Postbuckling Behavior Research Articles

Functionally graded graphene origami-enabled auxetic metamaterial beams with tunable buckling and postbuckling resistance

Auxetic metamaterials have emerged as novel advanced materials with unique physical and mechanical properties that conventional materials do not possess. This paper examines the buckling and postbuckling properties of functionally graded (FG) graphene origami (GOri)-enabled auxetic metallic metamaterial (GOEAM) beams. The beam is comprised of multiple GOEAM layers with GOri content varying in layer-wise patterns to realize gradient-changing Poisson’s ratio and stiffness coefficient through the beam thickness. The material properties of each GOEAM layer are estimated by the genetic programming (GP)-assisted micromechanical models. The first-order shear deformation theory and von Kármán type nonlinearity are employed to derive the nonlinear governing equations that are numerically solved by the differential quadrature method (DQM). Numerical investigations are carried out with the main focus on the impacts of GOri content, distribution pattern, folding degree, and temperature on the buckling and postbuckling behaviors of FG metamaterial beams. The theoretical results show that GOri is capable of contributing to the formation of auxetic metal metamaterial, leading to the tunable bucking and postbuckling properties of FG beams, which sheds significant insights into the design of high-performance structures.

Post-buckling evolution of compressed thin films adhered to rigid substrates

Compressing a thin film adhered to a substrate can yield delaminated blisters, if the interface adhesion is relatively weak. In this work, through the combination of theoretical modeling and numerical simulations, we investigate the mechanics governing the buckle-delamination and post-buckling evolution of compressed thin films adhered to rigid substrate, where the interface adhesion is attributed by van der Waals interactions. The deformed configurations of thin films can be categorized into five deformation modes depending on the material, geometric and interfacial parameters and the in-plane compressive strain. Through the energy-variational approach, large-deformation beam models are constructed to describe the deformed configurations and energy variations of the arc-shape and self-contact buckling modes. The phase diagrams related to the equilibrium buckling state are constructed by energy analysis, based on which the evolution of the post-buckling behavior is analyzed. Our results may find potential applications in the controlled compressive-buckling assembly of micro-/nano-scale thin films.

Three-dimensional dynamics and synchronization of two coupled fluid-conveying pipes with intermediate springs

A system of two coupled oscillating structures may experience dynamical synchronization under coupling effects. Dynamical synchronization refers to the process in which two vibrating structures under coupling effects may display similar dynamics and vibrational behaviors even though their important parameters are not equivalent. In this study, the three-dimensional dynamics and synchronization behaviors of two coupled fluid-conveying pipes connected with linear springs are investigated. In the analytical model, the geometric nonlinearities due to axial elongation of the two pipes and the extension of linear springs are considered. The flow velocities of the internal fluid of the two pipes may be steady or pulsatile. In the case of the flow velocities of the two pipes being steady, the lowest several natural frequencies and the post-buckling behavior of the two-pipe system are obtained by changing the flow velocities in the two pipes. When one pipe conveys pulsatile fluid and the other conveys steady fluid, the dynamical bifurcation and synchronization behaviors of the two-pipe system are analyzed, with several typical synchronization patterns being explored by means of time traces, phase portraits, power spectral density (PSD) diagrams, Poincaré maps, etc. It is shown that making use of synchronization characteristics of the two-pipe system may provide a possible way to control the vibrations of the two-pipe system and to achieve some special engineering objectives.

