In-plane Edge Loads Research Articles

Interaction of ultrasonic guided waves with interfacial debonding in a stiffened composite plate under variable temperature and operational conditions

Stiffeners play a vital role in strengthening thin panels in a wide range of engineering constructions by reducing additional structural weight. However, these structures are vulnerable to issues such as interlayer delamination or skin-stiffener interfacial debonding due to high stress levels developed from external environmental conditions and operational loadings. In contrast, ultrasonic-guided wave (UGW) techniques exhibit an efficient and precise approach for monitoring discontinuities or damages in composite structures. There is a lack of research on understanding the characteristics of the interaction between UGW and interfacial debonding when in-plane edge loading and environmental factors are simultaneously taken into account. Therefore, this study is motivated by the need to develop a multiphysics numerical model which employs a commercially available finite element software, COMSOL Multiphysics®, to simulate UGW propagation in a stiffened composite plate with debonding at the plate-stiffener interface through a piezoelectric transducer under the combined influence of in-plane edge load and hygrothermal environment. The stiffened plate and piezoelectric patches are modelled with the tetrahedral element, and the bottom surface of the attached stiffener has a through-width 0.1 mm deep groove simulated for debonding. The developed FE model is validated against the results of the conducted experiments and those found in the available literature through the correlation coefficient. Further, the study conducts a comprehensive parametric investigation on stiffened cross-ply (0/90/0) laminated plates, considering variations in debonding size, in-plane load, and hygrothermal load intensity through the excitation of A0 mode. The acquired response is processed to compare the peak amplitude of various modes and energy of the waveform. Additionally, statistical indices such as normalised correlation moment (NCM) and variance of the continuous wavelet transform (CWT) peak are estimated to understand the impact of various parameters on waveform. The results show that the presence of a 90° lamina in the cross-ply laminate generates a low amplitude S0 mode in the scattered response. Moreover, a mode conversion from A0 to S0 mode is observed due to perfect bonding between the plate and the stiffener, providing insights into the bonding state in the panel. Furthermore, it is found that the magnitude of the in-plane loading marginally affects the peak amplitude of various modes in the scattered response. Additionally, when temperature intensity rises, the energy and amplitude of the UGW signals acquired through piezoelectric patches positioned in a direct line with the actuator gradually increase. The NCM value enhances with debonding regardless of exposed hygrothermal condition and reduces with increasing temperature intensity. In addition, the variance of the CWT peak reduces with debonding. The findings of this research are expected to be helpful for the development of efficient algorithms for detecting damages for structural health monitoring of stiffened composite panels.

Buckling Behavior of Laminated Shell Panels Under Linearly Variable Edge Load

This paper reports a detailed numerical investigation on the buckling aspects of laminated composite shell panels of three forms (cylindrical, spherical and hyperbolic paraboloid) with five support conditions exposed to linearly varying in-plane edge load employing eight nodded isoparametric finite element formulation. The impacts of different parameters including ply orientation, load factor, shell forms, aspect ratio, modulus ratio, curvature ratio, width-to-thickness ratio and angle of lamination on the buckling load of shell panels are examined. It is found that the various parameters addressed in this study have a remarkable impact on the buckling phenomena of laminated composite shell panels. Further, a comparison is made showing the effects of five types of compressive edge loads like uniform, triangular, parabolic, partial edge load and point load on the buckling phenomena of laminated shell panels with respect to five support conditions.

Parametric Instability of Stiffened Plates with Cutouts

The dynamic instability behaviour of stffined plates with cutout is of great technical importance for understanding the behaviour of the system. In the present study, dynamic instability behaviour of stffined plates with cutouts are investigated by using theJinite element method. The method of Hill's infinite determinants is applied to analyze the dymmic instability regions. Rectangular stilfened plates with cutouts possessing diferent boundary conditions, aspect ralios, varying mass and stffiess properties and varying number of stiffeners have been analyzed for dynamic instability. The influence of various parameters like effects of size of cutout, load position, aspect ratios, and boundary condition, other paramelers of stffined plates have been studied. The method highlights interesting effects when stffined plate with cutout is subjected to uniftrm in-plane edge load. In the structural modelling, the plate and the stffiner are treated as separate elements where the compatibility between these two types of elements are maintained. The present approach is moreflexible than any otherJinite element modelling in that the mesh division is independent of the location of the stiffeners.

