Non-uniform Edge Loading Research Articles

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.

Buckling of laminated plates subjected to non-uniform distributed in-plane force

The buckling and vibration of laminated plates with various types of non-uniform compressive edge loads is analyzed. This problem requires that first the elasticity problem be considered to obtain the distribution of in-plane stresses, and after that, the buckling problem is solved. The method of solution is based on the R-functions theory and the variational Rayleigh-Ritz method. The results are compared with the results obtained by other authors to confirm the reliability of the method. The behavior of a complex form plate with a cutout has been investigated.

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.

Vibro-acoustics response of an isotropic plate under non-uniform edge loading: An analytical investigation

Analytical studies carried out on the vibro-acoustic response behavior of an isotropic plate under non-uniform edge loads subjected to steady-state mechanical excitation is presented. An analytical method based on the energy approach is used to calculate the buckling load (Pcr). Free and forced vibration responses of the plate are obtained using an analytical method based on Reddy's third-order shear deformation theorem (TSDT) while sound radiation behavior is analyzed using Rayleigh Integral. Results revealed that Pcr is significantly influenced by the nature of non-uniform edge load. Similarly, natural frequencies reduce with an increase in axial compressive load due to a reduction in structural stiffness. Vibration and acoustic resonant amplitudes are affected by the intensity of the compressive load. Sound transmission loss reduces with an increase in compressive load magnitude and the effect is significant in the stiffness dominant region.

Analytical investigation on free vibration frequencies of polymer nano composite plate: Effect of graphene grading and non-uniform edge loading

An analytical investigation carried out on free vibration characteristics of functionally graded graphene reinforced nanocomposite (FG-GRC) plate under different non-uniform edge loads is presented. Graphene nano-platelets (GPLs) are homogeneously dispersed and graded by varying weight fraction through the thickness. An analytical method based on strain energy approach is adopted to estimate the buckling load. Natural frequencies of the FG-GRC plate are attained using analytical solutions derived based on Reddy's third-order shear deformation theorem (TDST). Results revealed that buckling and free vibration behavior of the plate is influenced by the GPLs dispersion pattern and weight fraction under non-uniform edge loads. It is also observed that buckling mode and the fundamental vibration mode of the plate under combined tensile-compression load is entirely different from the other non-uniform edge load cases.

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.

A Parametric study on the effect of elliptical cutouts for buckling behavior of composite plates under non-uniform edge loads

The solution for the buckling problem of composite laminates is usually based on the assumption that the panels are subjected to uniform edge loads without any discontinuity or damage. Nevertheless, in practice, the panels are provided with various sized opening subjected to non-uniform edge loads. Hence in this work, a finite element analysis is carried out to study the buckling behaviour of composite laminates with various sized elliptical openings under non-uniform edge loads. In the FE formulation, the panels are discretised by adopting 9-noded heterosis element and incorporating the effect of shear deformation and rotary inertia. The correctness of the formulation is validated by correlating the results with available literature. The influence of different sized elliptical cutouts on the buckling behaviour is investigated under the compressive and tensile type of non-uniform in-plane edge loads. The effect of various parameters such as cutout size, ply-orientation, ellipse-orientation, boundary condition and panel thickness is included in this work. It is observed from this work that the elliptical vertical cutout has higher buckling resistance as compared to that of the elliptical horizontal cutout.

Buckling Characteristics of Laminated Functionally-Graded CNT-Reinforced Composite Plate under Nonuniform Uniaxial and Biaxial In-Plane Edge Loads

In this paper, the buckling response of laminated functionally-graded CNT-reinforced composite (FG-CNTRC) plate structure is predicted under various types of non-uniform edge compression loading. For the finite element (FE) discretization of the plate, a nine degree of freedom (DOFs)-type polynomial-based higher-order shear deformation theory (HSDT) is considered. The application of non-uniform edge load causes the in-plane stress distribution to be non-uniform. Hence, the in-plane stresses need to be evaluated prior to the buckling analysis. These in-plane stresses are calculated using the in-plane stress analysis method by FE approach or the in-plane elasticity approach. The differential equations are obtained by employing the Lagrange equation of motion and solved as a general eigenvalue problem, after the differential equations are converted into homogeneous equations by means of FE procedure. The accuracy and adaptability of the present model are validated by comparing the present result with the available literature. Further, the impact on the buckling response of the laminated FG-CNTRC plate is investigated by various parameters such as span thickness ratio, aspect ratio, various edge constraints, and different types of non-uniform edge load, CNT fiber gradation and temperature dependency material properties.