Modelling the fluid–structure interactions of a capsule using a nonlinear thin shell model: Effect of wall thickness

We address the question of the modelling of the fluid–structure interactions for a microcapsule enclosed by a finite-thickness wall, and of the prediction of the buckling behaviour when it is subjected to large displacements and deformations. Specifically, we model the strong coupling between the solid (the wall dynamics) and fluid (the flow inside and outside the capsule) mechanics, for a wall material that can be strain-hardening or softening, while accounting for the bending resistance due to thickness. The fluid flow is assumed to be inertialess on the capsule scale, which allows the use of the boundary integral formulation for the fluid velocity. We discuss the different simplifications that are made when designing a fluid–shell interaction model for large deformations, and present a shear-membrane-bending (SMB) shell model that allows for a non-linear wall stretching law. The performance of the model, as compared to a simple membrane model where bending resistance is neglected, is illustrated on a generic example: we consider an initially ellipsoidal capsule, freely suspended in a plane hyperbolic flow, that is subjected to such stringent deformation, that its short axis becomes the long one. We show that the simple membrane model predicts reasonably well the overall shape of the capsule, but cannot capture the detailed post buckling behaviour, for which a robust shell model is necessary. The SMB shell model complies with dominant membrane effects, remains stable even under large deformation and avoids numerical locking. It allows predicting post-buckling behaviour, which depends on the material constitutive law.

Study on compressive bearing capacity and efficiency of composite stiffened panels with different cross-sections

The post-buckling behavior and load bearing efficiency of composite panels with three kinds of stiffeners subject to the axial compression load were investigated. The progressive damage analysis models considering the failure of fiber/matrix, debonding of adhesive interface and stiffness degradation were established. The buckling load, failure load and failure process for three types of stiffened panels were derived. The numerical results coincide with tests, and the relative errors are all less than 6%, which indicates the FEM model given in this paper is reliable and practical. Further, the load bearing efficiency of these three kinds of stiffened panels with the same weight was discussed. Results reveal that the bearing capacity and efficiency of I-shape stiffened panel is the highest, and its bearing efficiency reaches 154.5N/g. The results of the paper provide a reference for engineering application of composite stiffened structure design.

Influence of Intermediate Stiffeners on Axial Capacity of Thin-Walled Built-Up Open and Closed Channel Section Columns

This paper investigates the post-buckling behaviour and axial capacity of thin-walled steel stiffened single-channel sections (ISSCS) and back-to-back stiffened channel sections (BISCS). BISCS were connected using fasteners at a spacing of 200 mm and with an edge distance of 100 mm. Under axial compression, 10 new ISSCS and BISCS columns with fixed-ended conditions were tested. In the experimental tests, the back-to-back channel sections failed due to a combination of local and global buckling, whereas the single-channel sections generally failed as a result of local buckling. The behaviour of both ISSCS and BISCS shows a 20% increase on an average in axial capacity through adding stiffeners at the junction of the flange and the web, in addition to stiffeners in the web. A nonlinear finite element model (FEM) with material and geometric nonlinearities was then developed. The FE model was validated against the experimental results. A comprehensive parametric study comprising 64 face-to-face intermediate stiffened channel sections (FISCS) was then conducted to study the influence of stiffener length on its axial capacity. The axial capacity obtained from the tests and FEA shows that design in accordance with the Direct Strength Method (DSM) is accurate and conservative by only 4% on average.

On compacting pattern control of finite-size 2D soft periodic structures through combined loading

2D soft periodic structures can produce reversible compaction and relaxation behaviors with complex and diverse compacting patterns during loading and unloading. The (trans)formation and control of the recoverable and adjustable compacting pattern are essential in their wide range of application prospects. In addition to the traditional schemes, this paper proposes and discusses the possibility of using combined loading to control the compacting pattern of finite-size 2D soft periodic structures. Based on the detailed analysis and design of the loading schemes, the buckling and post-buckling behaviors of four kinds of 2D soft periodic structures with finite size under biaxial loading and combined normal and shear loading are investigated by a validated ABAQUS finite element analysis. The buckling model shapes of each structure under each loading combination are discussed and classified in detail to determine the Representative Compacting Patterns (RCPs) and the Critical Points of model transformation Controlled by Loading (CPCL) on the loading plane. Of course, some new modes have been found. Moreover, the CPCL between RCPs without conversion relationship is found to have a corresponding relationship with the variation of the difference between the first eigenvalue and the second eigenvalue of the structure on the loading plane. By using the rotation angle of the hole walls as an index, the compacting behavior and characteristics of each pattern in the process of post-buckling are analyzed quantitatively. The research ideas and results of this paper will provide a new design space for various application fields of finite-size 2D soft periodic structures.

