Post-buckling Problems Research Articles

梁の３次元大変形の有限要素法解析 （第２報） 解析結果

In Part 1, the authors presented a new nonlinear finite element formulation of a spatial beam element, and obtained an unsymmetric tangent stiffness matrix. In this paper, firstly we solve two examples of plane and spatial deformations and show that the use of artificially symmetrised tangent stiffness matrix does not decrease the accuracy of the solution. Then, three numerical examples of spatial post-buckling problems are presented, indicating the accuracy and the efficiency of the proposed method.

Non-linear analysis of unsymmetrically laminated moderately thick shallow spherical shells

A Fourier-Bessel series solution is formulated for the non-linear vibration of an unsymmetrically laminated moderately thick shallow spherical shell with its edge elastically restrained against rotation. The effects of transverse shear deformation and rotatory inertia are included in the analysis. The equations of motion are reduced to a set of non-linear ordinary differential equations for the time-dependent coefficients in the series by making use of the Galerkin method. These equations are solved by the method of harmonic balance for undamped free vibrations. The corresponding postbuckling problem is treated as a special case. Numerical results in non-linear free vibration, buckling and postbuckling are presented for various ratios of base radius to thickness and shell rise to thickness, number of layers, material properties and boundary conditions. The present results are compared with available data.

Computer Simulation of Facial Expression. 2nd Report, Identification of In Vivo Mechanical Properties and Analysis of Wrinkling and Finite Deformation of Hyper Elastic Soft Tissue by the Finite-Element Method.

It is the purpose of the investigation to clarify mechanisms of facial expressions and formation of wrinkles on a face with a hyper elastic skin. In the present study, parameters of mechanical properties are identified by using the results of both in vivo experiments and calculation by the finite-element method (FEM). We also carried out in vitro experiments for formation of wrinkles with a rabbit skin and conducted analysis of wrinkles, in which formulations of hyper elasticity and large deformation for a skin-fat composite were incrementally made as buckling and post-buckling problems, using the same model of the experiments. Furthermore, we carried out in vivo experiments and FEM analysis of wrinkles in the forehead skin of a human being. The validity of the analysis method proposed in this study and the usefulness of the developed FEM programs are demonstrated by comparison with the series of experimental results.

Postbuckling Analysis of Plates Under Combined Loads by a Mixed Finite Element and Boundary Element Method

The postbuckling behavior of thin plates under combined loads is studied in this paper by using a mixed boundary element and finite element method. The transverse and the in-plane deformation of the plates are analyzed by the boundary element method and the finite element method, respectively. Spline functions were used as the interpolation functions and shape functions in the solution of both methods. A quadratic rectangular spline element is adopted in the finite element procedure. Numerical results are given for typical problems to show the effectiveness of the proposed approach. The possibilities to extend the method developed in this paper to more complicated postbuckling problems are discussed in the concluding section.

Sensitivity analysis in post-buckling problems of shell structures

In this study, a method to evaluate sensitivities in post-buckling analysis is developed. The gradients of load values obtained under several constraint conditions are discussed. Unlike other methods for sensitivity analysis, the present method can cope with both snap-through and bifurcation problems without eigenvalue analysis. In addition, the load sensitivity at arbitrary points on an equilibrium path can be calculated. Using the MITC4 shell element, several numerical examples, such as a bifurcation buckling and a snap-through analysis of shell structures, are solved to show the performance of this sensitivity analysis method.

A theoretical and experimental investigation of buckling induced delamination growth

A theoretical and experimental investigation of a laminated plate with an embedded, initially circular delamination, loaded in uniaxial compression is described. The main issue concerns the combined postbuckling and crack growth behaviour in general situations at arbitrary crack contours and applied to fibre-reinforced carbon-epoxy composite plates. In a numerical procedure a standard FE-code is used to solve the structural postbuckling problem and extended to account also for contact effects. To predict interlaminar crack growth an automatic finite element procedure was employed for the purpose. Delamination growth in laminates was detected by the aid of acoustic emission. After successive crack increments, the evolution of the delamination front was mapped using an ultrasonic C-scan technique and also for comparison by X-ray. Predicted and observed results are compared as regards postbuckling behaviour and in particular the magnitude and shapes of delamination growth fronts.