Dynamic instability of damped composite plates with embedded delaminations

A layerwise finite element formulation based on B-spline function is used to study dynamic instability of damped laminated composite plates with embedded delaminations in this article. The out-of-plane variation of in-plane displacement components are interpolated by Linear Lagrange interpolation function and the in-plane variation of in-plane and out-of-plane displacement fields are interpolated by B-spline functions. Delamination between layers of composite plate is modeled by jump discontinuities in displacement fields using Heaviside functions. In the delaminated region, virtual springs are added to prevent interpenetration of lower and upper sublaminates. The local buckling of thinner sublaminate is observed when delamination is placed near the top/bottom interface. The dynamic instability regions are computed by Bolotin's method. The effects of delamination lengths and its position on the natural frequencies, mode shapes, buckling loads and dynamic instability regions of the composite plates with various boundary conditions are examined. Both uniform and non-uniform periodic edge loading in the form of concentrated and partial edge loading are considered. Furthermore, the effect of damping on the dynamic instability regions of composite plate is studied. From the linear and nonlinear responses in the stable and unstable regions, various characteristics such as beats, its dependence on forcing frequency and damping are observed.

Effect of Reinforced Cutouts and Ply-Orientations on Buckling Behavior of Composite Panels Subjected to Non-Uniform Edge Loads

Buckling loads of laminated panels calculated by analytical approaches are usually based on the assumptions that the panels are subjected to uniform in-plane 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 different sized cutouts. The main objective of this paper is to study the effects of reinforced/unreinforced circular cutouts and non-uniform in-plane edge loads on the buckling behavior of composite panels with different ply-orientations by the finite element technique. Furthermore, it addresses the effects of different boundary conditions and thickness of panels. To carry out the analyses, a nine-noded heterosis plate element and a compatible three-noded beam element are developed, including the effect of shear deformation and rotary inertia for both the plate and the stiffeners. In structural modeling, the plate and the stiffener elements are treated separately, with their displacement compatibility maintained using transformation matrices. It has been illustrated in this study that presence of larger-sized reinforced cutouts predominantly increases the buckling strength of the panel as compared to those with smaller sized cutouts.

Stochastic dynamic stability analysis of composite curved panels subjected to non-uniform partial edge loading

Abstract The stochastic dynamic stability analysis of laminated composite curved panels under non-uniform partial edge loading is studied using finite element analysis. The system input parameters are randomized to ascertain the stochastic first buckling load and zone of resonance. Considering the effects of transverse shear deformation and rotary inertia, first order shear deformation theory is used to model the composite doubly curved shells. The stochasticity is introduced in Love's and Donnell's theory considering dynamic and shear deformable theory according to the Sander's first approximation by tracers for doubly curved laminated shells. The moving least square method is employed as a surrogate of the actual finite element model to reduce the computational cost. The results are compared with those available in the literature. Statistical results are presented to show the effects of radius of curvatures, material properties, fibre parameters, and non-uniform load parameters on the stability boundaries.

Effect of boundary conditions and non-uniform edge loads on buckling characteristics of laminated composite panels with and without cutout

ABSTRACTIn this work, the influence of uniaxial and biaxial nonuniform edge loads on buckling characteristics of composite laminates with and without cutout is investigated by using a finite element method. Furthermore, this investigation addresses the effects of different degree of edge restraints on buckling analysis. Regarding this, a nine node heterosis plate element has been adapted by employing the effects of shear deformation and rotary inertia. The influence of different parameters, such as ply-orientations, nonuniform edge loads, boundary conditions, and cutout sizes, has been considered and studied in detail. It is clearly illustrated that cutout sizes under different loading conditions have remarkable effects on buckling behavior of composite laminates.

