Sage Journals: Discover world-class research

Abstract

In this article, investigations on the influence of openings and additional mass on free vibration analysis of laminated composite sandwich skew plates using improved higher order shear deformation theory (IHSDT) have been done. The IHSDT model satisfies the interlaminar shear stress continuity at the layer interfaces and also ensures zero transverse shear stress conditions at the top and bottom of the plate. The piecewise parabolic shear stress variation across the thickness of each layer is considered. No shear correction factors are required. The 2-D C₀ finite element (FE) model has been developed by authors based on IHSDT. FE model based on IHSDT has been coded in FORTRAN. The problem of C₁ continuity requirement associated with the IHSDT is overcome using an appropriate C₀ FE formulation. The free vibration frequencies of laminated composite and sandwich plates obtained using the present 2-D FE model are in good agreement with the 3-D elasticity results. The influence of the side-to-thickness ratio, skew angles, boundary conditions, and mode shapes is taken into consideration for the present study.

Keywords

Sandwich plate finite element method improved higher order shear deformation theory skew angle mode shape

Introduction

In the era of the present scenario, composites and sandwich plates are becoming one of the most crucial materials for construction in Civil Engineering. Composites are those materials which are tougher and lighter and can be derived from the properties that can be customized to meet particular design requirements. Its property by virtue of which, the ease with which complex shapes can be feigned, helps in the complete innovation of an established design in which can often be advantageous to both cheaper and better solutions. A free vibration response, a mechanical behavior of the laminated composite sandwich plates, is the focus of this research.

Cutouts, in the design of composite structures, are used for easy access, assessment for passing hydraulic, fuel, and electric lines, and also to reduce the weight of the structures.¹ In many cases, the vibration responses of such perforated structural components are risky in nature for the purpose of which, to prevent premature failure and utilization of full strength, a complete knowledge of the stability performance of such panels is vital.

Due to the complex nature of composites, various theories have been proposed by researchers till date. The first plate theory developed was Kirchhoff’s popularly known as classical laminate plate theory.² The first attempt was made by Ashton and Whitney³ to analyze laminated composite plates using the classical plate theory for modeling of plate deformation. But the theory failed in predicting accurate results for thick plates due the neglection of the transverse shear strains by the present theory.

Reissner^4,5 had first provided a steady theory to include the shear deformation effects. The basic assumption made in the research gives a consistent representation of stress distribution across the thickness, which resulted in a through-thickness linear variation of in-plane displacements (i.e. two terms of Taylor’s series expansion) and constant normal deflection across the plate thickness. The same degree of approximation has been employed by Mindlin⁶ on kinematic assumptions of the displacement fields given by Reissner^4,5 without introducing corresponding stress distribution assumptions and obtained the governing equations from a direct method.

Yang et al.⁷ had derived a 2-D linear theory of motions from the 3-D theory of elasticity for heterogeneous plates. The transverse shear strains and rotatory inertia were taken into account during the analysis. This is broadly coming to be known as the first-order shear deformation theory (FSDT) for laminated plates. But the theory was restricted with the use of complex shear correction factors. Pagano⁸ later on extended the work of Yang et al. to develop a theory including shear deformation and rotatory inertia as given in Mindlin’s theory. For the analysis of laminated composite plates, Reddy⁹ developed a simple higher order theory. The theory contains the same independent unknowns as in FSDT and accounts for the parabolic distribution of transverse shear strains through the thickness of the plate without the involvement of the shear correction factors in computing the shear stresses.

Thus, higher shear deformation theory (HSDT) came into existence to remove the complex factor, as it was free from shear correction factors. The origin of HSDTs goes back to the work of Hildebrand et al.¹⁰ who made significant contributions by dispensing all the assumptions of Kirchoff. Lo et al.^11,12 presented a plate deformation theory taking into account the effects of transverse shear and normal deformation in addition to the nonlinear distribution of the in-plane displacement with respect to thickness coordinate. Kant et al.¹³ derived the complete set of equations of an isotropic version of the Lo et al. theory and presented extensive numerical results with a proposed numerical integration technique. Khdeir and Reddy¹⁴ proposed a second-order shear deformation theory for the analysis of laminated composite plates. To access the transverse stresses in laminated composite and sandwich laminates, higher order theories were proposed by Kant and Manjunatha.¹⁵ The theory consisted of C₀ isoparametric finite elements (FEs) and exact 3-D equations. Furthermore, a refined higher order model is proposed by Swaminathan and Patil¹⁶ to solve the natural frequency of simply supported antisymmetric angle-ply composites and sandwich plates. Some of the other higher order theories from the literature that included the unknown terms in the assumed kinematic model were: seven unknowns (Pandya and Kant¹⁷) and nine unknowns (Ren¹⁸; Pandya and Kant¹⁹; Nanda and Bandyopadhyay²⁰).

Till current years, many analytical methods have been proposed for the dynamic analysis of laminated plates using different theories as mentioned earlier. Ferreira,²¹ Natarajan et al.,²² and Ferreira et al.,^23,24 have exercised radial basis functions for the static and dynamic analyses of composite and sandwich laminated plates. Kulkarni and Kapuria²⁵ had proposed a new improved discrete Kirchhoff quadrilateral element based on the third-order zigzag theory analysis of composite and sandwich plates. Kant and Swaminathan²⁶ had used higher order refined theory for analytical formulations and solutions to obtain the natural frequencies of simply supported laminated composite sandwich plates. Fiedler et al.²⁷ had studied the vibration behavior of rectangular laminated composite plates subject to in-plane loading employing a global higher order theory.

For the study of vibration of plates with rectangular or circular cutout, the Rayleigh–Ritz method or the FE method was used by Ali and Atwal²⁸ to study the vibration effect on rectangular plates with rectangular cutouts. For the free vibrations with a central circular cutout on a rectangular elastic plate, Hegarty and Ariman²⁹ proposed the method of least squares point matching, which had shown that the relation between cutout size and minimum frequency was also dependent on both the boundary conditions as well as Poisson’s ratio.

The forced and free vibrations of simply supported and symmetric stacked composite plates with cutouts had been investigated by Rajamani and Prabhakaran^30,31 with an assumption that the cutout is almost equivalent to an external loading on the plate. With the aid of FE method, the hole diameter effect is investigated by Chai,³² on the natural frequencies for fiber-reinforced composite plates with a circular hole. Sheikh et al.³³ developed a high precision thick element, which was applied on laminated plates for free vibration analysis. Makhecha et al.³⁴ studied the effects of free vibration on natural frequencies of thick laminated plates using higher order theory. HOSDT has been used by Asadi and Fariborz³⁵ to assess composite plates’ natural frequencies. Sahoo and Singh³⁶ studied the free vibration of laminate composite plates using inverse hyperbolic and trigonometric zigzag theory.

