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In this article, natural frequencies of variable stiffness composite laminate (VSCL) plates with curvilinear fibers submerged in fluid with different depths of submersion are studied. In each layer of VSCL plate, the fiber-orientation angle changes linearly with respect to the horizontal coordinate. The problem is solved by using the hierarchical finite element method (HFEM) (p-version), with seven degrees of freedom per node and the higher-order shear deformation theory (HSDT) for C⁰ continuity. The plate theory ensures a zero shear–stress condition at the top and bottom surfaces of plate and does not require a shear correction factor. The fluid/structure interaction is examined in this study, in which the effect of fluid is assumed as pressure acting on the plate. The mathematical model of fluid is given by employing velocity potential and Bernoulli equation. The obtained frequency parameters shown an excellent agreement when compared with experimental and analytical results of available solutions in literature. Accurate results are obtained by increasing the degree of polynomial shape function of HFEM. In order to examine the effect of fluid on VSCL plate, different cases are studied to find out the effect of plate thickness, curvilinear fiber angle and depth of submerged plate.

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... They also employed a neural network approach to investigate the stochastic vibration of composite plates reinforced with curvilinear fibers. To the best of the authors' knowledge, the interaction of fluid with either VSCL or HCL plates has only been studied in [41]. Bendahmane et al. [41] investigated the free vibration of a VSCL plate submerged in fluid domain. ...

... To the best of the authors' knowledge, the interaction of fluid with either VSCL or HCL plates has only been studied in [41]. Bendahmane et al. [41] investigated the free vibration of a VSCL plate submerged in fluid domain. In their analysis, the plate was located at the bottom side of the fluid domain, and fluid sloshing influences were ignored. ...

... It must be mentioned that the elastic coefficients Q ij (k) appearing in Eq. (4) are functions of the fiber path since φ depends on the spatial coordinates. In the current research, the fiber path is assumed to vary linearly along the x-axis, and it is given as [36,41]. ...

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.

... The classical theory-based analytical and FE solution was presented to optimize the structural geometry of the layered plates to obtain the maximum fundamental frequency using a genetic algorithm [58]. The modal responses of laminate having variable stiffness immersed in the fluid were analyzed using a FE approach based on HSDT [59]. A nonpolynomial HSDT based FE method was implemented to investigate the nonlinear frequency of the layered pates using Green-Lagrange stain [34]. ...

... The classical theory-based analytical and FE solution was presented to optimize the structural geometry of the layered plates to obtain the maximum fundamental frequency using a genetic algorithm [58]. The modal responses of laminate having variable stiffness immersed in fluid are analyzed using the FE approach based on HSDT [59]. The influences of viscoelastic layers on the vibration behaviour of the fiber metal laminate was examined using HSDT in which the geometrical and material nonlinearity was introduced using von-Karman's strain and Jones-Nelson theory, respectively [166]. ...

This article reported the overall review on modeling (mathematical/simulation), analysis (linear/nonlinear), and prediction of structural responses (static, vibration, transient, etc.) of cutout abided laminated composite structure under thermo-mechanical load. In general, the discussion has been made available for a different kinds of mid-plane kinematics adopted in the past, including the cutout parameters (shape, size, position, and orientation). A few selected research articles and the knowledge gap (past and present) have been discussed critically. Additionally, the subsequent steps considered by different researchers to bridge the gap have also been indicated. Finally, the state of the art of cutout-borne layered composite structures, including the recommendations, are provided to give a future research aspect and a helpful resource to the researchers who want to expand their study into these areas. This article includes research trends in the last ten years and a few earlier works related to cutout-borne structure.

... The influence of different ply orientation in square and rectangular laminated plates was studied for different cases of dynamic loading. In [8][9][10], free vibrations of laminated plates under different boundary conditions were studied. The propagation of elastic waves in laminated composite with special emphasis on Lamb waves was also analyzed [11][12][13]. ...

... The geometric and material characteristics of the composite plate are given (8). Static analysis was performed for two cases of ply orientation of symmetrical cross ply laminate (0°/90°/90°/0° и -45°/+45°/+45°/-45°) by using the finite element method. ...

The subject of analysis in this paper are laminated composites as an always popular topic in the field of composite materials. The introductory part of the paper presents a brief overview of the current state of research in this area and points out further directions of research related to this very interesting topic. The classification of laminated composite plates from the aspect of different criteria are described in detail. As the subject of the paper is stress-strain analysis of laminated composite, the expression for the generalized Hooke's law of orthotropic class of material symmetry is given. The terms of the constitutive matrix are expressed through engineering constants. At the end, the bending analysis of the rigidly supported laminated plate under a sinusoidal load was performed. The mentioned analysis was conducted using software based on the finite element method. A comparative analysis of the plate with the different ply orientation (0°/90°/90°/0° and -45°/45°/45°/-45°) was done. The obtained results were analyzed and appropriate conclusions were made.

... The vibration behavior of VS composites is signi¯cantly in°uenced by altering VS¯ber orientation, which can be due to the spatial variation in the¯ber orientation. 52,53 It is veri¯ed that even small changes in the¯ber orientations can lead to signi¯cant changes not only in the linear vibration modes but also can lead to very di®erent nonlinear dynamic response, where control on modal interactions are possible. 54 It has also been demonstrated that curvilinear ber paths can have a noticeable e®ect on the natural frequencies and corresponding mode shapes of vibration of cylindrical shells, much larger than what occurs in plates. ...

