Higher-order theories for the free vibrations of doubly-curved laminated panels with curvilinear reinforcing fibers by means of a local version of the GDQ method

The aim of this paper is to investigate the dynamic behavior of singly and doubly-curved panels reinforced by curvilinear fibers. The Variable Angle Tow (VAT) technology allows the placement of fibers along curvilinear paths with the purpose of improving dynamic performance of plates and shells. The effect of the variation of constants which define analytically the fiber orientation is also investigated by several parametric studies. The Carrera Unified Formulation (CUF) with different thickness functions along the three orthogonal curvilinear directions is the basis of the present theoretical model. Various doubly-curved laminated panels reinforced by curvilinear fibers are analyzed using several structural theories. The Local Generalized Differential Quadrature (LGDQ) method is employed to solve numerically free vibration problems. Compared to the well-known GDQ method from which it descends, the LGDQ is characterized by banded matrices instead of full ones, since the current technique considers only few points of the whole domain. Therefore, the solution of the equation system needs a lower computational effort.     

Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation

The aim of this work is the application of Radial Basis Function (RBF) method to a General Higher-order Equivalent Single Layer (GHESL) formulation for the free vibrations of doubly-curved laminated composite shells and panels. The theoretical development of the present paper is based on the well-known Carrera Unified Formulation. In particular, the fundamental nuclei of a multi-layered doubly-curved shell structure are deducted and explicitly defined. The Differential Geometry (DG) tool has been used to geometrically define each of the structures under consideration: doubly-curved, singly-curved and degenerate shells. The 2D free vibration shell problems are numerically solved using RBFs, where the shape parameters have been optimized using two different algorithms. In fact, the shape parameters of RBFs depend not only on the choice of the radial basis functions, but also on how the points are located on the given computational domain. Modifying the positions of such points, the shape parameters change too. It has been discovered that, once the shape parameters have been optimized for a given grid distribution, they can be rounded off and used for every kind of structure. This is a very important aspect because when, using a fixed parameter, the RBF method becomes a “parameter free” numerical technique. In order to demonstrate the accuracy, stability and reliability of the present methodology, many comparisons are presented with reference to literature results, that are obtained by using Generalized Differential Quadrature method. The above numerical applications of this study are also compared with finite element method solutions.     

Radial basis functions based on differential quadrature method for the free vibration analysis of laminated composite arbitrarily shaped plates

The conventional strong form collocation approach known as Differential Quadrature (DQ) method has been applied in the past to a vast type of engineering problems. It is well-known that its application is strictly limited to regular regions where derivatives are approximated along mesh lines. Generally, its accuracy increases when the number of collocation points is large and the method tends to be stable. However, for some numerical problems several points are needed in order to obtain an accurate solution. Changing the basis functions another numerical technique was developed called Radial Basis Functions (RBFs) method, which has the advantage of approximating derivatives using irregular point distributions and the basis functions depend on the mutual radial distance of the grid points. In order to extend the idea of DQ method to a general case a Radial Basis Function based on Differential Quadrature (RBF-DQ) method has been recently developed. This method merges the advantages of both techniques. Furthermore, this work proposes the application of RBF-DQ when a domain decomposition technique is considered. In this way it will be shown that, using some kind of basis functions the number of grid points per element can be reduced compared to other classical approaches. Furthermore, once the shape parameter is fixed for one case, it is not needed to calculate it again for other applications.     

Thermal uncertainty quantification in frequency responses of laminated composite plates

The propagation of thermal uncertainty in composite structures has significant computational challenges. This paper presents the thermal, ply-level and material uncertainty propagation in frequency responses of laminated composite plates by employing surrogate model which is capable of dealing with both correlated and uncorrelated input parameters. The present approach introduces the generalized high dimensional model representation (GHDMR) wherein diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression is utilized to ensure the hierarchical orthogonality of high dimensional model representation component functions. The stochastic range of thermal field includes elevated temperatures up to 375 K and sub-zero temperatures up to cryogenic range of 125 K. Statistical analysis of the first three natural frequencies is presented to illustrate the results and its performance.     