The impact of the additional mass on the vibration analysis of the laminated composite sandwich plates was not considered for all of the literature mentioned earlier. In some of the modern era applications, a mass is needed to be attached to a plate that may be subjected to vibration, such as, electronic boards, aerospace and naval applications. Coupled free vibration analysis using Rayleigh–Ritz method has been done on a cantilever plate with an attachment of spring mass system on any arbitrary point by Chen et al.³⁷ Watkins et al.³⁸ investigated elastically point supported isotropic plate, for the various natural frequencies and modes shapes with attached masses under the effect of impulsive loadings.

According to the literature review, most of the vibration studies of sandwich plates are without skew angle and without cutouts and concentrated mass. Hence, there is no result, based on improved higher order shear deformation theory (IHSDT) to analyze the effect of openings and additional mass on natural frequencies of composite and sandwich laminated skew plates.

Taking into account all the abovementioned aspects, a nine-noded C₀ 2-D FE model based on IHSDT developed by authors is used to study the effect of openings and additional mass on free vibration frequencies of laminated composite sandwich skew plates. To overcome C₁ continuity problem related to the theory, a C₀ isoparametric FE has been developed. The element proposed is having seven unknowns at each node with a total number of nine nodes. Many numerical examples of angle-ply, cross-ply laminates and skew angles with different thicknesses and different boundary conditions are therefore investigated here.

In the present study, using IHSDT, vibration analysis of thin and thick laminated composite and sandwich skew plates is explored. Several stacking sequences are considered. The accuracy of the proposed 2-D FE model is assessed for earlier researches.

Mathematical model

For the present analysis, the following equation of in-plane displacement fields is being adopted and is also shown in Figure 1(a)

u_{α} = u_{α}^{0} + \sum_{k = 0}^{n u - 1} S_{α}^{k} (z - z_{k}^{u}) H (z - z_{k}^{u}) + \sum_{k = 0}^{n u - 1} T_{α}^{k} (z - z_{k}^{l}) H (- z + z_{k}^{l}) + ξ_{α} z^{2} + ϕ_{α} z^{3}

where $u_{α}^{0}$ denotes in-plane displacement, subscript α shows the coordinate directions (α being 1, 2, i.e., x, y in this case), nu is number of upper layers, $H (z - z_{k}^{u})$ and $H (- z + z_{k}^{l})$ are unit step functions, $T_{α}^{k}$ and $S_{α}^{k}$ are kth layer slopes subsequent to lower and upper layers, respectively, and ξ_α and ϕ_α are higher order unknown terms.

For constant transverse displacement over the plate thickness, an assumption is considered, that is

u_{3} = w (x, y)

According to Sanders’ approximation, the linear strain–displacement relations are as follows

\begin{array}{l} ε_{x} = \frac{\partial u_{1}}{\partial x}, ε_{y} = \frac{\partial u_{2}}{\partial y}, γ_{x y} = \frac{\partial u_{2}}{\partial x} + \frac{\partial u_{1}}{\partial y} \\ γ_{x z} = \frac{\partial u_{1}}{\partial z} + \frac{\partial u_{3}}{\partial x}, γ_{y z} = \frac{\partial u_{2}}{\partial z} + \frac{\partial u_{3}}{\partial y} \end{array}

Conditions at layer interfaces

The laminated composite is being transformed to Cartesian coordinate system; x, y are adopted to be the laminate reference for the bottom plane, and z is considered for thickness coordinate. Using this coordinate system, now z_k represents the interface coordinate of k and suffix with ⁽⁻⁾ or ⁽⁺⁾ is used to define the stresses above and below the interface. Using Pagano’s⁸ elascticity theory, the conditions as listed in the following are required for accurate prediction of interlaminar stresses. The following contact conditions are included in the present model at the interfaces of adjacent layers by transverse shear stress

τ_{x z}^{(+)} = {\underline{τ_{x z}^{(+)}}}_{(z = z_{k}^{+})} = τ_{x z}^{(-)} = {\underline{τ_{x z}^{(-)}}}_{(z = z_{k + 1}^{-})}

τ_{y z}^{(+)} = {\underline{τ_{y z}^{(+)}}}_{(z = z_{k}^{+})} = τ_{y z}^{(-)} = {\underline{τ_{y z}^{(-)}}}_{(z = z_{k + 1}^{-})}

The zero stress boundary conditions at the $({\underline{}}^{u})$ upper and $({\underline{}}^{l})$ lower free surfaces of the laminate are included in the present model

{\underline{τ_{x z}}}^{u} = {\underline{τ_{y z}}}^{u} = 0; {\underline{τ_{x z}}}^{l} = {\underline{τ_{y z}}}^{l} = 0

The present model and element also mete out the following kinematic contact conditions

{\underline{u_{α}}}_{(z = z_{k}^{+})} = {\underline{u_{α}}}_{(z = z_{k + 1}^{-})}

where u_α is the elastic displacement’s component.

For typical lamina (kth), the constitutive relations with respect to the material axis are given as follows

{\begin{matrix} σ_{1} \\ σ_{2} \\ τ_{12} \\ τ_{13} \\ τ_{23} \end{matrix}}_{k} = {[\begin{matrix} Q_{11} & Q_{12} & 0 & 0 & 0 \\ Q_{12} & Q_{22} & 0 & 0 & 0 \\ 0 & 0 & Q_{66} & 0 & 0 \\ 0 & 0 & 0 & Q_{44} & 0 \\ 0 & 0 & 0 & 0 & Q_{55} \end{matrix}]}_{k} {\begin{matrix} ε_{1} \\ ε_{2} \\ γ_{12} \\ γ_{13} \\ γ_{23} \end{matrix}}_{k}

or, it may also be represented as

{σ}_{k} = {[Q]}_{k} {ε}_{k}

where $Q_{11} = E_{1} / (1 - ν_{12} ν_{21})$ , $Q_{12} = ν_{12} E_{2} / (1 - ν_{12} ν_{21})$ , $Q_{22} = E_{2} / (1 - ν_{12} ν_{21})$ , $Q_{66} = G_{12}$ , $Q_{44} = G_{13}$ , $Q_{55} = G_{23}$ , and $\frac{ν_{12}}{E_{1}} = \frac{ν_{21}}{E_{2}}$ .