A von Kármán-type model for two-layer laminated glass plates, with applications to buckling and free vibration under in-plane edge loads

This article presents a layerwise geometrically nonlinear two-dimensional model for the undamped dynamical response of thin two-layer plates with partial shear interaction. The model is tailored for the analysis of laminated glass plates of the type most commonly used in building structures (two glass layers bonded by a polymeric interlayer). The geometrical nonlinearities are introduced via the so-called von Kármán strains. The process of dimensional reduction is based on Podio-Guidugli’s method of internal constraints. An unconventional choice of generalised displacements highlights several direct similarities with the single-layer plate theories of von Kármán and Mindlin.The model is used to study the flexural buckling and free vibration of two-layer plates under in-plane edge loads. Analytical solutions are obtained for rectangular plates with (i) all edges simply supported without shear restraint and (ii) a two-parameter system of uniform in-plane normal edge loads. These solutions degenerate into those of single-layer von Kármán plates in the two limiting cases of zero and full shear interaction. A conforming rectangular finite element is formulated and shown to perform effectively. When using uniform meshes, the observed asymptotic rate of convergence for the finite element approximation of the lowest buckling load and vibration frequency is quadratic.

Nonlinear dynamic instability of laminated composite stiffened plates subjected to in-plane pulsating loading

A nonlinear finite element dynamic instability analysis of laminated composite stiffened plates subjected to in-plane harmonic edge loading is presented in this article along with the linear and nonlinear dynamic response study. The eight-noded isoparametric degenerated shell element and a compatible three-noded curved beam element are used to model the stiffened plates. Bolotin method is applied to analyze the dynamic instability regions in linear case. Incremental Harmonic Balance (IHB) method is applied to solve the nonlinear frequency response equations and Newmark-β method is used to solve the linear and nonlinear time history response equations.

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.

Dynamic instability analysis of perforated stiffened laminated composite panels subjected to non-uniform in-plane edge load under hygrothermal condition

A robust and computationally efficient finite element (FE) formulation has been developed in an effort to understand the parametric instability behaviour of a perforated stiffened composite panel subjected to a pulsating non-uniform edge load under a hygrothermal environment. The panel is modelled using a 9-noded heterosis element to avoid shear locking, whereas the stringer uses a 3-noded isoparametric beam element with a displacement compatibility condition employed at the skin–stringer​ interface. Due to the non-uniform distribution of stress in the perforated panel as a result of exposure to environmental and mechanical loading conditions, a unique dynamic technique has been implemented to account for the instability performance by employing two types of boundary conditions. In the current investigation, an unstiffened panel is initially obtained based on the highest value of the dynamic instability opening (DIO) and the unstable excitation frequency of the panel for different lamina schemes, cutout shapes, and loading conditions. The obtained unstiffened panel is further investigated for various stiffener configurations to determine a highly unstable perforated stiffened panel based on the DIO value and instability zone frequency range. Detailed parametric studies are then carried out considering the cutout sizes, cutout orientation, panel’s aspect ratio, and stiffener depth to understand the parametric instability behaviour of the panel in a hygrothermal environment. It is found that moisture significantly influences the dynamic instability behaviour of the perforated stiffened panel compared to the temperature exposure by shifting the instability zone towards the lower frequency. The width of the instability region of the stiffened panel broadens abruptly as the magnitude of the applied static in-plane edge load enhances. Also, for certain combinations of the dynamic and static load factors, the lower bound of the instability region reduces to zero, resulting in fluttering. In general, it is observed that stiffer the panel, wider is the instability zone with a higher unstable excitation frequency range.

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.

Effect of trapezoidal shapes and non-uniform edge loads on buckling behaviour of plates with cutouts

Crucial buckling loads of composite laminates are typically determined by employing analytical techniques based on the presumptions that the plates are susceptible to uniform edge loads, ignoring the fact that the practical assemblies are regularly exposed to different non-uniform edge loads. The current research paper is centered on the buckling behaviour of different patterns of trapezoidal composite laminates with cutouts under the application of non-uniform in-plane edge loads by employing finite element technique. To achieve this, the panel is discretized by using 9-noded heterosis plate element by employing the impact of shear deformation and rotary inertia. For a given load and the shape of panel, the stress concentration within the panel is highly non-uniform in nature. Hence, a dynamic approach has been employed to calculate the buckling loads. The accuracy of the method is verified by relating the obtained results with existent research. The impact of various variables including the size of the cutout, non-uniform edge loads, boundary conditions, ply alignments and panel trapezoidal shape ratio has been considered to examine the buckling behavior of trapezoidal panels. Buckling load factor optimization using CCD and RSM has also been carried out 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.