The Post-Buckling Analysis of Porous Sandwich Cylindrical Shells with Shape Memory Alloy Wires Reinforced Layers Under Mechanical Loads

The main purpose of this paper is to analyze the nonlinear post-buckling behavior of three-layer porous sandwich cylindrical shells with shape memory alloy (SMA) wires reinforced outer layers under the mechanical loads. Considering the large deformations, the governing differential equations were extracted and then, using the Ritz energy method and considering the Airy function, the analytical expression for the critical stress of the system was extracted. Three types of porosity distributions were considered in the shell core and the mechanical properties of SMA-reinforced layers were determined using the law of mixtures. After validating the results, the effect of various parameters such as porosity distribution, volume fraction of SMA wires and geometric characteristics on nonlinear buckling loads and post-buckling behavior of these structures has been investigated. According to the results, it can be seen that the difference in critical buckling stress for three different porosity distribution types is significant, which reveals the need to determine the optimal porosity distribution for composite shells reinforced with SMA wires. The results show that the SMA wires increase the equivalent stiffness of the structure and their presence has a significant effect on improving the critical axial stress of the shells. According to the results, for the use of 1% volume fraction of SMA wires, the axial stress of sandwich shells can be increased up to about 51%. Therefore, the use of SMA wires is a good method to increase the load-bearing capacity of this type of structures.

On the post-buckling behavior of laterally constrained multilayers

Material layering occurs in natural, biological, geological and synthetic load-bearing structures. The bucking and post-buckling behavior of such structures is a subject of great interest. While many studies were devoted to the case of a thin film attached to a thick substrate, little is available on the behavior of multilayers. We explore this subject using a model system consisting of 2-D, linearly elastic hard/soft multilayer under uniform edge displacement. The lateral edges of the multilayer are constrained by rigid, frictionless walls. The number of stiff layers n and layer-to-interlayer thickness and elastic moduli ratios, h and E, are systematically varied. The deformation and stresses in the multilayer are obtained with the aid a FEA. Buckling initiates in the interior layers and spreads to the outer ones. The layers adjacent to the constraining walls tend to flatten as the load is increased. For relatively large h or small E, this causes large membrane stresses that may lead to a buckling mode transition. As many as ≈ 50 stiff layers are needed in order for the multilayer to approach a laminate configuration (n ≫ 1). The multilayer as a whole displays nonlinear elastic stress–strain response. By a proper choice of the system variables n, h and E, a variety of strain hardening effects as well as large energy absorption can be achieved. The results of this study provide useful insights into the effects of b.c., material choice and load level on the response of multilayers in general.

Buckling of Conventional and High-Strength Vanadium Steel Single- and Double-Angle Compression Members and Truss Subassemblies: Experimental and Computational Correlation Study

High-strength, low-alloy vanadium (HSLA-V) steel offers higher strength and toughness than conventional steel. The resulting lighter weight and more slender structural members are more susceptible to buckling in compression. This study establishes an understanding of buckling in this material and the ability to predict it analytically. A series of conventional ASTM A572 Grade 50 steel and HSLA-V (nominal Grade 80) steel angle compression members were tested at Lehigh University’s Advanced Technology for Large Structural Systems (ATLSS) laboratory. A general-purpose finite element (FE) software was used in this study to simulate the buckling and post-buckling behavior of the structural members. The objective of these simulations was to establish confidence in the ability to accurately predict buckling response. The influence of the following modeling parameters on the accuracy of the compression angle member simulation results was investigated: variation in material stress-strain relationship, residual stresses, and the shape and magnitude of geometric imperfections. For the truss subassembly simulations, the influence of the following parameters was also investigated: bracing and boundary element stiffness and design code assumptions of end conditions.

