Post-buckling Stiffness Research Articles

Multiobjective optimisation of laminated I-beams for maximum crippling, buckling and postbuckling strength

The present study deals with the optimal design of uniaxially loaded laminated I-beams for a maximum combination of crippling and buckling load, in the first instance, and a maximum combination of buckling load and postbuckling stiffness, in the second instance. The method of solution involves defining a design index comprising a weighted average of the objective functions and identifying candidate configurations which have to be optimised and compared to determine the best stacking sequence. This multiobjective approach leads to improved crippling, buckling and postbuckling strength. The multiobjective results are compared to single objective results, and the effect of various problem parameters on the optimal designs are numerically studied.

Optimal laminate configurations with symmetric lay-ups for maximum postbuckling stiffness

Optimal designs of symmetrically laminated angle-ply plates are obtained with the objective of maximizing the initial postbuckling stiffness. The design involves optimization over the ply angles and the stacking sequence to obtain the best laminate configuration. The stacking sequence is chosen from amongst five candidate designs. It is shown that the best configuration depends on the ratio of the in-plane loads in the x and y directions. Results are also given for two additional configurations which do not exhibit bending-twisting coupling.

Behaviour of axially loaded lipped channel

Thin plates have a stable post buckling behaviour so that its failure load is materially higher than its buckling load. This characteristics makes thin plates very suitable structural elements in lighweight structures. For a thin-walled channel which consists of a system of connected thin plates, the structure will develop local, torsional, flexural buckling or a combination of these modes under axially compressive loads. Either buckle of an individual plate component or buckle as a system of connected plates may cause a reduction in the stiffness of the section. Hence, a lower load carrying capacity is expected. In this paper, the finite strip method is employed to investigate the buckling behaviour of axially loaded lipped channels. In the formulation, by modifying the displacement function, the buckling solution of other end support conditions besides the simply supported case is obtained. Furthermore, utilizing the eigenvectors from the elastic buckling solution, the lower bound method is used to evaluate the initial post buckling stiffness of an adequately edged stiffened thin-walled channel.

Effect of Member Snap on Space Truss Collapse

A procedure is presented for the ultimate‐capacity analysis of space trusses whose members have either buckled or yielded. Unlike previous analyses, which carry out a static analysis to account for member postbuckling behavior, the possibility of a dynamic response due to snap‐through behavior of buckled struts is considered. Whether a dynamic response does occur depends on the stiffness of the degraded space truss vis‐à‐vis the negative member postbuckling stiffness. Studies of a simple truss model and a large double‐layer space truss indicate that neglect of the dynamic response due to member snap‐through leads to a significant overestimate of the structure's capacity.

Postbuckling behaviour of optimized rectangular composite laminates

In the literature, the linear buckling load of rectangular plate elements is maximized by optimizing fibre orientation and thickness in a preselected lamination sequence. The paper considers the effects of laminate optimization on the postbuckling behaviour of biaxially compressed, specially orthotropic laminates and suggests modifications of the lamination parameters to improve postbuckling performance. Postbuckling studies, in general, involve the formal derivation of nonlinear equilibrium paths using, say, the methods of Galerkin, finite element, etc. In contrast, a simpler but effective method is proposed whereby the initial postbuckling stiffness, defined by the slope of the postbuckling load-end shortening relation at bifurcation, is adopted as a qualitative index characterizing postcritical behaviour. An explicit solution is described for simply supported plates. It is established that the postbuckling stiffness and linear buckling load are governed by completely different functions: the former depends exclusively on inplane (membrane) stiffness while the latter is a function of bending stiffness only. It has been suggested that unfavourable postbuckling performance usually follows optimization of the buckling load. A detailed parametric study substantiates this proposition by illustrating that laminates, optimal in linear buckling, do exhibit inferior postbuckling characteristics due to reduced stiffness. Improved laminate designs are suggested that exhibit superior performance in both pre- and postbuckling.

Postbuckling analysis of layered composites using p‐ version finite strips

AbstractThis paper presents the results of a study on the buckling and postbuckling analyses of layered shear‐deformable composite plates using p‐version finite strips. The plates are considered to be sufficiently long so that the effect of boundary conditions along the shorter edges is of little significance; this makes it possible to use the ‘exact’ trigonometric functions in the longitudinal direction. The displacement field in the transverse direction is discretized by p‐version finite strips and convergence is studied by p‐extension, i.e. keeping the number of strips relatively small whilst increasing the polynomial order till convergence of the buckling load or the postbuckling stiffness is achieved.Detailed convergence studies are presented and comparison with results of earlier investigations are shown wherever appropriate. The results demonstrate the superior convergence characteristics of the p‐version approach.

Postbuckling Response Simulations of Laminated Anisotropic Panels

Numerical simulations are used in conjunction with experiments to study the buckling and postbuckling responses and failure initiation of flat, unstiffened composite panels. The numerical simulations are conducted using two‐dimensional shear‐flexible finite elements. The effect of the laminate stacking sequence on the buckling and postbuckling responses is studied. Correlation between numerical and experimental results is good through buckling, but the numerical models overestimate the postbuckling stiffness of the panels when nominal values of the material properties are used. To explain the discrepancies in the postbuckling stiffnesses, analytic sensitivity derivatives are calculated and used to study the sensitivity of the buckling and postbuckling responses to variations in different material and lamination parameters. Experimental results indicate that failure occurs along a nodal line. Numerical results show that the location of failure initiation corresponds to that of the maximum transverse shear‐...

Large Deflections of Orthotropic Plates in Compression

Rectangular specially orthotropic plates subjected to uniaxial in-plane compression are analyzed into the large deflection regime, using the computer program ELDAROP. All four edges are simply supported against out-of-plane movement, and opposite pairs of edges are maintained straight and parallel. Some plates analyzed are initially flat. Others have a geometrical imperfection shaped as a single half-sine wave in both directions. Particular attention is given to the effects on plate behavior of the modular ratio Ex/Ey, and the geometrical imperfection amplitude ΔO/t. For flat plates, the initial postbuckled stiffness depends entirely on the membrane rigidities and the aspect ratio a/b. However, on further loading, the bending rigidities then have a significant effect, provided the membrane behavior is not close to being isotropic.

Buckling and postbuckling behavior of compression-loaded isotropic plates with cutouts

An experimental study of the buckling and postbuckling behavior of square and rectangular compression loaded aluminum plates with centrally located circular, square, and elliptical cutouts is presented. Experimental results indicate that the plates exhibit overall trends of increasing buckling strain and decreasing initial postbuckling stiffness with increasing cutout width. Corresponding plates with circular and square cutouts of the same width buckle at approximately the same strain level, and exhibit approximately the same initial postbuckling stiffness. Results show that the reduction in initial postbuckling stiffness due to a cutout generally decreases as the plate aspect ratio increases. Other results presented indicate that square plates with elliptical cutouts having a large cutout-width-to-plate-width ratio generally lose prebuckling and initial postbuckling stiffness as the cutout height increases. However, the plates buckle at essentially the same strain level. Results also indicate that postbuckling stiffness is more sensitive to changes in elliptical cutout height than are prebuckling stiffness and buckling strain.

Imperfection sensitivity of a beam on a nonlinear elastic foundation

The imperfection sensitivity of a beam on a nonlinear elastic foundation is studied. The nonlinear equilibrium equations are based on a high-order nonlinear kinematic model which takes into account transverse shear deformation. The resulting differential equations are solved by Newton's method and a special finite-difference scheme. A parametric study of the effects of imperfection shape and amplitude is presented for a compression-loaded isotropic beam on a hardening and softening elastic foundation. The results indicate that the beam on a hardening foundation exhibits substantial postbuckling stiffness as compared to the beam with a softening foundation. Moreover, it is shown that the overall behavior can be characterized by wavelength changes of the buckled pattern as the loading parameter increases.

Local buckling strength of plate outstands

An elastic analysis is developed for a rectangular plate under uniform longitudinal compression. The plate is simply supported on the loaded ends, rotationally restrained by an elastic medium and also fully supported against out-of-plane movements along one unloaded edge, while the fourth edge is completely free. The energy method is used, based on an approximate expression for the critical buckling mode, and the plate is assumed to contain an initial imperfection in this same mode. The analysis determines the elastic critical buckling stress, the elastic post-buckling stiffness and the distribution of membrane, bending and torsional shear stresses in the plate. Various locations of first yield are studied to determine an approximate failure criterion. Finite element solutions are described which provide data for comparison with the predicted elastic and ultimate strength behaviour and very good agreement is achieved.

Magnetoelastic delamination buckling of thin ferromagnetic plates

The magnetoelastic delamination buckling and growth of thin ferromagnetic plates are analyzed theoretically. The thin ferromagnetic plate with surface normal to a uniform magnetic field is considered to adhere to a non-magnetic substratum. There is assumed to be an initial delamination at the edge or inner part of the plate. The Amperian current model for the magnetization of the ferromagnetic plate is used and the geometrical nonlinearity is included in order to determine the effect of postbuckling stiffness on the delamination growth process. The problem is solved by an asymptotic method, that is, Koiter's initial postbuckling theory. The delamination growth is determined by an energy balance, that is, a Griffith-type criterion. It is found that the magnetoelastic delamination buckling and the growth process depend on the initial delaminated area and the bond strength. Furthermore, the postbuckling behaviour of the initially delaminated ferromagnetic plate is unstable or stable, according to the edge or internal delamination, respectively. The initial imperfections of the buckling mode have a noticeable effect on the postbuckling behaviour and the delamination process of the plate with the edge delamination.

Large deflection analysis of orthotropic plates—adaptation of Coan's method

The problem considered is the elastic large deflection analysis of thin, rectangular, specially orthotropic plates with two axes of symmetry. Coan's method for allowing the prescription, for isotropic plates, of the in-plane tensile or compressive loading along two opposite edges, is adapted to deal with orthotropic plates. This analysis has been incorporated into an existing computer program (ELDAROP), and the results agree well with previous work on isotropic plates. Results are given for the post-buckling of square orthotropic plates under uniaxial in-plane compression. These confirm that if the unloaded edges are maintained straight, the post-buckled stiffness increases, and this effect decreases as the modular ratio increases.

Theoretical and Experimental Results on the Post-Buckling of Ring-Stiffened Cylinders

ABSTRACT The post-buckling behavior of ring-stiffened cylinders subjected to lateral pressure is studied theoretically and experimentally. The theoretical analysis uses the energy method and terms of up to and including fourth order of the buckling displacements are considered. The results of experiments on ten machined ring-stiffened cylinders subjected to lateral pressure and used to verify the theoretical analysis. The theory shows the post-buckling behavior to be stable-symmetric for both the shell and general instability modes. The experimental results are in reasonable agreement with the theory, although they indicate, in general, a lower post-buckling stiffness. For the range of parameters considered, no instability or imperfection sensitivity is found in either the theoretical or the experimental studies.

Pantographing Self-Adaptive Gap Elements

Abstract A pantographing self-adaptive gap element type contact strategy is presented. Due to the manner of its formulation, it can handle large deformations in the contact zone, contact initiation in a structure that has either positive or indefinite stiffness characteristics, kinematic and material nonlinearities, as well as self-adaptive load (time) stepping. Contact in pre- and post-buckling structures can be treated in this context. Several illustrative benchmark problems are given. These include coming into contact with a fiat surface, and involve large deformation kinematics and inelastic behavior as well as pre- and post-buckling stiffness characteristics.

Pantographing self adaptive gap elements

This paper develops a so-called pantographing self adative gap element type contact strategy. Due to the manner of formulation, the scheme has the capability to handle; large deformations in the contact zone; contact initiation in structure exhibiting either positive or indefinite stiffness characteristics; kinematic and material nonlinearity as well as; self adaptively adjusts load/time stepping. In this context, contact in pre and postbuckling structure can be treated. To illustrate the scheme, several benchmark problems are presented. These include contacting structure involving large deformation kinematics, inelastic behavior as well as pre and postbuckling stiffness characteristics.

Delamination Buckling and Growth in Laminates

The phenomenon of delamination buckling and growth in a compressively loaded, two-layer circular plate is analyzed. Geometrical nonlinearity is included in the model in order to determine the effect of postbuckling stiffness on the delamination growth process, and the influence of imperfections in the form of transverse loads is also examined.

The buckling and post-buckling behaviour of composite plates under biaxial loading

The response to applied load of general composite plates exhibits a coupling between the bending and extensional modes of deformation which may be significant when shear or compressive loads are applied in-plane. The additional deformation modes may affect the nature of the buckling behaviour, reduce the buckling load or change the post-buckled stiffness. This paper considers the stiffness immediately after buckling of a rectangular panel of angle-ply type in which coupling effects occur and which undergoes bifurcational buckling when biaxial load is applied in directions parallel to the panel edges. Expressions are derived for the buckling loads and for the in-plane stiffness of the panel immediately after the instant of buckling, it being found that the coupling terms affect the stiffness at buckling mainly via the associated change in buckling mode shape.

Ultimate Strength Analysis of Double Bottom Structure by “Idealized Structural Unit Method”

In a previous paper, two of the authors have developed an effective method for the ultimate strength analysis of large size structures. The method is named as “Idealized Structural Unit Method”.In the method, a large size element with idealized nonlinear character was necessary and an example element, “Girder Element”, was developed. In this paper, this method is applied to double bottom structures. The state of two-dimensional stress in the tank top and the bottom shell plating is considered. Conditions for buckling of these plates are established and its post buckling stiffness is determined. The condition for their ultimate strength and stiffness at the ultimate strength are determined in connection with the girder element.An example structure is analysed and the results of the analysis are presented.

168. Precipitation in microscopic cracks.: G. R. Pulliam, J. Amer. Ceram. Soc., 42, 447–482, Oct. 1959

The magnetoelastic delamination buckling and growth of thin ferromagnetic plates are analyzed theoretically. The thin ferromagnetic plate with surface normal to a uniform magnetic field is considered to adhere to a non-magnetic substratum. There is assumed to be an initial delamination at the edge or inner part of the plate. The Amperian current model for the magnetization of the ferromagnetic plate is used and the geometrical nonlinearity is included in order to determine the effect of postbuckling stiffness on the delamination growth process. The problem is solved by an asymptotic method, that is, Koiter's initial postbuckling theory. The delamination growth is determined by an energy balance, that is, a Griffith-type criterion. It is found that the magnetoelastic delamination buckling and the growth process depend on the initial delaminated area and the bond strength. Furthermore, the postbuckling behaviour of the initially delaminated ferromagnetic plate is unstable or stable, according to the edge or internal delamination, respectively. The initial imperfections of the buckling mode have a noticeable effect on the postbuckling behaviour and the delamination process of the plate with the edge delamination.