Vibration and buckling analyses of laminated panels with and without cutouts under compressive and tensile edge loads

In this study, the influence of centrally placed circular and square cutouts on vibration and buckling characteristics of different ply-oriented laminated panels under the action of compressive and/or tensile types of nonuniform in-plane edge loads are investigated. The panels are inspected under the action of uniaxial compression, uniaxial tension and biaxial, compression-tension, loading configurations. Furthermore, the effects of different degrees of edge restraints and panel aspect ratios are also addressed in this work. Towards this, a nine-node heterosis plate element has been adopted which includes the effect of shear deformation and rotary inertia. According to the results, the tensile buckling loads are higher than that of compressive buckling loads. However, the tensile buckling load continuously reduces with the increased cutout sizes irrespective of ply-orientations. This is also true for compressive buckling loads except for some particular ply-orientations with higher sized cutouts.

Post-Buckled Vibration Characteristic of Composite Cylindrical Shell Panels Under Parabolic In-Plane Edge Compression

In this paper, post-buckled vibration of cross-ply thick laminated cylindrical shell panels subjected to nonuniform (parabolic) uniaxial and biaxial compression is studied. The mathematical model is based on a higher order shallow shell theory incorporating von Kármán-type geometric nonlinearities and initial geometric imperfections. In the first step, the plate membrane problem is solved to evaluate the stress distribution within the plate in the pre-buckling range as the applied in-plane edge load is nonuniform. By using the above stress distributions, the governing shell panel post-buckling equations are derived through Hamiltonian principle. The governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations for post-buckling analysis and nonlinear ordinary differential equations in the case of free vibration analysis using Galerkin's method. The equilibrium paths through limit points are traced using Newton–Raphson method in conjunction with Riks approach. The free vibration frequency of pre-bucked and post-buckled cylindrical panels loaded with uniaxial or biaxial nonuniform in-plane edge load are studied. Free vibration frequency for symmetric (0/90/0) cross-ply laminated cylindrical shell panels under uniaxial and biaxial parabolic in-plane load with initial imperfections are presented for different post-buckled deflections. Limit loads and snap-through behavior of shell panels and corresponding free vibration results are co-related for better understanding of the problem.

3D B-spline finite element nonlinear elasticity buckling analysis of rectangular FGM plates under non-uniform edge loads, using a micromechanical model

Three-dimensional buckling analysis of the functionally graded plates has not been investigated under non-uniform in-plane compressive loads so far. The only available work in this field has been performed based on the first-order shear deformation theory, employing the simple rule of mixtures. In the present paper, a non-linear three-dimensional energy-based elasticity analysis is developed for buckling investigation of functionally graded plates subjected to non-uniform in-plane compressions. The comparative studies performed in the present paper reveal that the rule of mixtures is generally inappropriate for determination of the effective material properties. For this reason, a micromechanics-based model is used instead. In contrast to the common displacement-based numerical or semi-analytical elasticity analyses (with C0-continuity), present formulations are C2-continuous due to using the proposed 3D cubic B-spline element. Buckling analysis is accomplished through a two-step procedure wherein the prebuckling stresses are determined first. A non-linear weighted-residuals-based finite element solution is used. The buckling load associated with each of the adopted 8 types of the load distribution patterns is detected based on a generalized geometric stiffness concept. Various and comprehensive parametric studies are accomplished and various boundary conditions are considered to extract more precise conclusions.

Dynamic instability of composite plates subjected to non-uniform in-plane loads

In this paper, the dynamic instability of a shear deformable composite plate subjected to periodic non-uniform in-plane loading is studied for four sets of boundary conditions. The static component and the dynamic component of the applied periodic in-plane loading are assumed to vary according to either parabolic or linear distributions. Initially, the plate membrane problem is solved using the Ritz method to evaluate the plate in-plane stress distributions within the prebuckling range due to the applied non-uniform in-plane edge loading. Subsequently using the evaluated stress distribution within the plate, the equations governing the plate instability boundaries are formulated via Hamilton's variational principle. Employing Galerkin's method, these partial differential equations are reduced into a set of ordinary differential equations (Mathieu type of equations) describing the plate dynamic instability behaviour. Following Bolotin's method, the instability regions are determined from the boundaries of instability, which represents the periodic solution of the differential equations with period T and 2 T to the Mathieu equations. The instability regions are determined for uniform, linear and parabolic dynamic in-plane loads using first-order and second-order approximations. Numerical results are also presented to bring out the effects of span to thickness ratio, shear deformation, aspect ratio, boundary conditions and static load factor on the instability regions.

