and radial basis point interpolation. Numerical examples of two-dimensional free and forced vibration analyses show that by using the modified ELM, more stable and accurate results would be obtained in comparison with the direct collocation method.     

Hydrodynamic coefficients for vertical plane wave motion problems

In this paper a numerical method for calculating the hydro-dynamic coefficients for vertical plane wave motion problems in deep water is described.This procedure is developed by using the wave source method based on Green‘s theorem.The applications of the method to the cases of semi-circular and rectangular section bodies subjected to linear waves are pre-sented here,and the computed results are compared with the earlier experimental data of Vugts.     

Theoretical prediction on corrugated sandwich panels under bending loads

In this paper, an aluminum corrugated sandwich panel with triangular core under bending loads was investigated. Firstly, the equivalent material parameters of the triangular corrugated core layer, which could be considered as an orthotropic panel, were obtained by using Castigliano’s theorem and equivalent homogeneous model. Secondly, contributions of the corrugated core layer and two face panels were both considered to compute the equivalent material parameters of the whole structure through the classical lamination theory, and these equivalent material parameters were compared with finite element analysis solutions. Then, based on the Mindlin orthotropic plate theory, this study obtain the closed-form solutions of the displacement for a corrugated sandwich panel under bending loads in specified boundary conditions, and parameters study and comparison by the finite element method were executed simultaneously.     

Energy theorems and the J-integral in dipolar gradient elasticity

Within the framework of Mindlin’s dipolar gradient elasticity, general energy theorems are proved in this work. These are the theorem of minimum potential energy, the theorem of minimum complementary potential energy, a variational principle analogous to that of the Hellinger–Reissner principle in classical theory, two theorems analogous to those of Castigliano and Engesser in classical theory, a uniqueness theorem of the Kirchhoff–Neumann type, and a reciprocal theorem. These results can be of importance to computational methods for analyzing practical problems. In addition, the J-integral of fracture mechanics is derived within the same framework. The new form of the J-integral is identified with the energy release rate at the tip of a growing crack and its path-independence is proved.The theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory considered here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. The latter case is also treated in the present study.     

Solutions for Quasilinear Nonsmooth Evolution Systems in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>

We prove that nonsmooth quasilinear parabolic systems admit a local solution in L ^p strongly differentiable with respect to time over a bounded three-dimensional polyhedral space domain. The proof rests essentially on new elliptic regularity results for polyhedral Laplace interface problems for anisotropic materials. These results are based on sharp pointwise estimates for Greens function, which are also of independent interest. To treat the nonlinear problem, we then apply a classical theorem of Sobolevskii for abstract parabolic equations and recently obtained resolvent estimates for elliptic operators and interpolation results. As applications we have in mind primarily reaction-diffusion systems. The treatment of such equations in an L ^p context seems to be new and allows (by Gauss theorem) the proper definition of the normal component of currents across the boundary.     

Energy theorems for solving equations of deflections

In this paper the unit-dummy-load method is generalized on the basis of Castigliano’s Theorem. On these grounds the general eguations of deflection surfaces of the structures, such as a kind of beams,plates and shells, are directly derived by the force method.We derived the eguations of the deflection surfaces of the rectangular thin plates and thick plates considering the effect of transverse shearing deformations with the inhomogeneous displacement boundary conditions. At the same time we give the equations of deflection axes of the corresponding straight beams.The applications of the reciprocal theorem are also generalized.Three simple calculated examples are given.     

角点悬空弯曲厚矩形板的边界积分法

在边界积分法中引用了拟基本系统矩形板,在该拟基本系统与实际系统之间应用功的互等定理,得到一挠曲面方程的积分表达式,只要对此表达式进行极简单的积分便可得到该挠曲面方程,这比直接求解Reissner挠度控制方程要简单,边界积分法的求解过程概念清晰,计算伊始便给出了挠曲面方程的总体表达式.以Reissner厚板理论为基础,应用边界积分法研究了角点悬空厚矩形板的弯曲问题,给出了在集中荷载作用下两邻边固定另两邻边自由且角点悬空弯曲厚矩形板的封闭解析解,并给出了相应的数据和图表以供工程上的应用和参考.     