Multistable laminates have been actively studied in recent years due to its potential applications in morphing and energy harvesting devices. Variable stiffness (VS) bistable laminates provide opportunities for further improvements in design space in comparison with constant stiffness bistable laminates. The snap-through process involving shape transition between the stable configurations is highly nonlinear in nature and exhibits rich dynamics. Exploiting the dynamic characteristics during the snap-through transition is of considerable interest in designing the morphing structural components. In this paper, we present a semi-analytical model based on Rayleigh–Ritz approach in conjunction with Hamilton’s principle to predict the natural frequencies of bistable VS laminates. The obtained results are compared with the results from the full geometrically nonlinear finite element (FE) model. The proposed FE model is further extended to study the dynamics of VS laminates subjected to external forces with different amplitudes. Subsequently, a parametric study is performed to explore the effect of different curvilinear fiber alignments on natural frequencies, mode shapes, free vibration characteristics and forced vibration characteristics (single-well and cross-well vibrations).

... Hosseini-Hashimi et al. [17] then investigated the effect of finite fluid depth levels on free vibration of Mindlin plates, which was extended by Cho et al. [18] in a study of stiffened plates. Extensive work was also found in more complex fluid-plate scenarios such as considering the effect of a crack on a submerged plate [19,20,21], and plates of different material compositions [22,23,24]. ...

Predictions of the vibroacoustic response of a point-force excited baffled thin rectangular plate immersed in a heavy fluid and near a free surface are presented using an analytical model. The equations of motion are solved by Fourier analysis, where the eigenfunctions of plate vibration form the basis of spatial expansion for fluid loading. Vibroacoustic indicators, including the plate velocity, acoustic pressure, and acoustic power, are predicted using the analytical approach and verification is performed by comparison with finite element simulations. The results have shown that variations in the height of the free surface can have a significant effect on these indicators. From the vibration response, added mass effect due to heavy fluid loading is altered and further investigated with the explicit evaluation of an added mass ratio for different free surface heights for the first five plate modes. For a given height of a free surface, standing waves can form between the free surface and baffled plate at specific excitation frequencies and slightly alters the acoustic pressure spectra. This condition also presents an effect on the acoustic power, where the first standing wave frequency dictates the efficient sound radiation to the far field.

... The individual layers of the laminates may be orthotropic, transversely isotropic and anisotropic, depending on the micromechanical parameters. Some of the researches carried out in the field of deflection [10][11][12]76], buckling [13][14][15], vibration [16][17][18][19][20][21][22][23][24][25], impact [26][27][28][29][30][31][32][33][34][35][36], and failure analyses [37][38][39][40][41][42][43][44][45] of such composite plates are hereby mentioned. A critical review of such investigations reveals that the majority of such research activities focus on the development of laminate theories and experimental techniques deterministically. ...

The present paper proposes a surrogate-assisted moment-independent stochastic sensitivity analysis of laminated composite plates for establishing a unified measure in the case of multi-objective performances. With the advancements in artificially engineered structural systems spanning across different length scales, it has become more common to design composite structures for multi-objective performances like the criteria of deflection, buckling and vibration of multiple modes, different impact parameters, and failure. Normally sensitivity analysis is carried out separately and individually for different such performance parameters. This paradigm is no more suitable for advanced multi-functional structures like composite laminates. In this article, we propose an efficient unified sensitivity analysis approach based on weighted relative importance of different performance parameters by introducing the notion of engineering judgment. A moment-independent sensitivity analysis is proposed here based on finite element modeling of composites in conjunction with the Least Angle Regression assisted Polynomial Chaos Expansion (PCE) to achieve computational efficiency without compromising the outcome. Such surrogate-assisted finite element approaches are particularly crucial for computationally intensive multi-objective systems like composites. The layer-wise unified sensitivity quantification of laminated composites considering multi-functional objectives, as presented here, would lead to more optimized designs and better quality control while manufacturing the complex advanced structural systems.

... Watts et al. [33] proposed the semi-analytical method that relied on the Galerkin to investigate the vibration response of the non-rectangular bottom of tanks filling with liquid. Bendahmane et al. [34] employed the hierarchical finite element method based on the higher-order shear deformation theory to study the natural frequency of composite plates with variable stiffness submerged in fluid. Khorshidi et al. [35e37] used the Rayleigh-Ritz method with finite Fourier series to determine the natural frequencies of square plates in contact with a bounded fluid. ...

This paper for first time proposes an isogeometric analysis (IGA) for free vibration response of bi-directional functionally graded (BDFG) rectangular plates in the fluid medium. Material properties of the BDFG plate change in both the thickness and length directions via power-law distributions and Mori-Tanaka model. The governing equation of motion of BDFG plate in the fluid-plate system is formulated basing on Hamilton's principle and the refined quasi three-dimensional (3D) plate theory with improved function f(z). The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to determine the added mass. The discrete system of equations is derived from the Galerkin weak form and numerically analyzed by IGA. The accuracy and reliability of the proposed solutions are verified by comparing the obtained results with those published in the literature. Moreover, the effects of the various parameters such as the interaction boundary condition, geometric parameter, submerged depth of plate, fluid density, fluid level, and the material volume control coefficients on the free vibration behavior of BDFG plate in the fluid medium are investigated in detail. Some major findings regarding the numerical results are withdrawn in conclusions.