The relationship of stress versus strain for a lamina with reference to the structure axis system (x, y, and z) after doing the necessary transformation may be given as follows

{\begin{matrix} σ_{x} \\ σ_{y} \\ τ_{x y} \\ τ_{x z} \\ τ_{y z} \end{matrix}}_{k} = {[\begin{matrix} {\bar{Q}}_{11} & {\bar{Q}}_{12} & {\bar{Q}}_{16} & 0 & 0 \\ {\bar{Q}}_{12} & {\bar{Q}}_{22} & {\bar{Q}}_{26} & 0 & 0 \\ {\bar{Q}}_{16} & {\bar{Q}}_{26} & {\bar{Q}}_{66} & 0 & 0 \\ 0 & 0 & 0 & {\bar{Q}}_{44} & {\bar{Q}}_{45} \\ 0 & 0 & 0 & {\bar{Q}}_{45} & {\bar{Q}}_{55} \end{matrix}]}_{k} {\begin{matrix} ε_{x} \\ ε_{y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{matrix}}_{k}

For the FE implementation, it is required by the in-plane displacement fields to have C₁ continuity of the transverse displacement. To circumvent the difficulties related to C₁ continuity, the first-order derivatives of w are considered as independent variables given by

\frac{\partial w}{\partial x} = w_{1} and \frac{\partial w}{\partial y} = w_{2}

With the help of which all the variables can be defined as C₀ continuous.

So by utilizing the transverse shear stress free boundary condition, that is $(τ_{x z} |_{z = \pm h / 2} = 0, τ_{y z} |_{z = \pm h / 2} = 0)$ at the top and bottom of the plate, the components ϕ ₁, ϕ ₂, ξ ₁, and ξ ₂ could be expressed as

\begin{array}{l} ϕ_{1} = \frac{- 4}{3 h^{2}} {\frac{\partial w}{\partial x} + \frac{1}{2} (\sum_{k = 0}^{n u - 1} S_{1}^{k} + \sum_{k = 0}^{n l - 1} T_{1}^{k})}, ϕ_{2} = \frac{- 4}{3 h^{2}} {\frac{\partial w}{\partial y} + \frac{1}{2} (\sum_{k = 0}^{n u - 1} S_{2}^{k} + \sum_{k = 0}^{n l - 1} T_{2}^{k})} \\ and ξ_{1} = \frac{- 1}{2 h} (\sum_{k = 0}^{n u - 1} S_{1}^{k} - \sum_{k = 1}^{n l - 1} T_{1}^{k}), ξ_{2} = \frac{- 1}{2 h} (\sum_{k = 0}^{n u - 1} S_{2}^{k} - \sum_{k = 1}^{n l - 1} T_{2}^{k}) \end{array}

Substituting equation (11) in equation (1), ( $S_{1}^{k}$ , $S_{2}^{k}$ , k = 1, nu − 1) and ( $T_{1}^{k}$ , $T_{2}^{k}$ , k = 0, nl − 1), the kth layer values for slopes, may be represented in terms of $S_{1}^{0} = ψ_{1}$ , $S_{2}^{0} = ψ_{2}$ , $\frac{\partial w}{\partial x}$ , and $\frac{\partial w}{\partial y}$ by applying the conditions of transverse shear stress continuity at every layer interface, as shown in the following

\begin{array}{l} S_{1}^{k} = a_{11}^{k} (\frac{\partial w}{\partial x} + ψ_{1}) + b_{11}^{k} \frac{\partial w}{\partial x} + a_{12}^{k} (\frac{\partial w}{\partial y} + ψ_{2}) + b_{12}^{k} \frac{\partial w}{\partial y} \\ S_{2}^{k} = a_{21}^{k} (\frac{\partial w}{\partial x} + ψ_{1}) + b_{21}^{k} \frac{\partial w}{\partial x} + a_{22}^{k} (\frac{\partial w}{\partial y} + ψ_{2}) + b_{22}^{k} \frac{\partial w}{\partial y}, where k = 1, n_{u - 1} \\ T_{1}^{k} = c_{11}^{k} (\frac{\partial w}{\partial x} + ψ_{1}) + d_{11}^{k} \frac{\partial w}{\partial x} + c_{12}^{k} (\frac{\partial w}{\partial y} + ψ_{2}) + d_{12}^{k} \frac{\partial w}{\partial y} \\ T_{2}^{k} = c_{21}^{k} (\frac{\partial w}{\partial x} + ψ_{1}) + d_{21}^{k} \frac{\partial w}{\partial x} + c_{22}^{k} (\frac{\partial w}{\partial y} + ψ_{2}) + d_{22}^{k} \frac{\partial w}{\partial y}, where k = 0, n_{l - 1} \end{array}

where ψ ₁ and ψ ₂ represents the rotations of normal with respect to the midplane about the y and x axes, respectively, whereas $a_{α γ}^{k}, b_{α γ}^{k}, c_{α γ}^{k}, and d_{α γ}^{k}$ (α, γ = 1, 2) represent the constants depending upon the properties of material of the layers adjacent to the kth interface.

Thus, under the influence transverse load (q), the laminate’s potential energy (V) can be given as follows

V = \iint {ε}^{T} [D] {ε} d x d y - \iint w q d x d y

Using abovementioned equation, we can derive element stiffness matrix, [k_e ]

\begin{array}{l} [k_{e}] = \sum_{i = 1}^{n u + n l} ∭ {[B]}^{T} {[H]}^{T} [\bar{Q}] [H] [B] d x d y d z \\ = {\iint [B]}^{} [D] [B] d x d y \end{array}

In the abovementioned equation, matrices [B], $[\bar{Q}]$ , and [H] represent matrix of strain, matrix of the transformed material constant, and matrix having terms containing z and material properties related terms, respectively.

where [D] = \sum_{i = 1}^{n u + n l} \int {[H]}^{T} [\bar{Q}] [H] d z

The strain field vector is given by

{\bar{ε}} = [H] {ε}

where ${\bar{ε}}$ is of size 5 × 1 representing the standard strain field vector and {ε} is of size 17 × 1 representing the modified strain vector at the reference plane (i.e. at the midplane). {ε} is the midplane strain vector may be expressed as

\begin{array}{l} {\bar{ε}} = {\frac{\partial u_{1}^{0}}{\partial x} \frac{\partial u_{2}^{0}}{\partial y} \frac{\partial u_{2}^{0}}{\partial x} + \frac{\partial u_{1}^{0}}{\partial y} \frac{\partial w_{1}}{\partial x} \frac{\partial w_{2}}{\partial y} \frac{\partial w_{2}}{\partial x} \frac{\partial w_{1}}{\partial y} \frac{\partial ψ_{1}}{\partial x} \frac{\partial ψ_{2}}{\partial y} \frac{\partial ψ_{2}}{\partial x} \frac{\partial ψ_{1}}{\partial y} ψ_{1} ψ_{2} \frac{\partial w}{\partial x} \frac{\partial w}{\partial y} w_{1} w_{2}} \end{array}

Now the relationship of strain displacement may be given as follows

{ε} = [B] {δ}

In the abovementioned equation, [B] represents the matrix of strain displacement having size 17 × 63, whereas {δ} is of size 63 × 1 representing the nodal unknown vector.

