SECOND ORDER APPROXIMATION SOLUTION OF NONLINEAR LARGE DEFLECTION PROBLEMS OF YONGJIANG RAILWAY BRIDGE IN NINGBO

The solution and computational aspects on nonlinear deflection of Yongjiang Railway Bridge in Ningbo were investigated. An approximate iteration algorithm on nonlinear governing equation was presented, and the obtained results show that, if altitude difference and span of the riverbanks are taken as 5 meters and 100 meters, respectively, the maximum gradient in the middle of the bridge exceeds 5%, much larger than maximum allowance gradient in railway design code. Therefore, a new solution scheme for decreasing gradient of the bridge is put forward, that is, the altitude difference between two riverbanks can be decreased to about 1/10 of the initial magnitude by building roadbeds with 0.5% gradient and 1 kilometer length at two riverbanks. As a direct result, the deflection gradient of the railway bridge is much reduced and the value is between 0.5% similar to 0.6%.

Large deflection of curved elastic beams made of Ludwick type material

The large deflection of an axially extensible curved beam with a rectangular cross-section is investigated. The elastic beam is assumed to satisfy the Euler-Bernoulli postulation and be made of the Ludwick type material. Through reasonably simplified integration, the strain and curvature of the axis of the beam are presented in implicit formulations. The governing equations involving both geometric and material nonlin- earities of the curved beam are derived and solved by the shooting method. When the initial curvature of the beam is zero, the curved beam is degenerated into a straight beam, and the predicted results obtained by the present model are consistent with those in the open literature. Numerical examples are further given for curved cantilever and simply supported beams, and the couplings between elongation and bending are found for the curved beams.

BIPARAMETRIC PERTURBATION SOLUTIONS OF LARGE DEFLECTION PROBLEM OF CANTILEVER BEAMS

The large deflection problem of cantilever beams was studied by means of the biparametric perturbation method and the first order derivative substitution from pseudolinear analysis approach. This kind of substitution can transform the basic equation, an integral differential equation into nonlinear algebraic ones, thus simplify computational process. Compared with present results, it indicates that the large deflection problem solved by using pseudolinear analysis can lead to simple and precise results.

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.

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.

Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end

This work studies large deflections of slen- der, non-prismatic cantilever beams subjected to a combined loading which consists of a non-uniformly distributed con- tinuous load and a concentrated load at the free end of the beam. The material of the cantilever is assumed to be non- linearly elastic. Different nonlinear relations between stress and strain in tensile and compressive domain are considered. The accuracy of numerical solutions is evaluated by com- paring them with results from previous studies and with a laboratory experiment.

Large deflection of flexible tapered functionally graded beam

In this paper the semi-analytical analyses of the flexible cantilever tapered functionally graded beam under combined inclined end loading and intermediate loading are studied.In order to derive the fully non-linear equations governing the non-linear deformation,a curvilinear coordinate system is introduced.A general non-linear second order differential equation that governs the shape of a deflected beam is derived based on the geometric nonlinearities,infinitesimal local displacements and local rotation concepts with remarkable physical properties of functionally graded materials.The solutions obtained from semi-analytical methods are numerically compared with the existing elliptic integral solution for the case of a flexible uniform cantilever functionally graded beam.The effects of taper ratio,inclined end load angle and material property gradient on large deflection of the beam are evaluated.The Adomian decomposition method will be useful toward the design of tapered functionally graded compliant mechanisms driven by smart actuators.

FURTHER STUDY ON LARGE DEFLECTION OF CIRCULAR SANDWICH PLATES

In this paper, a nonlinear solution is first presented for a circular sandwich plate with the flexure rigidity of the face layers taken into account. In solving the nonlinear bending equations, a modified power series method is proposed. The uniformly distributed loading and the clamped but sliding boundary condition are also assumed. Then our results are compared with those from Liu Ren-huai and Shi-Yun-fang[15]. The present solution can be used ax a more accurate basis in engineering applications.

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.