Geometric and Material Nonlinear Analyses of Trusses Subjected to Thermomechanical Loads

In this article, geometric and material nonlinear analyses of planar and space trusses subjected to thermo-mechanical loads are performed using the Total Lagrangian formulation. The material properties are Thermally InDependent (TID). By obtaining the equilibrium path of these structures, their buckling and post-buckling behaviors are predicted under thermal and mechanical loads. To consider material nonlinearity, a mixed isotropic–kinematic hardening criterion is developed in this formulation. The behavior of material is assumed to be bilinear. To solve nonlinear equations iteratively, the Generalized Displacement Control Method (GDCM) is used. The authors’ scheme is general and can be employed for different types of planar and spatial truss structures having complicated geometries and behaviors. Several popular benchmarks are considered to show the accuracy and correctness of the proposed formulation. Corresponding equilibrium paths and stress–strain curves are reported for each problem.

Post-buckling failure analysis of composite stiffened panels considering the mode III fracture

Fully utilizing the post-buckling capacity of stiffened composite panels is important to reduce the weight of aeronautical composite structures. However, owing to the complex post-buckling failure modes and mechanisms, there are no effective analysis methods. This paper explores the impacts of the mode III fracture energy on the damage behavior of the stiffener–skin interface of composite stiffened panels under post-buckling conditions. To characterize the bonding interface damage, an interface damage initiation criterion considering the effects of through-thickness compression on the shear failure was used and combined with the Reeder damage extension criterion, which considers the mode III fracture. The mode III fracture toughness characteristics of the damage evolution of the stiffener–skin interface were obtained using the edge crack torsion (ECT) test method. Moreover, the compressive deformation response, buckling load, buckling modes, failure load, and failure modes of the panel were obtained by a compression test of I-shaped stiffened panels. Based on the aforementioned experimental data and numerical model, the post-buckling behavior of the I-shaped stiffened composite panels with an open cross-section stiffener was studied. It was found that the mode III fracture energy caused by the longitudinal shear stress τ31 played a major role in the stiffener–skin interface damage extension of the stiffened panels. Therefore, the mode III fracture behavior should not be ignored in the post-buckling analysis. The numerical analysis method developed can accurately predict the damage initiation and evolution processes of the composite panel interface. The method can be effectively used for post-buckling analysis of aeronautical composite panels.

On the thermal buckling and postbuckling responses of temperature-dependent graphene platelets reinforced porous nanocomposite beams

In this work, thermal buckling and postbuckling behaviors of graphene platelet (GPL) reinforced porous nanocomposite beams are studied with inclusion of temperature-dependent material properties for the first time. The Halpin-Tsai micromechanics model and extended rule of mixture as well as open-cell metal foam model are applied to determine the effective properties of nanocomposites. A high order shear deformation theory with the aid of von Kármán nonlinearity is implemented to yield governing equations corresponding to thermal buckling and postbuckling problems. The numerically stable admissible functions constructed through Gram-Schmidt procedure are developed to describe end restraints of beams. An eigenvalue equation without geometric nonlinearity is solved to obtain critical buckling temperatures, while an iterative methodology is implemented to find the temperature-dependent thermal postbuckling paths of beams. An extensive numerical study is performed to check the influences of internal pores and GPL reinforcements on the thermal buckling and postbuckling responses of GPL-reinforced metal foam beams. It is found that the temperature-dependency of material properties plays important roles on thermal buckling/postbuckling behaviors of GPL-reinforced porous beams, and dispersing more GPLs on the upper and lower surfaces and fabricating more internal pores near central portion of beams can improve thermal instability-resistance capability of composite structures.