FE formulation

In this article, C₀ isoparametric elements having nine nodes, with seven unknowns per node (Figure 1(b)), that is, $u, u_{2}, u_{3}, ψ_{1}, ψ_{2}, w_{1}$ , and w ₂ are used for the proposed FE model. Using the following relations, the generalized displacements included in the present theory can be expressed as follows

Figure 1.

(a) General lamination layup and in-plane displacement across the cross section of a plate and (b) nine-noded curved isoparametric element with typical node numbering.

\begin{array}{l} u_{1} = \sum_{i = 1}^{9} N_{i} u_{i}, u_{2} = \sum_{i = 1}^{9} N_{i} v_{i}, u_{3} = \sum_{i = 1}^{9} N_{i} w_{i}, ψ_{1} = \sum_{i = 1}^{9} N_{i} ψ_{1}_{i} \\ ψ_{2} = \sum_{i = 1}^{9} N_{i} ψ_{2}_{i}, w_{1} = \sum_{i = 1}^{9} N_{i} w_{1 i}, w_{2} = \sum_{i = 1}^{9} N_{i} w_{2 i} \end{array}

In the abovementioned equation, the shape function of the related node is represented by N_i .

After knowing the nodal unknown vector within an element, the mid-surface strains at any point in the plate, in terms of global displacements can be expressed in the matrix form as follows

{\bar{ε}} = \sum_{i = 1}^{9} [B_{i}] {d_{i}}

In the equation, [B_i ] is used to express the differential operator matrix of the shape function developed from equation (20).

For an element, the element stiffness matrix (say, eth), including the transverse shear effects, flexure, and membrane can be given as

[K_{e}] = \int_{- 1}^{1} \int_{- 1}^{1} {[B]}^{T} [D] [B] | J | d r d s

For all numerical integrations, a 3 × 3 Gaussian quadrature format has been used. Then, the element matrices are grouped together to attain the global stiffness matrices, [K], in accordance with the typical procedure of the FE method as considered by Bathe.³⁹

Free vibration analysis

In this analysis, the governing equation (22) is solved using the simultaneous iteration technique of Corr and Jennings (1976) for the computation of eigenvalues and eigenvectors.

[K] {φ} = ω^{2} [M] {φ}

In this method, [K] represents the positive definite and can be decomposed into Cholesky factors as

[K] = [L] {[T]}^{T} and {{[L]}^{- 1} [M] {[L]}^{- T}} {[L]}^{T} {δ} = \frac{1}{ω^{2}} {[L]}^{t} {δ}

Numerical application and results

New problems are tied to be solved in this article, for natural frequencies of free vibration for laminated composite and sandwich skew plates considering the effect of openings and additional mass. The mode shapes are also presented for the same using IHSDT. For the execution of the abovementioned FE model, a FORTRAN code is developed based upon the abovementioned mathematical formulation.

Boundary condition

The boundary conditions mainly applied in following examples are clamped boundary condition (CCCC) and simply supported boundary condition (SSSS) which are as follows

CCCC:

at x = 0, a and at y = 0, b

u_{1} = u_{2} = u_{3} = ψ_{1} = ψ_{2} = w_{1} = w_{2} = 0

SSSS:

at x = 0, a

u_{2} = u_{3} = ψ_{2} = w_{2} = 0

at y = 0, b

u_{1} = u_{3} = ψ_{1} = w_{1} = 0

Other than these four, more support conditions are used in different examples:

CCFF = Clamped, Clamped, Fixed, Fixed

CCSS = Clamped, Clamped, Simply supported, Simply supported

CFCF = Clamped, Fixed, Clamped, Fixed

CSCS = Clamped, Simply supported, Clamped, Simply supported

Engineering properties (geometrical and material)

For all the further investigations, unless mentioned otherwise, the composites with the following properties are taken: E ₁ = 25E ₂, E ₂ = 1, G ₁₂ = G ₁₃ = 0.5E₂, G ₂₃ = 0.2E _2, ρ = 1, and ν ₁₂ = 0.25. The properties for sandwich case taken in off-axis system for the core and face sheets are as follows: Face: E ₁ = 40E ₂, E ₂ = 1, G ₁₂ = G ₁₃ = E ₂, G ₂₃ = E ₂, ρ = 1, and ν ₁₂ = 0.25 and Soft Core: E ₂/E_c = 11.945, G ₁₂/E_c = G ₁₃/E_c = 1.173/6.279, G ₂₃/E_c = 2.415/6.279, ρ/ρ_c = 0.6818, and ν ₁₂ = 0.25. The geometrical properties taken are a/h = 5, 10, 100 and a = 1 and b = 1 in. Face sheet and core thicknesses are taken as 0.2h and 0.8h, respectively. The nondimensional frequency for composites is taken as $\bar{ω} = (ω a^{2} / h \sqrt{ρ / E_{2}})$ and for sandwich as $\bar{ω} = (100 ω a \sqrt{ρ_{c} / E_{11}})$ . The value of additional mass is given by the following equations: $\bar{\underset{˙}{M}} = \frac{M}{ρ h a^{2}}$

Convergence and comparison studies

Convergence study is done to determine the required mesh size N × N at which the dimensionless free vibration fundamental frequencies values may converge. After the variation of mesh sizes from 8 to 22, it may be concluded from Table 1 that the values of nondimensional fundamental frequencies have converged at N = 20. Therefore, for all subsequent analyses, mesh size of 20 × 20 (Full) is taken into consideration. The numbering of edges in plate is shown in Figure 3(a). The converged results are found to be in good accordance with the results of Chakrabarti and Sheikh⁴⁰ based upon refined higher shear deformation theory (RHSDT).

Table 1.

Convergence study of nondimensional fundamental frequencies $\bar{ω} = (\frac{ω a^{2}}{h \sqrt{ρ / E_{2}}})$ with a/h = 10 for a simply supported cross-ply square laminated plate (0°/90°/0°/90°) _t E ₁/E ₂ = 40, G ₁₂ = G₂₃ = 0.6E ₂, G ₂₃ = 0.5E ₂, and ν ₁₂ = 0.25.

References	Theory	Mode numbers
References	Theory	1	2	3	4
Present (8 × 8)	IHSDT	15.4858	32.6192	32.6855	43.4010
Present (10 × 10)	IHSDT	15.4853	32.6022	32.6891	43.4040
Present (16 × 16)	IHSDT	15.4847	32.5890	32.6849	43.3973
Present (18 × 18)	IHSDT	15.4845	32.5875	32.6837	43.3956
Present (20 × 20)	IHSDT	15.4845	32.5866	32.6828	43.3942
Present (22 × 22)	IHSDT	15.4845	32.5866	32.6828	43.3942
Chakrabarti and Sheikh⁴⁰	RHSDT	14.7160	32.1820	32.1850	42.6710

IHSDT: improved higher order shear deformation theory; RHSDT: refined higher shear deformation theory.