Postbuckling analysis of plates on an elastic foundation by the boundary element method

The postbuckling behavior of plates on an elastic foundation is studied by using the boundary element method (BEM). A new fundamental solution of lateral deflection is derived through the resolution theory of a differential operator, and a set of boundary element formulae in incremental form is presented. By using these formulae, the BEM solution procedure becomes relatively simple. The results of a number of numerical examples are compared with existing solutions and good agreement is observed. It shows that the proposed method is effective for solving the postbuckling problems of plates with arbitrary shape and various boundary conditions.

Localized analysis of thin-walled structure's buckling/initial post-buckling and its accuracy

A method of localization is proposed to lower the high order of equations in FEM calculation for the stability of a complex thin-walled structure. The localized analysis enables us to obtain both the upper and lower limits for the bifurcating point in a whole linear-elastic structural system, as well as an approximate solution to asymptotic post-buckling problem. Some numerical examples are included.

BEM of postbuckling analysis of thin plates

A new fundamental solution of deflection for plate buckling problems is first derived through the separation of in-plane stresses into an essential part and a disturbed part and by use of the resolution theorem of differential operator. A set of new boundary element formulas for the analysis of post-buckling problems of elastic plates are presented. On the basis of these formulas in-plane and out-of-plane displacements can be decoupled. The solution procedure for the boundary value problems of the in-plane displacement becomes very simple. Some examples of a circular plate are considered, and their results are in good agreement with the known ones. It appears that the proposed method is very effective for solving the postbuckling problem of plates with arbitrary geometry.

Koiter's stability theory in a computeraided engineering (CAE) environment

The development of “DISDECO”, the Delft Interactive Shell Design Code is described. The purpose of this project is to make the accumulated theoretical, numerical and practical knowledge of the last 20 years readily accessible to users interested in the analysis of buckling sensitive structures. With this open ended, hierarchical, interactive computer code the user can access from his work-station successive programs of increasing complexity.Included are modules that contain Koiter's imperfection sensitivity theory extended to anisotropic shell structures under combined loading. The nonlinear Donnell-type anisotropic shell equations in terms of the radial displacement W and the Airy stress function F are used. The circumferential dependence is eliminated by Fourier decomposition. The resulting sets of ordinary differential equations are solved numerically via the “Shooting Method”. Thus the specified boundary conditions can be enforced rigorously not only in the pre-buckling but also in the buckling and post-buckling problem. Initial results indicate that in order to obtain reliable results for anisotropic shells rigorous enforcing of the edge restraint and of the boundary conditions is indeed a must.

Axisymmetric buckling and post-buckling of bimodulus annular plates

The buckling and post-buckling analysis of bimodulus circular and annular plates is presented. Annular finite elements are used to solve the problem. In the case of bimodulus materials, the elastic constants in tension and compression are different. The constitutive matrix based on the ‘Jones model’ is used for the analysis. The post-buckling problem is solved as an eigen-value problem with deflexion at a point, as an independent variable. Numerical results are compared for particular cases.

A simple discrete element method for the initial postbuckling of elastic structures

AbstractIn this paper a discrete element matrix formulation for the analysis of linear and non‐linear bifurcation problems of some elastic structures is introduced. This, combined with a discrete perturbation technique, is implemented into a computer code to solve buckling and initial postbuckling problems of elastic struts.

Effect of transverse shear on nonlinear vibration and postbuckling of anti-symmetric cross-ply imperfect cylindrical shells

Based on the Timoshenko-Mindlin kinematic hypothesis, the Donnell-type shell theory is extended to include transverse shear and rotary inertia for the nonlinear analysis of an anti-symmetrically laminated cross-ply circular cylindrical shell. The resulting governing equations are expressed in terms of a stress function, two rotations and transverse displacement. The shell is assumed to be all-simply-supported and all-clamped. A solution is formulated by a multi-mode approach and the method of harmonic balance for nonlinear vibrations. The corresponding postbuckling problem is treated as a special case. The results are compared with available data.

