Pre-buckling Stresses Research Articles

First-order and linear buckling analysis of thin-walled cylinders using new GBT deformation modes

Conventional Generalised Beam Theory represents the displacement field of a thin-walled member as a linear combination of deformation modes classified in three main families: Vlasov, Shear and Transverse Extension modes. Recently the author proposed a unified family of deformation modes for the linear buckling analysis of thin-walled cylinders. The main advantage of a single family of deformation modes is the reduced number of degrees of freedom. The proposed formulation was limited to the linear buckling analysis of members under constant pre-buckling stresses along the members’ longitudinal axis (i.e., uniform compression, bending and torsion). In this paper, the novel formulation is extended to perform for arbitrary loading: (i) first-order (pre-buckling) analysis, and (ii) linear buckling analysis using variable pre-buckling stresses.

Postbuckled vibration behaviour of skew sandwich plates with metal foam core under arbitrary edge compressive loads using isogeometric approach

In the present work, nonuniform rational B-spline (NURBS) based isogeometric formulation in conjunction with refined higher-order theory is used to investigate the linear buckling, post-buckling, and post-buckled vibration behaviour of initially imperfect skew sandwich plates. The face sheets are functionally graded carbon nanotube-reinforced composite (FGCNTRC), and the core layer is made up of aluminium foam. The effects of three types of CNT distributions (uniform, FGX and FGO) in the face sheets, two types (uniform, symmetric) of porosity distribution functions for the core layer and five types of in-plane compressive loads are examined in the present investigation. The pre-buckling stresses are calculated using static analysis to evaluate accurate, critical loads. The post-buckling paths are traced using the modified Riks method. The accuracy of the present results is ascertained by comparing the results for critical loads and post-buckling paths with those available in the literature. Subsequently, the influence of CNT distribution functions, porosity functions, compressive loads, skew angle and the side-to-thickness ratio is studied on the nonlinear stability and free vibration behaviour of the post-buckled skew sandwich plates. The obtained results highlight that the buckling strength can be improved by increasing the skew angle, increasing the concentration of CNTs towards the surface of the face sheets and by using a metal foam core with nonuniform porosity distribution.

Pre-buckling vibration and buckling analyses of composite skew plate: An analytical investigation

In this work, the analytical investigation for pre-buckling vibration and buckling analyses of a composite skew plate subjected to parabolically and linearly varying in-plane edge load are presented. When the composite skew plate is subjected to parabolic (non-uniform) in-plane edge loading, the pre-buckling stresses within the composite skew plate are not known a priori. To estimate the pre-buckling stresses within the composite skew plate for which the in-plane elasticity problem is solved by minimizing the membrane strain energy using the Ritz method. It is observed that the rate of diffusion of applied parabolic in-plane edge load within the skew plate to a state of uniform in-plane stress is faster in the case of isotropic material than composite material. Using estimated pre-buckling stresses, the total energy functional is derived from the total strain energy, potential energy, and kinetic energy. The total energy functional is reduced into sets of an ordinary differential equation and algebraic equation, respectively for pre-buckling vibration and buckling problems using the Ritz method in conjunction with BCOPs. The associated linear eigenvalue problems are solved to compute the pre-buckling vibration frequency and buckling load of the stressed skew plate. The outcome of the study may provide crucial inputs in the design of skewed bridge decks, ship structures, and aircraft wing design.

Influence of porosity and nonuniform in-plane edge loads on vibration and buckling response of power law and sigmoid function based FG sandwich plates with geometrical discontinuities

