Nonlinear Postbuckling Analysis Research Articles

Structural buckling analysis of pre-twisted strips

This article focuses on the buckling response of columns formed by pre-twisting a flat, narrow and straight material strip of a rectangular cross section. The experimental analysis accomplished for birch plywood strips covers twisting up to 90 degrees. The corresponding computational analysis covers the twisting process (geometrically nonlinear analysis) and the subsequent compression (linear buckling analysis, including the residual stresses from the twisting process) and includes comparisons for 3D solid and 2D shell finite elements within orthotropic linear elasticity. A further finite element analysis provides findings up to twisting angles of 400 degrees and includes nonlinear post-buckling analyses for the compression subsequent to twisting, in addition to the linear buckling analyses. These sets of analyses reveal some new findings regarding this classical problem. Most importantly, there are four characteristic twisting angles which are associated to mode jumps in buckling, to a plateau in the curve relating the critical load and the twisting angle and, finally, to a loss of stability during the twisting process. The influence of the twisting-induced stresses on the buckling response is investigated as well.

In-plane nonlinear postbuckling and buckling analysis of Lee’s frame using absolute nodal coordinate formulation

Abstract In this study, four absolute nodal coordinate formulation (ANCF)-based approaches are utilized in order to predict the buckling load of Lee’s frame under concentrated load. The first approach employs the standard two-dimensional shear deformable ANCF beam element based on the general continuum mechanics (GCM). The second approach adopts the standard ANCF beam element modified by the locking alleviation technique known as the strain-split method. The third approach has the standard ANCF beam element with strain energy modified by the enhanced continuum mechanics formulation. The fourth approach utilizes the higher-order ANCF beam element based on the GCM. Two buckling load estimation methods are used, i.e., by tracing the nonlinear equilibrium path of the load–displacement space using the arc-length method and applying the energy criterion, which requires tracking eigenvalues through the dichotomy scheme. Lee’s frame with different boundary conditions including pinned–pinned, fixed-pinned, pinned-fixed, and fixed–fixed are studied. The complex nonlinear responses in the form of snap-through, snap-back, and looping phenomena during nonlinear postbuckling analysis are simulated. The critical buckling loads and buckling mode shapes obtained through the energy criterion-based buckling method are obtained. After the comparison, higher-order beam element is found to be more accurate, stable, and consistent among the studied approaches.

Nonlinear thermal post-buckling analysis of rectangular FG plates using CUF

In this study, a numerical analysis examines the thermal post-buckling response of rectangular functionally graded (FG) plates. Since the post-buckling is analyzed in a thermal environment, temperature distribution across the thickness is computed by an accurate numerical method. Here, for the geometrically nonlinear analysis because of assuming large displacement, FEM is utilized according to Carrera Unified Formulation (CUF), a precise formulation and higher-order deformation theory. The principle of virtual work is employed to achieve the nonlinear equilibrium equations. The governing equations are solved by operating the arc-length method. The influence of parameters such as geometric aspect ratio, length-to-thickness ratio, boundary conditions, type of temperature distribution, and volume fraction index on the FG plates’ temperature-deflection path has been investigated. In addition, a comparison among the present study method with another reference about thermal post-buckling and also linear buckling analysis (LBA) is performed. Nonlinear post-buckling analysis shows that the bifurcation point occurs only at fully clamped boundary conditions and low aspect ratios. It proved the results of LBA in the mentioned state are verifiable. The outcomes confirm the usefulness of utilizing CUF on thermal post-buckling since the temperature curves in the figures are at a lower level compared to the similar reference.

Post-buckling analysis of axially loaded two-dimensional quasicrystal cylindrical shells with initial geometric imperfection

In this paper, the nonlinear post-buckling analysis of imperfect two-dimensional quasicrystal cylindrical shells is performed. The nonlinear governing equations are established by considering initial geometric imperfection, higher-order shear deformation theory and phonon-phason fields coupling effect. The load-shortening curves and buckling modes are obtained by the Galerkin’s method. Numerical results are compared with the existing results and excellent agreements are obtained. Effects of geometrical parameters, initial imperfection and phonon-phason fields coupling effect on the critical load and imperfection sensitivity of two-dimensional quasicrystal cylindrical shells are discussed in detail.

