Variable Stiffness Composite Plates Research Articles

Thermoelastic geometrically nonlinear analysis and optimization of variable stiffness composite plates in presence of buckling

Thermoelastic geometrically nonlinear analysis and optimization of variable stiffness composite plates in presence of buckling

Frequency veering and nonlinear coupled vibration analysis of variable stiffness composite plates with curvilinear fiber paths

The coupling forced vibration behaviors of a variable stiffness plate with curvilinear fiber paths at mode veering regions caused by variations of fiber angle have been rarely found in previous papers, especially in the presence of considering the complex material nonlinearity of composite materials. In this paper, the variable stiffness plate of carbon fiber reinforced composites (CFRC) with curvilinear fiber paths is taken as the research object, and the influence of curvilinear fiber paths on its natural characteristics and vibration responses are studied in detail on the basis of considering the material nonlinearity of CFRC. Firstly, it is found that the variations of curvilinear fiber paths can cause complex frequency veering behavior of the variable stiffness plate, and lead to the appearance of irregular modal shapes formed by modal interaction. Then, the forced vibration response of the variable stiffness plate in the frequency veering region is analyzed methodically, and the variation rules of the vibration response under modal interaction is obtained. After analyzing the nonlinear vibration response of the variable stiffness plate under the combined influence of modal interaction and CFRC material nonlinearity, the mechanism of complex nonlinear dynamic behavior of variable stiffness plate in frequency veering region is further investigated.

A FEM Free Vibration Analysis of Variable Stiffness Composite Plates through Hierarchical Modeling

Variable Angle Tow (VAT) laminates offer a promising alternative to classical straight-fiber composites in terms of design and performance. However, analyzing these structures can be more complex due to the introduction of new design variables. Carrera’s unified formulation (CUF) has been successful in previous works for buckling, vibrational, and stress analysis of VAT plates. Typically, one-dimensional (1D) and two-dimensional (2D) CUF models are used, with a linear law describing the fiber orientation variation in the main plane of the structure. The objective of this article is to expand the CUF 2D plate finite elements family to perform free vibration analysis of composite laminated plate structures with curvilinear fibers. The primary contribution is the application of Reissner’s mixed variational theorem (RMVT) to a CUF finite element model. The principle of virtual displacements (PVD) and RMVT are both used as variational statements for the study of monolayer and multilayer VAT plate dynamic behavior. The proposed approach is compared to Abaqus three-dimensional (3D) reference solutions, classical theories and literature results to investigate the effectiveness of the developed models. The results demonstrate that mixed theories provide the best approximation of the reference solution in all cases.

Buckling analysis of variable stiffness composite plates with elliptical cutouts using an efficient RPIM based on naturally stabilized nodal integration scheme

Buckling resistance of variable stiffness composite (VSC) plates can be significantly improved by designing fiber orientation distributions, which affect structural stiffness and stress distribution. In this paper, a meshless model of VSC plates is developed using a radial basis point interpolation method (RPIM) based on the naturally stabilized nodal integration (NSNI) scheme, and the buckling behavior of the VSC plates with elliptical cutouts is investigated. In this model, a set of nodes is used to discretize the problem domain, which makes the loading response of the VSC plates can be calculated by its exact fiber angle. And the mechanical behavior of the VSC plates with various complex geometries and cutouts can be accurately simulated. This meshless method is verified by comparing with the finite element method (FEM) and semi-analytical solutions. And compared with the traditional RPIM based on the Gaussian integration (GI), NSNI-based RPIM achieves considerable enhancement in computational efficiency while maintaining high accuracy. Employing this efficient meshless method, the buckling behavior of the VSC plates with elliptical cutouts for different boundary conditions is studied. And the effects of size and location of the cutout on the buckling response of the VSC plates are analyzed. The research on the VSC plates with cutouts in this paper has practical implications for the design of cutouts for VSC plates and provides a new method for the numerical research of VSC plates.

Vibration of variable stiffness composite laminate and hybrid composite laminate plates coupled to sloshing fluid

In the present article, Reddy shear deformation plate theory is employed to investigate the free vibration of vertical laminated composite plates coupled to sloshing liquid. Two different types of composite plate are considered; hybrid composite laminate (HCL) plate, which consists of two types of fiber, and variable stiffness composite laminate (VSCL) plate, which is made of curvilinear fibers. The applied Reddy shear deformation theory introduces a nonlinear through-thickness distribution for the transverse shear stress, and it satisfies the zero stress conditions at the bottom and top surfaces of the plate. The sloshing fluid, which is partly in contact with the plate, is modeled as ideal, and the bulging and sloshing modes related to that are calculated using the fluid velocity potential. The interaction between fluid and solid is modeled through the continuity equation and boundary conditions at the fluid–solid interface. To extract the vibrational characteristics of the structure coupled to fluid, a polynomial approximation based on the Rayleigh-Ritz method is employed. After validating the proposed methodology, a parametric study is conducted to show the effects of various parameters associated with the fluid and structure on the natural frequencies of the system.

Free vibrations analysis of cracked variable stiffness composite plates by the eXtended Ritz method

Variable stiffness composite laminates show advantageous structural features related to their enlarged design space. They are attractive candidates for advanced engineering applications where the assessment of static and dynamic behavior and strength in the presence of cracks is often required. In the present work, a single-domain extended Ritz formulation is proposed to study the free vibrations of cracked variable stiffness composite plates. The plate model is based on the first-order shear deformation theory whose primary variable, i.e. displacements and rotations, are approximated via a set of orthogonal polynomial trial functions enriched with a set of special crack functions. These functions are able to inherently account for crack opening and crack tip singular fields. The plate governing equations are deduced by the stationarity of the energy functional and the formulation has been implemented in a computer code. The method has been validated by comparing the present results with literature solutions for cracked isotropic plates and uncracked variable stiffness plates as, to the best of author’s knowledge, no data on cracked variable stiffness plates free vibrations are available. An explicative and representative study on the free vibrations of variable angle tow composite laminates is finally presented with the aim of illustrating the approach capabilities, providing benchmarck results and identifying distinctive features and opportunities of the variable stiffness concept for the design of advanced damage tolerant structures.

An enhanced reduced-order model based optimization method for buckling and postbuckling performances of variable stiffness composite plates

The great potential of variable stiffness laminates in maximizing buckling load is obvious over straight fiber laminates, however the optimum design of such VAT laminates which has much more design variables and larger computational efforts, is still a challenge even for the modern computational method and capability. In addition, the postbuckling performance and the correlations with buckling resistance still need to be further studied for VAT laminates. This work proposes an enhanced reduced-order model based on the improved Koiter perturbation theory for buckling and postbuckling analysis of variable stiffness composite plates. The VAT laminas composed by “two-segment” fibers and “four-segment” fibers are considered, respectively, in construction of the reduced-order finite element model. The nonlinear prediction for initial postbuckling response of VAT laminate can be achieved, by solving the small-scaled reduced-order model. The nonlinear buckling load, postbuckling stiffness and postbuckling stiffness residual factor, are accurately and efficiently extracted from the initial postbuckling predictor, and then subsequently to be selected as inputs to the GA based optimization process for the design of VAT laminates. Extensive numerical investigations validate that the proposed method can find a combination of several varied angle fiber layers with constant thicknesses such that the combination results in an optimal stiffness distribution and provides a more favorable structural performance in both buckling and postbuckling regime for different demands of structure design.

Buckling Sensitivity of Tow-Steered Plates Subjected to Multiscale Defects by High-Order Finite Elements and Polynomial Chaos Expansion.

It is well known that fabrication processes inevitably lead to defects in the manufactured components. However, thanks to the new capabilities of the manufacturing procedures that have emerged during the last decades, the number of imperfections has diminished while numerical models can describe the ground truth designs. Even so, a variety of defects has not been studied yet, let alone the coupling among them. This paper aims to characterise the buckling response of Variable Stiffness Composite (VSC) plates subjected to spatially varying fibre volume content as well as fibre misalignments, yielding a multiscale sensitivity analysis. On the one hand, VSCs have been modelled by means of the Carrera Unified Formulation (CUF) and a layer-wise (LW) approach, with which independent stochastic fields can be assigned to each composite layer. On the other hand, microscale analysis has been performed by employing CUF-based Mechanics of Structure Genome (MSG), which was used to build surrogate models that relate the fibre volume fraction and the material elastic properties. Then, stochastic buckling analyses were carried out following a multiscale Monte Carlo analysis to characterise the buckling load distributions statistically. Eventually, it was demonstrated that this multiscale sensitivity approach can be accelerated by an adequate usage of sampling techniques and surrogate models such as Polynomial Chaos Expansion (PCE). Finally, it has been shown that sensitivity is greatly affected by nominal fibre orientation and the multiscale uncertainty features.

Investigation of buckling characteristics of cracked variable stiffness composite plates by an eXtended Ritz approach

Variable Angle Tow (VAT) composite plates are characterized by in-plane variable stiffness properties, which opens to new concepts of stiffness tailoring and optimization to achieve higher structural performance for advanced lightweight structures where damage tolerance consideration are often mandatory. In this paper, a single-domain eXtended Ritz formulation is proposed to study the buckling behaviour of variable stiffness laminated cracked plates. The plate behaviour is described by the first order shear deformation theory whose generalized displacements, namely reference plane translations and rotations, are expressed via suitable admissible trial functions. These consist of a set of regular terms, built using orthogonal polynomials, augmented with special functions able to describe the crack opening and the singular behaviour at the crack tips; boundary functions are used to ensure the required homogeneous essential boundary conditions. Governing equations are inferred via the stationarity of the energy functional an solved to carry out an extensive study on the buckling behaviour of variable angle tow homogeneous and layered composite plates. The results obtained for homogeneous plates evidence that the crack presence strongly influences the buckling behaviour depending on its length and inclination and plate boundary conditions with a meaningful variability with respect to the fibre paths configuration, which can arrange for better performances with respect to the straight fibre case. Also for cracked laminates the results show that there are several fibre path able to provide higher buckling loads and higher overall axial stiffness with respect to the straight fibre case. In the framework of damage tolerant engineering applications, this allows to select fibre paths that guarantee predefined design levels of buckling load and axial stiffness even in presence of cracks. Finally, this study highlights the potential of the proposed approach for the analysis of the buckling behaviour of cracked composite variable stiffness plates, which provides an efficient analysis tool for the damage tolerant design and optimization of advanced variable stiffness structures.

Optimization of open-hole variable stiffness composite plates under tensile loading using curved continuous fibers

This paper describes optimization of variable stiffness composite plates (VSCPs) with open-holes under tensile loading. Curved continuous fibers are used to optimize distributions of fiber orientation and the fiber volume fraction during the optimization for the VSCPs. The modeling of VSCPs is performed by means of fiber trajectories, which are aligned in the direction of maximum principal stress. The progressive failure of the VSCPs is modeled by a material property degradation method, taking into account the variable distributions of material properties to predict the ultimate load. It is demonstrated that the ultimate load for the composite plates is increased using the change in a reinforcement structure and a transition from unidirectional to curvilinear reinforcement of fibers. Thus, application of VSCPs with curved fibers compared with unidirectional composites results in more effective use of composite materials in various fields of industry.

Approximate Bayesian assisted inverse method for identification of parameters of variable stiffness composite laminates

The uncertainties of parameters are important for the performance of Variable Stiffness (VS) composite laminate plates. When parameters of VS composite laminate are given, it is easy to investigate physical characteristics of VS composite laminate. However, it is difficult to identify values of parameters due to a typical ill-posed inverse problem. In order to address this problem, an innovative approximate Bayesian computation (ABC) is suggested to identify composite parameters by considering uncertainties in this study. Compared with traditional Bayesian framework, ABC can avoid the calculation of likelihood which makes inverse procedure more complex or even intractable in practice. Generally, four advanced techniques are integrated to satisfy demands of ABC method in this study. In the suggested framework, a powerful Auto-Encoder (AE) is used to reduce calculation of the response with little information loss. Sequentially, Tikhonov regularization is integrated into ABC. Furthermore, to reduce computational cost, a high sample accepted rate-adaptive nested sampling method and Neural Network (NN) used to construct the mapping between parameters and responses are utilized. Finally, the efficiency and flexibility of the proposed method are validated by three cases.

Energy harvesting in variable stiffness composite piezolaminated plates

Harnessing ambient vibrational energy to power small and medium power electronic devices has been greatly explored in the past decade. Most of the available energy harvester models use straight fibers. Use of curvilinear fiber in bimorph energy harvester is explored in this computational study and its effect on tailoring the resonance frequency and extracted power is also investigated. The variable stiffness composite laminate (VSCL) bimorph energy harvester is modelled using first order shear deformation theory (FSDT) and equations are derived using a nine noded isoparametric finite element. Curvilinear fiber based composite and PZT-5A materials are utilized for the analysis of harvester. Curvilinear fiber model is validated by comparing the natural frequency of VSCL plate with the already published results. Power frequency response function (PFRF), voltage frequency response function (VFRF) and relative tip motion frequency response function are plotted for a fixed curvilinear fiber orientation. Effects of plate aspect ratio on the peak power is also investigated.

Nonlinear flutter of tapered and skewed cantilevered plates with curvilinear fiber paths

Aeroelastic stability of tapered/skew variable stiffness composite cantilevered plates are considered in the current study at flutter and post-flutter regions. The variable stiffness behavior is obtained by altering the fiber angles continuously according to a selected curvilinear fiber path function in the composite laminates. Flutter speed, limit cycle oscillations (LCOs), and bifurcation diagrams of tapered/skewed plates are obtained at two different fiber path functions. Nonlinear structural model is utilized based on the virtual work principle. Fully nonlinear Green’s kinematic strain relations are used to account for the geometric nonlinearities and the first order shear deformation theory (FSDT) is employed to generalize the formulation for the case of moderately thick plates including transverse shear effects. One prominent target of the present study is to determine how the variable stiffness parameters affect the nonlinear behavior. Consequently, one may find the best fiber path with improved flutter and post-flutter characteristics for tapered/skew plates in supersonic flow. First order linear piston theory is used to model the aerodynamic loading. In order to get a reliable solution, the Generalized Differential Quadrature (GDQ) method is employed. Moreover, time integration of the equations of motion is carried out using the Newmark’s average acceleration technique. It will be shown that taperness/skewness as well as variable stiffness lamination parameters have significant effects on the aeroelastic stability margins. In addition, the post-critical behavior is found to be periodic or quasi-periodic at all the presented simulations with no specific route to chaos.

Fundamental frequency optimization of variable stiffness composite skew plates

In this study, natural frequencies and vibrational mode shapes of variable stiffness composite skewed plates are optimized applying a genetic algorithm. The variable stiffness behavior is obtained by altering the fiber angles continuously according to two selected curvilinear fiber path functions in the composite laminates. Fundamental frequency and related mode shapes of the plates are optimized for two different fiber path functions using the structural model obtained based on the virtual work principle. A three-layer composite skewed plate with four types of boundary conditions and different plate geometries is considered as case study in this research. Diverse sweptback angles as well as different aspect ratios are considered as various plate geometries. The present study aims to calculate the best fiber path with maximized fundamental frequency or in-plane strengths for a composite skewed plate. The generalized differential quadrature method of solution is employed to solve the governing equations of motion. Moreover, the linear kinematic strain assumptions are used, and the first-order shear deformation theory is employed to generalize the formulation for the case of moderately thick plates including transverse shear effects. Numerical results demonstrate the effect of the fiber angles, boundary conditions, and diverse geometries on the natural frequencies of the composite plate. The optimal fiber angles of each layer are presented for the above cases in free vibration analysis. It is verified that the application of optimized curvilinear fibers instead of the traditional straight fibers introduces a higher degree of flexibility, which can be used to adjust frequencies and mode shapes.

Stability and bifurcations in oscillations of composite laminates with curvilinear fibres under a supersonic airflow

Nonlinear flutter of variable stiffness composite plates—using a reduced-order model benefiting from a large-enough number of modes or degrees of freedom—is the subject of this research. Dynamic of such plates is enriched by different scenarios including period-n (un)stable limit cycle oscillations, quasi-periodic or chaotic solutions, along with Hopf, symmetry breaking and period doubling bifurcations. The fibre path orientation, in any individual ply of these rectangular plates, changes linearly from the left edge to the right one. The plates are subjected to a supersonic airflow. The plate’s displacement field is modelled using a third-order shear deformation theory; a p-version finite element is used to discretize the displacement components. Aerodynamic pressure due to the airflow is approximated using linear piston theory. The self-exciting vibrational full-order model (FOM) is formed by employing the principle of virtual work, and then, the ROM is extracted from the FOM using two reduction techniques, namely static condensation and modal summation method. The inclusion of a large-enough number of modes of vibration in vacuum in the latter technique is critical for the observation of, often overlooked, complex dynamic behaviour of these plates. The equations of motion representing the ROM are solved using the shooting method (to obtain stable and unstable LCOs) and using Newmark method (for stable or non-oscillatory solutions). The rich dynamics of these plates are explored with the help of vibration and flutter mode shapes, phase-plane plots, bifurcation diagrams, fast Fourier transform diagrams and Poincare sections.

An element-free study of variable stiffness composite plates with cutouts for enhanced buckling and post-buckling performance

In this paper, an efficient element-free modeling framework for variable stiffness composite laminates with cutouts is developed. In comparison with other numerical methods or semi-analytical methods, element-free meshless methods offer many advantages in modeling variable stiffness materials and structures, e.g. complex geometries with cut-outs, providing accurate stress computations and stable nonlinear behavior modeling. In this study, an element-free Galerkin (EFG) method based modeling framework is developed to model variable stiffness composite plates containing circular holes, specifically for buckling and post-buckling analysis. Weighted orthogonal basis functions are used to further reduce the computational costs of the meshless method. Modeling accuracy for variable stiffness composite structures using node-based fiber angle definitions in this element-free framework are validated against FEM results obtained from ABAQUS. The critical buckling loads of linearly varying fiber composite laminates with different sizes of central circular holes are then computed. Based on the proposed element-free modeling scheme, nonlinear post-buckling equilibrium paths of variable stiffness composite plates are traced using an incremental loading step control method, which enables efficient post-buckling analysis. Changes in both critical buckling loads and nonlinear post-buckling performance with respect to hole size and the linear variation of fiber angles are investigated with the results presented in the form of parametric studies. The numerical results show that stress redistribution remains the major factor driving improvements in the buckling and post-buckling performance of variable stiffness composite plates with cut-outs. The research work presented in this paper for composite plates with linearly varying stiffness provides a basis for the analysis of general variable stiffness soft materials in the future.

Variable Stiffness Composites: Optimal Design Studies

This research work has two main objectives, being the first related to the characterization of variable stiffness composite plates’ behavior by carrying out a comprehensive set of analyses. The second objective aims at obtaining the optimal fiber paths, hence the characteristic angles associated to its definition, that yield maximum fundamental frequencies, maximum critical buckling loads, or minimum transverse deflections, both for a single ply and for a three-ply variable stiffness composite. To these purposes one considered the use of the first order shear deformation theory in connection to an adaptive single objective method. From the optimization studies performed it was possible to conclude that significant behavior improvements may be achieved by using variable stiffness composites. Hence, for simply supported three-ply laminates which were the cases where a major impact can be observed, it was possible to obtain a maximum transverse deflection decrease of 11.26%, a fundamental frequency increase of 5.61%, and a buckling load increase of 51.13% and 58.01% for the uniaxial and biaxial load respectively.

Stability of variable stiffness composite laminates under compressive and shearing follower forces

Dynamic stability characteristics of variable stiffness composite rectangular panels subjected to non-conservative compressive or shear follower loads are investigated here using the shear-deformable finite element method. At the beginning, the veering of modal frequencies are studied by the eigenvalue analysis to classify the divergence and flutter type of instabilities and evaluate the critical loads for isotropic and variable stiffness composite plates under both types of follower forces. Thereafter, the governing finite element equations are reduced into a set of Mathieu-Hill equations employing the modal transformation technique and the method of multiple scales is used to determine the regions of instability of composite laminates under pulsating follower forces. Limited numerical examples are presented to demonstrate the simple and combination (additive/difference) type of parametric instabilities of edge supported and cantilever panels with curvilinear fibers under compressive or shear pulsating follower forces.

Stability behavior of variable stiffness composite panels under periodic in-plane shear and compression

Dynamic stability characteristics of variable stiffness composite plates under periodic in-plane loads are investigated here using a shear-deformable sixteen-node plate bending element. The modal transformation technique is used to reduce the equations of motion for the flexural vibration problem of a rectangular panel into a set of Mathieu-Hill equations. The diagonal and off-diagonal dominance of the parametric coefficient matrix is studied for the representative cases of in-plane loading, i.e., periodic compression/shear in presence or absence of static compression for the isotropic and composite plates. The multi-modal response (participating and accompanying modes) of the shear panels under periodic edge traction is demonstrated through the time history analysis. Then, the method of multiple scales is used to identify the dynamic instability regions corresponding to the simple (single mode flexural vibration) or combination (two mode flexural vibrations) type of resonances in various (isotropic and laminated composite made of straight or curvilinear fibers) shear panels, for which results are not available in the literature.

Nonlinear buckling analysis of variable stiffness composite plates based on the reduced order model

Abstract The variable-stiffness fiber composite plates which have an enhanced design flexibility, largely rely on laminate optimizations to maximize the buckling performance. The corresponding computational efficiency becomes a key issue, in particular when the nonlinear structural behavior is considered. The finite element method based on a full nonlinear analysis is a standard technique for nonlinear structural analysis, however the high computational complexities generated from both the incremental-iterative procedure and the very refined mesh needed for the discrete modeling of curved fibers, are still a decisive cost factor on modern computers. In this work, the Koiter-Newton method is further extended to nonlinear buckling analysis, including the pre and post buckling stage, of variable stiffness composite plates. A four-node quadrilateral element based on the classical laminated plate theory is developed in framework of the von Karman kinematics, for the finite element implementation of the proposed asymptotic method . The reduced order model , with or without imperfections, is constructed using the improved Koiter’s asymptotic expansion , for both the symmetrical and unsymmetrical laminates. The nonlinear response curve of loaded structure can be traced automatically, using the nonlinear predictor and corrections both generated from the reduced order model. This leads to a fairly large step size in the path-tracing process, compared to that for the classical Newton method . The reduced order model largely reduces the computational burden produced by the high-density FE mesh for the varied fiber path. Numerical results indicate the overall high quality and efficiency of the proposed method.