EFFECT OF HARMONIC IN-PLANE EDGE LOADING ON DYNAMIC STABILITY OF STIFFENED SHELL PANELS WITH CUTOUTS

The effect of non-uniform in-plane pulsating edge loading on dynamic instability behavior of perforated stiffened shell panels is presented in this paper using finite element method. The eight-noded isoparametric degenerated shell element and a compatible three-noded curved beam element are used to model the shell panels and the stiffeners, respectively. Bolotin method is applied to analyze the dynamic instability regions. Numerical results of convergence studies are presented and comparison is made with the published results from the literature. The effects of various parameters like shell geometry, loading type, cutout size and dynamic load factors are considered in dynamic instability analysis of stiffened shell panels with cutout. The buckling results of the cutout stiffened panels are also presented.

Multi-mode parametric excitation of a simply supported plate under time-varying and non-uniform edge loading

Abstract In this study, multi-mode parametric excitation of a simply supported plate under time-varying and non-uniform edge loading is modeled and the solution is found. Equations for multi-mode parametric excitation of a simply supported plate are derived using stress distributions within the plate as well as on the edges, considering both the effects of non-uniform edge loading and the non-linearity caused by the large deflection. The multi-mode equations are coupled by first-order linear terms, even in the case of simply supported boundary conditions, due to the non-uniform edge loading. The perturbation solutions of two-mode parametric excitation are examined by the method of multiple scales. For the edge loading, which consists of a uniform term as well as a non-uniform one, equations could be coupled or de-coupled by parametric excitation terms, and the numbers and values of the resonance frequencies of the parametric excitations could also differ, depending on whether the non-uniform term of the edge loading is time-varying or not. In addition to the resonant frequencies of the case when only the uniform term of the edge loading is time-varying, there are additional combination resonances at the vicinity of the sum of two natural frequencies of each mode when the non-uniform term of the non-uniform edge loading is time-varying.

Stress distribution and wrinkling of webs under non-uniform and asymmetric edge loading

Non-uniform edge loading can cause unwanted wrinkling of a web. Here, Airy stress functions are obtained by a modified Fourier series method for webs under non-uniform and different edge loading on each side. The stress distributions of a web under non-uniform edge loading only on one side are subjected to the effect of Saint-Venant’s principle. For the web of small aspect ratio under non-uniform edge loading, the stress distribution converges to a uniform distribution or a linear distribution with both the resultant force and the resultant moment being equal to those of the applied loading. Conditions of wrinkling and the corresponding wrinkling modes of a web are found when the edge loadings are different on each side. The non-uniformities and shapes of the applied edge loadings result in different wrinkling conditions. Moreover, the non-uniformities together with the shapes of the applied edge loadings on each side affect the wrinkling conditions.

Combination Resonance Instability of Curved Panels with Cutout Subjected to Nonuniform Loading with Damping

The dynamic instability of doubly curved panels with a centrally located circular cutout, subjected to nonuniform compressive in-plane harmonic edge loading is investigated. The present work deals with the problem of the occurrence of combination resonances in contrast to single (mode) resonances in parametrically excited doubly curved panels with a central circular cutout. The method of multiple scales is used to obtain analytical expressions for the single (mode) and combination resonance instability regions. It is shown that other cases of the combination resonance can be of major importance and yield a significantly enlarged instability region in comparison to the principal instability region. The effects of nonuniform edge loading, centrally located circular cutout, damping, the static, and dynamic load factors on dynamic instability behavior of simply supported doubly curved panels are studied. The results show that under localized edge loading, combination resonance zones are as important as single (mode) resonance zones. The effects of damping show that there is a finite critical value of the dynamic load factor for each instability region below which doubly curved panels cannot become dynamically unstable. A central circular cutout has the destabilizing effect on the dynamic stability behavior of doubly curved panels subjected to nonuniform edge loading. This example of simultaneous excitation of two modes, each oscillating steadily at its own natural frequency, may be of considerable interest in vibration testing of actual structures.