기판에 사전변형률이 인가된 박막의 구조 해석

Stretchability enables the device to be patched to a curved surface or to be folded several times to maximize usability. Among many methods, the pre-strain method is advantageous in that the stretchability as much as the pre-strain applied to the substrate is guaranteed even without material improvement. When the pre-strain is restored to its original state, the thin film gets wrinkled or the substrate gets buckled. Wrinkles and buckling that appear in this way are affected by the physical properties and dimensions of the substrate, and it is necessary to analyze their effect. In this study, a theoretical approach was used and a nonlinear post-buckling analysis was performed using a finite element method. The analysis was divided into two steps: the pre-strain step and the recovery step. According to the analysis results, it was possible to predict and analyze the wrinkle and buckling behavior due to pre-strain according to the physical properties and dimensions of the substrate. The pre-strain analysis method can be applied to multi-layer structures with three or more layers and can be used as a method to analyze wrinkle suppression and wrinkle shape control in future studies.

Nonlinear postbuckling analysis of magneto-electro-thermo-elastic laminated microbeams based on modified couple stress theory

In this article, the nonlinear postbuckling behavior of the magneto-electro-thermo-elastic (METE) laminated microbeams is presented. According to the modified couple stress theory (MCST) and Reddy's three-order shear deformation theory (RTSDT), in conjunctions with the von Karman geometric nonlinearity, the nonlinear static model of METE laminated microbeam is established. The nonlinear governing equations and the corresponding boundary conditions are derived by using the principle of virtual work principle. Afterwards, using an analytical method, the nonlinear postbuckling behavior of METE laminated microbeam with simple supported and clamped boundary conditions at the both ends is described. Moreover, numerical examples are exhibited to reveal the effects of the material length scale parameter, slenderness ratio, temperature rise, magneto-electric potential on the critical buckling load and the nonlinear postbuckling response. Also the distribution law of magneto-electric potential through the thickness direction of the microbeam with various lay-up modes is discussed.

Size-dependent nonlinear post-buckling analysis of functionally graded porous Timoshenko microbeam with nonlocal integral models

Strain-driven (ɛD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) are applied to study the size-dependent nonlinear post-buckling behaviors of functionally graded porous Timoshenko microbeams. The differential governing equations and boundary conditions of motion are derived on the basis of the principle of minimum potential energy and expressed in nominal form. The integral relation between local and nonlocal stresses is transformed unitedly into equivalent differential form with constitutive constraints for ɛD- and σD-TPNIMs. Bending deflection and cross-sectional rotation are derived explicitly with Laplace transformation for linear buckling. Taking into account boundary conditions and constitutive constraints, a general eigenvalue problem is obtained to determine linear buckling mode shape (LBMS) and nominal buckling load (NBL). Local and nonlocal LBMS based Ritz–Galerkin methods and general differential quadrature method (GDQM) based Newton–Raphson’s method are utilized to obtain the numerical solutions for linear and nonlinear NBLs. Numerical results show that nonlocal LBMS based Ritz–Galerkin method and GDQM would lead to same prediction for linear NBLs as analytical method, and nonlocal LBMSs based Ritz–Galerkin and GDQM based Newton–Raphson’s method would lead to exact same prediction for nonlinear post-buckling loads. However, local LBMS based Ritz–Galerkin method is failed to provide accurate prediction for both linear NBLs and nonlinear post-buckling loads, through the difference between local and nonlocal LBMSs is not significant. The influence of nonlocal parameters, porous distribution patterns and boundary conditions on the linear buckling and nonlinear post-buckling behaviors are investigated numerically.

An analytical approach of nonlinear buckling behavior of torsionally loaded auxetic core toroidal shell segments with graphene reinforced polymer coatings

The buckling and nonlinear postbuckling analysis of toroidal shell segments with auxetic-core layer and Graphene-reinforced polymer coatings under torsional loads is reported in the present research. The functionally graded Graphene-reinforced polymer coatings are considered with three distribution laws, and the auxetic cores are designed in the honeycomb lattice forms. An auxetic homogenization technique is applied to establish the stiffness terms of the auxetic core. The combination of nonlinear von Karman Donnell shell theory and the Stain and McElman approximate is applied to formulate the nonlinear equilibrium equations of shells with the shallow longitudinal curvature considering the two-parameter foundation model. The deflection of shells is assumed to be a three-term form corresponding to the pre-buckling, linear and nonlinear postbuckling behaviors, and the Galerkin method is employed for three terms of defection. The torsion-deflection and torsion-twist angle postbuckling behaviors can be obtained in explicit forms. The numerical examinations validate the large effects of honeycomb lattice auxetic core, the functionally graded Graphene-reinforced polymer coatings, the parameters of shell’s geometric and foundation on the nonlinear buckling behaviors of shells.