Porosity defects can emerge during the manufacturing process of Functionally Graded Sandwich Plates (FGSPs). In practice, FGSPs have geometric discontinuities/cutouts and are exposed to Nonuniform In-plane Edge Loads (NIELs). The presence of porosity, cutout and NIELs causes a significant reduction in the stiffness of the plate affecting its vibration and buckling behavior. Therefore, in this paper vibration and buckling results obtained using the Finite Element (FE) method hitherto not reported in the literature are presented for porous FGSPs with cutouts and subjected to NIELs. Four types of porosity distribution models are explored, and the porosity imperfections are modeled as the criteria of stiffness reduction. Porosity-dependent material properties of FGSPs are evaluated using modified power law and sigmoid function. The current study incorporates two distinct kinds of sandwich configuration in such a way that there is no material mismatch along the thickness direction. The application of different cases of NIELs on the plate with cutouts leads to the development of nonuniform stresses. Hence a novel dynamic approach has been proposed to evaluate buckling loads by implementing two sets of boundary conditions. The first set of boundary conditions calculates pre-buckling stress, while second set calculates critical buckling loads. The generated results from the current FE formulation are compared with existing results in the literature to ensure accuracy. Finally, the effect of porosity distribution, NIELs, cutout ratio and its positions, support conditions, volume fraction exponents and geometric parameters on the vibration and buckling response are studied and discussed to arrive at appropriate conclusion.

Post-buckling and vibration analysis of randomly distributed CNT reinforced fibre composite plates under localised heating

Present study aims to investigate the non-linear stability and vibration characteristics of randomly distributed carbon nanotube reinforced fiber composites (RD-CNTRFC) under localized heating. The effective material properties of RD-CNT reinforced polymer is determined using Eshelbhy-Mori-Tanaka (EMT) approach and Chamis method is used to estimate the material properties of the RD-CNTRFC plates considering properties to be temperature-dependent. The strain-displacement fields of plates are developed using Reddy’s higher order shear deformation theory and von-Kármán non-linearities. The in-plane pre-buckling stresses within the plates due to localized heating are estimated using Airy’s stress function by satisfying compatibility equation. Galerkin’s approach is adopted to simplify the partial differential equations (PDEs) into non-linear algebraic equations for stability analysis and ordinary differential equations (ODEs) for vibration analysis. The critical temperature of the plates can be obtained using standard eigen value approach and thermal non-linear equilibrium path of plate can be traced using iterative eigen value approach. The present approach is initially verified with the results of published literature. Moreover various parametric studies like agglomeration effect of CNTs, mass fraction of CNT, aspect ratio, width-to-thickness ratio are also carried-out to elucidate their influence on thermal stability and vibration characteristics of RD-CNTRFC plates.

Postbuckling and postbuckled vibration behaviour of imperfect trapezoidal sandwich plates with FG-CNTRC face sheets under nonuniform loadings

The present work investigates the postbuckling, and postbuckled vibration behaviour of initially imperfect trapezoidal sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) face sheets and FG porous metal foam core under the influence of non-uniform edge compression. The plate's kinematic assumptions are based on a refined higher order theory and the strain-displacement relations include von Karman assumptions for geometrical nonlinearity. The weak form of governing equations derived using Hamilton's principle is transformed into a discretized form of algebraic equations using the element free Galerkin (EFG) method in conjunction with moving kriging (MK) interpolation functions. The pre-buckling stresses are determined using static analysis to evaluate accurate critical buckling loads. Modified Riks technique is used to trace nonlinear equilibrium paths. Parametric studies include the effect of CNT distribution in face sheets, porosity distribution in the core layer and edge loading conditions on the nonlinear stability and vibration behaviour of sandwich plates. New results on trapezoidal sandwich plates with initial imperfections, hitherto not found in the literature, are presented for the first time, which can be used as benchmark solutions for further research.

A semi-analytical framework for non-linear dynamic response of multi-phase laminated composite plate under the transverse patch and in-plane pulse localized loadings

The non-linear dynamic response of a randomly distributed carbon nanotube fiber-reinforced laminated composite (CNTFRLC) plate subjected to various in-plane pulse localized loadings and transverse patch loading is presented semi-analytically. Implementing higher-order shear deformation theory (HSDT) along with von-Kármán non-linearity, the non-linear kinematics of the CNTFRLC plate are modeled. The nonuniform pre-buckling stresses ( and ) were developed implementing Airy’s stress equilibrium approach, and subsequently utilized while deriving the non-linear partial differential equations (PDEs) of the plate through Hamilton’s principle and solved via. Galerkin’s method. Finally, the non-linear dynamic responses of the CNTFRLC plate are traced using Newmark’s method.