To show the performance of different plate theories, a cross-ply (0°/90°/90°/0°) square laminate under the effect of free vibration is taken for consideration in this example. In Table 2, the analysis of full plate is done with different mesh divisions and different a/h ratios. As from literature review, it is much clearer that there are no results available on the free vibration analysis of the composite and sandwich laminated skew plates with cutout and concentrated mass based on IHSDT; the results obtained are hence evaluated with reference to those obtained by Whitney and Pagano,⁴¹ Reddy and Phan,⁴² Senthilnathan et al.,⁴³ and Kant and Swaminathan.²⁶ The percentage errors shown in Table 2. have been evaluated with respect to the analytical solution of Kant and Swaminathan.²⁶ Table 2 shows the utility of the present FE model in predicting nondimensional fundamental frequencies close to exact analytical results given by Kant and Swaminathan.²⁶ The different elastic modulus ratios have been considered in Table 3, and the results of the present formulation have been compared with those of 3-D elastic solutions of Noor⁴⁴ and other results of Reddy and Phan⁴² and Kant and Swaminathan.²⁶

Table 2.

Validation of nondimensional fundamental frequencies $\bar{ω} = (\frac{ω a^{2}}{h \sqrt{ρ / E_{2}}})$ with a/h for a simply supported cross-ply square laminated plate (0°/90°/90°/0°) _t E ₁/E ₂ = 40, G ₁₂ = G ₂₃ = 0.6E ₂, G ₂₃ = 0.5E ₂, and ν ₁₂ = 0.25.^a

Lamination and number of layers	Source	Theory	a/h
Lamination and number of layers	Source	Theory	4	10	20	50	100
(0°/90°/90°/0°)	Present	IHSDT	9.1388	15.0081	17.6483	18.6797	18.8892
	Kant and Swaminathan²⁶	HODT	9.2870	15.1048	17.6470	18.6720	18.8357
	Senthilnathan et al.⁴³	SHOSDT	10.2032	15.9405	17.9938	18.7381	18.8526
	Reddy and Phan⁴²	HOSDT	9.3235	15.1073	17.6457	18.6718	18.8356
	Whitney and Pagano⁴¹	FSDT	9.3949	15.1426	17.6596	18.6742	18.8362
	Error (%)		1.600	0.640	0.007	0.041	0.28

IHSDT: improved higher order shear deformation theory; FSDT: first-order shear deformation theory; HODT: Higher order deformation theory; HOSDT: Higher order shear deformation theory.

^a%Error calculated with respect to Kant and Swaminathan²⁶

Table 3.

Validation of nondimensional fundamental frequencies $\bar{ω} = (\frac{ω a^{2}}{h \sqrt{ρ / E_{2}}})$ with a/h = 5 for a simply supported cross-ply square laminated plate (0°/90°/90°/0°) _t G ₁₂ = G ₂₃ = 0.6E ₂, G ₂₃ = 0.5E ₂ ν ₁₂ = 0.25.^a

Lamination and number of layers	Source	Theory	E ₁/E ₂
Lamination and number of layers	Source	Theory	10	20	50	100
(0°/90°/90°/0°)	Present	IHSDT	8.1442	9.4325	10.2685	10.7252
	Kant and Swaminathan²⁶	HODT	8.1482	9.4675	10.2733	10.8221
	Noor⁴⁴	3-D elastic	8.1445	9.4055	10.1650	10.6798
	Reddy and Phan⁴²	HOSDT	8.1954	9.6265	10.5348	11.1716
	Error (%)		0.004	0.287	1.01	0.425

IHSDT: improved higher order shear deformation theory.

^a%Error calculated with respect to 3-D elastic solution.

To show the efficiency of the present model, all edges clamped laminated composite plate (0°/90°/90°/0°) have been analyzed. The frequencies for different modes of vibration and a/h ratios are compared with the results of Kulkarni and Kapuria²⁵ based upon 3-D FE ABAQUS model and zigzag theory as shown in Table 4. The results based upon present formulation when compared are found to be in good accordance with the results of Kulkarni and Kapuria.²⁵ The material properties adopted for the present analyses are E ₁ = 181 GPa, E ₂ = 10.3 GPa, G ₁₂ = G ₁₃ = 7.17 GPa, G ₂₃ =2.87 GPa_, ρ = 1578, and ν ₁₂ = 0.28.

Table 4.

Frequency parameter $\bar{ω} = (\frac{ω a^{2}}{h \sqrt{ρ / E_{2}}})$ of all edges clamped, cross-ply (0°/90°/90°/0°) square laminate.

Modes of Frequency
a/h	Reference	Theory	First frequency	Second frequency	Third frequency	Fourth frequency	Fifth frequency	Sixth frequency
5	Kulkarni and Kapuria²⁵	3-D FE ABAQUS	11.4860	18.2956	19.3319	24.2519	27.3411	28.9117
	Kulkarni and Kapuria²⁵	Zigzag theory	12.0965	19.3528	21.1266	26.2179	29.0997	33.1200
	Present	IHSDT	11.5986	17.9676	19.8273	23.7762	25.5415	29.3734
10	Kulkarni and Kapuria²⁵	3-D FE ABAQUS	17.7502	28.2032	32.4505	39.6697	43.7521	50.4570
	Kulkarni and Kapuria²⁵	Zigzag theory	18.1118	29.0729	33.5629	41.0151	45.6649	52.7698
	Present	IHSDT	17.6571	28.1875	32.5686	39.5497	43.7373	50.7545
20	Kulkarni and Kapuria²⁵	3-D FE ABAQUS	23.7212	36.3153	49.6061	57.6707	58.2579	74.4577
	Kulkarni and Kapuria²⁵	Zigzag theory	23.8689	36.6889	50.2908	58.4074	59.3371	75.6414
	Present	IHSDT	23.3434	35.8892	49.2007	57.1508	58.0143	73.9477

IHSDT: improved higher order shear deformation theory; FE: finite element.

Another example of laminated composite plate having lamination scheme as (0°/90°/90°/0°) has been analyzed. The nondimensional frequencies for various aspect ratio have been computed and compared with different theories of Cho et al.,⁴⁵ Wu and Chen,⁴⁶ Matsunaga,⁴⁷ and Fiedler et al.²⁷ based upon individual layer theory, HSDT, and RHSDT, respectively, and shown in Table 5. For all aspect ratios, frequencies predicted by the present model are closer to the results predicted by other theories.

Table 5.