Nonlinear Buckling and Postbuckling Analysis of Tow-Steered Composite Cylinders with Cutouts

The buckling and postbuckling behavior of two composite tow-steered shells with cutouts of different sizes is assessed using nonlinear finite element analysis and compared to experimental measurements. The cylindrical shells are manufactured using an automated fiber placement system, where the shells’ fiber orientation angles vary continuously around the shell circumference from on the crown and keel to on the sides. One shell features thickness variations due to tow overlaps that result from application of all 24 tows during each pass of the fiber placement system. The second shell uses the system’s tow drop/add capability to achieve a more uniform wall thickness without overlaps. Unreinforced cutouts of two different sizes—the first smaller cutout representing a passenger door on a commercial aircraft and the second larger cutout a cargo door—were machined into each of the two cylinders resulting in a total of four test cases. These cylinders were tested in axial compression and buckled elastically in previous work and are now analyzed using nonlinear finite element models to compare bifurcation buckling loads as well as the load–displacement response in the prebuckling and postbuckling regimes. For all four shells analyzed, the prebuckling stiffness, buckling load, and deformation mode sequence throughout the loading–unloading cycle is accurately reproduced by the models. In particular, the shells first buckle locally around the cutouts in a stable (supercritical) manner with only a slight decrease in axial stiffness, which occurs due to the favorable load redistribution facilitated by tow steering. The shells then buckle globally in an unstable (subcritical) manner with diamond-shaped buckles forming to the left and right (circumferential direction) of the cutouts. The buckling load of all shells with cutouts is at least 82% of the buckling load of the pristine shells without cutouts. Overall, the ability to sustain local buckling phenomena, and the relatively small reductions in global buckling load compared to pristine shells without cutouts, demonstrates the great potential of using tow steering to mitigate the adverse effects of cutouts in axially compressed shell structures.

Refined Unified Formulation for Efficient Folding and Unfolding Analyses of Slender Thin-Walled Structures

We present a high-fidelity refined unified nonlinear finite element formulation for the efficient and robust analysis of slender thin-walled bodies during highly nonlinear deformation. Our formulation utilizes an independent discretization of the displacement field along the beam axis and over the cross section. By matching different refinements in different cross sections, it is able to apply higher-order beam theories in highly deformed regions, to capture complex buckling and postbuckling behavior, whilst retaining the computational efficiency offered by lower refinements elsewhere. The exemplar structure for this paper is the tape spring—a commonly proposed component of deployable structures with ability to combine self-deployment, via a release of stored strain energy, with locking into a relatively stiff geometric configuration with a curved cross section. To simulate localized folds due to a flattening of the cross section, the arc-length method with an automatic increment technique is employed.

Lateral Buckling of Subsea Pipelines Triggered by Sleeper with a Nonlinear Pipe–Soil Interaction Model

Buckle-initiation techniques, such as sleepers, are usually installed to trigger lateral buckling at pre-designated locations to release the axial compressive forces induced by thermal loading. Taking the nonlinear pipe–soil interaction model into account, a mathematical model is proposed to investigate the lateral buckling of subsea pipelines triggered by a sleeper. The numerical solution is validated by comparing the model with solutions in the literature, and the model shows good agreement. The discrepancy between them is analysed by presenting the effect of mobilisation distance during buckling. The influence of the breakout resistance, sleeper height, and sleeper friction coefficient on the buckled configuration, post-buckling behaviour, and minimum critical temperature difference is discussed parametrically. The results show that the deformation of the buckled pipeline shrinks, and both the minimum critical temperature difference and the maximum stress along the buckled pipeline enlarge when the nonlinear pipe–soil interaction model is incorporated. However, the influence of the nonlinear pipe–soil interaction reduces with increasing sleeper height.

Higher order theories for the buckling and post-buckling studies of shallow spherical shells made of functionally graded materials

For composite shallow spherical shells (including functionally graded materials) the buckling and post-buckling behaviour is strongly affected by the material characteristics of a shell wall constructions. Using a power series expansion for a parameter (z/h) different variants of 2-D shell formulations are discussed herein starting from the Love-Kirchhoff theory and finishing on the 9th order theory (stretching effects). The solutions are searched for with the use of the Bubnov-Galerkin method and the finite element analysis (the h- and p-approximations). Due to geometrical limits of shallowness shell theory the investigations can be carried out for structures having the moderate thickness – the h/R ratio should be less than 0.15.