Global buckling and wrinkling of variable angle tow composite sandwich plates by a modified extended high-order sandwich plate theory

The advanced Automated Fibre Placement (AFP) manufacturing technologies make the synthesis of Variable Angle Tow (VAT) composites, which enable the design of lightweight sandwich structures to possess variable stiffness facesheets. However, both global and local instability phenomena of VAT sandwich plates under in-plane compressive loads are rarely explored till now. The objective of this article is to fill this gap by developing a Rayleigh–Ritz procedure based on a modified extended high-order sandwich plate theory (EHSAPT). The original two-dimensional EHSAPT proposed by Phan et al. (2012) is extended to a three-dimensional case with a minor modification that the first-order shear deformation theory instead of the classical Kirchhoff–Love hypothesis is adopted for each facesheet, which brings several merits such as the conciseness in derivation process and the easiness in program implementation. The Rayleigh–Ritz approach combined with the principle of minimum potential energy is employed to derive the eigenvalue equation that governs the instability problem of VAT sandwich plates under in-plane compressive loads. Both global buckling and wrinkling patterns of VAT sandwich plates can be captured under this proposed analytical model framework. Before instability analysis, the nonuniform prebuckling stresses over the entire sandwich plate are determined under the assumption of membrane prebuckling state. The usage of Lagrange multiplier method in the prebuckling regime removes the restrictions inherent in conventional Rayleigh–Ritz formulation and thus provides a general way to model in-plane boundary conditions. The accuracy and effectiveness of the developed Rayleigh–Ritz procedure are validated by comparing with previously published results and FE solutions given by ABAQUS. Effects of core thickness, core orthotropy, and fibre orientation angle of the facesheets on the instability behaviour of VAT sandwich plates are investigated. Finally, the mechanism of steering the fibre trajectory over the facesheets to improve the buckling resistance for the sandwich plate is studied and discussed.

Instability characteristics of variable stiffness laminated composite curved panels under non-uniform periodic excitation

The instability characteristics of variable stiffness laminated composite (VSLC) shell panels subjected to non-uniform in-plane periodic excitations are explored in the present article. The non-constant orientation of fiber angle in the VSLC panel induces non-uniform stress distributions within the shell panel under uniform as well as non-uniform in-plane loadings. Initially, the pre-buckling stress distributions within the shell panels are evaluated by minimizing the membrane strain energy. Subsequently, a displacement based Ritz method is employed to obtain the matrix representation of the governing equations for static and dynamic instability problems. Bolotin’s method is used to determine the dynamic instability regions. The numerical results depicted in the article show the influence of fiber angle orientations, non-uniform distribution of in-plane loadings and damping on the dynamic instability characteristics of VSLC shell panels.

A semi‐analytical framework for time history response of a carbon nanotube‐reinforced polymer damped composite plate: Influence of instability regions

AbstractThe influence of dynamic instability regions (DIRs) (stable and unstable) on the time history response of a randomly distributed carbon nanotube‐reinforced polymer composite (RD‐CNTRPC) plate under partial edge (nonuniform) in‐plane loading is presented semi‐analytically in this study. Kinematics of the RD‐CNTRPC plate is expressed based on higher‐order shear deformation theory (HSDT) and includes von‐Kármán nonlinearity. The effective isotropic mechanical properties of the RD‐CNTRPC plate (CNT mixed with epoxy) are executed using the Eshelby‐Mori‐Tanaka technique. The prebuckling stresses generated within the plate due to the partial‐edge in‐plane loading is computed using Airy's stress approach. The computed pre‐buckling stresses are used while applying Hamilton's principle to develop the governing nonlinear partial differential equation (PDEs). The PDEs are solved via. Galerkin's method to convert into the nonlinear ordinary differential equations (ODEs). These ODEs are solved using Newmark's method to trace the time history responses of the RD‐CNTRPC plate. The boundaries of DIRs are estimated using Bolotin's method. The effect of frequencies corresponding to upper, lower, and unstable regions of DIRs on the time history response and phase plot of the RD‐CNTRPC plate with and without damping are examined in detail.