Static stability and vibration behavior of graphene platelets reinforced porous sandwich cylindrical panel under non-uniform edge loads using semi-analytical approach

Buckling and free vibration characteristics of sandwich cylindrical panel with porous functionally graded graphene platelets (FG-GPL) core are investigated using semi-analytical approach. The effective mechanical properties are obtained by using properties of open cell foams and Halpin–Tsai micro mechanical model. The governing equations are obtained using Hamilton’s principle, considering a higher order theory to account the transverse shear and solved by Galerkin’s method. Effects of nature of in-plane edge load, distribution of porosity and GPL, porosity coefficient, GPL loading, core to total thickness ratio are analyzed in detail. It is shown that for a FG-GPL core sandwich cylindrical panel with high core thickness, even at higher amount of porosity the buckling resistance and free vibration frequency can be improved by properly tailoring both the GPL and porosity distribution. Moreover, a much variation in buckling and free vibration response with the type of in plane loading is observed and evident mode shape changes are observed with increase in aspect ratio. The cylindrical sandwich panel having a core with D-PD porosity variation and I-GPL-P pattern of GPL distribution has the maximum buckling resistance and free vibration frequency value.

Dynamic Instability Analysis and Design Charts of Curved Panels with Linearly Varying Periodic In-Plane Load

The present paper reports an extensive study on dynamic instability characteristics of curved panels under linearly varying in-plane periodic loading employing finite element formulation with a quadratic isoparametric eight nodded element. At first, the influences of three types of linearly varying in-plane periodic edge loads (triangular, trapezoidal and uniform loads), three types of curved panels (cylindrical, spherical and hyperbolic) and six boundary conditions on excitation frequency and instability region are investigated. Further, the effects of varied parameters, such as shallowness parameter, span to thickness ratio, aspect ratio, and Poisson’s ratio, on the dynamic instability characteristics of curved panels with clamped–clamped–clamped–clamped (CCCC) and simply supported-free-simply supported-free (SFSF) boundary conditions under triangular load are studied. It is found that the above parameters influence significantly on the excitation frequency, at which the dynamic instability initiates, and the width of dynamic instability region (DIR). In addition, a comparative study is also made to find the influences of the various in-plane periodic loads, such as uniform, triangular, parabolic, patch and concentrated load, on the dynamic instability behavior of cylindrical, spherical and hyperbolic panels. Finally, typical design charts showing DIRs in non-dimensional forms are also developed to obtain the excitation frequency and instability region of various frequently used isotropic clamped spherical panels of any dimension, any type of linearly varying in-plane load and any isotropic material directly from these charts without the use of any commercially available finite element software or any developed complex model.

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.

Dynamic behaviour of perforated laminated panels under biaxial non-uniform edge loads based on FE approach

Buckling loads of laminated panels are calculated by analytical approaches are usually based on the assumptions that the panels are subjected to uniform in-planes edge loads without cutouts, despite of the fact that real structural components are subjected to various kinds of non-uniform in-plane edge loads along with the different sized cutouts. The aim of this work is to investigate the effect of biaxial non-uniform in-plane edge loads on the vibrations and buckling behaviour of panel with and without circular cutouts by using finite element technique. Further, the effect of different ply-orientations, boundary conditions and cutout sizes are also addressed in this work. Towards this, a nine-noded heterosis plate element has been adopted to model the panel by incorporating the effect of shear deformation and rotary inertia. Finally, it is shown that the non-uniform edge loading, boundary condition and cutout sizes have a remarkable effect on the vibration and buckling behavior of panel.

Buckling behaviour of composite laminates of trapezoidal panel with cutout subjected to non-uniform in-plane edge loads

Abstract Critical buckling loads of composite laminates are usually calculated by using analytical solutions based on the assumptions of uniform in-plane loads, despite of the fact that real structures are often subjected to various nonuniform loads. The present work is focused on the buckling behavior of composite trapezoidal laminates panel with cutouts, subjected to various nonuniform in-plane loads. The effect of variation is size of trapezoidal plate on the buckling behavior has been studied using finite element method. The governing differential eigenvalue equations have been solved using subspace algorithm. The analytical solution for such problems is very difficult. Hence in this work, a finite element software FORTRAN is used to study the buckling behavior of thin and thick laminated panels with cutouts subjected to various kinds of non-uniform edge loads. The panel is modeled by using eight node elements with five degrees of freedom at each node. The accuracy of the model is confirmed by comparing present results with available literature. The effect of various parameters such as cutout size, boundary conditions and ply-orientations of panel are included in this work. It is observed from this study that the non-uniform edge loads and cutout size has remarkable effect on the buckling behavior of said panels.