... The effect of types of boundary conditions on the critical velocities of plates subjected to infinite-height flowing fluid was evaluated by their study. Bendahmane et al. [40] modeled the influence of fluid-depth on the dimensionless vibrational frequencies of composite laminate rectangular plates submerged in fluid. They concluded that enlarging the degree of orthotropy of the plate leads to an increment of system dimensionless natural frequencies. ...

For the first time in the present investigation, aiming to improve the performance of mono-gyroscopic systems, the stability and vibrational behavior of flowing fluid coupled functionally graded (FG) porous rectangular plate resting on different foundations subjected to thermal environments are studied. Applying the higher-order shear deformation theory (HSDT) incorporated in nonlocal elasticity theory results in the governing equations of the system. Also, to explain and analyze the effect of various key factors such as nonlocal parameter, power-law index of FG material, the volume fraction of porosity, the height and densities of fluid, structure aspect ratio, boundary conditions, types of substrates, and thermal environments on the dynamic characteristics and critical velocities of the system, a comprehensive parametric study is performed. By utilizing the generalized differential quadrature method (GDQM), the governing size-dependent equations are solved numerically, and the natural frequencies in addition to divergence and flutter corresponding velocities are extracted. To validate the present work, a comparative study is performed between the obtained results and outcomes reported in the engineering literature.

... Serdoun and Hamza-cherif [29], Hachemi [30], studied the vibration of sandwich composite plates reinforced with parabolic fibers based on the HSDT-C 1 and HSDT-C theories respectively where both used the p-version of finite element method. Bendahmane et al. [31] employed the HFEM for improving the performance of the VSCL plates immersed in fluid based on higher-order C theory. Researches on the vibration of composite beams with variable stiffness are very limited. ...

In this paper, the free vibration analysis of composite laminated beams reinforced with parabolic fibers are studied on the basis of the equivalent single layer theory (ESLT) using the isogeomtric analysis method. In the composite material with variable stiffness (VSCL), each layer is reinforced by curvilinear fiber, while in the traditional composite materials with constant stiffness (CSCL) each layer is reinforced by straight fiber. The Equivalent Single Layer Theory (ESLT) given by the Continuum-based Timoshenko beam theory (CTBT) is combined with the isogeometric analysis, in which twisting and stretching effects are considered. The differential equations of motion governing the dynamics of stretching, shearing, bending and twisting composite beam are derived using the Hamilton principle. A new isogeometric composite beam element with six degrees of freedom per control point is developed and used to find natural frequencies of variable stiffness composites beams with parabolic fibers. In this new model, the effects of transverse shear deformation, rotary inertia, and the coupling effect due to the lamination of composite layers are included. Results are obtained for a number constant stiffness composite beam. The results confirm that the solutions converge as the number of elements or the degrees of basic functions are increased. Highly accurate values are obtained with the use of a very few degrees of freedoms, in which h-, p- and k-refinement are used in the convergence analysis. New numerical results of comparison study between the variable stiffness composite beams and constant stiffness composite beams are investigated. Next, parametric study is presented to investigate the impact of orientation angle of parabolic fiber, the stacking sequences, number of layers, boundary conditions, modulus ratio and length to mean diameter ratios on the natural frequencies of the variable stiffness composite beams. The solutions of variable stiffness composite beams are provided as benchmark for future studies.

The effect of boundary conditions on the vibrations of variable stiffness composite laminates (VSCL) with curvilinear fibers is particularly interesting because fiber orientations at the plate edges and in the plate’s domain can differ. Since real boundaries are not the ideal limit cases usually assumed in the literature, a more correct model can be achieved by representing boundaries as elastically restrained edges. This approach is followed in this work, which investigates the combined effect of fiber orientation and boundary stiffness on the geometrically non-linear forced vibrations of rectangular VSCL plates. The boundary stiffness is adjusted using experimental data.

This work presents a refinement of the finite strip method (FSM) based on the Carrera unified formulation (CUF) and is applied to free vibration analysis of variable stiffness composite laminated (VSCL) plates. VSCL plates considered here are composed of layers with curvilinear fibers in which their orientation angle changes linearly with respect to one of the in-plane coordinates. The FSM permits the plate to be divided into some finite strips connected along the so-called nodal lines. The conventional finite strip method involves an approximation of the solution of a plate problem by using continuously harmonic series in the longitudinal direction along the nodal lines and simple polynomial shape functions (the usual beam shape functions) in the transverse direction of the plate. However, refined finite strip model that presented in this paper allows one to express the unknown variables as a set of thickness functions that only depend on the thickness coordinate and the corresponding variable that depends on the in-plane coordinates which involve continuously harmonic series and polynomial shape functions. Thanks to this new finite strip model, the shape functions used in the transverse direction can be chosen as 1D bar shape functions, and also, all three components of the displacement field are considered in each nodal line. Moreover, based on this approach, three-dimensional displacement field is approximated in a compact form as a generic N-order expansion. Therefore, explicit expressions of fundamental nuclei of finite strip matrices are obtained in a compact form and presented here. The results obtained by the present formulation for VSCL plates are compared with those obtained from published data. The results show that this kind of strip is efficient and useful. Further, several numerical examples are presented, and the effect of the order of expansion, the number of strips, curvilinear fiber angle, and various boundary conditions are examined.