Effect of elliptical cutouts on buckling and vibration characteristics of stiffened composite panels under non-uniform edge loads

This study is motivated by the need to understand the effect of elliptical cutouts on the buckling and vibration characteristics of stiffened composite panels under the influence of various types of non-uniform in-plane edge loads. The study is carried out by using a Finite Element (FE) formulation wherein the plate, and the stiffener are discretized by employing a 9-noded heterosis plate element and a 3-noded isoparametric beam element, respectively. The contributions of shear deformation and rotary inertia are considered in the FE formulation. Since the stress distribution within the panel is highly non-uniform for a given loading and geometric imperfection, a dynamic approach has been adopted to solve the buckling problem wherein two sets of boundary conditions are used to calculate the pre-buckling stresses and the buckling load of the stiffened panel. A new structured mesh pattern is proposed to house the stiffeners attached to the centrally located elliptical cutout panel in the form of a grid so that the stiffener can be attached anywhere and in any direction as per the requirements. The influences of various parameters, such as ply orientation, cutout size, cutout orientation, boundary conditions and depth to width ratio of the stiffener, are examined in detail. From the analysis, it is noticed that the depth of the stiffener has a significant effect on the buckling and vibration characteristics of the panel.

Influence of local stiffeners and cutout shapes on the vibration and stability characteristics of quasi-isotropic laminates under hygro-thermo-mechanical loadings

Abstract s The perforated stiffened panel is generally found as a sub-component of sophisticated structures. The fundamental purpose of this panel is to withstand against buckling under complicated loading and environmental conditions. Hence, an accurate knowledge of critical buckling behaviour of stiffened panels is very much essential for a reliable and lightweight structural design. In this paper, the focus is on quasi-laminated panels with different cutout shapes of various sizes and their responses to hygrothermal environments under nonlinearly varying edge loads and is compared with the locally stiffened panels. Towards this, the modelling of the panel and stiffener is done by adopting nine-noded heterosis plate elements and three noded beam elements respectively. The stiffener formulation is suitably modified in order to take the torsional effect also into consideration along with the effect of shear deformation. Initially, the plate and the stiffener elements are treated separately, and then the displacement compatibility is maintained between them by using the transformation matrix. For a given loading and geometric discontinuity, the stress distribution within the perforated panel is highly non-uniform in nature and hence a dynamic approach has been used to calculate buckling loads by adopting two sets of boundary conditions, one set for pre-buckling stress analysis and the second set for buckling analysis. Four different quasi-isotropic stacking sequences are deliberated in this work by varying different ply-orientation in each scheme. The study also addresses the effect of various parameters such as nonlinear loads, hygro-thermal loads, cutout size and shapes, position of cutout, stiffener parameters, stacking sequences, thickness of plate and boundary conditions.

Effects of different interlaminar hybridization and cutout sizes on the vibration and buckling characteristics of fiber metal composite laminates under partial edge loads

Fiber reinforced polymer composites are quickly gaining market share in structural applications, but further growth is limited by their lack of toughness. Fiber hybridization is a promising strategy to toughen composite materials. By combining two or more fiber types, these hybrid composites offer a better balance in mechanical properties than non-hybrid composites. This study presents the effect of different interlaminar hybridization and cutout sizes on the vibration and buckling characteristics of hybrid fiber metal laminates (HFMLs) subjected to partial edge loads by developing finite element formulation. Since the stress distribution within the plate element is highly non-uniform in nature, the dynamic approach has been used to solve the buckling problems wherein two sets of boundary conditions are used, one for pre-buckling stresses and other one for buckling load calculations. A 9-noded heterosis plate element has been used to discretize the plate by taking into account the effect of shear deformation and rotary inertia. The present study consists of aluminum metal face sheets bound with four layered symmetric hybrid cross-ply laminate schemes. In this scheme, six different hybrid combinations have been considered. The performance of each hybrid combination is investigated under various loading combinations. Further, the effect of different parameters such as boundary condition, cutout ratio, width and position of localized edge loads, plate aspect ratio, and hybrid configurations are included in this work. It is revealed from the study that the hybrid configurations under different cutouts significantly affect the vibration and buckling characteristics under different localized edge loads.