Post-buckling of elastic annular plates at multiple eigenvalues

The buckling and post-buckling problem of elastic annular plates is treated at multiple eigenvalues making strong use of symmetry properties of the problem in deriving bifurcation equations (amplitude equations of the critical modes) from the von Karman plate equations by means of the Ljapunov-Schmidt method. Further, a restricted generic unfolding of the bifurcation equations is made and some practically interesting results concerning the post-buckling behaviour are presented.

Global-local methodologies and their application to nonlinear analysis

An assessment is made of the potential of different global-local analysis strategies for predicting the nonlinear and postbuckling responses of structures. Two postbuckling problems of composite panels are used as benchmarks and the application of different global-local methodologies to these benchmarks is outlined. The key elements of each of the global-local strategies are discussed and future research areas needed to realize the full potential of global-local methodologies are identified.

Postbuckling analysis using a general-purpose code

A new capability for solving postbuckling problems in shell structures is described. The matrix theory to adapt Newton's method to nonlinear finite element shell analysis is outlined first. The matrix theory is directed at writing consistent linear algebratic equations for problems where the tangent stiffness matrix is singular or nearly singular. The matrix theory suggests a change of variables as part of the usual iterative procedure in Newton's method. The change of variables is shown to be feasible for introduction into the algorithm programmed in general purpose codes for finite element analysis of structures. Numerical results from a new option that has been programmed in an existing general purpose code are presented. The analysis of shell structures for collapse and for branching at bifurcation loads is illustrated by the numerical examples.

Imperfection sensitivity of axially compressed laminated flat plates due to bending-stretching coupling

This paper investigates the effects of the bending-stretching coupling on the imperfection sensitivity of axialty compressed laminated, thin, rectangular flat plates. In particular, it is found that under certain circumstances this coupling phenomenom results in a nonzero cubic term of the potential energy, so that the structure may be imperfection sensitive depending on the sign of the imperfection. The analysis considers a nonlinear prebuckling state due to bending-stretching coupling of the structure. The buckling and initial postbuckling problem is solved using Koiter's theory of elastic stability.

Post-buckling of a pressurized elastic sheet on a rigid surface

An infinite elastic sheet, subject to axial forces, is pressurized on one side and constrained on the other side by a rigid flat surface. The post-buckling problem is solved by perturbation analysis and numerical integration. We find the buckling force of a perfect sheet is infinite. As displacement increases, the sheet undergoes three stages. In Stage I the sheet does not touch itself and is in unstable equilibrium for constant axial force. The sheet in Stage II has one point in contact with itself and is stable. Stage III is neutrally stable with one segment collapsed. A finite stability criterion is defined.

Thermal stress analysis of skew plates by finite element method

Thermal stresses are induced in general due to nonuniform temperature distribution or due to the boundary restriction. Most of the work reported so far deals with either plates with edges clamped in plane of the plate or plates with stress free edges. While studying buckling or post-buckling problems, one should ideally analyse the plates with mixed in-plane boundary conditions. Hence, in the present analysis, thermal stress analysis of skew plates with mixed in-plane boundary conditions using finite element approach is attempted. In addition, the effect of in-plane boundary conditions on the thermal stresses is also discussed.

Analysis of impact response of buckled composite laminates

The dynamic response of buckled composite plates impacted by a hard object is studied. A nine-node isoparametric finite element, developed based on the Mindlin plate theory and von Karman large deflection assumptions, is used for both postbuckling and impact analyses. In order to evaluate the contact force, an experimentally established contact law which accounts for the permanent indentation is employed. The static postbuckling problem of laminated plates is first considered. The impact responses include contact force histories, dynamic deflections, and dynamic strains in the plate for various buckling conditions. In addition, free vibration of buckled laminated plates is also solved to determine the range of natural frequencies which are needed to choose a proper time increment for time integration in the impact analysis.