AN ANALYSIS OF THE LARGE DEFLECTION OF AN ELASTIC-PLASTIC CANTILEVER SUBJECTED TO AN INCLINED CONCENTRATED FORCE

Based on the Plastica theory (see ref. [12]), the large deflection of an elastic-perfectly plastic cantilever subjected to an inclined concentrated force at its tip, before the unloading in the plastic region occurs, is analyzed in this paper. The emphasis of the analysis is put on the effects of the angle of inclination of the concentrated force upon the deformed shape, the load-deflection relationship and the length of the plastic region. Both analytical and computed results are given.

A nonlinear mathematical model for large deflection of incompressible saturated poroelastic beams

Nonlinear governing equations are established for large deflection of incompressible fluid saturated poroelastic beams under constraint that diffusion of the pore fluid is only in the axial direction of the deformed beams. Then, the nonlinear bending of a saturated poroelastic cantilever beam with fixed end impermeable and flee end permeable, subjected to a suddenly applied constant concentrated transverse load at its free end, is examined with the Gaierkin truncation method. The curves of deflections and bending moments of the beam skeleton and the equivalent couples of the pore fluid pressure are shown in figures. The results of the large deflection and the small deflection theories of the cantilever poroelastic beam are compared, and the differences between them are revealed. It is shown that the results of the large deflection theory are less than those of the corresponding small deflection theory, and the times needed to approach its stationary states for the large deflection theory are much less than those of the small deflection theory.

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.

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.

THE SOLUTION OF RECTANGULAR PLATES WITH LARGE DEFLECTION BY SPLINE FUNCTIONS

In this paper, Von Karman's set of nonlinear equations for rectangular plates with large deflection is divided into several sets of linear equations by perturbation method, the dimensionless center deflection being taken as a perturbation parameter. These sets of linear equations are solved by the spline finite-point (SFP) method and by the spline finite element (SFE) method. The solutions for rectangular plates having any length-to-width ratios under a uniformly distributed load and with various boundary conditions are presented, and the analytical formulas for displacements and deflections are given in the paper. The computer programs are worked out by ourselves. Comparison of the results with those in other papers indicates that the results of this paper are satisfactorily better.

Large deflection response-based geometrical nonlinearity of nanocomposite structures reinforced with carbon nanotubes

This paper deals with the nonlinear large deflection analysis of functionally graded carbon nanotube-reinforced composite(FG-CNTRC)plates and panels using a finite element method.Based on the first-order shear deformation theory(FSDT),the proposed model takes into account the transverse shear deformations and incorporates the geometrical nonlinearity type.A C^0 isoparametric finite shell element is developed for the present nonlinear model with the description of large displacements and finite rotations.By adopting the extended rule of mixture,the effective material properties of FG-CNTRCs are approximated with the introduction of some efficiency parameters.Four carbon nanotube(CNT)distributions,labeled uniformly distributed(UD)-CNT,FG-VCNT,FG-O-CNT,and FG-X-CNT,are considered.The solution procedure is carried out via the Newton-Raphson incremental technique.Various numerical applications in both isotropic and CNTRC composite cases are performed to trace the potential of the present model.The effects of the CNT distributions,their volume fractions,and the geometrical characteristics on the nonlinear deflection responses of FG-CNTRC structures are highlighted via a detailed parametric study.

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 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 PROBLEM OF CIRCULAR PLATES WITH VARIABLE THICKNESS UNDER THE ACTION OF COMBINED LOADS

By using the modified iteration method of large deflection theory of plates with variable thichness[1], we solve the problem of circular plates with variable thickness subjected to combined loads under the boundary conditions of the clamped edges and get comparatively more accurate second-order approximate analytical solution. If the results of this paper are degraded into the special cases, the results coinciding with those of papers [1,2] can be obtained. In this paper, the characteristic curves are plotted and some comparisons are made. The results of this paper are satisfactory.

A NEW TECHNIQUE FOR SOLVING NON-AXIALLY SYMMETRICALLARGE DEFLECTION OF ELASTIC AND THIN CIRCULAR PLATES

A new technique for solving large deflection problem of circular plates flexural non-axisymmetrically is proposed in this paper. The large deflection problem of a circular plate with built-in edge under non-axisymmetrical load is taken as an example to clarify the principle and procedure of the technique mentioned here. The technique given here can also be used to solve large deflection problem of circular plates under other non-axisymmetrical loads and boundary conditions.