Buckling analysis of symmetrically laminated composite plates including the effect of variable pre-stress field using the Ritz method

A buckling solution capable of accounting for prebuckling stress field heterogeneities is proposed. Oftentimes the same structural elements that restrict the out-of-plane displacement of the edges of a plate may also constrain their in-plane movements and cause heterogeneous prebuckling stress fields. The effect of these constraints and the consequent prebuckling stress field heterogeneity is vastly overlooked in the context of approximate analytical buckling solutions. The novelty of this work lies in basing the eigenvalue buckling analysis on a calculated prebuckling in-plane stress field. The in-plane stress field is determined by conducting a displacement-based linear elasticity analysis. In both elasticity and eigenvalue buckling analyses, the deformations of the anisotropic laminate were approximated using a first-order shear deformation theory. The equilibrium and instability equations were derived using the principle of stationary total potential energy and by employing the Ritz method. The derived formulation along with arbitrarily selected sets of boundary conditions was incorporated into a Matlab program and used for demonstrating the effect of in-plane constraints on prebuckling and buckling behavior of plates. Through a parametric study, the effect of the geometric configuration of plates on their buckling strength for selected symmetrical laminates was investigated.

Dynamic instability of trapezoidal composite plates under non-uniform compression using moving kriging based meshfree method

Meshfree formulation based on the element free Galerkin method (EFGM) in conjunction with moving kriging (MK) shape function is employed to investigate buckling and parametric instability behaviour of shear deformable isotropic and laminated composite trapezoidal plates subjected to different types of non-uniform periodic edge compressive loads. Hamilton’s principle is used to derive the governing equations, which are transformed into the discretized form using the EFG method. The actual pre-buckling stresses are determined from static analysis to evaluate the accurate buckling loads of isotropic and laminated composite trapezoidal plates under non-uniform edge compression. The ordinary differential equations of Mathieu–Hill type are solved using Bolotin’s method to determine regions of dynamic instability. The accuracy of the present formulation is examined first by comparing results with those available in the literature. Thereafter, the influence of geometric parameters, lamination scheme, boundary conditions, static pre-load, and various types of non-uniform edge compression on the critical buckling loads and dynamic instability behaviour of both isotropic and laminated composite trapezoidal plate is investigated. The new results on dynamic stability behaviour of trapezoidal plates under non-uniform edge loads are presented for the first time, which may serve as benchmark results for future research. Furthermore, the time history response and corresponding phase plots are also presented for a better understanding of the dynamic behaviour of the trapezoidal plates.

Wrinkling of liquid-crystal elastomer disks caused by light-driven dynamic contraction.

The trigger time of light-driven wrinkling is very critical for accurate active control in photo-powered machines. In this paper, the wrinkling of liquid-crystal elastomer disks caused by light-driven dynamic contraction is theoretically studied, and the critical times for appearance and disappearance of the wrinkles are numerically calculated. The light-driven prebuckling stress can be significantly adjusted by changing the contraction coefficient, while controlled within a certain limit by tuning the light intensity and illumination time. There exists a critical contraction coefficient for triggering the wrinkling of the disk, and the second-order mode of wrinkling is the most unstable mode, which is most easily induced for the illumination radius ratio 0.69. The critical times for the appearance and disappearance of wrinkling can be significantly changed by the contraction coefficient, while regulated only within a certain range by the light intensity and the illumination radius ratio. These results have potential applications for accurate active control in the fields of soft robotics, active microlens, smart windows, and tunable surface patterns.