Frequency parameter $\bar{ω} = (\frac{ω a^{2}}{h \sqrt{ρ / E_{2}}})$ of a simply supported, cross-ply (0°/90°/90°/0°), square laminate. E ₁ = 40E₂, E ₂ = 1, G ₁₂ = G ₁₃ = 0.6E ₂, G ₂₃ = 0.5E ₂, ρ = 1, and ν ₁₂ = 0.25.

References	Cho et al.⁴⁵	Wu and Chen⁴⁶	Matsunaga⁴⁷	Fiedler et al.²⁷	Present
Theory	ILT	ILT	HSDT	RHSDT	IHSDT
a/h ratio	ILT	ILT	HSDT	RHSDT	IHSDT
2	5.9230	5.3170	5.3211	5.3291	5.0333
4	—	9.1930	9.1988	9.2110	9.1388
5	10.6730	10.6820	10.6876	10.7001	10.6649
10	15.0660	15.0690	15.0721	15.0806	15.0881
12.5	—	16.1340	16.1367	16.1433	16.1531
20	17.5350	17.6360	17.6369	17.6403	17.6484
25	18.0540	18.0550	18.0557	18.0580	18.0647
50	18.6700	18.6700	18.6702	18.6708	18.6797
100	18.8350	18.8350	18.8352	18.8354	18.8892

ILT: individual layer theory; RHSDT: refined higher shear deformation theory; IHSDT: improved higher order shear deformation theory; HSDT: higher shear deformation theory.

In Table 6, the results are calculated for laminated sandwich plate (45°/−45°/0°/90°/45°/0°/−45°/90°/core/90°/−45°/0°/45°/90°/0°/−45°/45°) with the help of present FORTRAN code and FE tool ABAQUS. A square cutout of size 0.4a has been taken at the center of the plate for the present analysis. In Table 6, nondimensional frequencies for different modes of vibrations are calculated for different values of skew angles and a/h ratios. The results predicted from both methods are found to be nearly the same for any condition.

Table 6.

Validation of nondimensional fundamental frequencies $\bar{ω} = (ω a^{2} \sqrt{ρ_{c} / E_{1}})$ for a simply supported cross-ply square laminated sandwich plate (45°/−45°/0°/90°/45°/0°/−45°/90°/core/90°/−45°/0°/45°/90°/0°/−45°/45°).

		Nondimensional frequencies from FORTRAN code						Nondimensional frequencies from ABAQUS
Skew angle	a/h	First mode	Second mode	Third mode	Fourth mode	Fifth mode	Sixth mode	First mode	Second mode	Third mode	Fourth mode	Fifth mode	Sixth mode
30°	100	7.4136	8.7418	9.0300	12.3294	12.5827	16.2107	7.3196	8.5367	8.9002	11.9163	12.3789	15.8795
30°	20	16.6286	17.6274	19.3695	22.5317	23.2515	27.0868	15.5829	16.4572	18.1863	20.9736	21.8564	25.4066
60°	100	18.1304	18.8063	19.5986	21.5845	23.2122	26.7329	17.7948	18.3326	19.2790	20.8415	22.6472	25.9849
60°	20	32.5244	33.2006	34.4081	36.9008	37.6335	41.2441	29.2076	29.6863	31.0727	32.8305	33.7056	36.8077

Novel results

After validating the present FE model based upon the abovementioned theory through comparison studies, new problems are worked out to analyze the effect of openings and additional mass on free vibration of laminated composite sandwich skew plates. For the following examples, various laminated composite skew plates having different lamination schemes with different boundary conditions have been analyzed for free vibration. The geometrical and material parameters used for analysis are defined in the previous sections. The ply numbering scheme, as shown in Figure 2, is in such a way that the counting of lamina is done from bottom to top. The skew layup is shown in Figure 3(b).

Figure 2.

Typical front view of four-layer laminated plate with ply numbering.

Figure 3.

(a) Numbering of edges of plate and (b) a skew plate having a mesh of m × n.

The comparison of nondimensional frequencies is shown in the following examples for the composite and sandwich plates with additional mass ( $\bar{M} = \frac{M}{ρ h a^{2}}$ = 0.5, 1, and 2) and square cutout (0.2a, 0.4a, and 0.6a), concentrated at the center for different boundary conditions using four different skew angle schemes, viz., 0°,15°, 30°, 45°, and 60°. The variation of a/h ratio is used for changing geometry of plates.

Laminated skew composite plates

Laminated skew composite plates with additional mass

For a particular value of additional mass, it was found that on increasing the skew angles the value of frequencies is increasing irrespective of change in a/h values as shown in Figure 4(a). But for a particular a/h ratio, it was found that irrespective of change in additional mass value, the frequencies other than fundamental frequency are almost the same for a particular skew angle. On decreasing the a/h value, the frequencies are decreasing with their corresponding additional mass and skew angle.

Figure 4.

Variation of nondimensional frequencies with (a) skew angle for different additional mass $(\bar{M})$ and a/h ratios for 0°/90°/0°/90° ply layup and CCCC boundary condition, (b) different ply layups for different additional mass $(\bar{M})$ and skew angles for CCCC boundary condition, and (c) different boundary conditions for different ply layups, additional mass $(\bar{M})$ , and skew angles.

The values of frequencies for a/h ratio = 5 have been compared for different ply layup and skew angles as shown in Figure 4(b). The variation of frequencies was almost the same for different ply layups for skew angle other than 60°.

The variation was also analyzed for different boundary conditions for which CCFF and CFCF only show variations in nature of frequencies for any skew angle as shown in Figure 4(c).

The mode shape plotted in Figure 5(a) and (b) shows the variation on the basis of boundary condition. The CCCC and SSSS boundary conditions were taken for a fixed a/h ratio, skew angle, and additional mass. It shows that the pattern of mode shape remains the same only its value changes. It may be due to the fact that the variation of skew angle only effects the mode shape as it is shown in Figure 6(a) and (b). The pattern of mode shape of fundamental frequency remains the same on the variation of skew angles but for other mode shapes it reverses.

Figure 5.

Mode shapes of skew (60°) laminated composite plate (45°/0°/−45°/0°/−45°/90°/0°/45°/0°/90°/−45°/0°/−45°/0°/45°) _t with (a) lumped mass ( $\bar{M} = 1$ ) (CCCC) (a/h = 100) and (b) lumped mass ( $\bar{M} = 1$ ) (SSSS) (a/h = 100).

Figure 6.

(a) Mode shapes of skew (60°) laminated composite plate (45°/0°/−45°/0°/−45°/90°/0°/45°/0°/90°/−45°/0°/−45°/0°/45°) _t with (a) lumped mass ( $\bar{M} = 1$ ) (CCCC) (a/h = 20). (b) Mode shapes of skew (15°) laminated composite plate (45°/0°/−45°/0°/−45°/90°/0°/45°/0°/90°/−45°/0°/−45°/0°/45°) _t with lumped mass ( $\bar{M} = 1$ ) (CCCC) (a/h = 20).