Extremalprinzipe der Thermodynamik und Mechanik für stabile Gleichgewichtslagen von Kontinua

Übersicht In dieser Arbeit soll der Zusammenhang zwischen den Extremalprinzipien der Thermodynamik und der Mechanik für stabile Gleichgewichtszustände eines Kontinuums studiert werden. Als Nebenprodukt ergeben sich dabei zwanglos Verallgemeinerungen der klassischen Extremalprinzipe der Elastizitätstheorie (des Minimum-Energie-Prinzips und des Prinzips von Castigliano) auf multipolare und chemisch reagierende Materialien.

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Summary The aim of this paper is to study the relations between the variational principles of thermodynamics and mechanics for stable equilibrium states of a deformable body. As a by-product one obtains generalisations of the classical principles of elasticity-theory (the minimum-energy-principle and Castigliano's theorem) to multipolar and chemically reacting materials.

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Herrn Prof. Dr.-Ing. K. Klotter zum siebzigsten Geburtstag gewidmet.     

High order behavior of sandwich plates with free edges––edge effects

The edge effects of a sandwich plate with a “soft” core and free edges, i.e. the plate is supported only at the lower face-sheet, and the upper face-sheet and the core are free of stresses at their edges, using the high order approach (HSAPT), are presented. The two-dimensional analysis consists of a mathematical formulation that uses the classical thin plate theory for the face-sheets and a three-dimensional elasticity theory for the core. The governing equations and the required boundary conditions are derived explicitly through variational principals, yielding a system of eight partial differential equations. The non-homogeneous differential equations system is numerically solved using a modification of the extended Kantorovich method (MEKM). The model presented enables a two-dimensional solution of the stress and displacement fields when subjected to a general scheme of loads. It is applicable to any type of boundary conditions that can be applied separately on each face-sheet and on the core. A numerical study is presented, and it examines the behavior and the two-dimensional stress field of a sandwich plate with free edges, at the upper face-sheet and core, subjected to thermal and uniformly distributed loads, for various boundary conditions at the lower face-sheet. For completeness, the MEKM solution of the two-dimensional high order model is verified through comparison with a three-dimensional Finite Element model revealing good correlation. Furthermore, the problems involved in the construction of an appropriate three-dimensional FE model of a full scale sandwich plate that require large computer resources are discussed.The numerical study yields that the peeling (normal) stresses, which reach their maximum values at the edges of the sandwich plate, using a one-dimensional analysis, varies also in the transverse direction from a maximum value in the middle of the edge, descending towards the corners. Moreover, the nature of variation along the boundaries strongly depends on the type of loading and the transverse boundary conditions. The substantial variation of the stress field in the transverse direction clearly shows the necessity of a two-dimensional analysis and the inefficiencies of the one-dimensional model.     

弹性地基上各向异性板的静力分析

根据弹性地基上各向异性矩形板弯曲挠度的微分方程精确的求得了适用于各种载荷的非齐次解和各类齐次解。其中由三角函数和双曲线函数组成的齐次解能满足四个边为任意边界条件的问题;由代数多项式和双正弦级数组成的齐次解能满足四个角为任意边界条件的问题。通过适当选取建立了满足任意边界条件和任意载荷作用的一般解。解中的积分常数完全由边界条件来决定。以四边简支承受均布载荷和局部分布载荷的对称迭层复合材料方板为例进行了计算和分析。其结果与已有文献结果是一致的。由于集中载荷不能求得作用点的弯矩,故在例题中改用局部分布载荷因而求得了最大弯矩。     

Lower bounds to fundamental frequencies and buckling loads of columns and plates

A general derivation of expressions for lower bounds to fundamental frequencies and buckling loads is given for the class of structures governed by linear elastic theory in the prebuckling state. These expressions involve two Rayleigh quotients both of which are upper bounds for the fundamental frequency under a prescribed load. The displacement trial functions must satisfy force and kinematic continuity but no other conditions are required. Thus, if appropriate high order base functions are used, the finite element procedure can be used to systematically narrow the difference between the upper and lower bounds.The theory is illustrated with several column and plate problems. The finite element method is applied to uniform and nonuniform columns with a representative set of boundary conditions. Elementary trial functions are used to show that reasonable bounds can also be obtained for plates subjected to known states of stress. Since the lower bound is obtained with a variation of the classical technique of Rayleigh, these results indicate that the method may be suitable for conservatively estimating buckling loads and fundamental frequencies of engineering structures.