Influences of shear stresses on the dynamic instability of exponentially graded sandwich cylindrical shells

The aim of present study is to investigate the dynamic instability of exponentially graded (EG) sandwich cylindrical shells under static and time dependent periodic axial loadings using the shear deformation theory (SDT). The modified Donnell-type dynamic instability equations of EG sandwich cylindrical shells based on the SDT are deduced. Then are reduced to Mathieu-Hill equation and by solving the expressions for the boundaries of instability regions of EG sandwich cylindrical shells are obtained. The similar expressions for EG single-layer shell, ceramic-rich shell and metal coated sandwich cylindrical shell on the basis of SDT and classical shell theory (CST) are obtained in a special case. The numerical illustrations concern the influences of compositional profiles of coating layers, shear stresses and geometrical parameters of sandwich cylindrical shells on the boundaries of instability regions. As a check on the accuracy of the present study, the values of the lower and upper boundaries of instability regions are compared with those in the literature.     

Optimisation of sandwich panels with functionally graded core and faces

The distributions of properties across the thickness (core) and in the plane (face sheets) that minimise the interlaminar stresses at the interface with the core are determined solving the Euler–Lagrange equations of an optimisation problem in which the membrane and transverse shear energy contributions are made stationary. The bending stiffness is maximised, while the energy due to interlaminar stresses is minimised. As structural model, a refined zig-zag model with a high-order variation of displacements is employed. Simplified, sub-optimal distributions obtainable with current manufacturing processes appear effective for reducing the critical interfacial stress concentration, as shown by the numerical applications.     

Experimental validation of modal strain energies based damage identification method for a composite sandwich beam

A vibration-based damage identification method, based on changes in modal strain energies before and after occurrence of damage, is presented for a composite sandwich beam. Experiments were performed to obtain the natural frequencies and mode shapes for validation of the presented method. The observed changes in modal strain energies were used for the prediction of existence and location of damage in a composite sandwich beam. Subsequently, these changes were also used to predict damage extents in the two stages. In the first stage, the proposed method is used to approximate the damage extents in the face and core. These were updated to obtain more accurate values of damage extent at the second stage by using the model updating theory. Experimental and numerical results were presented to demonstrate and verify the effectiveness of this method for several single and multiple damage cases. These damage cases also include interactive damage modes. Results indicate that the proposed method is capable of identifying the location and extent of damage in the faces and core of a composite sandwich beam.     

Compressive behaviour of thin-skin stiffened composite panels with a stress raiser

The in-plane compressive behaviour of thin-skin stiffened composite panels with a stress concentrator in the form of an open hole or low velocity impact damage is examined analytically. Drop weight impact in laminated polymer composites causes matrix cracking, delaminations and fibre breakage, which together can seriously degrade the laminate compressive strength. Experimental studies, using ultrasonic C-scan images and X-ray shadow radiography, indicated that the overall damage resembles a hole. Under uniaxial compression loading, 0° fibre microbuckling surrounded by delamination grows laterally (like a crack) from the impact site as the applied load is increased. These local buckled regions continued to propagate, first in discrete increments and then rapidly at failure load. The damage pattern is very similar to that observed in laminated plates with open holes loaded in compression. Because of this resemblance, a fracture mechanics model, developed initially to predict notched compressive strength, was applied to estimate the compression-after-impact (CAI) strength of a stiffened panel; in the analysis the impact damage is replaced with an equivalent open hole. Also, the maximum stress failure criterion is employed to estimate the residual compressive strength of the panel. The unnotched compressive strength of the composite laminate required in the analysis is obtained from a three-dimensional stability theory of deformable bodies. The influence of the stiffener on the compressive strength of the thin-skin panel is examined and included in the analysis. A good agreement between experimental measurements and predicted values for the critical failure load is obtained.     

Dynamic analysis and damping of composite structures embedded with viscoelastic layers

In recent years, it has been found that composites co-cured with viscoelastic materials can enhance the damping capacity of a composite structural system with little reduction in stiffness and strength. Because of the anisotropy of the constraining layers, the damping mechanism of co-cured composites is quite different from that of conventional structures with metal constraining layers. This paper presents an analysis of the dynamic properties of multiple damping layer, laminated composite beams with anisotropic stiffness layers, by means of the finite element-based modal strain energy method. ANSYS 4.4A finite element software has been used for this study. The variation of resonance frequencies and modal loss factors of various beam samples with temperature is studied. Some of these results are compared with the closed-form theoretical results of an earlier published work. For obtaining optimium dynamic properties, the effects of different parameters, such as layer orientation angle and compliant layering, are studied. Also, the effect of using a combination of different damping materials in the system for obtaining stable damping properties over a wide temperature range is studied.     