The optimal stacking sequence design for the maximum frequencies of the lowest three modes of symmetric variable stiffness composite laminated beams in air and in water is investigated for the first time using a layer-wise optimization method. Two fiber orientation angles in each layer are considered as design variables. The shifted path technique is used to construct the fiber paths. The p-version of the finite element method is used in conjunction with the Euler–Bernoulli beam theory coupled with torsion to calculate the frequencies. The numerical results are validated by comparison with published results for a single-layer constant stiffness composite beam with rectilinear fibers in air and in water. Extensive results are presented, which may serve as a benchmark for future research. Furthermore, a parametric study is performed showing the effects of geometrical and mechanical parameters and boundary conditions on the optimal solutions.

The p-version of the finite element method based on the first-order shear deformation theory (FSDT) is applied, to the free vibration of symmetric variable stiffness composite laminated (VSCL) elliptical plates, with curvilinear fibers. A curved hierarchical rectangular finite element is developed and used to model the elliptical plate. The geometry is described accurately by the blending function method. The element stiffness and mass matrices are integrated numerically using the Gauss-Legendre quadrature. The equations of motion are derived employing Lagrange’s method. The numerical results are validated by means of convergence tests and compared with available data in the literature, for isotropic and constant stiffness composite laminated (CSCL) circular and elliptical plates. The contour plots of the fundamental frequency as a function of the fiber orientation angles are presented for clamped and soft simply supported symmetric VSCL elliptical plates. The effect of thickness ratio, aspect ratio, boundary conditions, and mechanical parameters on the frequencies of the lowest five modes is investigated and discussed. The mode shapes are examined by considering the effect of the fiber orientation angles and stacking sequences.

Partly filled rectangular containers are widely used for liquid transportation and storage in many industrial and civil applications. Undesirable liquid sloshing in these structures can highly degrade their reliable and safe operations. This paper presents a rigorous 3D hydro-elasto-dynamic analysis for suppression of transient liquid sloshing in a rigid-walled rectangular parallelepiped container that is equipped with a smart piezo-sandwich free-floating rectangular panel. The problem formulation is based on the linear water wave theory, the thin piezo-sandwich floating plate model, the pertinent structure/liquid compatibility condition, and the active damping control strategy. The controller gain parameters are systematically tuned using a standard MOPSO algorithm with conflicting objective functions. The key hydro-electro-elastic response parameters are calculated and discussed for three different external excitations, namely, a bidirectional seismic event, an oblique planar base acceleration, and a distributed impulsive floating roof excitation. Effectiveness of proposed active floating panel control configuration in remarkable suppression of the main hydro-elastic parameters is established, essentially regardless of liquid depth and loading configuration. Limiting cases are considered and validity of model is demonstrated against available data as well as by comparison with the results of a general finite element package.

AbstractThis paper presents the free vibration analysis of composite thick rectangular plates coupled with fluid, The governing equations for a thick rectangular plate are analytically based on Reddys higher order shear deformation theory (HSDT). The plate theory ensures a zero shear-stress condition at the top and bottom surfaces of the plate and do not requires a shear correction factor. Although the plate theory is quite attractive but it could not be used in the finite element analysis. This is due to the difficulties associated with the satisfaction of the C1 continuity requirement. To overcome this problem associated with Reddys HSDT, a new C1 HSDT p-element with eight degrees of freedom per node is developed and used to find natural frequencies of thick composite plates.
Whereas the velocity potential function and Bernoullis equation are employed, to obtain the fluid pressure applied on the free surface of the plate. The simplifying hypothesis that the wet and dry mode shapes are the same, is not assumed in this paper.
A comparison is made with the published experimental and numerical results in the literature, showing an excellent agreement. natural frequencies of the plate are presented in graphical forms for different fluid levels, aspect ratios, thickness to length ratios and boundary conditions.

This paper presents a novel and advanced finite element formulation of the structural-acoustic problem involving thin and thick multilayered composite plates coupled with a cavity. Exploiting the Carrera’s unified formulation, many plate and fluid-structure interface elements based on different kinematic models including higher-order equivalent single-layer and layerwise theories are developed within a single mathematical framework. Accordingly, a large number of vibro-acoustic models can be easily obtained and selected according to the accuracy requirements of the application. In particular, it is shown that refined models can be adopted in those cases where models relying on traditional or low-order plate theories fail in providing the correct estimation of the fluid-structure coupling. The proposed formulation is also validated with respect to some reference cases available in the literature.

Large deflection and stresses of variable stiffness composite laminated (VSCL) plates with curvilinear fibres are studied. In each ply of these plates, the fibre-orientation angle changes linearly with respect to the horizontal coordinate. The manufacturing restrictions that exist regarding the fibre curvatures in this type of laminates are taken into account. To carry out the analyses, a new p-version finite element, which follows third-order shear deformation theory, is employed. Deflections, normal and transverse (with constitutive and equilibrium equations) stresses are determined as functions of tow-orientation angles in the non-linear regime.

