Vibration and Buckling of Skew Plates UnderLinearly Varying Edge CompressionAbhinav Kumar, Sarat Kumar Panda and Sekhar Chandra DuttaDepartment of Civil Engineering, Indian School of Mines, Dhanbad-826004, India.

Rajesh KumarDepartment of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur-721302, India.

(Received 21 February 2016; accepted 30 June 2016)

Pre-buckling vibration and buckling behaviour of composite skew plates subjected to linearly varying in-plane edgeloading with different boundary conditions are studied. The total energy functional of the skew plate mapped fromphysical domain to computational domain over which a set of orthonormal polynomials satisfying the essentialboundary conditions is generated by Gram-Schmidt orthogonalization process. Using Rayleigh-Ritz method inconjunction with Boundary Characteristics Orthonormal Polynomials, the total energy functional is converted intosets of algebraic equations for static stability problems and ordinary differential equation for free vibration problem.Pre-buckling vibration frequencies of the stressed skew plate are obtained by solving associated linear eigen valueproblem for free vibration and solution of the eigen value problem for static case results critical buckling load.From different parametric study, it is observed that the pre-buckling vibration frequency and critical buckling loadincrease with the increase of skew angle and edge restraint.

NOMENCLATUREa, b Plan-form dimension of platea/b Aspect ratio of plateNξξ and Nηη Applied edge forces along ξ and η axesNxcr Buckling loadu, v In-plane displacementw Out-of-plane displacementx, y, z Cartesian coordinatesϕ1, ϕ2 Rotations of the reference surface

about the ξ2 and ξ1 axesMxx, Myy , Mxy Moment resultantNxx, Nyy , Nxy Force resultantPxx, Pyy , Pxy Additional moment resultantS Strain energyΦ Stress functionNs Static load factorNt Dynamic load factorSm Membrane energySb Bending EnergySab Additional bending energySs Shear energyψ Skew angleω∗ Non-dimensional fundamental frequencyω Fundamental frequencyΠ Total potential energy

1. INTRODUCTION

Composite skew plates are widely used as structural ele-ments in many engineering applications such as aircraft wings,marine structures, ship structures, and bridges. These compos-ite plates are generally preferred because of their high strength-to-weight ratio and ability to be moulded according to any de-sired orientation. In a complex structural system, the loadingcoming on it may not be uniform. For a few simple examples,

the web of an I- beam, aircraft wings, and the stiffened platesin ship structures are generally subjected to other than uniformin-plane loading. There is literature available on the stability ofrectangular plates subjected to non-uniform in-plane loading.But so far as authors’ knowledge, studies on the pre-bucklingvibration and static stability of a skew plate under non-uniformor linearly varying in-plane loading is not available in the openliterature.

Literatures are available on stability and vibration studiesof skew plates under uniform in-plane loading. Durvasulaworked on buckling problems of simply supported skew plateswith in-plane stresses represented in terms of orthogonal com-ponents using the Rayleigh-Ritz method (RRM), employinga double Fourier sine series in oblique coordinates.1 Liewand Lam used two-dimensional orthogonal plate functions asan admissible deflection function for the study of the flexu-ral vibration of skew plates with the RRM.2 Authors followedthe Gram-Schmidt orthogonalization process to generate two-dimensional orthogonal plate functions. Wang et al. performedthe buckling analysis using the RRM with pb-2 Ritz functions,which consist of the product of a two-dimensional polynomialfunction and a basic function.3 The first five frequencies havebeen determined for the transverse vibration of a rectangularor a skew plate under different boundary conditions by us-ing boundary characteristic orthogonal polynomials (BCOP)in two variables by Singh and Chakraverty.4 Wang presented aB-spline RRM based on first-order shear deformation (FOSD)plate theory for a buckling study of composite skew plates.5

Mizusawa et al. presented a method for evaluating the bucklingof stiffened skew plates by using theRRM with B-spline func-tions as co-ordinate functions.6 Taj and Chakrabarti studiedthe static and dynamic analysis of functionally-graded skewplates by using the finite element method (FEM) based on thethird-order shear deformation (TOSD) theory.7 Kitipornchaiet al. followed the principle of stationary total potential en-

International Journal of Acoustics and Vibration, Vol. 24, No. 2, 2019 (pp. 271–283) https://doi.org/10.20855/ijav.2019.24.21215 271

A. Kumar, et al.: VIBRATION AND BUCKLING OF SKEW PLATES UNDER LINEARLY VARYING EDGE COMPRESSION

ergy in conjunction with the pb-2 RRM for a buckling anal-ysis of skew plates of various aspect ratios, skew angles, andboundary conditions.8 Farag and Ashour used the Kantorovichmethod, a fast-converging, semi-analytical method for a vibra-tion analysis of thin orthotropic skew plates.9 In this method,the finite strip method is modified to reduce the complexityof the problem. Ganapathi and Prakash studied the thermalbuckling of functionally-graded simply supported skew platesusing FOSD theory with the FEM.10 The authors consideredlinear and nonlinear temperature variation across the thick-ness. Wang et al. proposed a modified version of the differen-tial quadrature method (DQM) for analysing the static stabilityof different skew plates.11 Numerical results indicate that themodified DQM is faster in convergence study and gives betterresults than the old DQM version. Daripa and Singha workedon the stability characteristics of composite skew plates sub-jected to in-plane compressive load using the shear deformablefinite element approach.12 The effect of pre-buckling stressesat the corner regions of isotropic and composite skew plateshas been studied. Karami et al. followed the DQM for static,free vibration, and stability analysis of thin trapezoidal com-posite skew plates.13 The authors used the general transforma-tion scheme for transferring the variation of the variables in thecomputational domain to the physical domain and vice versa.Zhou and Zheng employed the moving least square (MLS)method for analysing the vibration characteristic of skew plateswith large skew angles.14 The MLS technique in conjunctionwith the Ritz method is applied to derive the governing eigen-value equation for the skew plate. The boundary conditions ofthe plate are satisfied through a point substitution techniquethat makes the MLS-Ritz trial function satisfy the essentialboundary conditions along the plate edges. Singha and Gana-pathi investigated the large amplitude free flexural vibration ofthin laminated composite skew plates within the frame workof the FEM.15 Lagrange’s equations of motion and von Kar-man’s geometric nonlinearity is used to derive the governingdifferential equation. Taj and Chakrabarti worked on the staticand dynamic analysis of functionally graded material (FGM)skew plates under mechanical loading by using the FEM. Theauthors employed the FEM formulation based on the TOSDtheory.7 Eftekhari and Jafari proposed modified mixed Ritz-differential quadrature (Ritz-DQ) methodology for a vibrationand buckling study of rectangular skew plates.16 All the natu-ral boundary conditions are exactly implemented by modifiedmixed Ritz-DQ formulation. The method resulted in accuratesolutions for the natural frequencies of thick rectangular platesinvolving adjacent free edges and skew plates using a smallnumber Ritz terms and DQ sampling points. Vibration anal-ysis of skew plates with different boundary conditions by us-ing the new version of the DQM is investigated by Wang etal. (2014).17 The authors observed good convergence of fre-quency for the simply supported skew plate with a large skewangle.