An effective thickness to estimate stresses in laminated glass beams under dynamic loadings

Finite element models for estimating stresses and displacements in laminated glass elements under dynamic loadings are very time-consuming because (1) many small 3D elements are needed to model accurately all the layers of the sandwich element and (2) the core usually shows a time and temperature dependent behaviour. In the last years, the concept of effective thickness using a quasi-elastic solution has got the attention of the research community because of its simplicity and reasonable level of accuracy achieved in the calculation of laminated glass elements under static loadings. In this paper, a dynamic effective thickness to estimate stresses in laminated glass beams under dynamic loadings in the frequency domain is derived using the correspondence principle. The analytical equations are validated by experimental tests carried out on simply supported and free–free laminated glass beams at different temperatures in the range 20–40 °C.     

Optimal design and parameter estimation of frequency dependent viscoelastic laminated sandwich composite plates

Recent developments in optimization and parameter estimation of frequency dependent passive damping of sandwich structures with viscoelastic core are presented in this paper. A finite element model for anisotropic laminated plate structures with viscoelastic frequency dependent core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach, by considering a higher order shear deformation theory (HSDT) to represent the displacement field of the viscoelastic core, and a first order shear deformation theory (FSDT) for the displacement fields of adjacent laminated face layers. The complex modulus approach is used for the viscoelastic material behaviour, and the dynamic problem is solved in the frequency domain, using viscoelastic material data for the core, assuming fractional derivative constitutive models. Constrained optimization of passive damping is conducted for the maximisation of modal loss factors, using the Feasible Arc Interior Point Algorithm (FAIPA). Identification of the frequency dependent material properties of the sandwich core is conducted by estimating the parameters that define the fractional derivative constitutive model. Optimal design and parameter estimation applications in sandwich structures are presented and discussed.     

Effects of local damage on vibration characteristics of composite pyramidal truss core sandwich structure

The effects of local damage on the natural frequencies and the corresponding vibration modes of composite pyramidal truss core sandwich structures are studied in the present paper. Hot press molding method is used to fabricate intact and damaged pyramidal truss core sandwich structures, and modal testing is carried out to obtain their natural frequencies. A FEM model is also constructed to investigate their vibration characteristics numerically. It is found that the calculated natural frequencies are in relatively good agreement with the measured results. By using the experimentally validated FEM model, a series of numerical analyses are conducted to further explore the effects of damage extent, damage location, damage form on the vibration characteristics of composite pyramidal truss core sandwich structures as well as the influence of boundary conditions. The conclusion derived from this study is expected to be useful for analyzing practical problems related to structural health monitoring of composite lattice sandwich structures.     

Hybrid FE–IDQ–CG method for dynamic parameters estimation of multilayered half-space

As a first endeavor, a hybrid finite element (FE)–incremental differential quadrature (IDQ) method together with the discrepancy principle and the conjugate gradient method (CGM) is used to develop an inverse algorithm for the parameters estimation of the axisymmetric multilayered half-spaces. The approach is based on the measurement of the dynamic transverse displacement at some boundary points of the half-space to estimate the unknown parameters of its layers. Using the accuracy and unconditional stability of the hybrid FE–IDQ method, the direct problem is solved to get the dynamic transverse displacements. After adding some random errors to the obtained results, they are considered as the measured responses by sensors. Then, the conjugate gradient method as a general and robustness optimization technique is employed to minimize the error between the measured and calculated dynamic surface responses at sensor locations. The sensitivity analysis of the displacement field is performed using a semi-analytical method. The applicability and correctness of the proposed hybrid algorithm is demonstrated through different examples by considering the influence of the layers arrangement, the measurement errors and sensor numbers.     

Enhanced Effective Thickness of multi-layered laminated glass

The stiffness and strength of laminated glass, a composite of glass layers bonded together by polymeric interlayers, depends upon shear coupling between the glass plies through the polymer. In the design practice, this effect is commonly considered by defining the effective thickness of laminated glass, i.e., the thickness of a monolith with equivalent bending properties. Various theories have been proposed to calculate such a value for a package of two layers of glass and one polymeric interlayer, but extrapolation to a higher number of layers gives in general inaccurate results. Here, the Enhanced Effective Thickness method, previously proposed for two-glass-layer composites, is extended to the case of laminated glass beams made (i) by three layers of glass of arbitrary thickness, or (ii) by an arbitrary number of equally-thick glass layers. Comparisons with numerical experiments confirm the accuracy of the proposed approach also in these cases.     