A skew p-element is developed for the nonlinear free vibration of variable stiffness symmetric skew laminates. The governing equations are based on thin plate theory and Von Karman strains. The fundamental frequencies and normal modes are computed for fully clamped edge conditions. The equations of motion are derived using Lagrange's method. By employing the harmonic balance method, the transformation from time to frequency domain is facilitated. The nonlinear equations are solved using the iterative technique known as the linearized updated mode method. The numerical results are validated with the help of convergence tests and comparisons with published data. New results are presented for variable stiffness symmetric skew laminates with different fiber configurations showing the effects of variation in skew angle on frequency, normal mode, and degree of hardening.

The dynamic response of flat horizontal plates vibrating in air and under water has been investigated experimentally and analytically. The effect of the boundary conditions and the depth of submergence of the plates has been studied. Excellent agreement was obtained between the analytical solution and the experimental results. An approximate expression for the evaluation of the modal added masses for rectangular plates has been derived. This can be used to estimate the natural frequencies of plating constituting the internal structure within the cargo spaces in tankers and liquid cargo carriers.

Deflection and damage onset of variable stiffness composite laminated (VSCL) rectangular plates with curvilinear fibres, under static and dynamic loads, are investigated. The VSCLs here studied are characterized by the fact that, in each ply, the fibre-orientation angle changes linearly with respect to the horizontal coordinate. A recently developed p-version finite element, which follows third-order shear deformation theory (TSDT), is employed. Large deflections are considered, hence, the analysis is in the geometrically non-linear regime. To predict damage onset, Tsai-Wu criterion and an associated damage onset index, described in this paper as a safety factor, are used. It is shown that geometrical non-linearity affects stresses and the location of damage onset. VSCL and CSCL (constant stiffness composite laminated) plates subjected to static, dynamic, and impact loads, are compared.

A new p-version finite element based on a zig-zag layerwise theory is developed. With the proposed model, one can accurately determine the behaviour of laminated composites with different material properties, straight or curvilinear fibres. Comparative studies with published results and with results from finite element software Abaqus are performed to verify the new model. Deflections in the linear and non-linear regimes are then calculated for several constant and variable stiffness composite laminates (VSCL), in which the fibre orientation angle varies linearly in each layer. In a few cases, VSCL plates show better performance in comparison to CSCL plates both in the linear and non-linear regimes. It is also found that it is possible to take advantage of VSCL by varying fibre orientations only in some plies of a laminate, hence reducing manufacturing costs, as well as the amount of gaps and overlaps.

The geometrically nonlinear free vibration of laminated composite rectangular plates with curvilinear fibers is investigated. The assumptions of Von Karman's nonlinear thin plate theory are made. The problem is solved numerically using the hierarchical finite element method. The nonlinear equations of free motion are mapped from the time domain to the frequency domain using the harmonic balance method. The resultant nonlinear equations are solved iteratively using the linearized updated mode method. Results for the fundamental linear and nonlinear frequencies and associated mode shapes are obtained for fully clamped laminated composite square plates composed of shifted curvilinear fibers. The efficiency and accuracy of the hierarchical finite element technique is demonstrated through convergence and comparison studies. Contour plots of fundamental linear and nonlinear frequencies as a function of fiber orientation angles are presented. The fiber orientation angles and layup sequence are shown to affect the degree of hardening and mode shapes.

Recently developed inverse hyperbolic shear deformation theory by the authors is extended to analyze the free vibration response of laminated composite and sandwich plates. Euler–Lagrange equations are derived employing the principle of virtual work for the dynamic problem. A Navier type and finite element solutions are proposed to obtain the free vibration response of laminated composite and sandwich plates. A C0-continuous isoparametric biquadratic-quadrilateral serendipity element is used for the finite element solution of generalized higher order shear deformation theory so as to ensure its applicability to general laminates subjected to different combinations of boundary conditions. Higher modes of vibration are obtained for laminated cross-ply and angle-ply plates and efficiency of the theory is ensured by comparing the results with the existing results. It is observed that both analytical and finite element solutions with the present theory are capable for accurate prediction of the free vibration response.

This paper investigates the effects of surrounding boundaries on the free vibration response of fully and partially submerged cantilevered composite plates and how these effects change due to material anisotropy. The results show that added mass significantly reduces the natural frequencies of cantilevered marine structures as a function of relative submergence depth, more so for composite plates than for steel plates because of the much lower ratio of effective structural mass to hydrodynamic added mass. Added mass effects are most dramatic for partially submerged plates as the plates move from being above to completely beneath the free surface. Free surface effects are shown to become negligible for a fully submerged plate parallel to the free surface when the depth of submergence exceeds 50% of the plate length. Solid boundaries are found to have limited effects for fully-submerged plates near a wall, where maximum decreases in resonance frequencies due to increases in added mass are only a few percent.

It is the objective of this work to analyze vibrations of variable stiffness composite laminated plates (VSCL), and investigate the differences between the oscillations of these plates and traditional laminates. The analysis is based on numerical experiments and a new p-version finite element with hierarchic basis functions, which follows first order shear deformation theory and considers geometrical non-linearity, is derived. Considering first linear oscillations, the natural frequencies and mode shapes of different VSCL are computed and compared with the ones of constant stiffness laminates. The linear natural frequencies of the present model are also compared with the ones computed using a recently developed higher-order model for VSCL. After, numerical tests are carried out in the time domain and, for the first time in VSCL, taking geometric non-linearity into account, to investigate the response to external forces. The non-linear ordinary differential equations of motion are solved by Newmark’s method. It is verified that the variation of the fibre orientation can lead to significant differences in the amplitudes of the non-linear response.

