ANALYSIS OF THE LARGE DEFLECTION DYNAMIC PLASTIC RESPONSE OF SIMPLY-SUPPORTED CIRCULAR PLATES BY THE“MEMBRANE FACTOR METHOD”

Based on energy equilibrium,a new procedure called the Membrane Factor Method is devel- oped to analyze the dynamic plastic response of plates with deflections in the range where both bending mo- ments and membrane forces are important.The final deflection of a simply -supported circular rigid-plastic plate loaded by a uniformly distributed impulse is obtained.In comparison with other approximate solutions, the present results are found to be simpler and in better agreement with the corresponding experimental values reoorded by Florence.

OPTIMIZATION OF THE MAGNETIC DEFLECTION SYSTEM BY THE METHOD OF ORTHOGONAL DESIGN

By using the expressions of the field parameters and with the aid of the Orthogonal DesignMethod,we optimized the design of the saddle delection yoke with a ferromagnetic shield.The results ob-tained agree with the practical model.It is pointed out that the end ear and the distribution of the fieldparameter B0（z）have important effects on the deflection aberrations.An example is given,in which the elec-tron optical properties of the deflection coils having cosine distribution and finite length and that of the de-flection coils having distributed winding are compared.

A Novel Deflection Method for Measuring the Growth Stress of Thermally Growing Oxide Scales

A kind of new deflection technique has been developed for measuring the growth stress of thermally growing oxide scales during high temperature oxidation of alloys. The average growth stresses in oxide scales such as Al2O3, NiO and Cr2O3 formed on the surface of the superalloys can be investigated by this technique. Unlike the comventional deflection method, the novel method does not need to apply a coating for preventing one main face of thin strip specimen from oxidizing and can be used under the condition of longer time and higher temperature.

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION AND THE ASYMPTOTICS OF SOLUTIONS(Ⅱ)

This paper is a continuation of part (Ⅰ), on the asymptotics behaviors of the series solutions investigated in (Ⅰ). The remainder terms of the series solutions are estimated by the maximum norm.

ON THE METHOD OF ORTHOGONALITY CONDITIONS FOR SOLVING THE PROBLEM OF LARGE DEFLECTION OF CIRCULAR PLATE

In this paper, we reexamine the method of successive approximation presented by Prof. Chien Wei-zangfor solving the problem of large deflection of a circular plate, and find that the method could be regarded as the method of strained parameters in the singular perturbation theory. In terms of the parameter representing the ratio of the center deflection to the thickness of the plate, we make the asymptotic expansions of the deflection, membrane stress and the parameter of load as in Ref. [1], and then give the orthogonality conditions (i.e. the solvability conditions) for the resulting equations, by which the stiffness characteristics of the plate could be determined. It is pointed out that with the solutions for the small deflection problem of the circular plate and the orthogonality conditions, we can derive the third order approximate relations between the parameter of load and the center deflection and the first-term approximation of membrane stresses at the center and edge of the plate without solving the differential equations. For some special cases (i.e. under uniform load, under compound toad, with different boundary conditiors), we deduce the specific expressions and obtain the results in agreement with the previous ones given by Chien Wei-zang, Yeh kai-yuan and Hwang Chien in Refs. [1 - 4J.

ITERATIVE METHOD FOR LARGE DEFLECTION NONLINEAR PROBLEM OF LAMINATED COMPOSITE SHALLOW SHELLS AND PLATES

Based on the results by Wang,in this paper, the iterative method is presented for the study of large deflection nonlinear problem of laminated composite shallow shells and plates. The rectangular laminated composite shallow shells have been analyzed. The results have been compared with the small deflection linear analytical solution and finite element nonlinear solution. The results proved that the solution coincide with small deflection linear analytical solution in the condition of the low loads and finite element nonlinear solution in the condition of the high loads.

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION ANDTHE ASYMPTOTICS OF SOLUTIONS (Ⅰ)

In this paper, the modified method of multiple scales is applied to study the bending problems for circular thin plate with large deflection under the hinged and simply supported edge conditions. Theseries solutions are constructed, the boundary layer effects are analysed and their asymptotics are proved.

Large Deflection Dynamic Response Analysis of Flexible Hull Beams by the Multibody System Method

In this paper the large deflection dynamic problems of Euler beams are investigated. The vibration control equations are derived based on the multibody system method. A numerical procedure for solving the resulting differential algebraic equations is presented on the basis of the Newmark direct integration method combined with the Newton-Raphson iterative method. The sub beams are treated as small deformation in the convected coordinate systems, which can greatly simplify the deformation description. The rigid motions of the sub beams are taken into account through the motions of the convected coordinate systems. Numerical ex- amples are carried out, where results show the effectiveness of the proposed method.