Comparative Analysis of Nonlinear Deformation and Buckling of Thin Elastic Shells of Step-Variable Thickness

A comparative analysis of finite element models and methods for solving complex problems of geometrically nonlinear deformation, buckling and post-buckling behavior of thin shells of stepwise variable thickness is carried out. An approach based on the use of the moment scheme of finite elements is considered. The features of using the software suite LIRA and integrated software system SCAD for solving the assigned problems are also provided. Thin and medium thickness shells are considered. They can have different geometric features in thickness and be under the action of static thermomechanical loads. A technique for solving these problems with the help of an efficient refined approach is presented. The technique is based on the general methodological positions of the three-dimensional theory of thermoelasticity and the use of the finite element moment scheme. With this approach, the approximation through the shell thickness is carried out by a single universal spatial finite element. The element can be modified in different portions of the shell with a step-variable thickness. It can be located eccentrically relative to the middle surface of the casing and can change its dimensions in the direction of the shell thickness. Such a unified approach made it possible to create a unified designed finite element model of a shell of an inhomogeneous geometric structure under the combined action of a thermomechanical load. A comparative analysis of the application of three finite element approaches for problems of geometrically nonlinear deformation and buckling of shells of stepwise variable thickness is carried out.

Thermo-torsional buckling and postbuckling of thin FGM cylindrical shells with porosities and tangentially restrained edges

This paper aims to analyze the effects of porosities, tangential constraints of boundary edges and elevated temperature on the buckling and postbuckling behaviors of thin functionally graded cylindrical shells subjected to uniform torsion. The porosities are assumed to exist within functionally graded material (FGM) according to even and uneven distributions. The properties of constituents are assumed to be temperature dependent and effective properties of porous FGM are determined using a modified rule of mixture. Mathematical formulations are established within the framework of classical shell theory taking into account von Kármán–Donnell nonlinearity and elasticity of tangential edge constraints. Multi-term analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is used to determine buckling torsional load and load–deflection relation for postbuckling analysis. Numerical results show that porosities have remarkably deteriorative influences on the torsional buckling resistance and postbuckling load capacities of FGM cylindrical shells. This study also reveals that tangential edge constraints have pronouncedly beneficial and detrimental effects on the postbuckling strength of torsion-loaded porous FGM cylindrical shells at room and elevated temperatures, respectively.

Optimization cross-sectional dimensions for cold-formed steel lipped channel columns

Parametric optimization problem of cross-sectional sizes for cold-formed C-profiles subjected to central compression has been considered by the paper. Parametric optimization problem for cross-sectional sizes of cold-formed C-profiles has been formulated as follow: to define optimum cross-sectional sizes taking into account post-buckling behavior and structural requirements when stripe width and thickness as well as type of the cold-formed profile are constant and defined by the designer.
 Criterion of the profile load-bearing capacity maximization has been assumed as purpose function. The latter has been presented in the form of linear convolution of the resistance to central compression taking into account flexural, torsional and torsional-flexural buckling of thin-walled structural member determined according to the requirements EN 1993-1-3:2012 and EN 1993-1-5:2012. Searching for the optimum cross-sectional sizes has been performed taking into account a possibility of post-critical buckling behavior of the structural member based on the local buckling of the web and flanges and/or distortional buckling of the edge fold stiffeners.
 Formulated parametric optimization problem has been solved using software OptCAD. Update gradient method of the purpose function projection on the active constraints hyperplanes with simultaneous liquidations of the residuals in the constraints has been implemented by the software.
 As optimization results cold-formed C-profiles have been obtained. With the same stripe width optimum profiles have higher load-bearing capacity level taking into account buckling resistance under central compression comparing with the cold-formed C-profiles proposed by Ukrainian manufacturers. Besides, torsional-flexural buckling resistance of the cold-formed C-profile is determinative for all optimum cross-sectional decisions.