A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells

An exact free vibration analysis of doubly-curved laminated composite shallow shells has been carried out by combining the dynamic stiffness method (DSM) and a higher order shear deformation theory (HSDT). In essence, the HSDT has been exploited to develop first the dynamic stiffness (DS) element matrix and then the global DS matrix of composite cylindrical and spherical shallow shell structures by assembling the individual DS elements. As an essential prerequisite, Hamilton’s principle is used to derive the governing differential equations and the related natural boundary conditions. The equations are solved symbolically in an exact sense and the DS matrix is formulated by imposing the natural boundary conditions in algebraic form. The Wittrick–Williams algorithm is used as a solution technique to compute the eigenvalues of the overall DS matrix. The effect of several parameters such as boundary conditions, orthotropic ratio, length-to-thickness ratio, radius-to-length ratio and stacking sequence on the natural frequencies and mode shapes is investigated in details. Results are compared with those available in the literature. Finally some concluding remarks are drawn.     

Experimental and numerical investigation of carbon fiber sandwich panels subjected to blast loading

The objective of this paper is to investigate the structural response of carbon fiber sandwich panels subjected to blast loading through an integrated experimental and numerical approach. A total of nine experiments, corresponding to three different blast intensity levels were conducted in the 28-inch square shock tube apparatus. Computational models were developed to capture the experimental details and further study the mechanism of blast wave-sandwich panel interactions. The peak reflected overpressure was monitored, which amplified to approximately 2.5 times of the incident overpressure due to fluid-structure interactions. The measured strain histories demonstrated opposite phases at the center of the front and back facesheets. Both strains showed damped oscillation with a reduced oscillation frequency as well as amplified facesheet deformations at the higher blast intensity. As the blast wave traversed across the panel, the observed flow separation and reattachment led to pressure increase at the back side of the panel. Further parametric studies suggested that the maximum deflection of the back facesheet increased dramatically with higher blast intensity and decreased with larger facesheet and core thickness. Our computational models, calibrated by experimental measurements, could be used as a virtual tool for assessing the mechanism of blast-panel interactions, and predicting the structural response of composite panels subjected to blast loading.     

The influence of correlating material parameters of gradient foam core on free vibration of sandwich panel

For the sandwich panel with mass density gradient (DG) foam core, the Young's modulus of the core varies with the mass density along the thickness direction. To characterize the correlative effect of Young's modulus and mass density of the DG closed-cell foam material, a simplified formula is presented. Subsequently, based on a high-order sandwich plate theory for sandwich panel with homogeneous core, a new gradient sandwich model is developed by introducing a gradient expression of material properties. Finite element (FE) simulation is carried out in order to verify this model. The results show that the proposed model can predict well the free vibration of composite sandwich panel with the gradient core. Finally, the correlating effects of material parameters of the DG foam core on the natural frequencies of sandwich panel are investigated. It is found that the natural frequencies of sandwich panels decrease as the gradient changes of the DG foam cores increase under the condition of that the core masses keep constant.     

High-order free vibration analysis of sandwich plates with both functionally graded face sheets and functionally graded flexible core

Free vibration analysis of functionally graded material sandwich plates is studied using a refined higher order sandwich panel theory. A new type of FGM sandwich plates, namely, both functionally graded face sheets and functionally graded flexible core are considered. The functionally graded material properties follow a power-law function. The first order shear deformation theory is used for the face sheets and a 3D-elasticity solution of weak core is employed for the core. On the basis of continuities of the displacements and transverse stresses at the interfaces of the face sheets and the core, equations of motion are obtained by using Hamilton’s principle. The accuracy of the present approach is validated by comparing the analytical results obtained for a degradation model (functionally graded face sheets and homogeneous flexible core) with ones published in the literatures, as well as the numerical results obtained by finite element method and good agreements are reached. Then, parametric study is conducted to investigate the effect of distribution of functionally graded material properties, thickness to side ratio on the vibration frequencies.     

Shear coupling effects of the core in curved sandwich beams

We consider a composite package formed by two curved external Euler-Bernoulli beams, which sandwich an elastic core with negligible bending strength but providing the shear coupling of the external layers. This coupling considerably affects the gross response of the composite structure. There is an extensive literature on straight sandwich beams of this type, but very little attention has been paid to the effects of curvature. Here, an analytical linear elastic model is proposed for beams with arbitrary variable curvature. Equilibrium equations and boundary conditions are obtained through a variational approach. Useful simplifications are possible for the case of moderately curved beams and beams with constant curvature.