Laminated skew composite plates with square cutout

The square cutout in the plates is taken from the center and it is with the ratio of a. For a particular value of cutout, it was found that on increasing the skew angles the value of frequencies are increasing for a fixed a/h values, whereas on decreasing the a/h values the values of frequencies decreases as shown in Figure 7(a).

Figure 7.

Variation of nondimensional frequencies with (a) skew angle for different cutout area and a/h ratios for 0°/90°/0°/90° ply layup and CCCC boundary condition, (b) different ply layups for different skew angles and a/h for CCCC boundary condition, and (c) different boundary conditions for different ply layup and skew angles.

In Figure 7(b), the variation of frequencies is shown for different ply layups. It can be seen from the figure that for skew angle 60° the larger ply layups show a different trend of variation of frequencies other than that, on increasing the number of plies there is no much rise in the frequencies value with increase in ply numbers.

The variation was also analyzed for different boundary conditions for which CCFF and CFCF only show variations in nature of frequencies for any skew angle as shown in Figure 7(c).

Laminated skew sandwich plates

Laminated skew sandwich plates with additional mass

The behavior of sandwich plates is similar to composites but the values of frequencies are very less in comparison with composites with the same a/h ratio and additional mass value. For a particular value of additional mass, it was found that on increasing the skew angles the value of frequencies is increasing irrespective of change in a/h values as shown in Figure 8(a). On decreasing the a/h value, the frequencies are decreasing with their corresponding additional mass and skew angle.

Figure 8.

Variation of nondimensional frequencies with (a) skew angle for different additional mass and a/h ratios for (−45°/45°/core/−45°/45°) ply layup and CCCC boundary condition, (b) different ply layups for different additional mass and skew angles for CCCC boundary condition, and (c) different boundary conditions for different ply layups for given additional mass ( $\bar{M}$ = 1) and skew angle = 30°.

In Figure 8(b), the variation of frequencies is analyzed for different ply layups. All the ply layups have almost the same frequencies other than 0°/90°/0°/90°/0°/core/0°/90°/0°/90°/0° for any skew angle.

The variation was also analyzed for different boundary conditions for which CCFF and CFCF only show variations in nature of frequencies for any skew angle as shown in Figure 8(c). All these trends are similar for any ply layup and on increasing the number of plies.

Laminated sandwich plates under square cutout

The square cutout in the plates is taken from the center and it is with the ratio of a. The behavior of sandwich plates is similar to composites but the values of frequencies are very less in comparison with composites with the same a/h ratio and cutout value as shown in Figure 9(a). For a particular value of cutout, it was found that on increasing the skew angles, the value of frequencies is increasing irrespective of change in a/h values. On decreasing the a/h value, the frequencies are increasing with their corresponding cutout area and skew angle.

Figure 9.

Variation of nondimensional frequencies with (a) skew angle for different additional mass ( $\bar{M}$ ) and a/h ratios for (−45°/45°/core/−45°/45°) ply layup and CCCC boundary condition, (b) different ply layups for different skew angles for CCCC boundary condition, and (c) different boundary conditions for different ply layups for given cutout area, a/h, and skew angle.

In Figure 9(b), variation of nondimensional frequencies with different ply layups for different skew angles for CCCC boundary condition has been analyzed which shows the similar trend for variation in frequencies for different ply layups

The variation was also analyzed for different boundary conditions for which CCFF and CFCF only show variations in nature of frequencies for any skew angle as shown in Figure 9(c). All these trends are similar for any ply layup and on increasing the number of plies.

The mode shape plotted in Figure 10(a) and (b) shows the variation on the basis of boundary condition. The CCCC and SSSS boundary conditions were taken for a fixed a/h ratio, skew angle, and cutout. It shows that the pattern of mode shape remains the same only its value changes. It may be due to the fact that the variation of skew angle only affects the mode shape as it is shown in Figure 11(a) and (b). The pattern of mode shape of fundamental frequency remains the same on the variation of skew angles but for the other mode its shape reverses.

Figure 10.

Mode shapes of skew (60°) laminated sandwich plate (45°/−45°/0°/90°/45°/0°/−45°/90°/core/90°/−45°/0°/45°/90°/0°/−45°/45°) _t with (a) central cutout (0.4a) (CCCC) (a/h = 100) and (b) central cutout (0.4a) (SSSS) (a/h = 100).

Figure 11.

(a) Mode shapes of skew (60°) laminated sandwich plate (45°/−45°/0°/90°/45°/0°/−45°/90°/core/90°/−45°/0°/45°/90°/0°/−45°/45°) _t with central cutout (0.4a) (CCCC) (a/h = 20). (b) Mode shapes of skew (15°) laminated sandwich plate (45°/−45°/0°/90°/45°/0°/−45°/90°/core/90°/−45°/0°/45°/90°/0°/−45°/45°) _t with central cutout (0.4a) (CCCC) (a/h = 20).

Conclusion

Various cases have been investigated with additional mass and cutout on laminated composite and sandwich skew plates. The variation of boundary condition and ply layup has been taken into consideration. Various results of frequencies other than fundamental frequencies have been shown for the first time for the case of sandwich plates. Along with frequencies, various mode shape has been plotted to study its variation. The results show that skew angle plays a vital role in the value of frequencies as for a fixed skew angle, variation of additional mass does not affect the higher mode of frequencies except the fundamental frequency. The 2-D FE C₀ model based on IHSDT solutions presented in this article is in good agreement with the analytical solutions and, hence, quite elegant enough to explore the effect of openings and additional mass on free vibration behavior of laminated composite sandwich skew plates.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ajay Kumar

References

Soni

Singh

Mitra

. Buckling behavior of composite laminates (with and without cutouts) subjected to nonuniform in-plane loads. Int J Struct Stab Dyn 2013; 13(8): 1350044.

Love

AEH

. The small free vibrations and deformation of a thin elastic shell. Philos Trans R Soc A Math Phys Eng Sci 1888; 179: 491–546.

Ashton

Whitney

. Theory of laminated plates. 4th ed. Pennsylvania: Technomic Publishing Co. Inc, 1970, p. 153.

Reissner

. The effect of transverse shear deformation on the bending of elastic plates. Trans ASME J Appl Mech 1945; 12: 69–77.

Reissner

. On bending of elastic plates. Q Appl Math 1947; 5(1): 55–68.

Mindlin

. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech Asme 1951; 18(1): 31–38.

Yang

Norris

Stavsky

. Elastic wave propagation in heterogeneous plates. Int J Solids Struct 1966; 2: 665–684.

Pagano

. Exact solutions for rectangular bidirectional composites and sandwich plates. J Compos Mater 1970; 4(1): 20–34.