There are many research articles available in open literatureon the static stability analysis of rectangular plates subjectedto non-uniform in-plane loading or linearly varying in-planeloading. Leissa and Kang (2002) and Kang and Leissa (2001,2005) presented exact solutions for a plate subjected to differ-ent types of linearly varying in-plane loading with differentboundary conditions.18–20 Zhong and Gu presented the ex-act solution of the buckling of moderately thick simply sup-

ported plates with symmetric cross-ply laminates under differ-ent linearly varying in-plane edge loads.21 The authors usedthe FOSD theory for obtaining the buckling load for variousaspect ratios of the composite plates. Lai and Xiang presentedthe discrete singular convolution method for solving bucklingand vibration problems of rectangular plates with elasticallyrestrained edges and linearly varying uni-axial in-plane load-ing.22 Authors presented accurate buckling and vibration solu-tions of plates having two opposite edges elastically restrainedand the other two sides clamped. Lopatin and Morozov pre-sented the analytical solution of the buckling problem for anorthotropic rectangular plate having two parallel edges sim-ply supported, one edge clamped, and the remaining edge free(the SSCF plate).23 The plate considered is subjected to alinearly varying in-plane load, and the solution technique in-volves the Kantorovich procedure in conjunction with the gen-eralised Galerkin method. The buckling problems are solvedfor isotropic and orthotropic plates with various aspect ratios.The buckling of thin rectangular plates with nonlinearly dis-tributed loadings along two opposite plate edges is analysedusing the DQM by Wang et al. (2006, 2007).24, 25 The au-thors first solved the plane elasticity problem to obtain the dis-tribution of in-plane stresses, and then the buckling problemwas solved. The DQM was proved an accurate and computa-tionally efficient numerical method by Wang et al. (2006).26

Authors used this method to study the vibration and bucklingof an SS-C-SS-C rectangular plate loaded by linearly vary-ing in-plane stresses.26 Tang and Wang analysed symmetri-cally laminated rectangular plates with parabolic distributedin-plane compressive loadings along two opposite edges usingthe RRM in conjunction with the classical laminated plate the-ory.27 Stress functions satisfying all stress boundary conditionsare constructed based on the Chebyshev polynomials. It wasobserved that double sine series displacement for simply sup-ported symmetrically laminated plates overestimate bucklingloads. Recently, Kumar et al. studied the parametric resonanceof a composite skew plate under non-uniform in-plane loadingusing the RRM in conjunction with BCOP functions.28

So far, to the best of the authors’ knowledge, there is nowork available in open literature on the pre-buckling vibrationand buckling of composite skew plates based on higher-ordershear deformation theory (HSDT) under non-uniform or lin-early varying in-plane loading. In the present investigation, thepre-buckling vibration and buckling of composite skew platessubjected to linearly varying in-plane loading is considered fordifferent boundary conditions. The linearly varying in-planeedge compression causes same linearly varying stress in theskew plate stress. The total energy functional is formulated byusing the in-plane stress distribution. The total energy func-tional is transformed from the physical domain to the compu-tational domain using transformation relations. The orthonor-mal polynomials are generated by using the Gram-Schmidt or-thogonalization process, which satisfies the essential boundaryconditions of skew plates in computational domain. Using theRRM in conjunction with BCOPs, the total energy functionalis converted into sets of algebraic equations for the static sta-bility problem and into ordinary differential equations for thefree vibration problem. Pre-buckling vibration frequencies ofthe stressed skew plate are obtained by solving the associatedlinear eigenvalue problem for free vibration. The solution oftheeigenvalue problem for the static case results in the critical

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A. Kumar, et al.: VIBRATION AND BUCKLING OF SKEW PLATES UNDER LINEARLY VARYING EDGE COMPRESSION

Figure 1a. Geometry and linearly varying in-plane loading of the skew plate.

Figure 1b. Various types of linearly varying loadings.

buckling load. The effects of skew angles, boundary condi-tions, shear deformations, aspect ratios, and loadings on thepre-buckling vibration and buckling behaviour of compositeskew plates are presented.

2. FORMULATION

In this study, composite skew plates are analysed havinglength “a,” width “b,” and “ln” layers of equal thickness lami-nates as shown in Fig. 1a. The z-axis is in the direction perpen-dicular to the ξ−η plane. Different types of linearly varying in-plane edge loadings are considered for finding the pre-bucklingfrequencies and static stability of the composite skew plate.The linearly varying in-plane edge loadings are expressed as:Nxx = N0(1 − γ(ηb )). For different values of γ, we obtaindifferent in-plane load distributions: uniform (γ = 0), trape-zoidal (γ = 0.5), triangular (γ = 1), partial tension (γ = 1.5),and in-plane bending (γ = 2). The linearly varying in-planeloads are shown in Fig. 1b.

Higher-order shear deformation theory (HSDT) is imple-mented for composite plates as proposed by Reddy and Liu.29

In this theory, the displacements of the middle surface are ex-pressed as cubic functions of the thickness coordinate, and thetransverse displacement is assumed to be constant through thethickness. For this displacement field, the distribution of trans-verse shear stress through the thickness is of parabolic varia-tion. The transverse shear stress is zero at both the top and bot-tom free surface and is maximum at the middle surface. The

displacement fields may be written as:

u(x, y, z, t) = uo(x, y, t) + zϕ1(x, y, t)+

z3(4/3h2)[−ϕ1(x, y, t)− wo,x(x, y, t)]

v(x, y, z, t) = vo(x, y, t) + zϕ2(x, y, t)+

z3(4/3h2)[−ϕ2(x, y, t)− wo,y(x, y, t)]

w(x, y, z, t) = wo(x, y, t); (1)

The above displacement fields can be rearranged as Soldatosdisplayed:30

u(x, y, z, t) = uo(x, y, t)− zwo,x(x, y, t) + f(z)φ1o(x, y, t)

v(x, y, z, t) = vo(x, y, t)− zwo,y(x, y, t) + f(z)φ2o(x, y, t)

w(x, y, z, t) = wo(x, y, t); (2)

where φ01(x, y, t) = ϕ1(x, y, t) + w0,x(x, y, t);

φ02(x, y, t) = ϕ2(x, y, t) + w0,y(x, y, t)

and f(z) = z[1− (4/3)(z/h)2]; (3)

Here, the displacement components u, v, and w are in direc-tions along the x-, y-, and z-axes, respectively, at a distance zaway from mid plane, and u0(x, y, t), v0(x, y, t) and are dis-placement components of a generic point on the middle sur-face. The expressions φ01(ϕ1+w0

,x) and φ02(ϕ2+w0,y) represent

the total rotations (transverse shear strains) of the normal to themiddle surface about the y- and x-axes, respectively. The vari-able h is plate thickness, and (),x represents the differentiationwith respect to x. The strain-displacement relations at a dis-tance z away from the mid-plane of a plate can be expressedas:

εx(x, y, z) = εox(x, y, t)− zwo,xx(x, y, t) + f(z)φ1o,x(x, y, t)

εy(x, y, z, t) = εoy(x, y, t)− zwo,yy(x, y, t) + f(z)φ2o,y(x, y, t)

γxy(x, y, z, t) = γoxy(x, y, t)− 2zwo,xy(x, y, t)+

f(z)φ1o,y(x, y, t) + f(z)φ2

o,x(x, y, t)

γxz(x, y, z, t) = u,z(x, y, z, t) + w,x(x, y, z, t) =

f′(z)φ1

o(x, y, t)

γyz(x, y, z, t) = v,z(x, y, z, t) + w,y(x, y, z, t) =

f′(z)φ2

o(x, y, t); (4)

Both εox(x, y), εoy(x, y) and γoxy(x, y) are reference surfacestrains and are defined as:

εoy(x, y, t) = vo,y(x, y, t), εox(x, y, t) = uo,x(x, y, t),

γoxy(x, y, t) = uo,y(x, y, t) + vo,x(x, y, t); (5)

The different components of strain energy in which mem-brane energy (Sm), bending energy (Sb), additional bendingenergy due to additional change in curvature (Sab), and shearenergy(SS) are related respectively to membrane strainsε0

T

={ε0x(x, y) ε0y(x, y) γ0xy(x, y)

}, bending strains

κT ={−wo,xx(x, y, t) − wo,yy(x, y, t) − 2woxy(x, y, t)

},

additional bending strains κaT

={φo1,x(x, y, t) φo2,y(x, y, t) φo1,y(x, y, t) + φo2,x(x, y, t)},and transverse shear strains φ0T

=

International Journal of Acoustics and Vibration, Vol. 24, No. 2, 2019 273

A. Kumar, et al.: VIBRATION AND BUCKLING OF SKEW PLATES UNDER LINEARLY VARYING EDGE COMPRESSION{φ02(x, y, t) φ01(x, y, t)

}, through the following consti-

tutive relations:

Sm =1

2

∫A

(ε0

T

Aε0 + ε0T

Bκ+ ε0T

Cκa)dA; (6)

Sb =1

2

∫A

(κT Bε0 + κT Dκ+ κT Eκa

)dA; (7)

Sab =1

2

∫A

(κaT

Cε0 + κaT

Eκ+ κaT

Fκa)dA; (8)

Ss =1

2

∫A

(φ0T

Hφ0)dA; (9)

The sum of all strain energy of the composite skew plate iscalculated as:

S = Sm + Sb + Sab + Ss; (10)

In the present investigation, the bold upright letters and bolditalic letters are used to denote matrices and vectors, respec-tively. The extensional stiffness (A), coupling stiffness (B),bending stiffness (D), and shear stiffness (H) matrices are de-fined in terms of the transformed lamina stiffness Q as:

(AijBijDij) =

∫ h/2

−h/2Qij(1, z, z

2)dz

=

ln∑k=1

∫ zk

zk−1

Qij(1, z, z2)dz i = 1, 2, 6 and j = 1, 2, 6

(Cij , Eij , Fij) =

∫ h/2

−h/2Qij(1, z, f(z))f(z)dz

=

ln∑k=1

∫ zk

zk−1

Qij(1, z, f(z))f(z)dz i = 1, 2, 6 and j = 1, 2, 6

Hij =

∫ h/2

−h/2Qijf

′(z)f

′(z)dz

=

ln∑k=1

∫ zk−1

zk

Qijf′(z)f

′(z)dz i = 4, 5 and j = 4, 5;

(11)

where, ln is the number of orthotropic layer; Aij , Bij , Cij ,Dij , Eij , Fij and Hij represent plate stiffness; Qij(i, j =1, 2, 6) are the transformed plane stress reduced stiffness co-efficients; and Qij(i, j = 4, 5) are the transformed through-thickness shear stiffness coefficients.

The potential energy due to in-plane loading can be ex-pressed as:

P = −1

2

∫ a

0

∫ b

0

{ ∂w0

∂x∂w0

∂y

}T [Nxx NxyNxy Nyy

]{

∂w0

∂x∂w0

∂y

})dxdy; (12)

where Nxx, Nyy and Nxy are the in-plane loads in the x-direction, the in-plane loads in the y-direction, and the in-planeshearing loads, respectively. The expression for kinetic energy

Figure 2. Mapping of the skew plate domain into a unit square plate domain.

may be expressed as:

TE =1

2

∫ a

0

∫ b

0

∂u0

∂t∂v0

∂t∂w0

∂t∂φ0

2

∂t∂φ0

1

∂t

T ρh 0 0 0 00 ρh 0 0 00 0 ρh 0 0

0 0 0 ρh3

12 0

0 0 0 0 ρh3

12

∂u0

∂t∂v0

∂t∂w0

∂t∂φ0

2

∂t∂φ0

1

∂t

dxdy; (13)

where ρ is the density. The energy expression Sm, Sb, Sab, Ss

and P is transformed from the orthogonal co-ordinate systemto the oblique co-ordinate system using transformation rela-tions. Mapping of a skew plate having length “a,” breadth “b,”and thickness “h,” into a unit square plate domain in the ξ − ηplane is shown in Fig. 2. The transformation relations are givenby:

x = aξ + bη sinψ; y = bη cosψ; (14)

The following transformation relations are followed for themapping and function of the functional from the x-y plane tothe ξ − η plane:[ ∂

∂x∂∂y

]=

[1a 0

− tanψa

secψb

] [ ∂∂ξ∂∂η

]; (15)

∂2

∂x2

∂2

∂y2

∂2

∂x∂y

=

1a2 0 0

tan2 ψa2

sec2 ψb2 − 2 tanψ secψ

ab

− tanψa2 0 secψ

ab

∂2

∂ξ2

∂2

∂η2

∂2

∂ξ∂η

; (16)

The transformed membrane strain energy S∗m, S∗b, S∗ab

and S∗s potential energy (P ∗) are kinetic energy (T ∗E) repre-sented as:

S∗m =1

2

∫ 1

0

∫ 1

0

<1(uo,ξ, uo,η, v

o,ξ, v

o,η) (17a)

cosψdξdη;

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A. Kumar, et al.: VIBRATION AND BUCKLING OF SKEW PLATES UNDER LINEARLY VARYING EDGE COMPRESSION

S∗b =1

2

∫ 1

0

∫ 1

0

<2(wo,ξξ, wo,ξη, w

o,ηη, φ

o1,ξ, φ

o1,η, φ

o2,ξ, φ

o2,η)

(17b)

cosψdξdη;

S∗ab =1

2

∫ 1

0

∫ 1

0

<3(wo,ξξ, wo,ξη, w

o,ηη, φ

o1,ξ, φ

o1,η, φ

o2,ξ, φ

o2,η)

(17c)

cosψdξdη;

S∗s =1

2

∫ 1

0

∫ 1

0

<4(φo1, φo2) cosψdξdη; (17d)

P ∗ =− 1

2

∫ 1

0

∫ 1

0

<5(nxx, nxy, nyy, wo,ξ, w

o,η) (17e)

cosψdξdη;

T ∗E

=1

2

∫ 1

0

∫ 1

0

<6(uo, vo, wo, φo1, φo2) cosψdξdη; (17f)

The expressions for <i i = 1, 2, . . ., 6) are given in Ap-pendix A.

2.1. Generation of Orthogonal PolynomialsTo generate orthogonal polynomials satisfying essential

boundary conditions over the unit square plate domain in theξ − η plane, the following displacement fields are used:

uo(ξ, η, t) =

N∗∑j=1

Uj∧αj (ξ, η) cos(ωt)

vo(ξ, η, t) =

N∗∑j=1

Vj∧βj (ξ, η) cos(ωt)

wo(ξ, η, t) =

N∗∑j=1

Wj

∧Θj (ξ, η) cos(ωt)

φ1o(ξ, η, t) =

N∗∑j=1

Kj∧γj (ξ, η) cos(ωt)

φ2o(ξ, η, t) =

N∗∑j=1

Lj∧δj (ξ, η) cos(ωt); (18)

where N∗ is the order of approximation to get the desired ac-curacy; Uj , Vj , Wj , Kj and Lj are unknown constants; and∧αj(ξ, η),

∧βj(ξ, η),

∧Θj(ξ, η),

∧γj(ξ, η)and

∧δj(ξ, η) are boundary

characteristics orthonormal polynomial (BCOP) functions.

The orthonormal polynomials (∧Θj(ξ, η) are generated for sat-

isfying the essential boundary conditions over the unit squareplate domain in the ξ, η plane by the Gram-Schmidt pro-cess explained by Singh and Chakraverty.4 In the presentstudy, the same orthonormal polynomials are considered for∧αj(ξ, η),

∧βj(ξ, η),

∧γj(ξ, η) and

∧δj(ξ, η) also. Orthonormal

polynomials (∧Θj(ξ, η)) over the region 0 ≤ ξ ≤ 1, 0 ≤ η ≤ 1

have been generated using the linearly independent set of func-tions Fj(ξ, η) = f(ξ, η)× fj(ξ, η). Both f(ξ, η) and fj(ξ, η)are two functional scalar quantities as given below:

f(ξ, η) = ξp(1− ξ)qηr(1− η)s; (19a)

fj(ξ, η) ={

1, ξ, η, ξ2, ξη, η2, ξ3, ξ2η, ξη2, η3, . . . .}. (19b)

The explicit expression for the orthogonal polynomials fordifferent types of boundary conditions are derived by using thefollowing equations:

Θj (ξ, η) = ξpηq(1− ξ)r(1− η)sfj(ξ, η). (20)

The value of ’p” depends on the boundary condition on ξ =0, 1. At ξ = 0, p = 0, 1, 2 respectively for free, simply sup-ported, and clamped boundary conditions. At ξ = 1, q =0, 1, 2 respectively for free, simply supported, and clampedboundary conditions. In a similar way, the values of r and sare chosen for η = 0, 1 as:

Θ1 (ξ, η) = F1 (ξ, η) ,Θj (ξ, η) = Fj (ξ, η)−j−1∑i=1

αjiΘi (ξ, η) ;

(21)

αji =< Fj (ξ, η) ,Θi (ξ, η) >

< Θi (ξ, η) ,Θi (ξ, η) >,

i = 1, 2, 3, . . . , (j − 1), j = 2, 3, 4, . . . , N∗. (22)

The inner product of the functions Θi (ξ, η) and Θj (ξ, η) isexpressed as:

< Θi (ξ, η) ,Θj (ξ, η) >=

∫ 1

0

∫ 1

0

Θi (ξ, η) Θj (ξ, η) dξdη.

(23)The norm of the function Θj (ξ, η) is expressed by:

‖Θj (ξ, η)‖ =< Θj (ξ, η) ,Θj (ξ, η) >1/2

=

[∫ 1

0

∫ 1

0

Θj (ξ, η) Θj (ξ, η) dξdη

]1/2. (24)

The normalisation of the orthogonal polynomial is essential foravoiding the illness of the stiffness matrix. The normalisationis done by:

∧Θj (ξ, η) =

Θj (ξ, η)

‖Θj (ξ, η)‖. (25)

In the present study for all cases of the skew plate, 36 num-bers of terms have been considered for getting the convergedpre-buckling free vibration frequency and buckling load. Theexplicit expressions for 36 polynomial terms for simply sup-ported boundary conditions (SSSS) are given in Appendix B.For the sake of brevity, the explicit expressions for polynomialterms for other boundary conditions are not presented.

2.2. Rayleigh-Ritz MethodThe total potential energy (Π) of the system is the

sum of the strain energy (S) and the potential energy(P ). The total potential energy function Π(S + P )and kinetic energy function (TE) are transformed fromthe x-y plane to the ξ − η plane by using Eqs. (14)and (15). The displacement and rotational components(uo(ξ, η, t), vo(ξ, η, t), wo(ξ, η, t), φ1o(ξ, η, t)andφ2o(ξ, η, t))are expressed in terms of unknown constants (Uj ,Vj , Wj , Kj , Lj) using the orthonormal polynomials

(∧αj(ξ, η),

∧βj(ξ, η),

∧Θj(ξ, η),

∧γj(ξ, η)and

∧δj(ξ, η)). Further,

these displacement and rotational components are substituted

International Journal of Acoustics and Vibration, Vol. 24, No. 2, 2019 275

A. Kumar, et al.: VIBRATION AND BUCKLING OF SKEW PLATES UNDER LINEARLY VARYING EDGE COMPRESSION

in the total potential energy function and the kinetic energyfunction to get the energy function in the ξ−η plane. The totalenergy (TL) is the addition of the kinetic energy (T ∗E) to thetotal potential energy (Π). This involves different unknowns,Uj , Vj , Wj , Kj and Lj , and can be minimised with respect toconstants as follows:

∂TL∂Cj

= 0 for j = 1, 2, 3, . . . , 5N∗; (26a)

where,

C1

C2

.

.

.CN∗

CN∗+1

CN∗+2

.

.

.C5N∗

=

{u}{v}{w}{k}{l}

and {u} =

U1

U2

.

.

.UN∗

,

{v} =

V1V2...VN∗

, {w} =

W1

W2

.

.

.WN∗

,

{k} =

K1

K2

.

.

.KN∗

, {l} =

L1

L2

.

.

.LN∗

.

(26b)

The total number of constants is 5N∗. The above Eq. (26a)leads to the governing eigenvalue equation for finding the pre-buckling free vibration frequency (λf ) as given below:

∑[[KL − λKG]− λfM ]

{u}{v}{w}{k}{l}

= {0} ; (27a)

where [M] is the mass matrix; [KL] is the linear elastic stiff-ness; and [KG] is the linear geometric stiffness matrix. Forfinding the buckling load (λbuck) in the static buckling prob-lem, the eigenvalue problem is reduced to the following:

∑[KL − λbuckKG]

{u}{v}{w}{k}{l}

= {0} . (27b)

Here, λ < λbuck. Equations (26) and (27) constitutes a set of5N∗ simultaneous algebraic equations referred to as the Ritzsystem.

Figure 3. Variation of non-dimensional frequency (ω∗) for different modesof vibration for three layered crosspl- [0/90/0] composite SSSS skew plates(a/h = 100) having different skew angles.

3. RESULTS AND DISCUSSIONS

The non-dimensional fundamental frequency (ω∗) ofisotropic skew plates (a/b = 1, a/h = 100) for different skewangles (ψ) and different support conditions are similar to openliterature and given in Table 1. For obtaining the convergenceof results, 36 numbers of terms (orthogonal polynomials) areconsidered for all cases of support conditions and skew angles.The present solution agrees well with the results obtained byusing the double series of beam characteristic function by Nairand Durvasula and the two-dimensional orthogonal plate func-tions by Liew and Lam.2, 31 Durvasula used the double Fouriersine series to find the non-dimensional fundamental frequencyof a simply supported skew plate.1

The influence of skew angles on the free transverse vibra-tion of simply supported composite [0/90/0] skew plates isshown in Fig. 3. As the skew angle increases 0 ≤ ψ ≤ 450,the frequency of the first three modes of vibration increases.This behaviour is expected sincebecause, when the skew angleincreases, the stiffness of the plate increases, which makes ahigher frequency for all three modes of vibration. Similar be-haviour is also observed for skew plates with more and moreedge restraints. Figure 4a shows the effect of boundary condi-tions on the fundamental frequency of a composite skew plate.As the four edges of the composite [0/90/0] skew plate supportchanges in boundary conditions from simply supported (S) toclamped (C) one after another, the stiffness increases, whichcauses a higher frequency value. The fundamental frequen-cies of the composite [0/90/0] skew plate are shown for thefollowing four boundary conditions: SSSS, CSSS, CSCS andCCCC. The pre-buckling vibration behaviour is presented asa plot of non-dimensional fundamental frequencies against di-mensionless in-plane loads for composite skew plates of differ-ent boundary conditions as shown in Fig. 4b. The pre-bucklingfundamental frequency and in-plane load of each plate are nor-malized with respect to the free vibration frequency and buck-ling load of an SSSS plate (a/b = 1; ψ = 30◦; a/h = 100).The result indicates that the fundamental pre-buckling fre-quency increases with the increase in the edge restraints of theskew plate. The frequency is always higher for a CSCS plateas compared to an SCSC plate.

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Table 1. Non-dimensional fundamental frequency (ω∗) of isotropic square skew plates (a/b = 1, a/h = 100) for different skew angles (psi) and differentsupport conditions.

Type of Supports Skew angle (ψ)Non-dimensional fundamental frequency (ω∗)

Present solution Durvasula1 Liew and Lam2

SSSS

0◦ 1.99 2.0 -15◦ 2.11 2.11 2.1130◦ 2.52 2.54 2.5245◦ 3.53 3.54 3.53

Nair and Durvasula31 Liew and Lam2

CSSS

0◦ 2.39 2.40 2.4015◦ 2.54 2.54 2.5530◦ 3.10 3.11 3.1245◦ 4.53 4.53 4.58

CCSS

0◦ 2.74 2.74 2.7415◦ 2.90 2.91 2.9130◦ 3.52 3.57 3.5245◦ 5.08 5.12 5.08

Note: ω∗ = ω[(b2/π2)

√ρh/D

]; D is flexural rigidity.

Figure 4a. Variation of non-dimensional fundamental frequency (ω∗) for dif-ferent boundary conditions for three layered cross-ply [0/90/0] composite skewplates (a/h = 100) having different skew angles.

Figure 5a shows the influence of aspect ratio on the freevibration frequency of skew plates. As the aspect ratio(a/b) increases, the non-dimensional fundamental frequencyincreases. This is because the plate becomes stiffer with theincrease of the aspect ratio. It is also observed that as theaspect ratio increases, the non-dimensional fundamental fre-quency increases more rapidly for plates having a higher skewangle. Figure 5b shows the effect of the aspect ratio on thesecond mode of transverse vibration of skew plates. The vari-ation of second mode frequency is similar to the fundamentalfrequency for a plate with an aspect ratio up to nearly 1.5 forall values of skew angles. However, the frequency variationdecreases beyond an aspect ratio of 1.5 for all values of skewangles. The pre-buckling vibration behaviour is presented asa plot of non-dimensional fundamental frequencies against di-mensionless in-plane loads for SSSS composite skew plates ofdifferent skew angles as shown in Fig. 5c. The pre-bucklingfundamental frequency and in-plane load of each plate are nor-malized with respect to the free vibration frequency and buck-ling load of an SSSS plate (a/b = 1; ψ = 0◦; a/h = 100). Theresult indicates that the stiffness of the skew plate increases

Figure 4b. The influence of skew angle on the pre-buckling vibration be-haviour of a composite skew plate a/b = 1; a/h = 100; ψ = 300) havingdifferent boundary conditions.

in a non-uniform manner with the increase of the skew angle.Also, in the pre-buckling range, frequency decreases with theincrease of in-plane load, reaching zero frequency at bucklingload.

Figure 6 reflects on the effect of shear deformation on trans-verse vibration of skew plates for different aspect ratios (a/b)of the three layered cross-ply [0/90/0] composite SSSS skewplates ψ = 30◦. The effect of shear deformation on frequencyis more for higher values of the aspect ratio. The frequencyof different modes of vibration increases with the increase ofaspect ratio until a certain value, and after that, it remains con-stant or slowly decreases for a plate of higher thickness inwhich a/h = 10 and 20. The non-dimensional fundamentalfrequencies increase with the increase of aspect ratio (a/b) forplates of a/h ratio 50 and 100. For plates of a/h ratio 10 and20, the non-dimensional fundamental frequency increases upto an aspect ratio (a/b) with values of 1.75 and 2.5, respec-tively, and then the frequency value slowly decreases.

Figure 7 shows the influence of different types of linearlyvarying in-plane loading on the pre-buckling vibration be-haviour of an SSSS composite plate (a/b = 1; ψ = 30◦;a/h = 100). The pre-buckling fundamental frequency and

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Figure 5a. Variation of non-dimensional fundamental frequency (ω∗) withrespect to the aspect ratio (a/b) for three layered cross-ply [0/90/0] compositeSSSS skew plates (a/h = 100) having different skew angles.

Figure 5b. Variation of non-dimensional frequency (ω∗) for the second modewith respect to the aspect ratio (a/b) for three layered cross-ply [0/90/0] com-posite SSSS skew plates (a/h = 100) having different skew angles.

in-plane load of each plate are normalized with respect to thefree vibration frequency and buckling load of the SSSS plate(a/b = 1; ψ = 30◦; a/h = 100). The result indicates thatthe pre-buckling fundamental frequency value increases withthe increase in the γ value. Also, in the pre-buckling range,frequency decreases with the increase of in-plane load, reach-ing zero frequency at buckling load. The buckling loads ob-tained from the present method of solution for skew platessubjected to uniform in-plane loading with boundary condi-tions in which all edges are simply supported and in which alledges are clamped are similar to open literature and given inTable 2. The mechanical properties used in the present analysisfor composite skew plate are: E11/E22 = 25, G12 = G13 =0.5E22, G23 = 0.2E22 and ν12 = 0.25. The variation of di-mensionless buckling load coefficients kc(=

Ncrb2

E22π2h3 ) for dif-ferent plate aspect ratios (a/b) of a simply supported cross-plycomposite [0/90/0] skew plate (a/h = 100, ψ = 30circ) sub-jected to different types of in-plane loading (γ = 0.5, 1.0, 1.5and 2.0) are shown in Fig. 8. It is observed that for uniformin-plane loading (γ = 0), the value of the buckling coefficient

Figure 5c. The influence of skew angle on the pre-buckled vibration behaviourof an SSSS composite skew plate (a/b = 1; a/h = 100).

Figure 6. Effect of shear deformation on non-dimensional fundamental fre-quency (ω∗) for different aspect ratios (a/b) of the three layered cross-ply[0/90/0] composite SSSS skew plates (ψ = 300).

(k) for the composite skew plate decreases up to an aspect ra-tio (a/b) of 1.75 and then slowly increases up to an aspect ratiovalue of 2.35 and then decreases slowly. Here, the aspect ra-tio at which the composite skew plate buckling mode shiftsfrom first mode to second mode is 2.35. For other types oflinearly varying in-plane loading when γ = 0.5, 1.0, 1.5 and2.0 the value of the buckling coefficient first decreases up toan aspect ratio (a/b) of 1.45, 1.10, 0.9 and 0.8, respectively.The mode-shifting occurs from first mode to second mode fordifferent types of linearly varying in-plane loading in whichγ = 0.5, 1.0, 1.5 and 2.0 for aspect ratios (a/b) of 2.25, 2.0,1.33, and 1.0, respectively. For all types of linearly varying in-plane loading, the value of the buckling coefficient decreasesafter mode-shifting from first mode to second mode.

Figure 9 shows the variation of the dimensionless bucklingload coefficient kc(=

Ncrb2

E22π2h3 ) for different plate aspect ratios(a/b) of a clamped cross-ply composite [0/90/0] skew plate(a/h = 100) subjected to in-plane bending loading (γ = 2.).For in-plane bending, the value of the buckling coefficient firstdecreases up to an aspect ratio of nearly 0.8 and then increasesup to an aspect ratio of nearly 1.0 for a skew plate having a

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Table 2. Critical buckling coefficient (ki) isotropic square skew plates (a/b = 1, a/h = 100) for different skew angles (ψ) under uniform in-plane loading.

Type of Supports Skew angle (ψ)Dimensionless buckling coefficient (ki)

Present solution Wang5 Babu and Kant32

Simply supported plate (SSSS)

0◦ 4 4.0 4.015◦ 3.826 3.824 3.83030◦ 3.323 3.316 3.33045◦ 2.559 2.525 2.557

Durvasula33 Wang et al.3

Clamped supported plate (CCCC)

0◦ 10.074 10.074 10.07415◦ 9.431 9.462 9.47930◦ 7.612 7.638 7.73445◦ 5.110 5.110 5.172

Note: ki = Ncrb2cos4ψ/π2D

Figure 7. The influence of different types of in-plane loading on the pre-buckled vibration behaviour of cboxan SSSS composite skew plate (a/b = 1;a/h = 100; ψ = 30◦).

skew angle of ψ = 30◦and45◦. For the skew angle of ψ = 30◦

and 15◦, the buckling coefficient value mostly remains con-stant between aspect ratio values of 0.8 and 1.0. The change ofmode occurs for all skew plates at an aspect ratio of 1.0, andthe further the buckling coefficient value decreases, the higherthe value of the aspect ratio.

In the present investigation, the following nine sets ofboundary conditions are considered: SSSS, SSCS, SCSS,CSCS, SCSC, SSCC, CCSC, CCCS, CCCC, where ’S” standsfor simply supported edge and ’C” for clamped edge. The let-ters indicate the boundary conditions on the edge of the platein the anti-clockwise fashion starting from the left-hand corner.Table 3 shows the dimensionless buckling coefficient of threelayered cross-ply [0/90/0] composite skew plates (ψ = 45◦,a/h = 100) with nine sets of boundary conditions subjected totrapezoidal in-plane loading (γ = 0.5). It is observed that themore edge restraint there is, the higher the buckling coefficientis.

Tables 4 and 5 show the dimensionless buckling coefficientfor three layered cross-ply [0/90/0] composite skew plates(ψ = 45◦, a/h = 100) with nine sets of boundary conditionssubjected to triangular in-plane loading and partial tension in-plane loading, respectively.

Figure 8. Variation of the buckling load coefficient (k) with respect to theaspect ratio (a/b) for three layered cross-ply [0/90/0] composite SSSS skewplates (ψ = 30◦, a/h = 100) subjected to different in-plane loadingcboxs.

4. CONCLUSIONS

The pre-buckling vibration and buckling behaviour of com-posite skew plates with different skew angles and boundaryconditions have been investigated under different types of lin-early varying in-plane loadings. The present analysis is basedon higher-order shear deformation theory (HSDT) for deter-mining more realistic behaviour of composite skew plates. Thetotal energy functional is derived using in-plane stress dis-tributions and transformed from the physical domain to thecomputational domain using transformation relations. Thisfunctional is solved using the Rayleigh-Ritz method (RRM)considering boundary characteristics orthonormal polynomials(BCOPs) functions. The pre-buckled free vibration frequencyand critical buckling loads for composite skew plates underdifferent linearly varying in-plane loadings are obtained fromthe solution of the associated linear eigenvalue problem. Thefree vibration frequency of the skew plate under both loadedand unloaded conditions increases with an increase in edge re-straint and skew angle. For different types of linearly varyingin-plane loading, the pre-buckling fundamental frequency in-creases with an increase in the γ value. For all types of linearlyvarying in-plane loading in static stability analysis, the valueof the buckling coefficient decreases after shifting the buck-ling mode from first mode to second mode. With the reductionof length to thickness ratio (thicker plate), the free vibration

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Table 3. Dimensionless buckling coefficient (kc) for cross-ply [0/90/0] composite skew plates (ψ = 45◦, a/h = 100) with nine sets of boundary conditionssubjected to trapezoidal (γ = 0.5) in-plane loading.

Aspect ratio SSSS SSCS SCSS CSCS SCSC SSCC CCSC CCCS CCCC0.5 12.85 23.67 13.49 43.19 13.97 23.83 24.67 43.27 44.29

0.75 7.44 12.50 8.46 21.56 9.48 12.87 14.17 21.88 22.871.0 5.74 8.70 7.31 13.95 9.11 9.36 11.29 15.10 15.59

1.25 5.26 7.18 7.63 10.55 10.47 8.24 10.92 12.08 12.671.5 5.43 6.70 8.76 8.93 10.55 8.27 11.03 11.17 11.74

1.75 6.05 6.88 9.25 8.29 10.69 8.78 10.81 11.03 11.982.0 6.71 7.55 9.79 8.29 11.67 8.80 12.02 12.12 12.60

Table 4. Dimensionless buckling coefficient (kc) for cross-ply [0/90/0] composite skew plates (ψ = 45◦, a/h = 100) with nine sets of boundary conditionssubjected to trapezoidal (γ = 0.5) in-plane loading.

Aspect ratio SSSS SSCS SCSS CSCS SCSC SSCC CCSC CCCS CCCC0.5 14.70 26.66 15.76 47.47 16.12 26.70 28.11 49.89 49.98

0.75 9.40 15.50 11.21 25.68 12.13 15.68 17.77 28.77 28.851.0 8.25 12.26 11.45 18.53 13.26 12.69 15.86 22.76 23.10

1.25 8.93 12.03 14.64 16.14 16.91 12.80 17.44 22.36 23.041.5 11.20 14.09 19.06 16.27 19.91 15.19 20.83 25.34 25.80

1.75 14.17 17.23 25.09 18.07 25.22 17.25 25.73 30.55 30.652.0 17.92 17.87 18.00 20.64 32.36 19.91 32.78 35.12 35.73

Figure 9. Variation of buckling load coefficient (k) with respect to the aspectratio (a/b) for three layered cross-ply [0/90/0] composite CCCC skew plateshaving different skew angles subjected to in-plane bending.

frequencies decrease with an increase in the aspect ratio. It isalso noted that the more edge restraint there is, the higher thebuckling coefficient is.

APPENDICES

Appendix A

<1(uo,ξ, uo,η, v

o,ξ, v

o,η) =

1

a2(A11 +A66 tan2 ψ)(uo,ξ)

2−

2

abA66 tanψ secψuo,ξu

o,η +

1

b2A66 sec2 ψ(uo,η)2−

2

a2(A11 +A66) tanψuo,ξv

o,ξ +

2

abA12 secψuo,ξv

o,η+

2

abA66 secψuo,ηv

o,ξ +

1

a2(A22 tan2 ψ +A66)(vo,ξ)

2−

2

abA22 tanψ secψvo,ξv

o,η +

1

b2A22 sec2 ψ(vo,η)2

(A.1)

<2(wo,ξξ, wo,ξη, w

o,ηη, φ

o1,ξ, φ

o1,η, φ

o2,ξ, φ

o2,η) =

1

a4(D11+

2D12 tan2 ψ + 4D66 tan2 ψ +D22 tan4 ψ)(wo,ξξ)2−

4

a3b(D12 + 2D66 +D22 tan2 ψ) tanψ secψwo,ξξw

o,ξη+

2

a2b2(D12 +D22 tan2 ψ) sec2 ψwo,ξξw

o,ηη+

4

a2b2(D66 +D22 tan2 ψ) sec2 ψ(wo,ξη)2−

4

ab3D22 tanψ sec3 ψwo,ξηw

o,ηη +

1

b4D22 sec4 ψ(wo,ηη)2−

1

a3(E11 + E12 tan2 ψ + 2E66 tan2 ψ)wo,ξξφ

o1,ξ+

2

a2bE66 tanψ secψwo,ξξφ

o1,η +

1

a3(E12 + 2E66+

E22 tan2 ψ) tanψwo,ξξφo2,ξ−

1

a2b(E12 + E22 tan2 ψ) secψwo,ξξφ

o2,η+

2

a2b(E12 + E66) tanψ secψwo,ξηφ

o1,ξ−

2

ab2E66 sec2 ψwo,ξηφ

o1,η−

2

a2b(E66 + E22 tan2 ψ) secψwo,ξηφ

o2,ξ+

2

ab2E22 tanψ sec2 ψwo,ξηφ

o2,η−

1

ab2E12 sec2 ψwo,ηηφ

o1,ξ+

1

ab2E22 tanψ sec2 ψwo,ηηφ

o2,ξ−

1

b3E22 sec3 ψwo,ηηφ

o2,η (A.2)

<3(wo,ξξ, wo,ξη, w

o,ηη, φ

o1,ξ, φ

o1,η, φ

o2,ξ, φ

o2,η) =

− 1

a3(E11 + E12 tan2 ψ + 2E66 tan2 ψ)wo,ξξφ

o1,ξ+

2

a2bE66 tanψ secψwo,ξξφ

o1,η+

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Table 5. Dimensionless buckling coefficient (kc) for three layered cross-ply [0/90/0] composite skew plates (ψ = 45◦, a/h = 100) with nine sets of boundaryconditions subjected to partial tension (γ = 1.5) in-plane loading.

Aspect ratio SSSS SSCS SCSS CSCS SCSC SSCC CCSC CCCS CCCC0.5 16.86 29.81 18.60 51.66 18.83 29.79 32.05 55.22 55.59

0.75 12.23 19.32 15.76 30.28 16.36 19.35 22.91 35.32 35.991.0 12.97 18.05 21.00 24.62 21.76 18.14 24.05 33.18 33.73

1.25 17.82 22.34 25.99 25.24 29.96 22.36 31.20 34.82 39.341.5 12.53 12.96 13.08 16.76 21.54 21.98 25.25 16.98 26.95

1.75 7.59 7.73 8.19 9.85 13.42 13.21 15.35 10.35 16.042.0 5.28 5.36 5.74 6.59 9.49 9.09 10.69 7.13 10.89

1

a3(E12 + 2E66 + E22 tan2 ψ) tanψwo,ξξφ

o2,ξ−

1

a2b(E12 + E22 tan2 ψ) secψwo,ξξφ

o2,η+

2

a2b(E12 + E66) tanψ secψwo,ξηφ

o1,ξ−

2

ab2E66 sec2 ψwo,ξηφ

o1,η−

2

a2b(E66 + E22 tan2 ψ) secψwo,ξηφ

o2,ξ+

2

ab2E22 tanψ sec2 ψwo,ξηφ

o2,η−

1

ab2E12 sec2 ψwo,ηηφ

o1,ξ+

1

ab2E22 tanψ sec2 ψwo,ηηφ

o2,ξ−

1

b3E22 sec3 ψwo,ηηφ

o2,η+

1

a2(F11 + F66 tan2 ψ)(φo1,ξ)

2−

2

abF66 tanψ secψφo1,ξφ

o1,η+

1

b2F66 sec2 ψ(φo1,η)2−

2

a2(F11 + F66) tanψφo1,ξφ

o2,ξ+

2

abF12 secψφo1,ξφ

o2,η+

2

abF66 secψφo1,ηφ

o2,ξ+

1

a2(F22 tan2 ψ +A66)(φo2,ξ)

2−

2

abF22 tanψ secψφo2,ξφ

o2,η +

1

b2F22 sec2 ψ(φo2,η)2 (A.3)

<4(φo1, φo2) = H44(φo2)2 +H55(φo1)2 (A.4)

<5(nxx, nxy, nyy, wo,ξ, w

o,η) =

(nxxa2− 2nxy

abtanψ +

nyyb2

tan2 ψ)(wo,ξ)2+

(2nxyab

secψ − 2nyyb2

tanψ secψ)wo,ξwo,η+

nyyb2

sec2 ψ(wo,η)2 (A.5)

<6(uo,t, vo,t, w

o,t, φ

o1,t, φ

o2,t) =

ρh[(uo,t)2 + (vo,t)

2 + (wo,t)2]+

ρh3

12[(φo1,t)

2 + (φo2,t)2] (A.6)

Appendix B

Θ1 = 30(−1 + η)η(−1 + ξ)ξ (B.1)

Θ2 = 30√

7(−1 + η)η(−1 + ξ)ξ(−1 + 2ξ) (B.2)

Θ3 = 30√

7(−1 + η)η(−1 + 2η)(−1 + ξ)ξ (B.3)

Θ4 = 30√

3(−1 + η)η(−1 + ξ)ξ(3− 14ξ + 14ξ2) (B.4)

Θ5 = 210η(1− 3η + 2η2)ξ(1− 3ξ + 2ξ2

)(B.5)

Θ6 = 30√

3(−1 + η)η(−1 + ξ)ξ(3− 14η + 14η2) (B.6)

Θ7 = 30√

77(−1+η)ηξ(1−9ξ+26ξ2−30ξ3+12ξ4), (B.7)

Θ8 = 30√

21η(1−3η+2η2)ξ(−3+17ξ−28ξ2+14ξ3) (B.8)

Θ9 = 30√

21η(−3+17η−28η2+14η3)ξ(1−3ξ+2ξ2) (B.9)

Θ10 = 30√

77η(1− 9η + 26η2 − 30η3 + 12η4)ξ(−1 + ξ)(B.10)

Θ11 =30√

182(−1 + η)ηξ(−1 + 13ξ − 57ξ2+

111ξ3 − 99ξ4 + 33ξ5) (B.11)

Θ12 =210√

11η(1− 3η + 2η2)ξ(1− 9ξ+

26ξ2 − 30ξ3 + 12ξ4) (B.12)

Θ13 =90η(−3 + 17η − 28η2 + 14η3)ξ(−3 + 17ξ−28ξ2 + 14ξ3) (B.13)

Θ14 =210√

11η(1− 9η + 26η2−30η3 + 12η4)ξ(1− 3ξ + 2ξ2) (B.14)

Θ15 =30√

182η(−1 + 13η − 57η2+

111η3 − 99η4 + 33η5)ξ(−1 + ξ) (B.15)

Θ16 =30√

42(−1 + η)ηξ(3− 53ξ+

325ξ2 − 935ξ3 + 1375ξ4 − 1001ξ5 + 286ξ6) (B.16)

Θ17 =210√

26η(1− 3η + 2η2)ξ

(−1 + 13ξ − 57ξ2 + 111ξ3 − 99ξ4 + 33ξ5) (B.17)

Θ18 =30√

231η(−3 + 17η − 28η2

+ 14η3)ξ(1− 9ξ + 26ξ2 − 30ξ3 + 12ξ4) (B.18)

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Θ19 =30√

231η(1− 9η + 26η2−30η3 + 12η4)ξ(−3 + 17ξ − 28ξ2 + 14ξ3) (B.19)

Θ20 =210√

26η(−1 + 13η − 57η2+

111η3 − 99η4 + 33η5)ξ(1− 3ξ + 2ξ2) (B.20)

Θ21 =30√

42η(3− 53η + 325η2−935η3 + 1375η4 − 1001η5 + 286η6)ξ(−1 + ξ)

(B.21)

Θ22 =30√

714(−1 + η)ηξ(−1 + 23ξ − 187ξ2+

737ξ3 − 1573ξ4 + 1859ξ5 − 1144ξ6 + 286ξ7),(B.22)

Θ23 =210√

6(1− 3η + 2η2)ηξ(3− 53ξ+

325ξ2 − 935ξ3 + 1375ξ4 − 1001ξ5 + 286ξ6),(B.23)

Θ24 =30√

546(−3 + 17η − 28η2 + 14η3)

ηξ(−1 + 13ξ − 57ξ2 + 111ξ3 − 99ξ4 + 33ξ5),(B.24)

Θ25 =2310η(1− 9η + 26η2 − 30η3+

12η4)ξ(1− 9ξ + 26ξ2 − 30ξ3 + 12ξ4) (B.25)

Θ26 =30√

546η(−1 + 13η − 57η2 + 111η3−99η4 + 33η5)ξ(−3 + 17ξ − 28ξ2 + 14ξ3), (B.26)

Θ27 =210√

6η(3− 53η + 325η2 − 935η3+

1375η4 − 1001η5 + 286η6)ξ(1− 3ξ + 2ξ2) (B.27)

Θ28 =30√

714η(−1 + 23η − 187η2 + 737η3−1573η4 + 1859η5 − 1144η6 + 286η7)(−1 + ξ),

(B.28)

Θ29 =30√

1254(−1 + η)ηξ(1− 29ξ + 301ξ2 − 1547ξ3+

4459ξ4 − 7553ξ5 + 7462ξ6 − 3978ξ7 + 884ξ8),(B.29)

Θ30 =210√

102η(1− 3η + 2η2)ξ(−1 + 23ξ − 187ξ2+

737ξ3 − 1573ξ4 + 1859ξ5 − 1144ξ6 + 286ξ7),(B.30)

Θ31 =90√

14η(−3 + 17η − 28η2 + 14η3)ξ(3− 53ξ+

325ξ2 − 935ξ3 + 1375ξ4 − 1001ξ5 + 286ξ6),(B.31)

Θ32 =210√

286η(1− 9η + 26η2 − 30η3 + 12η4)

ξ(−1 + 13ξ − 57ξ2 + 111ξ3 − 99ξ4 + 33ξ5),(B.32)

Θ33 =210√

286η(−1 + 13η − 57η2 + 111η3 − 99η4+

33η5)ξ(1− 9ξ + 26ξ2 − 30ξ3 + 12ξ4), (B.33)

Θ34 =90√

14η(3− 53η + 325η2 − 935η3 + 1375η4−1001η5 + 286η6)ξ(−3 + 17ξ − 28ξ2 + 14ξ3),

(B.34)

Θ35 =30√

1122/1303(−1 + η)ηξ

(1− 3ξ + 2ξ2)(−31− 154η + 1155η2 − 4004η3+

7007η4 − 6006η5 + 2002η6 + 988ξ − 8398ξ2+

31616ξ3 − 57798ξ4 + 50388ξ5 − 16796ξ6) (B.35)

Θ36 =30√

1254/417437(−1 + η)η(−1 + ξ)ξ(635 + 308η−3003η2 + 14014η3 − 35035η4 + 48048η5 − 34034η6+

9724η7 − 18088ξ + 176358ξ2 − 823004ξ3+

2057510ξ4 − 2821728ξ5 + 1998724ξ6 − 571064ξ7)(B.36)

REFERENCES1 Durvasula, S. Buckling of simply supported skew plates, J.

Eng. Mech., 97 (3), 967–979, (1971).2 Liew, K.M. and Lam, K.Y. Application of two-dimensional

orthogonal plate function to flexural vibration of skewplates, J. Sound. Vib, 139 (2), 241–252, (1990).https://dx.doi.org/10.1016/0022-460X(90)90885-4

3 Wang, C. M., Liew, K. M., and Alwis, W. A.M. Buckling of skew plates and corner conditionfor simply supported edges, J. Eng. Mech., 118 (4),651–662, (1992). https://dx.doi.org/10.1061/(ASCE)0733-9399(1992)118:4(651)

4 Singh, B. and Chakraverty, S. Flexural vibration of skewplates using boundary characteristic orthogonal polynomi-als in two variables, J. Sound. Vib., 173 (2), 157–178,(1994). https://dx.doi.org/10.1006/jsvi.1994.1224

5 Wang, S. Buckling analysis of skew fibre-reinforced com-posite laminates based on first-order shear deformationplate theory, Compos. Struct., 37 (1), 5–19, (1997).https://dx.doi.org/10.1016/S0263-8223(97)00050-0

6 Mizusawa, T., Kajita, T. and Naruoka, M. Buck-ling of skew plate structures using B-spline functions,Int. J. Numer. Meth. Eng., 15 (1), 87–96, (1980).https://dx.doi.org/doi.org/10.1002/nme.1620150108

7 Taj, G. and Chakrabarti, A. Static and dy-namic analysis of functionally graded skewplates, J. Eng. Mech., 139 (7), 848–857,(2012).https://dx.doi.org/10.1061/(ASCE)EM.1943-7889.0000523

282 International Journal of Acoustics and Vibration, Vol. 24, No. 2, 2019

A. Kumar, et al.: VIBRATION AND BUCKLING OF SKEW PLATES UNDER LINEARLY VARYING EDGE COMPRESSION

8 Kitipornchai, S., Xiang, Y., Wang, C. M. andLiew, K. M. Buckling of thick skew plates, Int.J. Numer. Meth. Eng., 36 (8), 1299–1310, (1993).https://dx.doi.org/10.1002/nme.1620360804

9 Farag, A. M. and Ashour, A. S. Free vibration of orthotropicskew plates, J. Vib. Acoust., 122 (3), 313–317, (2000).https://dx.doi.org/10.1115/1.1302085

10 Ganapathi, M. and Prakash, T. Thermal buck-ling of simply supported functionally gradedskew plates, Compos. Struct., 74 (2), 247–250,(2006).https://dx.doi.org/10.1016/j.compstruct.2005.04.004

11 Wang, X., Tan, M., and Zhou, Y. Buckling analyses ofanisotropic plates and isotropic skew plates by the newversion differential quadrature method, Thin Wall Struct.,41 (1), 15–29, (2003). https://dx.doi.org/10.1016/S0263-8231(02)00100-3

12 Daripa, R. and Singha, M. K. Influence of cornerstresses on the stability characteristics of composite skewplates, Int. J. Nonlinear Mech., 44 (2), 138–146, (2009).https://dx.doi.org/10.1016/j.ijnonlinmec.2008.10.003

13 Karami, G., Ali Shahpari, S., and Malekzadeh, P.DQM analysis of skewed and trapezoidal laminatedplates, Compos. Struct., 59 (3), 393–402, (2003).https://dx.doi.org/10.1016/S0263-8223(02)00188-5

14 Zhou, L. and Zheng, W. X. