Post-buckling Phase Research Articles

Buckling and post-buckling of anisogrid lattice-core sandwich plates with nanocomposite skins

This research offers an in-depth examination of the buckling and post-buckling responses observed in sandwich plates distinguished by the presence of an anisogrid lattice core and face layers composed of functionally graded carbon nanotube-reinforced composites (FG-CNTRC). To model this structure, a global continuous model for the lattice core is employed, while the displacement field is approximated using the First-Order Shear Deformation Theory (FSDT). Geometrically nonlinear strain relations based on the von-Kármán assumptions are incorporated to accurately capture the structural response. The homogenization of the FG-CNTRC skins is achieved through the modified rule of mixture, enabling estimations of effective material properties. To separate pre-buckling and post-buckling phases, we implement the adjacent-equilibrium criterion. Solving the linear (pertaining to buckling) and nonlinear (related to post-buckling) equations is based on the Generalized Differential Quadrature (GDQ) technique alongside an iterative method grounded in the displacement control strategy. Our analysis explores the influence of core and skin characteristics on the stability of these sandwich plates. By examining critical buckling thresholds, post-buckling equilibrium paths, and the underlying mechanisms driving these responses, we contribute insights into the structural stability and performance of such composite structures. This study not only enhances our understanding of buckling behavior in anisogrid lattice core sandwich plates but also holds promise for diverse engineering applications where structural stability is paramount.

On the Interaction of Damage Evolution and Thermal Buckling in Stepped Circular Bilaminates

Abstract The behavior and evolution of stepped circular bi-laminates with edge damage are studied for structures subjected to uniform thermal load. The problem is formulated as a moving intermediate boundaries problem in the calculus of variations, where the boundary of an evolving region of damage emanating from the edge of the smaller substructure, as well as the boundary of a progressing/regressing region of sliding contact adjacent to the intact region of the composite structure are each allowed to vary along with the displacements. This yields the associated transversality conditions that define the locations of the propagating boundaries that correspond to equilibrium configurations of the evolving composite structure, as well as the equations of equilibrium and the associated interior and exterior boundary conditions. Various configurations of contact of the detached segments of the composite structure and the associated behavior are considered, and the influence and progression of contact on the overall evolution of the composite structure are assessed. Closed-form analytical solutions to the geometrically nonlinear problem are obtained, and expressions for the critical buckling load are developed. The explicit forms of the total energy release rate along the delamination front, as well as of the conditions for propagation of the contact zone boundary, are obtained from the analytical solutions and the corresponding transversality conditions. Results of numerical simulations based on the analytical solutions are presented and are seen to unveil a rich evolution process involving contact progression/recession and metamorphosis, buckling, and detachment progression during the prebuckling phase, during sling-shot buckling, and during the postbuckling phase, depending on the material properties of the sublaminates, the geometry of the sublaminates, the initial size of the damage, and the strength of the interfacial bond. Characteristic behavior of damage propagation is found to be quite robust and is seen to include stable propagation, stable followed by unstable progression, arrest, and catastrophic propagation.

The Effects of Loading Conditions on the Behaviour of Fixed-Ended Plain Channel Columns.

The compressive behaviour of column members can be considerably affected by local buckling, material yielding and local end conditions. In this paper, the effects of the loading conditions at the ends of plain channel section columns subjected to uniformly compressed loading, and fixed conditions at the column ends with respect to global rotations, was examined. Finite element simulation was employed to look at the post-buckled response of thin-walled, plain channel section columns that covered the complete loading history of the compression columns from the onset of elastic local buckling through the nonlinear elastic and elastoplastic post-buckling phases of behaviour to final collapse and unloading. Two types of loading conditions were considered: the first was one that has been used practically in tests whereby one end is loaded with a moving top platen while the other end is fixed at the lower platen, but, for the second loading condition, both ends were loaded with equally moving top and lower platens. These two conditions were shown to lead to quite different characteristic interactive responses of the columns due to mode jumping in the buckling mode for the locally rotationally constrained case.

Interactional analysis of steel plate shear wall with thin plate using modified slope deflection method

Steel Plate Shear Walls have been discussed as a lateral load resisting system in various building and especially high-rise buildings. Application of this system in high-rise buildings requires a detailed analysis of the behavior of Steel Plate Shear Walls. This is due to the fact that in high-rise and mid-rise buildings, great axial deformations of columns and aggregation of their effects in height lead to domination of bending displacement. This study aims at analysis of steel plate shear walls using plate frame interaction and modified slope-deflection method (modified slope-deflection method with consideration of axial deformation effect). The plates in this system are equaled by diagonal springs. Plate - frame interaction method is used for calculating plate stiffness. The process of analysis, which aims at reduction of analysis duration, has been conducted in a programming language, thus the analysis of the system and only by changing the values of steel shear wall parameters, is possible with high accuracy and in a very short time. The results of analysis through this method, which uses plate frame interaction method for calculation of plate stiffness, indicate the incorrect assumption of this method in formation of diagonal tension field in post-buckling phase in comparison with the finite element analysis. After modification, the accuracy of this new method has been investigated by changing the parameters of panel width and thickness in this system. Analytical results, compared with finite element analysis, showed good coincidence. The main cause of difference is in the method of calculating the length of the plate which has not buckled and played any role in the post - buckling phase.

Buckling and post-buckling behaviour of moderately thick plates using an exact finite strip

An exact finite strip for the buckling and post-buckling analysis of moderately thick plates is presented in this paper by using First order Shear Deformation Theory (FSDT). In the buckling phase the Von-Karman’s equilibrium equation is solved exactly to obtain out-of-plane mode shapes and critical loads. The Von-Karman’s compatibility equation is solved exactly in the post-buckling phase with the assumption that the deflected form after buckling is a combination of obtained buckling modes (single or multiple mode shapes). The principle of minimum potential energy is invoked to solve for the unknown coefficients in the assumed out-of-plane deflections and rotation functions.

The in-plane shear failure of transversely stiffened thin plates

This paper examines the response of stiffened plates with plain flat outstands when subjected to in-plane shear loading in the form of applied in-plane shear displacement. The buckling and post-buckling failure capabilities of thin plates subjected to in-plane shear, can, of course, be improved through the introduction of stiffening elements whose flexural and torsional rigidities can contribute significantly towards a more stabilised structural system. This paper details appropriate suitable finite element modelling strategies and procedures to enable the determination of the post-buckled failure response of the stiffened shear panels and to highlight the significant influence of the stiffeners. The modelling procedures are able to describe the complete loading history of the stiffened panel structures from the onset of initial buckling through the elastic post-buckling phase of behaviour involving the considerable interaction between plate and stiffener and then through initial material yielding and yield propagation to ultimate conditions and elasto-plastic unloading.

An exact finite strip for the calculation of initial post-buckling stiffness of shear-deformable composite laminated plates

The first order shear deformation theory (FSDT) is used to present the buckling and initial post-buckling characteristics of symmetrically cross-ply laminated plates. In the buckling phase the Von-Karman’s equilibrium equation is solved exactly for the FSDT to obtain out-of-plane mode shapes and critical loads. The current post-buckling study is effectively a single-mode analysis, which is attempted by utilizing the so-called semi-energy method. The Von-Karman’s compatibility equation is solved exactly in the post-buckling phase with the assumption that the deflected form immediately after buckling is the same as that obtained for buckling. The Principle of Minimum Potential Energy is invoked to solve for the unknown coefficients in the assumed out-of-plane deflections and rotation functions.

Failure Mechanics of Uniformly Compressed Thin-Walled Box-Section Struts

It is well known that the structural performance of thin-walled compression members is subject to the effects of local buckling and material yielding. Due to these effects, the compressive carrying capability of short strut members can be significantly reduced. This paper employs finite element simulation to examine the post-buckled response of thin-walled box-sections that covers complete loading history of the compression struts from the onset of elastic local buckling through the nonlinear elastic and elasto-plastic post-buckling phases of behaviour up to final collapse and unloading. A detailed account of the growth and redistribution of stresses on the surfaces is given in the paper. The results from finite element simulations are shown to compare well with the analytical method of analysis.

Pre-buckling and post-buckling failure at web-flange junction of pultruded GFRP beams

Separation at the web-flange junction is a common failure mode of pultruded glass fiber-reinforced (GFRP) beams subjected to bending. The causes of this separation appear to depend on the presence of lateral supports to prevent lateral buckling. To clarify the driving mechanisms, four-point bending experiments were carried out on pultruded GFRP girders. Lateral buckling was prevented by using lateral supports. Web-flange separation failure due to exceeding the shear strength was observed before any buckling was seen. Furthermore, nonlinear FEA was performed to identify the critical stress states of GFRP beams from the research literature, without any lateral supports in the post-buckling phase. In this case, based on numerical calculation, the critical stress states and their locations depended significantly on the shape of initial imperfections. The ultimate loads, with or without lateral supports, were predicted by a modified von Mises criterion applied to the stress states at the web-flange junction.

The influence of post-local buckling mechanics on the stress variations, axial stiffness and ultimate failure strength of uniformly compressed thin-walled i-section struts

It is well known that thin-walled compression members are subject to the effects of local buckling and that due to these local effects the compressive carrying capability of short strut members can be significantly reduced. Finite element simulation is employed in this paper to examine the post-buckled response of thin-walled sections giving due consideration to the influence of geometric imperfections and to elasto-plastic material behaviour. The findings from this work highlight the complete loading history of the compression struts from the onset of elastic local buckling through the nonlinear elastic and elasto-plastic post-buckling phases of behaviour to final collapse and unloading. A detailed account of the growth and redistribution of stresses as well as the influence of yielding and yield propagation throughout loading is given in the paper. The results from the finite element simulations are shown to compare well with independent simulations using the finite strip method of analysis.

Role of atomic scale interfaces in the compressive behavior of carbon nanotubes in composites

Carbon nanotubes (CNT) are potentially promising fibers for ultra high strength composites. In order to fully harness the outstanding mechanical properties of carbon nanotubes as fiber reinforcements, it is essential to understand the nature of load transfer between fiber and matrix under various types of loading conditions that include tension, compression, torsion and a combination thereof. In this paper, we study the compressive behavior (buckling and post-buckling) of carbon nanotubes in the neat form, when they are embedded in polyethylene matrix and with interface chemical modifications using molecular dynamics simulations based on Tersoff–Brenner potential. It is observed that the critical load for buckling increases only very marginally for nanotubes embedded in polythene matrix (with non-bonded interface) compared to neat CNTs. When CNTs are chemically bonded to the matrix, the compressive behavior occurs in two phases; pre- and post-buckling phases. First, the critical stress for buckling is found to reduce because the change in chemical bonding induces deviation from perfect cylindrical structure. In the post-buckling phase, however, the nanotubes behave similar to short fibers and deform by crushing. The results are compared with continuum solutions, wherever applicable. It is shown that the continuum solutions should be applied carefully whenever the effect of nanoscale interfaces becomes a factor.

Free in-plane vibrations of highly buckled beams carrying a lumped mass

Free undamped in-plane vibrations of shear undeformable beams around their highly buckled configurations are investigated neglecting rotary inertia effects. The beams are inertially nonuniform since a lumped mass is rigidly clamped along the span. Two mechanical models are considered depending on the boundary conditions in the post-buckling phases. First, the beam is considered inextensible because it is hinged at one end and is acted upon by an axial compressive force on the other end, a roller support, both in the buckling and post-buckling phases. In the second model, the beam is extensible in the post-buckling phase because the roller support boundary is changed into a fixed hinged end. Free undamped vibrations are governed, in the first case, by a homogeneous integral-partial-differential equation and, in the second case, by two coupled partial-differential equations with variable coefficients. The solutions of the associated eigenvalue problems are found employing two approaches: a semi-analytical method based on Galerkin discretization and a finite element method. A close agreement in the outcomes is found. The leading differences relating to the natural frequencies and linear normal modes of the two pre-stressed curved beam models are discussed; in particular, the occurrence of the veering phenomenon and the crossovers are outlined.

On the linear normal modes of planar pre-stressed curved beams

The natural frequencies and mode shapes of planar shear undeformable beams around their curved pre-stressed post-buckling configurations are investigated neglecting rotary inertia effects. Two mechanical models are considered depending on the assumed boundary conditions in the buckling and post-buckling phases. With the first model, the beam is considered inextensible because it is hinged at one end and is acted upon by an axial compressive force on the other end, a sliding hinge. In the second case, the beam is assumed inextensible in the pre-stressed phase (same boundary conditions as above), whereas it is extensible in the subsequent free linear dynamic phase because the sliding hinged boundary is changed into a stationary hinged end. Linear vibrations are governed by partial-differential equations with non-constant coefficients and the solutions for the frequencies and mode shapes are found employing two approximate approaches: a fully numerical method based on a finite element formulation and a semi-analytical method based on a weak formulation (Galerkin method). The main results are compared and a close agreement in the outcomes is found. The leading mechanical differences in the linear normal modes of the two pre-stressed curved beam models are discussed.

Postbuckling Behavior of Axially Restrained and Axially Loaded Steel Columns under Fire Conditions

This paper presents the results of a theoretical study of the postbuckling behavior of an axially loaded steel column at elevated temperatures whose thermal expansion is restrained by the adjacent structural members. This study investigates the effects on the column failure temperature by various factors, including the restraint stiffness during the column loading (expansion) and unloading (contraction) phases, column slenderness, and the initially applied column load ratio. The column failure temperature is defined as the temperature at which the load in the column during the postbuckling phase returns to its initial load. Results of this study indicate that the column failure temperature can be much higher than that of the column at first buckling and the higher the column slenderness, the larger the difference between temperatures of column failure and first buckling. Consequently, the column postbuckling behavior should be considered. Nevertheless, for columns with light restraints (restraint stiffness to column stiffness less than 5%) or high load ratios (load ratio higher than 0.5), the failure temperature of the column with realistic unloading stiffness of the restraint is only slightly affected by the postbuckling behavior.

The Initial Post-buckling Behavior of Face-Sheet Delaminations in Sandwich Composites

Should an interface crack between the layers of the composite face-sheet or between the core and the composite face-sheet of a sandwich beam/plate exists, local buckling and possible subsequent growth of this interface crack (delamination) may occur under compression. In this study, the buckling, and initial post-buckling behavior is studied through a perturbation procedure that is based on the nonlinear beam equations with transverse shear included. Closed-form solutions for the load and midpoint delamination deflection versus applied compressive strain during the initial postbuckling phase are derived. Illustrative results are presented for several sandwich construction configurations, in particular with regard to the effect of material system and transverse shear.

Buckling and Initial Postbuckling Behavior of Sandwich Beams Including Transverse Shear

An asymptotic solution is presented for the buckling and initial postbuckling behavior of sandwich beams. The effect of transverse shear is included, and the shear correction is calculated from energy equivalency. The asymptotic procedure is based on the nonlinear beam equation (with transverse shear included), and closed-form solutions are derived for the critical load and for the load and midpoint deflection and axial shortening vs applied compressive load during the initial postbuckling phase. Illustrative results are presented for a few typical sandwich construction configurations, in particular, with regard to the effect of face sheet and core material system.

Delamination growth during pre- and post-buckling phases of delaminated composite laminates

The delamination growth during the pre- or post-buckling phases in composites with a single delamination (one-dimensional model) is investigated. For that purpose, a general nonlinear analysis, using variational principle, with a new set of relations for the decomposition of the total energy release rate, is used for an arbitrary one-dimensional composite laminate structure. A computer code is developed and used for the parametric study of delamination growth during buckling. The effects of a wide range of parameters, like imperfection, layup, location of delamination, theory type and length-to-thickness ratio, were intensively studied. It is found that a dominant parameter, in terms of the level of energy release rate, is the length-to-thickness ratio.

The stability of delamination growth in compressively loaded composite plates

The stability of growth of internal delaminations in composite plates subjected to compressive loading is investigated. Due to the compressive loading, these structures can undergo buckling of the delaminated layer and subsequently growth of the delamination. The study does not impose any restrictive assumptions regarding the delamination thickness and plate length (as opposed to the usual thin film assumptions). The growth characteristics of the delamination under monotonic compressive loading are obtained on the basis of a combined delamination buckling/postbuckling and fracture mechanics model. The postbuckling solution is derived through a perturbation procedure, which is based on an asymptotic expansion of the load and deformation quantities in terms of the distortion parameter of the delaminated layer, the latter being considered a compressive elastica. The closed form solutions for the energy release rate at the delamination tip versus applied compressive strain during the initial postbuckling phase are used to define the combinations of delamination length and applied strain that lead to unstable growth. This would practically cause either contained “jump” growth or complete (catastrophic) growth of the delamination. Estimates for the lower and upper bounds of the jump distance (unstable growth) are provided. Moreover, a study of the influence of the mode dependence of interface toughness on the conditions of initiation and extent of delamination growth is performed.

The Initial Post-buckling and Growth Behavior of Internal Delaminations in Composite Plates

The initial post-buckling and growth behavior of delaminations in plates is studied by a perturbation procedure. In this work, no restrictive assumptions regarding the delamination thickness and plate length are made, i.e., the usual thin film assumptions are relaxed. The perturbation procedure is based on an asymptotic expansion of the load and deformation quantities in terms of the distortion parameter of the delaminated layer, the latter being considered a compressive elastica. Closed-form solutions for the load and midpoint delamination deflection versus applied compressive displacement during the initial post-buckling phase are derived. Moreover, closed-form expressions for the energy release rate and the mixity ratio (i.e., Mode II versus Mode I) at the delamination tip are produced. A higher Mode I component is found to be present during the initial post-buckling phase for delaminations of increasing ratio of delamination thickness over plate thickness, h/T (i.e., delaminations further away from the surface). Moreover, the-energy release rate corresponding to the same applied strain is larger for a higher h/T ratio. The reduced growth resistance of these configurations is verified by experimental results on unidirectional composite specimens with internal delaminations.

Postbuckling behaviour of foam-filled, thin-walled steel beams

Sandwich construction is increasingly used as wall and roof cladding for building structures. Typically, a cladding panel may consist of two plane or profiled metal faces with a foamed plastic core. The core may be polyurethane, polyisocyanurate, polystyrene or phenolic resin. When such a panel is subject to static loading due, for instance, to wind, snow or temperature gradient, one face is compressed and becomes liable to local buckling. If this face has a trapezoidal or similar profile the failure mode is similar to that for profiled steel sheeting, though the failure stress is enhanced by the presence of the core. The compressed face element first forms a series of buckling waves which increase in amplitude in the postbuckling phase. Failure takes place when one buckle in the region of maximum bending moment cripples. In light gauge steel applications, the conventional design treatment for this phenomenon utilises the concept of effective width. In order to investigate the extension of the effective width concept to plate elements supported by plastic foam material, a series of tests were undertaken on foam-filled steel beams. This paper describes these tests and their interpretation in terms of an enhanced effective width formula.