Reddy

. A simple higher-order theory for laminated composite plates. J Appl Mech 1984; 51(4): 745–752.

10.

Hildebrand

Reissner

Thomas

. Notes on the foundations of the theory of small displacements of orthotropic shells, United States, 1 March 1949.

11.

Christensen

. A high-order theory of plate deformation—part 2: laminated plates. J Appl Mech 1977; 44(4): 669–676.

12.

Christensen

. A high-order theory of plate deformation—part 1: homogeneous plates. J Appl Mech 1977; 44(4): 663.

13.

Kant

Owen

DRJ

Zienkiewicz

. A refined higher-order C plate bending element. Comput Struct 1982; 15(2): 177–183.

14.

Khdeir

Reddy

. On the forced motions of antisymmetric cross-ply laminated plates. Int J Mech Sci 1989; 31(7): 499–510.

15.

Kant

Manjunatha

. On accurate estimation of transverse stresses in multilayer laminates. Comput Struct 1994; 50(3): 351–365.

16.

Swaminathan

Patil

. Analytical solutions using a higher order refined computational model with 12 degrees of freedom for the free vibration analysis of antisymmetric angle-ply plates. Compos Struct 2008; 82: 209–216.

17.

Pandya

Kant

. Higher-order shear deformable theories for flexure of sandwich plates—finite element evaluations. Int J Solids Struct 1988; 24(12): 1267–1286.

18.

Ren

. A new theory of laminated plate. Compos Sci Technol 1986; 26(3): 225–239.

19.

Pandya

Kant

. Finite element analysis of laminated composite plates using a higher-order displacement model. Compos Sci Technol 1988; 32: 137–155.

20.

Nanda

Bandyopadhyay

. Nonlinear free vibration analysis of laminated composite cylindrical shells with cutouts. J Reinf Plast Compos 2007; 26(14): 1413–1427.

21.

Ferreira

AJM

. A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates. Compos Struct 2003; 59(3): 385–392.

22.

Natarajan

Ferreira

AJM

Bordas

SPA

, et al. Analysis of composite plates by a unified formulation-cell based smoothed finite element method and field consistent elements. Compos Struct 2013; 105: 75–81.

23.

Ferreira

AJM

Roque

CMC

Neves

AMA

, et al. Buckling and vibration analysis of isotropic and laminated plates by radial basis functions. Compos B Eng 2011; 42: 592–606.

24.

Ferreira

AJM

Roque

CMC

Carrera

, et al. Analysis of thick isotropic and cross-ply laminated plates by radial basis functions and a unified formulation. J Sound Vib 2011; 330(4): 771–787.

25.

Kulkarni

Kapuria

. Free vibration analysis of composite and sandwich plates using an improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory. Comput Mech 2008; 42(6): 803–824.

26.

Kant

Swaminathan

. Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher order refined theory. Compos Struct 2001; 53(1): 73–85.

27.

Fiedler

Lacarbonara

Vestroni

. Buckling of composite laminated plates via a refined higher-order theory. In: Collection of technical papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Newport, Rhode Island, 1–4 May 2006, pp. 1784–1795. AIAA.

28.

Ali

Atwal

. Prediction of natural frequencies of vibration of rectangular plates with cutouts. Comput Struct 1980; 12: 819–823.

29.

Hegarty

Ariman

. Elasto-dynamic analysis of rectangular plates with circular holes. Int J Solids Struct 1975; 11: 895–906.

30.

Rajamani

Prabhakaran

. Dynamic response of composite plates with cut-outs, part I: simply-supported plates. J Sound Vib 1977; 54(4): 549–564.

31.

Rajamani

Prabhakaran

. Dynamic response of composite plates with cut-outs, part II: clamped-clamped plates. J Sound Vib 1977; 54(4): 565–576.

32.

Chai

. Free vibration of laminated composite plates with a central circular hole. Compos Struct 1996; 35(4): 357–368.

33.

Sheikh

Haldar

Sengupta

. Free flexural vibration of composite plates in different. J Vib Control 2004; 10: 371–386.

34.

Makhecha

Ganapathi

Patel

. Vibration and damping analysis of laminated/sandwich composite plates using higher-order theory. J Reinf Plast Compos 2002; 21: 559–575.

35.

Asadi

Fariborz

. Free vibration of composite plates with mixed boundary conditions based on higher-order shear deformation theory. Arch Appl Mech 2012; 82(6): 755–766.

36.

Sahoo

Singh

. Assessment of zigzag theories for free vibration analysis of laminated-composite and sandwich plates. Proc Inst Mech Eng G J Aerosp Eng 2015; 229: 1931–1949.

37.

Chen

Liew

Lim

, et al. Vibration analysis of symmetrically laminated thick rectangular plates using the higher-order theory and p-Ritz method. J Acoust Soc Am 1997; 102: 1600–1611.

38.

Watkins

Santillan

Radice

, et al. Vibration response of an elastically point-supported plate with attached masses. Thin-Walled Struct 2010; 48(7): 519–527.

39.

Bathe

. Finite element method. Hoboken, NJ: John Wiley & Sons, Inc.

40.

Chakrabarti

Sheikh

. Vibration of laminate-faced sandwich plate by a new refined element. J Aerosp Eng 2004; 17(3): 123–134.

41.

Whitney

Pagano

. Shear deformation in heterogeneous anisotropic plates 1. J Appl Mech 2013; 37: 1031–1036.

42.

Reddy

Phan

. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. J Sound Vib 1985; 98(2): 157–170.

43.

Senthilnathan

Lim

Lee

, et al. Vibration of laminated orthotropic plates using a simplified higher-order deformation theory. Compos Struct 1988; 26(10): 211–229.

44.

Noor

. Free vibrations of multilayered composite plates. AIAA J 1973; 11(7): 1038–1039.

45.

Cho

Bert

Striz

. Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory. J Sound Vib 1991; 145(3): 429–442.

46.

Chen

. Vibration and stability of laminated plates based on a local high order plate theory. J Sound Vib 1994; 177(4): 503–520.

47.

Matsunaga

. Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Compos Struct 2000; 48(4): 231–244.

Influence of openings and additional mass on vibration of laminated sandwich rhombic plates using IHSDT

Abstract

Keywords

Introduction

Mathematical model

Conditions at layer interfaces

FE formulation

Free vibration analysis

Numerical application and results

Boundary condition

Engineering properties (geometrical and material)

Convergence and comparison studies

Novel results

Laminated skew composite plates

Laminated skew composite plates with additional mass

Laminated skew composite plates with square cutout

Laminated skew sandwich plates

Laminated skew sandwich plates with additional mass

Laminated sandwich plates under square cutout

Conclusion

Footnotes

Funding

ORCID iD

References