DYNAMIC BUCKLING OF DOUBLE-WALLED CARBON NANOTUBES UNDER STEP AXIAL LOAD

An approximate method is presented in this paper for studying the dynamic buckling of double-walled carbon nanotubes （DWNTs） under step axial load. The analysis is based on the continuum mechanics model, which takes into account the van der Waals interaction between the outer and inner nanotubes. A buckling condition is derived, from which the critical buckling load and associated buckling mode can be determined. As examples, numerical results are worked out for DWNTs under fixed boundary conditions. It is shown that, due to the effect of van der Waals forces, the critical buckling load of a DWNT is enhanced when inserting an inner tube into a single-walled one. The paper indicates that the critical buckling load of DWNTs for dynamic buckling is higher than that for static buckling. The effect of the radii is also examined. In addition, some of the results are compared with the previous ones.

Nonlinear vibration and buckling of circular sandwich plate under complex load

The nonlinear vibration fundamental equation of circular sandwich plate under uniformed load and circumjacent load and the loosely clamped boundary condi- tion were established by von Karman plate theory, and then accordingly exact solution of static load and its numerical results were given. Based on time mode hypothesis and the variational method, the control equation of the space mode was derived, and then the amplitude frequency-load character relation of circular sandwich plate was obtained by the modified iteration method. Consequently the rule of the effect of the two kinds of load on the vibration character of the circular sandwich plate was investigated. When circumjacent load makes the lowest natural frequency zero, critical load is obtained.

Buckling analysis of shear deformable composite conical shells reinforced by CNTs subjected to combined loading on the two-parameter elastic foundation

The main objective of this study is to investigate the buckling analysis of CCSs reinforced by CNTs subjected to combined loading of hydrostatic pressure and axial compression resting on the twoparameter elastic foundation(T-P-EF).It is one of the first attempts to derive the governing equations of the CCSs reinforced with CNTs,based on a generalized first-order shear deformation shell theory(FSDST)which includes shell-foundation interaction.By adopting the extended mixing rule,the effective material properties of CCSs reinforced by CNTs with linear distributions are approximated by introducing some efficiency parameters.Three carbon nanotube distribution in the matrix,i.e.uniform distribution(U)and V and X-types linear distribution are taken into account.The stability equations are solved by using the Galerkin procedure to determine the combined buckling loads(CBLs)of the structure selected here.The numerical illustrations cover CBLs characteristics of CCSs reinforced by CNTs in the presence of the T-P-EF.Finally,a parametric study is carried out to study the influences of the foundation parameters,the volume fraction of carbon nanotubes and the types of reinforcement on the CBLs.

STACKING SEQUENCE OPTIMIZA-TION OF LAMINATED COMPOSITE CYLINDER SHELL FOR MAXIMAL BUCKLING LOAD

A new optimization method for the optimization of stacking of composite glass fiber laminates is developed. The fiber orientation and angle of the layers of the cylindrical shells are sought considering the buckling load. The proposed optimization algorithm applies both finite element analysis and the mode-pursuing sampling （MPS）method. The algorithms suggest the optimal stacking sequence for achieving the maximal buckling load. The procedure is implemented by integrating ANSYS and MATLAB. The stacking sequence designing for the symmetric angle-ply three-layered and five-layered composite cylinder shells is presented to illustrate the optimization process, respectively. Compared with the genetic algorithms, the proposed optimization method is much faster and efficient for composite staking sequence plan.

Axisymmetrical Nonlinear Bending and Buckling of a Circular Plate Under Large Load

With the terms of the exact series solution taken as trial functions, the method of point collocation was used to calculate the large deflection of a circular plate. The axisymmetrical bending formulae were developed for the calculation of a circular plate subjected to polynomial distributed loads, a concentrated load at the center, uniform radial forces and moments along the edge or their combinations. The support may be elastic. The buckling load was calculated. Under action of uniformly distributed load, central load or their compound load, solutions were compared with those obtained by other methods. Buckling beyond critical thrust was compared with that calculated by the power series method. The method presented in this paper has advantages of wide convergent range, high precision and short computing time. Moreover, the computing time is nearly independent of the complexity of the loads.

The critical loading for lateral buckling of continuous welded rail

The most significant differences between continuous welded rails （CWRs） and general split-type connectors are axial compression in the longitudinal direction, buckling stability and other issues generated under the influence of thermal effect. Under thermal effect, a dynamical behavior similar to that of a beam fixed on two sides occurs in the central locked area of the welded rail, as there is axial compression but no possibility of sliding. Continuous welded rails do not contract or expand, and are supported by the dynamical system made up of ballasts and rail clips. The rail-support system mentioned above has the features of non-uniform material distribution and uncertainty of construction quality. Due to these facts, the dynamics method based on the linear elastic hypothesis cannot correctly evaluate the rail＇s buckling conditions. This study is aimed at applying Finite Difference Method （FDM） and Monte Carlo Random Normal Variables Method to the analysis of welded rail＇s buckling behavior during the train＇s acceleration and deceleration, under thermal effect and uncertain factors of ballast and rail clips. The analysis result showed that buckling occurs under the combined effect of thermal effect and the train＇s deceleration force co-effect and the variance ratio of ballast and rail clips is over 0.85, or under the combined effect of thermal effect and the train＇s acceleration force when the ariance ratio is over 0.88.

TORSIONAL BUCKLING OF SPHERICAL SHELLS UNDER CIRCUMFERENTIAL SHEAR LOADS

By the aid of differential geometry analysis on the initial buckling of shell element, a set of new and exact buckling bifurcation equations of the spherical shells is derived. Making use of Galerkin variational method, the general stability of the hinged spherical shells with the circumferential shear loads is studied. Constructing the buckling mode close to the bifurcation point deformations, the critical eigenvalues, critical load intensities and critical stresses of torsional buckling ranging from the shallow shells to the hemispherical shell are obtained for the first time.

Post-Buckling Behavior of Laminated Composite Cylindrical Shells Subjected to Axial, Bending and Torsion Loads

In present work, post-buckling behavior of imperfect (of eigen form) laminated composite cylindrical shells with different L/D and R/t ratios subjected to axial, bending and torsion loads has been investigated by using an equilibrium path approach in the finite element analysis. The Newton-Raphson approach as well as the arc-length approach is used to ensure the correctness of the equilibrium paths up to the limit point load. Post-buckling behavior of imperfect cylindrical shells with different L/D and R/t ratios of interest is obtained and the theoretical knock-down factors are reported for the considered cylindrical shells.

Vibration and Buckling Approximation of an Axially Loaded Cylindrical Shell with a Three Lobed Cross Section Having Varying Thickness

On the basis of the thin-shell theory and on the use of the transfer matrix approach, this paper presents the vibrational response and buckling analysis of three-lobed cross-section cylindrical shells, with circumferentially varying thickness, subjected to uniform axial membrane loads. A Fourier approach is used to separate the variables, and the governing equations of the shell are formulated in terms of eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations are written in a matrix differential equation. The transfer matrix is derived from the non-linear differential equations of the cylindrical shells with variable thickness by introducing the trigonometric series in the longitudinal direction and applying a numerical integration in the circumferential direction. The natural frequencies and critical loads beside the mode shapes are calculated numerically in terms of the transfer matrix elements for the symmetrical and antisymmetrical vibration modes. The influences of the thickness variation of cross- section and radius

Analysis modeling for plate buckling load of vibration test

In view of the recent technological development, the pursuit of safehigh-precision structural designs has been the goal of most structural designers. To bridge the gapbetween the construction theories and the actual construction techniques, safety factors are adoptedfor designing the strength loading of structural members. If safety factors are too conservative,the extra building materials necessary will result in high construction cost. Thus, there has been atendency in the construction field to derive a precise buckling load analysis model of member inorder to establish accurate safety factors. A numerical analysis model, using modal analysis toacquire the dynamic function calculated by dynamic parameter to get the buckling load of member, isproposed in this paper. The fixed and simple supports around the circular plate are analyzed by thisproposed method. And then, the Monte Carlo method and the normal distribution method are used forrandom sampling and measuring errors of numerical simulation respectively. The analysis resultsindicated that this proposed method only needs to apply modal parameters of 7 X 7 test points toobtain a theoretical value of buckling load. Moreover, the analysis method of inequality-distanttest points produces better analysis results than the other methods.

On the Buckling of Euler Graphene Beams Subject to Axial Compressive Load

In this paper, we consider the buckling of an Euler-Bernoulli graphene beam due to an axial compressive load. We formulate the problem as a non-linear (eigenvalue) two-point boundary value problem, prove some properties of the eigenpairs and introduce a suitable numerical shooting method scheme for approximating them. We present the perturbation and the numerical approximations of the first and second buckling loads and the corresponding shapes.

A Symplectic Method of Numerical Simulation on Local Buckling for Cylindrical Long Shells under Axial Pulse Loads

In this paper,the local buckling of cylindrical long shells is discussed under axial pulse loads in a Hamiltonian system.Using this system,critical loads and modes of buckling of shells are reduced to symplectic eigenvalues and eigensolutions respectively.By the symplectic method,the solution of the local buckling of shells can be employed to the expansion series of symplectic eigensolutions in this system.As a result,relationships between critical buckling loads and other factors,such as length of pulse load,thickness of shells and circumferential orders,have been achieved.At the same time,symmetric and unsymmetric buckling modes have been discuss.Moreover,numerical results show that modes of post-buckling of shells can be Bamboo node-type,bending type,concave type and so on.Research in this paper provides analytical supports for ultimate load prediction and buckling failure assessment of cylindrical long shells under local axial pulse loads.