Analysing wrinkle interaction behaviour with Z-fold crease pattern in thin-film planar membrane reflector

Limited stowage volume of launch vehicles demands compact packaging of large deployable membrane structures. Employment of different folding approaches led to the unavoidable fold-lines(creases). Moreover, wrinkle reduction is essential in such thin-film solar sail and inflatable antenna structures. This article analyzes the combined effect of multiple creases and wrinkles on deployed thin Kapton’s surface accuracy. Authors have introduced a novel approach for employing membrane length shortening because of multiple creases and fold characteristics. Real-world logarithmic behaviour of crease relaxation and fold-region properties help in computing the crease curvature. A rhombus-shaped shell model with Z-folding is introduced and explored for the effect of the number of creases and their orientation. Stress concentration and wrinkle contours for applied edge tensile loading state that the fold lines affect the stress transfer path. Non-linear post-buckling analyses for the Kapton® membrane express higher wrinkle evolution in the central region over the loading edge. Besides, a discrepancy in wrinkle amplitude and wavelength reveals the effect of fold-line orientation on wrinkling. Sighting actual conditions for antenna structure, creasing profile normal to the loading direction was found preferable for maintaining reflector surface-effectiveness. Additionally, double creased membranes achieve maximum surface correctness for any angular position, as the mirror effect fold shapes reduce wrinkles the most. Results are validated by comparing with experimental outcomes.

Nonlinear buckling of higher-order shear deformable stiffened FG-GRC laminated plates with nonlinear elastic foundation subjected to combined loads

The nonlinear postbuckling analysis of functionally graded graphene reinforced composite (FG-GRC) plates taking into account the nonlinear effect of elastic foundation subjected to external pressure and axial compression load in thermal environment is analytically examined in this paper. Assuming that the FG-GRC laminated plates are stiffened by FG-GRC laminated stiffener system resting on a three-parameter nonlinear elastic foundation. The higher-order shear deformation plate theory (HSDPT) with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Moreover, the smeared stiffener technique is successfully improved for the higher-order shear deformable anisotropic stiffener system by using a homogeneous model of the laminated beam. Galerkin's method is used to achieve the expressions of critical buckling loads and postbuckling load-deflection curves in explicit form. The numerical values display the influences of nonlinear elastic foundation, temperature, laminated stiffeners, material, and geometrical features on nonlinear responses of plates.

Robust sizing optimization of stiffened panels subject to geometric imperfections using fully nonlinear postbuckling analyses

This work presents a new approach for deterministic and robust thickness sizing optimization of a stiffened panel subject to geometric imperfections. Fully nonlinear pre- and postbuckling analyses determine time-dependent reaction forces using the backward Euler integration method. The global buckling resistance is optimized by maximizing the sum of reaction forces in the pre- and postbuckling regions without the necessity to directly evaluate the buckling point. The robust design optimizations consider geometric imperfections as parametrized random fields using an approach based on Fourier series. The first-order second-moment method (FOSM) determines the mean and the variance of the objective function in each optimization iteration. Monte Carlo simulations validate the FOSM approach for the present optimization formulation. The robust optimized designs show significantly increased buckling loads and robustness compared to the initial design and deterministically optimized designs even for the deterministic use case.

Buckling Knockdown Factors of Composite Cylinders under Both Compression and Internal Pressure

The internal pressure of a thin-walled cylindrical structure under axial compression may improve the buckling stability by relieving loads and reducing initial imperfections. In this study, the effect of internal pressure on the buckling knockdown factor is investigated for axially compressed thin-walled composite cylinders with different shell thickness ratios and slenderness ratios. Various shell thickness ratios and slenderness ratios are considered when the buckling knockdown factor is derived for the thin-walled composite cylinders under both axial compression and internal pressure. Nonlinear post-buckling analyses are conducted using the nonlinear finite element analysis program, ABAQUS. The single perturbation load approach is used to represent the geometric initial imperfection of thin-walled composite cylinders. For cases with the axial compressive force only, the buckling knockdown factor decreases as the shell thickness ratio increases or as the slenderness ratio increases. When the internal pressure is considered simultaneously with the axial compressive force, the buckling knockdown factor decreases as the slenderness ratio increases but increases as the shell thickness ratio increases. The buckling knockdown factors considering the internal pressure and axial compressions are higher by 2.67% to 38.98% compared with the knockdown factors considering the axial compressive force only. The results show the significant effect of the internal pressure, particularly for thinner composite cylinders, and that the buckling knockdown factors may be enhanced for all the shell thickness ratios and slenderness ratios considered in this study when the internal pressure is applied to the cylinder.

Post-buckling optimization of bending-induced variable stiffness composite cylinders considering worst geometric imperfections

Variable stiffness (VS) composite cylinders have attracted much more attention in the aerospace industry, due to their significantly improved mechanical properties. Most of the existing studies on the optimization of VS composite cylinders are based on linear buckling analysis, without considering the imperfection sensitivity and ignoring the potential of the non-linear post-buckling bearing of cylindrical shells. Therefore, the imperfection sensitivity analysis of VS composite cylinders under bending load is performed in the non-linear post-buckling regime. It is illustrated that the optimized result based on the linear buckling analysis is not robust, if imperfections are taken into account. Moreover, an optimization framework considering imperfection sensitivity is established, wherein the Worst Multiple Perturbation Load Approach (WMPLA) is used to calculate the lower-bound limit moment of composite cylinders. The critical moment of the optimized result with the worst imperfections has an improvement of 13.81%, and the knockdown factor (KDF) is improved by 27.27%. It is demonstrated to be necessary to consider the imperfection sensitivity based on the non-linear post-buckling analysis during the optimization of VS composite cylinders, and the considerable performance with improved anti-imperfection ability can be realized by tailoring the fiber orientation of VS cylinders.

Electro-thermo-mechanical post-buckling of piezoelectric functionally graded cylindrical shells

Unlike the widely used eigen-buckling theory, the nonlinear post-buckling analysis provides a more accurate stability model of piezoelectric cylindrical shells and predicates more reliable instability characteristics. In this paper, an accurate post-buckling analysis of piezoelectric functionally graded cylindrical shells under combined electro-thermo-mechanical loadings is performed. The nonlinear large deflection governing equations involving the self-induced electric potential are established based on the higher-order shear deformation theory. The post-buckling equilibrium paths with mode-jumping phenomenon for both symmetric and asymmetric deformation modes are obtained by the Galerkin's method enriched with new displacement functions. Results indicate that the symmetric mode usually occurs for a thick shell and the asymmetric mode only occurs for an ultrathin shell. Furthermore, it is demonstrated that the effect of power law index highly depends on the external electro-thermal loadings. Therefore, the loading capacity of the piezoelectric FG cylindrical shells can be improved by an appropriately selected power law index.

Post-buckling of bamboo reinforced composite plates

Composite materials are preferred as today’s material for aerospace structural applications largely because composites provide high specific strength and stiffness. To their advantages, natural fibers such as bamboo, kenaf, jute and flax have been researched significantly for having high effective strength and stiffness and environmental friendly advantages in being renewable, biodegradable and sustainable. While many structures used in aerospace applications are thin and as such susceptible to buckling problem, this paper presents the study on the buckling and post-buckling of bamboo reinforced composite plates (BRCPs), applying the finite element method software of ANSYS APDL. A compressive load was given to the BFRC and linear buckling analysis was conducted first. The determined critical load from this analysis was then used in the non-linear post-buckling analysis to give the post-buckling path of the BRFC. In both linear and non-linear analysis, the effect of BRFC thickness was considered while both angle-ply and cross-ply orientations were used. The results of deflection of a composite plate showed an excellent agreement with past results. The study shows that as the length to thickness ratio (l/t) was increased, critical loads increased as well. Further, the critical loads for the angle-ply BFRCs were higher than the cross-ply counterparts of the same thickness. Similar pattern of behaviour can be seen for the post-buckling paths of the BFRCs.

On the NURBS-based isogeometric analysis for couple stress-based nonlinear instability of PFGM microplates

A porosity-dependent nonlinear postbuckling analysis for microplates prepared from a porous functionally graded material (PFGM) is performed based on the modified couple stress theory of elasticity (MCSTE). To accomplish this purpose, the modified couple stress-based nonlinear differential equations are derived by third-order shear deformation plate theory (TSDPT). To extract PFGM microplate effective mechanical properties, a power-law function is utilized which is capable of incorporating porosity dependency and material gradient, simultaneously. Thereafter, the non-uniform rational B-spline (NURBS)-based isogeometric analytical method is employed as an effective discretization method which could satisfy C−1 continuity condition. It is demonstrated that the gap between equilibrium paths relevant to various couple stress length scales reduces by going deeper in postbuckling regime. Also, one can see that for certain porosity and material property gradient index values, by boundary condition transformation from simply supported to clamped, the contribution of the effect of couple stress size in PFGM microplate postbuckling response becomes more significant, especially for a higher maximum deflection values.

In-plane nonlinear postbuckling analysis of circular arches using absolute nodal coordinate formulation with arc-length method

In this study, Absolute Nodal Coordinate Formulation (ANCF) in conjunction with Crisfield’s arc-length method is utilized in order to predict the nonlinear postbuckling behaviour of circular arches. The whole primary equilibrium path in load-displacement space of circular arches under central concentrated load is obtained. Three ANCF based approaches, i.e., the conventional two-dimensional fully parameterized shear deformable ANCF beam element based on the General Continuum Mechanics (GCM) approach, the same element modified by the Strain Split Method (SSM) approach and the ANCF planar Higher Order Beam Element (HOBE) with GCM approach are used. Circular arches with various geometric configurations and boundary conditions such as clamped-clamped, hinged-hinged, clamped-hinged and three-hinged arches are studied which exhibit nonlinear response in the form of snap-through, snap-back and looping phenomenon. The obtained results are compared with the analytical solutions, experimental result (where available in the literature) and numerical approximations (by using the commercially available FEM package). In this paper, the recently proposed ANCF based approaches are successfully implemented which validate and verify the utility of ANCF in nonlinear postbuckling analysis. The characteristics of the three approaches with regard to the adoptability of arc-length method are compared and discussed.

Wrinkling prediction of laminated composite panels under in-plane shear deformation

In this article, the wrinkling behavior of laminated orthotropic composite panels due to in-plane shear deformation is investigated using nonlinear finite element analysis. Our model of the different laminated composite panels was set up for in-plane shear deformation at one end, with the opposing end fully restrained. Initially, A static analysis was carried out to establish the shear induced stress distribution patterns in the elastic panels assuming no wrinkling. This was then followed by performing eigenvalue analysis to determine the potential buckling modes of the elastic panels under prescribed boundary conditions. Finally, a nonlinear post-buckling analysis was performed to show the evolution of shear-induced wrinkling. We studied the effect of lamina orientations upon the development of shear-induced compressive stresses in the transverse direction. Our model, which was verified with existing experiments, shows the resulting wrinkling patterns, their amplitude, and their wave lengths under the prescribed loads and boundary conditions. Our model predictions further reveal a sudden shift in wrinkling patterns near the fixed and free boundaries of the panels at a given critical load.

Influence of dynamic temperature variation and inplane varying loads over post-buckling and free vibration analysis of sandwich composite beam

In this study, nonlinear post-buckling and free vibration analysis of shape memory polymer sandwich composite (SMPSC) under dynamic temperature variation is performed. For the analysis, simplified Co continuity based on higher-order shear deformation theory (HSDT) has been adopted to perform finite element analysis (FEA). Numerical solutions are obtained by iterative Newton–Raphson method considering Von-Karman nonlinear kinematics. Material properties of SMPSC, with Shape Memory Polymer (SMP) as matrix and carbon fiber as reinforcements, have been calculated by theory of volume averaging. The effect of dynamic temperature variation and axial variable inplane loadings (AVIL) on SMPC and SMPSC has been evaluated for various parameters such as beam thickness ratio, layer variation, boundary conditions (BCs), position of core, thickness of core in sandwich structures for the first time. Apart from these, this study also clearly reveals the difference in magnitude of buckling and free vibration parameters between the shape memory polymer composites (SMPC) and SMPSC, before and after glass transition region of material.

Multiscale modelling of soft lattice metamaterials: Micromechanical nonlinear buckling analysis, experimental verification, and macroscale constitutive behaviour

Soft lattice structures and beam-metamaterials made of hyperelastic, rubbery materials undergo large elastic deformations and exhibit structural instabilities in the form of micro-buckling of struts under both compression and tension. In this work, the large-deformation nonlinear elastic behaviour of beam-lattice metamaterials is investigated by micromechanical nonlinear buckling analysis. The micromechanical 3D beam finite element model uses a primary linear buckling analysis to incorporate the effect of geometric imperfections into a subsequent nonlinear post-buckling analysis. The micromechanical computational model is validated against tensile and compressive experiments on a 3D-printed sample lattice structure manufactured via multi-material jetting. For the development and calibration of macroscale continuum constitutive models for nonlinear elastic deformation of soft lattice structures at finite strains, virtual characterization tests are conducted to quantify the effective nonlinear response of representative unit cells under periodic boundary conditions. These standard tests, commonly used for hyperelastic material characterization, include uniaxial, biaxial, planar and volumetric tension and compression, as well as simple shear. It is observed that besides the well-known stretch- and bending-dominated behaviour of cellular structures, some lattice types are dominated by buckling and post-buckling response. For multiscale simulation based on nonlinear homogenization, the uniaxial standard test results are used to derive parametric hyperelastic constitutive relations for the effective constitutive behaviour of representative unit cells in terms of lattice aspect ratio. Finally, a comparative study for compressive deformation of a sample sandwich lattice structure simulated by both full-scale beam and continuum finite element models shows the feasibility and computational efficiency of the effective continuum model.