Simulation of Steel Reinforcement on the Nonlinear Behaviour of Slender Glulam Beam Columns by Using the Newton-Raphson Method

The current theory in NF EN 1995-1-1/NA of Eurocode 5, which is based on maximum deflection, has been investigated on softwoods. Therefore, this theory is not adapted for slender glulam beam columns made of tropical hardwood species from the Congo Basin. This maximum deflection is caused by a set of loads applied to the structure. However, Eurocode 5 doesn’t provide how to predict this deflection in case of long-term load for such structures. This can be done by studying load-displacement (P-Δ) behaviour of these structures while taking into account second order effects. To reach this goal, a nonlinear analysis has been performed on a three-dimensional beam column embedded on both ends. Since conducting experimental investigations on large span structural products is time-consuming and expensive especially in developing countries, a numerical model has been implemented using the Newton-Raphson method to predict load-displacement (P-Δ) curve on a slender glulam beam column made of tropical hardwood species. On one hand, the beam has been analyzed without wood connection. On the other hand, the beam has been analyzed with a bolted wood connection and a slotted-in steel plate. The load cases considered include self-weight and a uniformly applied long-term load. Combinations of serviceability limit states (SLS) and ultimate limit states (ULS) have also been considered, among other factors. A finite-element software RFEM 5 has been used to implement the model. The results showed that the use of steel can reduce displacement by 20.96%. Additionally, compared to the maximum deflection provided by Eurocode 5 for softwoods, hardwoods can exhibit an increasing rate of 85.63%. By harnessing the plastic resistance of steel, the bending resistance of wood can be increased by 32.94%.

Investigation on behavior of crack penetration/deflection at interfaces in intelligent coating system

Based on the three-phase model, the propagation behavior of a matrix crack in an intelligent coating system is investigated by an energy criterion. The effect of the elastic mismatch parameters and the thickness of the interface layer on the ratio of the energy release rate for infinitesimal deflected and penetrated crack is evaluated with the finite element method. The results show that the ratio of the energy release rates strongly depends on the elastic mismatch al between the substrate and the driving layer. It also strongly depends on the elastic mismatch a2 between the driving layer and the sensing layer for a thinner driving layer when a primary crack reaches an interface between the substrate and the driving layer. Moreover, with the increase in the thickness of the driving layer, the dependence on a2 gradually decreases. The experimental observation on aluminum alloys monitored with intelligent coating shows that the established model can better explain the behavior of matrix crack penetration and can be used in optimization design of intelligent coating.

VIBRATION PROBLEMS OF FLEXIBLE CIRCULAR PLATES WITH INITIAL DEFLECTION

In this paper, the differential equations of flexible circular plates with initial deflection are derived. The stability of motion is investigated in phase plane. The periodical solutions of nonlinear vibration for circular plates with initial deflection are obtained by use of Galerkin method and Lindstedt-Poincare perturbation method. The effect of initial deflection on the dynamic behavior of the flexible plates are also discussed.

ON THE EVALUATION OF DEFLECTIONS OF THE VERTICAL USING FFT TECHNIQUE

There exist three types of convolution formulae for the efficient evaluation of gravity field convolution integrals,i.e.,the planar 2D convolution,the spherical 2D convolution and the spherical 1D convolution.The largest drawback of both the planar and the spherical 2D FFT methods is that,due to the approximations in the kernel function,only inexact results can be achieved.Apparently,the reason is the meridian convergence at higher latitudes.As the meridians converge,the Δ φ ,Δ λ blocks do not form a rectangular grid,as is assumed in 2D FFT methods.It should be pointed out that the meridian convergence not only leads to an approximation error in the kernel function,but also causes an approximation error during the implementation of 2D FFT in computer.In order to meet the increasing need for precise determination of the vertical deflections,this paper derives a more precise planar 2D FFT formula for the computation of the vertical deflections.After having made a detailed comparison between the planar and the spherical 2D FFT formulae,we find out the main source of errors causing the loss in accuracy by applying the conventional spherical 2D FFT method.And then,a modified spherical 2D FFT formula for the computation of the vertical deflections is developed in this paper.A series of numerical tests have been carried out to illustrate the improvement made upon the old spherical 2D FFT.The second part of this paper is to discuss the influences of the spherical harmonic reference field,the limited capsize,and the singular integral on the computation of the vertical deflections.The results of the vertical deflections over China by applying the spherical 1D FFT formula with different integration radii have been compared to the astro_observed vertical deflections in the South China Sea to obtain a set of optimum deflection computation parameters.

Buckling and large deflection behaviors of radially functionally graded ring-stiffened circular plates with various boundary conditions

The buckling and large deflection behaviors of axis-symmetric radially functionally graded （RFG） ring-stiffened circular plates are investigated by the dynamic relaxation （DR） method combined with the finite difference discretization technique. The material properties of the constituent components of the RFG plate are assumed to vary continuously according to the Mori-Tanalka distribution along the radial direction. The nonlinear governing equations are obtained in the incremental form based on the firstorder shear deformation plate theory （FSDT） and the von Karman relations for large deflection. In the buckling analysis, an external in-plane load is applied to the plate in- crementally so that, in each load-step, the incremental form of the governing equations can be solved by a numerical code prepared based on the DR method. After converging the DR code in the first increment, the latter load-step is added to the previous one, and the program is repeated again. The critical buckling load is determined from the compressive load-displacement curve obtained by solving the incremental form of the governing equations. Based on the present incremental form of formulation, a bending analysis can also be conducted if the whole load is applied simultaneously. Finally, a detailed parametric study is carried out to investigate the influences of various boundary conditions, grading indices, thickness-to-radius ratios, stiffener＇s positions and depths on the critical buckling load, and displacements and stresses resulted from the bending analysis. It is observed that the effect of the stiffener on the results is much greater in the functionally graded plate with higher material grading indices. The results also reveal that, by increasing the depth of the stiffer, the values of ascending the critical buckling load are approximately identical for both simply supported and clamped boundary conditions.