Effects of different interlaminar hybridization and localized edge loads on the vibration and buckling behavior of fiber metal composite laminates

This investigation presents the effect of different interlaminar hybridization and localized in-plane edge loads on the vibration and buckling characteristics of hybrid fiber metal laminates (HFMLs) by developing finite element formulation. A 9-noded heterosis plate element has been used to discretize the plate by taking into account the effect of shear deformation and rotary inertia. Since the stress distribution within the plate element is highly non-uniform in nature, the dynamic approach has been used to solve the buckling problems wherein two sets of boundary conditions are used, one for pre-buckling stresses and another for buckling load calculations. The present study consists of aluminum metal face sheets bound with four layered symmetric hybrid cross-ply and angle-ply laminate schemes. In each scheme, six different hybrid combinations have been considered. The performance of each hybrid combination is investigated under various loading combinations. Further, the effect of different parameters such as boundary condition, thickness of plate, width and position of localized edge loads, plate aspect ratio, and hybrid configurations are included in this work. The effect of each parameter is well investigated and explained by plotting mode shapes. It is revealed from the study that the hybrid configurations and localized edge loads significantly affect the vibration and buckling characteristics.

Buckling behavior of GLARE panels subjected to partial edge loads

The advancement in FRP laminates leads to the development of fibre metal laminates (FMLs) commonly found in aircraft components. The mechanical properties of FMLs are superior as compared to that of bare fiber laminates because of the infused strength combination of both fibers and metals. In this work, the Glass fiber reinforced aluminum/epoxy laminates (GLARE) are considered to investigate the effect of fibre-orientation on vibration and buckling characteristics under the action of partial edge loads by using finite element approach. A 9-noded heterosis plate element has been used to discretize the plate by incorporating the effect of shear deformation. For a given GLARE with ply-orientation and loading, the stress distribution within an element is highly non-uniform in nature. Therefore, dynamic approach has been used to calculate the buckling loads. Here, two boundary conditions are used, in which one for pre-buckling stress calculation and other one for buckling loads. The effect of various parameters such as ply-orientation in GLARE, thickness, boundary conditions, aspect ratios of plate and loading width and its position are considered to investigate the buckling characteristics in detail.

Non-linear stability analysis of CNT reinforced composite cylindrical shell panel subjected to thermomechanical loading

The present study explores the nonlinear stability characteristics of simply supported carbon nanotubes (CNTs) reinforced composite (CNTRC) cylindrical shell panel subjected to combined axial compressive loading and localized heating using semi-analytical approach. The thermomechanical properties of CNTRC panel are considered to be temperature-dependent and are evaluated using extended rule of mixture method. Higher order shear deformation theory and von Kármán type nonlinearity are employed to model the CNTRC cylindrical panel. Two types of localized heating profiles such as rectangular and circular are considered along with full heating. The pre-buckling stresses generated because of applied localized heating are determined using Airy’s stress function by satisfying the strain compatibility relations. The governing equations for stability problem of CNTRC panel are derived using variational principle incorporating the obtained pre-buckling stresses. The partial differential equations obtained are converted to a set of nonlinear algebraic equations by employing the Galerkin’s technique. The buckling temperature and nonlinear equilibrium paths are determined by solving the abovementioned equations using appropriate methods. The obtained results using present semi-analytical approach demonstrated the influence of various dispersion profiles of CNTs, CNT volume fraction and heating profiles on the buckling and post-buckling characteristics of CNTRC cylindrical shell panel subjected to thermomechanical loadings.

Tailoring the stability of an axially compressed circular-cylindrical shell using piezoelectric patch actuators

The ability to control equilibrium trajectory to buckling via uniform/non-uniform prebuckling stresses introduced using piezoelectric actuators creates interesting possibilities for designing smart structures. Recent innovative applications consider buckling as a favorable phenomenon in contrast to the conventional wisdom that considers the peak load carrying capacity as the sole design consideration. A circular-cylindrical shell under axial compression can be effectively used in many such applications if active tailoring of its buckling response is demonstrated. The effect of localized as well as distributed placement of piezoelectric patch actuators is analyzed in the present study based on the mechanics of deformation of a cylindrical shell under axial compression. The study establishes the possibility of active tailoring of buckling parameters such as peak load carrying capacity, initial stiffness, and first critical load. The results indicate that the principal curvatures of the cylindrical shell surface play a significant role in dictating the inter-actuator gap for the effective placement of discrete actuators.