The coupled equations of motion of plates carrying liquids, the free surfaces of which are parallel to the plate, are developed. The liquid is treated as incompressible, with free surface oscillations. For a simply supported rectangular plate, carrying liquid with reservoir conditions at its edges, a closed form solution of the natural frequencies and modes of the plate-liquid combination is developed. For each wavenumber set (m, n), two natural frequencies exist. The first corresponds to a liquid motion dominated natural mode, and the second, higher one to a plate motion dominated mode. The harmonic response of the plate-liquid system to a dynamic pressure distribution (and also point load) on the plate is expressed in terms of the orthogonal plate-liquid modes. Numerical examples are given.

The present investigation is concerned with free vibration analysis of composite plates in the presence of cutouts undergoing large amplitude oscillations. The Ritz finite element model using a nine-nodedC0continuity, isoparametric quadrilateral element along with a higher order displacement theory which accounts for parabolic variation of transverse shear stresses is used to predict the dynamic behavior. Results have been obtained for laminated plates with various cutout geometries such as square, rectangle, circle and ellipse in the large amplitude range. Backbone curves are drawn for various boundary conditions and aspect ratios of the cutout.

In this paper, natural frequencies and vibrational mode shapes of variable stiffness composite laminate (VSCL) plates with curvilinear fibers are studied. In each ply of this rectangular VSCL, the fiber-orientation angle changes linearly with respect to the horizontal coordinate. To define the modes of vibration of the laminates, a new p-version finite element, which follows third-order shear deformation theory (TSDT), is employed. The convergence properties of this new element are investigated. Taking manufacturing restrictions regarding the fiber curvatures into account, maps of natural frequencies as functions of tow-orientation angles are determined in demonstrative examples. It is verified that the use of curvilinear fibers instead of the traditional straight fibers introduces a greater degree of flexibility, which can be used to adjust frequencies and mode shapes.

The influence of the fluid medium in modifying the dynamic characteristics of composite plates considering fluid-structure coupling effect is discussed in this paper. The complete plate-fluid system is discretised as an assemblage of finite elements assuming fluid pressure and generalised displacements as the nodal unknowns for the fluid and structural domain, respectively. The resulting coupled equations of motions for these two domains in the absence of any viscous damping are then integrated using Newmark’s algorithm. Transient analysis of both submerged isotropic and composite plates are carried out to study the effect of submergence on the dynamic characteristics namely, natural frequencies as well as amplitudes and period of response.

The dynamic pressure distribution on a rectangular plate attached to a rigid wall and supporting an infinitely large extent of fluid subjected to a harmonic ground excitation is evaluated in the time domain. Governing equations for the fluid domain are set considering the compressibility of the fluid with negligibly small change in density and a linearized free surface. A far boundary condition for the three-dimensional fluid domain is developed so that the far boundary is truncated at a closer proximity to the structure. The coupled problem is solved independently for the structure and the fluid domain by transferring the acceleration of the plate to the fluid and pressure of the fluid to the plate in sequence. Helmholtz equation for the three-dimensional fluid domain and Mindlin's theory for the two-dimensional plate are used for the solution of the interacting domains. Finite element technique is adopted for the solution of this problem with pressure as nodal variable for the fluid domain and displacement for the plate. The time dependent equations are solved in each of the interacting domain using Newmark-β method. The effectiveness of the technique is demonstrated and the influences of surface wave, exciting frequency and flexibility of the plate on dynamic pressure are investigated.

A C0continuous finite element model having five- and seven-degres-of-freedom (DOF) has been developed for the free vibration analysis of laminated coposite plates, together with a higher order shear deformation theory (HSDT) to account for the parabolic variation of transverse shear stresses through the thickness and linear variation of the normal stresses. The displacement field has been presented in such a way that a C0continuous element would be sufficient to represent the plate behaviour. By choosing appropriate coefficients involved in the displacement field model, conformity with either the first order shear deformation theory (FSDT) or the HSDT can be achieved. Results are presented for rectangular, antisymmetric, angle-ply laminates, as obtained by using both FSDT and HSDT, highlighting the effect of material properties, fiber orientation, number of layers, aspect ratio and side to thickness ratio on the fundamental frequency. Computed results are compared with three-dimensional elasticity theory and closed form solution and are found to be in good agreement. A comparative study of both FSDT and HSDT results that FSDT is sufficient to represent the behavior of plates up to a side a thickness ratio of approximately 5·0.

The natural frequencies of annular plates on an aperture of an infinite rigid wall and in contact with a fluid on one side are theoretically obtained by using the added mass approach. The fluid is assumed to be incompressible and inviscid and the velocity potential describes its irrotational motion. The Hankel transform is applied to solve the fluid–plate coupled system; boundary conditions are expressed by integral equations. Mode shapes are first assumed not to be modified by the fluid. Accurate numerical results are given for different plate boundary conditions; they are suitable for engineering applications. The accuracy of the assumed-modes approach is theoretically studied by using the Rayleigh–Ritz method that removes the simplifying hypothesis that dry and wet mode shapes are the same. Eigenfunctions of the plate vibrating in vacuum are assumed as admissible functions and the Rayleigh quotient for coupled vibration is used to obtain a Galerkin equation. It was found that the fundamental mode and frequency, for all the plate boundary conditions considered, is well estimated by the assumed-modes approach; higher modes are computed with less accuracy by this formula and for some enhanced applications the Rayleigh–Ritz approach is necessary.

In this study Free vibration analysis of vertical rectangular Mindlin plates resting on Pasternak elastic foundation and fully or partially in contact with fluid on their one side is investigated for different combinations of boundary conditions. The plate is assumed to be one of vertical rectangular walls of a container in contact with fluid. In order to analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate are expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions in a Cartesian co-ordinate system. The method of separation of variables and the method of Fourier series expansion is used to model fluid and to obtain the exact expression of the motion of fluid in the form of integral equations. The fluid domain is finite in depth and width but infinite in the length direction. To demonstrate the accuracy of the present solution, convergence study is first carried out and then a few comparison studies are carried out with the available data in the literature. Finally, natural frequencies of rectangular plates are presented in tabular and graphical forms for different fluid levels, foundation parameters, aspect ratios, thickness to width ratios and boundary conditions.

In This study hydrostatic vibration analysis of a laminated composite rectangular plate partially contacting with a bounded fluid is investigated. Wet dynamic transverse displacements of the plate are approximated by a set of admissible trial functions which are required to satisfy the clamped and simply supported geometric boundary conditions. Fluid velocity potential satisfying fluid boundary conditions is derived and wet dynamic modal functions of the plate are expanded in terms of finite Fourier series for compatibility requirement along the contacting surface between the plate and the fluid. Natural frequencies of the plate coupled with sloshing fluid modes are calculated using Rayleigh–Ritz method based on minimizing the Rayleigh quotient. The proposed analytical method is validated with available data in the literature. Using numerical data provided, effect of different parameters including boundary conditions, aspect ratio, thickness ratio, fiber orientation, material properties of the laminas and dimensions of the tank on the plate natural frequencies are examined and discussed in detail.

The present study is concerned with the free vibration analysis of a horizontal rectangular plate, either immersed in fluid or floating on its free surface. The governing equations for a moderately thick rectangular plate are analytically derived based on the Mindlin plate theory (MPT), whereas the velocity potential function and Bernoulli’s equation are employed to obtain the fluid pressure applied on the free surface of the plate. The simplifying hypothesis that the wet and dry mode shapes are the same, is not assumed in this paper. In this work, an exact-closed form characteristics equation is used for the plate subjected to a combination of six different boundary conditions. Two opposite sides are simply supported and any of the other two edges can be free, simply supported or clamped. To demonstrate the accuracy of the present analytical solution, a comparison is made with the published experimental and numerical results in the literature, showing an excellent agreement. Then, natural frequencies of the plate are presented in tabular and graphical forms for different fluid levels, fluid densities, aspect ratios, thickness to length ratios and boundary conditions. Finally, some 3-D mode shapes of the rectangular Mindlin plates in contact with fluid are illustrated.

The presence of the liquid free surface has a significant effect upon the dynamic plate characteristics (frequencies) only when the surface is less than about one-half span length from the plate. The effect of the liquid free surface on the resonant frequency of a vertical surface peircing plate is highly dependent upon the relationship between the depth of immersion of the plate and the mode of vibration. The overall damping of cantilever plate vibration is increased significantly in water as compared with air. The results of this study would indicate that for those problem areas in the field of Naval dynamics wherein elastic plate vibrations are involved, and for which predictions of resonant frequencies and damping factors are derived, knowledge is either inadequate or unavailable. For cantilever plates, the present results should enable one to predict resonant frequencies in practical applications with some degree of confidence; but damping factors have been obtained only to within order-of-magnitude and to show trends. Clearly, similar information is very much needed for other plate geometries (i.e., skewed) and support conditions.

A bending theory for anisotropic laminated plates developed by Yang, Norris,and Stavsky is investigated. The theory includes shear deformation and rotary inertia in the same manner as Mindlin's theory for isotropic homogeneous plates. The governing equations reveal that unsymmetrically laminated plates display the same bending-extensional coupling phenomenon found in classical laminated plate theory based on the Kirchhoff assumptions. Solutions are presented for bending under transverse load and for flexural vibration frequencies of symmetrical and nonsymmetrical laminates. Good agreement is observed in numerical results for plate bending as compared to exact solutions obtained from classical elasticity theory. For certain fiber reinforced composite materials, radical departure from classical laminated plate theory is indicated. (Author-PL)

Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton's principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates.

For coupled vibration analyses of prismatic shell structures immersed in an infinite fluid medium, a composite element consisting of a semi-analytical infinite fluid element and a cylindrical shell strip element is proposed in this paper. The behaviour in the infinite direction can be accurately modelled with minimum effort and great saving in computational cost is achieved using this element. Unlike many other methods which tend to take advantage of axisymmetry to simplify the analysis, this method may be used to analyse the dynamic behaviour of prismatic shell structures with arbitrary cross-sections in offshore engineering.

Based on empirical added mass formulation, this work presents a simple procedure to determine the vibration frequencies and mode shapes of submerged cantilever plates. Once the added mass formulation is derived, the procedure can be used to analyze free vibration response easily. An analytical and numerical study is also performed for the vibrations of cantilever plates in air and in water, with these results compared with experimental and numerical data from pertinent literature. Besides, the frequency parameters of the submerged plate for various aspect ratios and thickness ratios are given in design data sheet form and are appropriate for engineering design applications.

The new improved discrete Kirchhoff quadrilateral element based on the third-order zigzag theory developed earlier by the
present authors for the static analysis of composite and sandwich plates is extended for dynamics and assessed for its performance
for the free vibration response. The element is free from the shear locking. The finite element formulation is validated by
comparing the results for simply supported plates with the analytical Navier solution of the zigzag theory. Comparison of
the present results for the natural frequencies with those of a recently developed triangular element based on the zigzag
theory, for composite and sandwich plates, establishes the superiority of the present element in respect of simplicity, accuracy
and computational efficiency. The accuracy of the zigzag theory is assessed for composite and sandwich plates with various
boundary conditions and aspect ratio by comparing the finite element results with the 3D elasticity analytical and finite
element solutions.

The approach developed in this paper applies to vibration analysis of rectangular plates coupled with fluid. This case is representative of certain key components of complex structures used in industries such as aerospace, nuclear and naval. The plates can be totally submerged in fluid or floating on its free surface. The mathematical model for the structure is developed using a combination of the finite element method and Sanders’ shell theory. The in-plane and out-of-plane displacement components are modelled using bilinear polynomials and exponential functions, respectively. The mass and stiffness matrices are then determined by exact analytical integration. The velocity potential and Bernoulli’s equation are adopted to express the fluid pressure acting on the structure. The product of the pressure expression and the developed structural shape function is integrated over the structure-fluid interface to assess the virtual added mass due to the fluid. Variation of fluid level is considered in the calculation of the natural frequencies. The results are in close agreement with both experimental results and theoretical results using other analytical approaches.

The interaction of an elastic bottom with the liquid exhibiting a free liquid surface has been investigated for a rectangular container. For this reason the container bottom was considered either as a flexible membrane or as a thin elastic rectangular plate. Furthermore the hydroelastic problem of a liquid in a rigid rectangular tank in which the free liquid surface was covered by a flexible membrane or a thin elastic plate has also been treated. In both cases the coupled frequencies of the structure-liquid system has been obtained. It was found that even structural modes couple with odd liquid modes and vice versa and that the coupled frequencies exhibit decreased magnitude compared with the uncoupled structural frequencies and increased magnitude compared to the uncoupled liquid frequencies. They decrease with decreasing tension of the membrane or decreasing stiffness of the plate.

Two new C0 assumed strain finite element formulations of Reddy's higher-order theory are used to determine the natural frequencies of isotropic, orthotropic, and layered anisotropic composite and sandwich plates. The material properties typical of glass fibre polyester resins for the skin and HEREX C70 PVC (polyvinyl chloride) foam materials for the core are used to show the parametric effects of plate aspect ratio, length-to-thickness ratio, degree of orthotropy, number of layers and lamination scheme on the natural frequencies. A consistent mass matrix is adopted in the present formulation. The results presented in this investigation could be useful for a better understanding of the behaviour of sandwich laminates under free vibration conditions and potentially beneficial for designers of sandwich structures.

The dry and wet dynamic characteristics of a vertical and a horizontal cantilever square plate [1] immersed in fluid are discussed from the viewpoint of a linear hydroelasticity theory [2–5]. The surface piercing vertical plate is partially immersed in the fluid and the influence of submerged plate length on the resonance frequencies investigated. For the horizontal plate the influence of submerged depth below the free surface on the resonance frequencies is examined. Incorporated into the theoretical model is a free surface boundary condition allowing wave disturbances to be present. The interaction existing between the vibrating cantilever plate and the free surface is clearly exhibited in the calculated curves describing the generalized hydrodynamic coefficients. A limited comparison between predictions and experimental data [1] is also included.

The transient response of orthotropic, layered composite sandwich plates is investigated by using two new C0 four and nine node finite element formulations of a refined form of Reddy's higher-order theory. This refined third order theory accounts for parabolic variation of the transverse shear stresses, and requires no shear correction factors. The assumed strain approach is employed to model both thin and thick plates without any major defects like shear locking and parasitic spurious zero energy modes. A consistent mass matrix formulation is adopted. The Newmark direct integration scheme is used to solve the governing equilibrium equations. The parametric effects of plate aspect ratio, length to thickness ratio, boundary conditions and lamination scheme on the transient response are investigated. The present results are in very close agreement with earlier published results in the literature and can serve as a benchmark for future investigators.

Summary of some of the results of a recent study of the reliability and range of validity of two-dimensional plate theories in application to low-frequency free vibration analysis of simply supported, bidirectional, multilayered plates consisting of a large number of layers. These results show that for composite plates the error in the predictions of the classical plate theory is strongly dependent on the number and stacking of the layers, in addition to the degree of orthotropy of the individual layers and the thickness ratio of the plate.

Natural frequencies of complex, free or submerged structures by the finite element method

- O C Zienkiewicz
- B M Irons
- B Nath

Dynamic Analysis of Isotropic and Laminated Reinforced Composite Plates Subjected to Flowing Fluid

- J Alireza