Numerical Solution of Membrane Forces for A Free-Free Floating Plate with Large Deflection

Considering that the thickness of a pontoon-type very large floating structure (VLFS) is very small in comparison with the length and width, VLFS can be modeled as a thin plate. In theory, the displacements and the membrane forces of a plate with large deflection are all the functions of the second-order differentials of the Ariy stress function. With these characteristics considered, the Ariy stress function of a floating free-free plate is calculated by setting the virtual values of three of the corner points. The finite difference method is chosen to solve the problem. When the Ariy stress function of the plate is obtained, the membrane forces can easily be calculated. Comparisons between the forces induced by the membrane forces and by the fluid are considered. It is shown that the membrane forces can not be neglected in many cases.

Analysis of rolls deflection of Sendzimir mill by 3D FEM

The deflection of rolls of Sendzimir mill with double AS-U-Roll was simulated by finite element method(FEM). The influences of rolling pressure, strip width and rolls-assignment on rolls deflection were analyzed. The results show that the work roll deflection increases with the increase of rolling pressure and the reduction of work roll radius, but the rigid displacement of work roll slightly changes; the work roll end might appear negative displacement for the narrow strip width and high rolling pressure that might cause the contact of work rolls. The research results are significant for guiding production and theoretical analysis of the rolls system of Sendzimir mill.

THE PREDICTION OF CASE FURNITURE DEFLECTION

In this paper we used finite element method(FEM)to prediet case furniture de-flection.We dispersed and analyzed furniture which made of particleboard by elastic plate elements.The corner joint of case furniture is a key problem during the calculation by FEM.We assumed thatsmall elements which located near corner joints were semi—rigid elements.Then we obtained the fab-ricated modulus of elasticity of the semi-rigid elements by individual corner joint experimcnts.Weused four kinds of particleboard and two kinds of corner joint t forms to make up eleven cases.Thedisplacement of some typical points under each load measured by the Furniture Strength Test Ma-chine.Then we compared the displacements of measurcments with that of calculations,the errorswere usually within 30 percent.Results of this study indicated that the deflection of case furniturecould be calculated by using FEM,and proved that the program could be used for predicting casefurniture deflection reliably.

NUMERICAL ANALYSIS OF THE LARGE DEFLECTION OF AN ELASTIC-PLASTIC BEAM

The layered approach was adopted to study the numerical procedure of the large deflection of art elastic-plastic Timoshenko' s beam, and the nonlinear equilibrium equation was derived by TL Formula. The solution was conducted by means of mNR method. The tangential stiffness matrix of the beam,vas introduced, and the solving procedures were presented in detail. The solution of the problem is satisfactory.

LARGE DEFLECTION PROBLEM OF A CLAMPED ELLIPTICAL PLATE SUBJECTED TO UNIFORM PRESSURE

In this paper, the perturbation solution of large deflection problem of clampedelliptical plate subjected to uniform pressure is given on the basis of the perturbationsolution of large deflection problem of similar clamped circular plate (1948)[1], (1954)[2]. The analytical solution of this problem was obtained in 1957. However, due to social difficulties, these results have never been published. Nash and Cooley (1959)[3] published a brief note of similar nature, in which only the case λ=a/b=2 is given. In this paper, the analytical solution is given in detail up to the 2nd approximation. The numerical solutions are given for various Poisson ratios v =0.25, 0.30, 0.35 and for various eccentricities λ= 1, 2, 3, 4, 5, which can be used in the calculation of engineering designs.

Large deflection of annular throttle-slices in shock absorbers

method was used to derive the stiffness curve equation of a single throttle-slice in shock absorbers. The analytical formula of large deflection for superposed throttle-slices was deduced directly and generalized. The undetermined coefficients of analytical for- mula were obtained through the finite element method （FEM） and curve fitting. Numerical results show that the analytical formula has satisfactory accuracy.

LARGE DEFLECTION PROBLEM OF THIN ORTHOTROPIC CIRCULAR PLATE ON ELASTIC FOUNDATION WITH VARIABLE THICKNESS UNDER UNIFORM PRESSURE

Basic equations for large deflection theory of thin orthotropic circular plate on elastic foundation with variable thickness under uniform pressure are derived in this paper. The second opproximation solutions are obtained by means of the modified iteration method. The relation curves of the nondimensional loading and foe deflection, as to the differential ε and μrθ and λ are shown in Figs. 2, 3, 4. In special circumstance, the results are in accordance with those in [1], [6].