Optimization for Maximum Nonlinear Buckling Load and Topics on Imperfection of Latticed Shells

The present paper represents comparison of continuum shells and latticed shells with qualitative analysis. For shells, the mechanical characteristics in the two perpendicular directions are continuous and related to each other, and any change in thickness will result in change in stiffness in any direction. In latticed shells, members are discrete and stiffnesses in two mutually perpendicular directions are discontinuous and independent of each other. Therefore, sensitivity of geometrical imperfection for buckling of latticed shells should be different from that of continuum shells. The author proposes a shape optimization method for maximum buckling load of a latticed shell. A single layer latticed dome is taken as a numerical example, and the results show that the buckling load parameter for full area loading case increases 32.75% compared to that of its initial shape. Furthermore, the numerical example demonstrates that an optimum latticed shell with maximum buckling load, unlike an optimum continuum shell, may not be sensitive to its geometrical imperfection.

THE APPROXIMATE ANALYTICAL SOLUTION FOR THE BUCKLING LOADS OF A THIN-WALLED BOX COLUMN WITH VARIABLE CROSS-SECTION

For a thin-walled box column with variable cross-section, the three governing equations for torsional-flexural buckling are ordinary differential equations of the second or fourth order with variable coefficients, so it is very difficult to solve them by means of an analytic method. In this paper, polynomials are used to approximate the geometric properties of cross-section and certain coefficients of the differential equations. Based on the energy principle and the Galerkin's method, the approximate formulas for calculating the flexural and torsional buckling loads of this kind of columns are developed respectively, and numerical examples are used to verify the correctness of the solutions obtained. The results calculated in this paper provide the basis for demonstrating the stability of thin-walled box columns with variable cross-section. This paper is of practical value.

Interval Analysis Method on the Buckling of Composite Laminate

A method named interval analysis method, which solves the buckling load of composite laminate with uncertainties, is presented. Based on interval mathematics and Taylor series expansion, the interval analysis method is used to deal with uncertainties. Not necessarily knowing the probabilistic statistics characteristics of the uncertain variables, only little information on physical properties of material is needed in the interval analysis method, that is, the upper bound and lower bound of the uncertain variable. So the interval of response of the structure can be gotten through less computational efforts. The interval analysis method is efficient under the condition that probability approach cannot work well because of small samples and deficient statistics characteristics. For buckling load of a special cross-ply laminates and antisymmetric angle-ply laminates with all edges simply supported, calculations and comparisons between interval analysis method and probability method are performed.

Predictive analysis of buckling distortion of thin-plate welded structures

The welding buckling distortions of thin plated structures were investigated based on finite element methods.An engineering treatment method for predicationg the buckling distortion was proposed.The equivalent applied thermal load was used to simulate the welding residual stress,thus the calculation of complex welding distortion can be transformed into 3D elastic structural applied load analyses,which can reduce the quantities of calculating work effectively.The validation of the method was verified by comparison of the numerical calculation with experimental results.The prediction of buckling distortion for side walled structures of passenger train was performed and the calculation was in agreement with measuring results in general.It is shown that the main factors for producing the buckling are the intermittent fillet and plug weld during welding the stiffened beams and columns to the panel.

Buckling analysis of functionally graded plates partially resting on elastic foundation using the differential quadrature element method

We extend the differential quadrature element method (DQEM) to the buckling analysis of uniformly in-plane loaded functionally graded (FG) plates fully or partially resting on the Pasternak model of elastic support. Material properties of the FG plate are graded in the thickness direction and assumed to obey a power law distribution of the volume fraction of the constituents. To set up the global eigenvalue equation, the plate is divided into sub-domains or elements and the generalized differential quadrature procedure is applied to discretize the governing, boundary and compatibility equations. By assembling discrete equations at all nodal points, the weighting coefficient and force matrices are derived. To validate the accuracy of this method, the results are compared with those of the exact solution and the finite element method. At the end, the effects of different variables and local elastic support arrangements on the buckling load factor are investigated.

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material(FGM)structures is one of the major threats to their service safety in engineering applications.This paper aims to clarify a long-standing controversy on the thermal instability type of simply-supported FGM beams.First,based on the Euler-Bernoulli beam theory and von K′arm′an geometric nonlinearity,a nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang’s two-variable method is formulated.Second,an approximate analytic solution to the nonlinear integro-differential boundary value problem under a thermal-induced inhomogeneous force boundary condition is obtained by using a semiinverse method when the coordinate axis is relocated to the bending axis(physical neutral plane),and then the analytical predictions are verified by the differential quadrature method(DQM).Finally,based on the free energy theorem,it is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes;furthermore,the nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect.These results are expected to provide new ideas and references for the design and regulation of FGM structures.

Postbuckling of Imperfect Stiffened Cylindrical Shells Under Combined External Liquid Pressure and Axial Compression

posthuckling analysis is presented for the stilTened cylindrical shell of finite length subjected to combined loading of external liquid pressure and axial compression. The formulations are based on a boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, nonlinear large deflections in the postbuckling range and initial geometrical imperfections of the shell. The 'smeared stifl'cner' approach is adopted for the stiffencrs. In the analysis a singular perturbation technique is used (o determine the interactive buckling loads and the postbuckling paths. Numerical examples cover the performance of perfect and imperfect, stringer and ring stiffened cylindrical shells. Typical results arc presented in the dimcnsionless graphical form.