Geometrically nonlinear analysis of doubly curved imperfect shallow shells made of functionally graded carbon nanotube reinforced composite (FG_CNTRC)

This article presents an analytical approach to investigate the geometrically nonlinear response of doubly curved imperfect shallow shells made of functionally graded carbon nanotube reinforced composite (FG_CNTRC) under in-plane and lateral loads. For all of the case studies, the edges of shells are assumed to be diaphragm supports, which are subjected to in-plane edge-loads. The governing equations are developed for thin shells based on the classical theory considering the geometric nonlinear terms using the Von Karman–Donnell sense. The Galerkin method is used to solve the extended nonlinear system of differential equilibrium and compatibility equations. The solution results in a closed-form formula representing the equilibrium path (load-deflection relationship) of the shell which is validated by numerical methods, such as finite element and mesh-free methods. The parametric studies are performed for doubly curved and cylindrical shells subjected to uniform external pressure and in-plane edge loads, respectively. The effects of various parameters such as volume fraction of FG_CNTRCs and geometrical properties on the equilibrium paths are examined in detail.

Meshfree radial point interpolation method for the vibration and buckling analysis of FG-GPLRC perforated plates under an in-plane loading

A meshfree radial point interpolation method (RPIM) based on a higher-order shear deformation theory (HSDT) is here developed to investigate the free vibration and buckling behavior of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) perforated plates under an in-plane edge loading. The non-uniform initial stresses in the multilayer FG-GPLRC perforated plates are evaluated accurately by solving the equilibrium equations derived from the same approach. The mixed collocation-Lagrange multiplayer technique is employed to enforce the various boundary conditions. The multilayer FG-GPLRC plates are built up as perfectly-bonded composite layers reinforced with a uniform distribution of graphene platelets (GPLs), randomly oriented. Perforated plates with three different cut-out shapes are analyzed. Several numerical examples are provided to demonstrate the high performance, the fast convergence rate and the high accuracy of the proposed method. Then, we study the influence of different GPLs distribution patterns, GPLs weight fraction, shape and size of cut-outs on the free vibration and buckling behavior of the multilayer FG-GPLRC perforated plates under an in-plane edge loading.

Buckling analysis of porous FGM sandwich plates under various types nonuniform edge compression based on higher order shear deformation theory

This study intends to model the effect of porosity type defect and analyze the consequences of porosity on the buckling characteristic of various types of FGM sandwich combinations. Here, we consider four types of FGM sandwich combinations considering the multiple arrangements of the constituent layers (i.e., two face sheets and a core). Further, the combinations of all kinds of sandwiches are considered in such a way that there is no material discontinuity or jump. The porosity defects are modeled in this work as the criteria of stiffness reduction and incorporated in the rule of mixture. Finally, results are calculated by considering two types of porosity distribution, evenly and unevenly, while modeling the FGM plate. The buckling response of FGM sandwich plates has been analyzed under a nonuniform in-plane edge load. The application of nonuniform in-plane loads causes in-plane stresses to be nonlinear. Hence, in-plane stresses need to be calculated apriori for the buckling study of the FGM plate under nonuniform load. Further, the effects of various parameters like porosity contents, sandwich combinations, aspect ratio, thickness ratio, boundary conditions on buckling load of FGM plate have been studied in this paper.

Buckling analysis of laminated composite plate due to localised in plane loading

Laminated composite plates are inevitable parts of structure due to its superior properties than that of conventional material. Many times, these structural elements are subjected to the partial loadings during various applications which arises buckling in the elements. This necessitates proper analysis of these types of structural member for better safety and stability. In the present paper buckling analysis of laminated composite stiffened plate under in-plane localized edge loading is presented. Buckling load for various cases of partial edge loading are investigated for different fiber orientations and loading extent. Finite element method-based MATLAB program is developed for the present analysis. Validation study is done, which shows well agreement with the results already published.

Parametric instability analysis of laminated composite plate subject to various types of non-uniform periodic in-plane edge load

In this paper, the dynamic instability behavior of laminated composite plate structure is predicted under different types of non-uniform harmonic edge compressive loading. A nine degree of freedom (DOF) type polynomial based higher order shear deformation theory (HSDT) is considered for the finite element discretization of the plate. The application of non-uniform harmonic in-plane edge load causes the in-plane stress variation to be non-uniform. Hence, the in-plane stresses need to be evaluated, prior to the instability analysis, for the evaluation of critical buckling load. These in-plane stresses are computed using in-plane stress analysis approach using finite element method. The differential equations of motion of the system are turned into a set of ordinary differential equations (Mathieu type equations) and solved as a general eigenvalue problem as per the procedure suggested by Bolotin. The accuracy and flexibility of the present model are validated by contrasting the present outcomes and the available solution. Further, the impact various parameters like span-thickness ratio, aspect ratio, diverse in edge constraints, different types of non-uniform periodic edge load, etc. on the dynamic instability behavior of the laminated composite plate are contemplated.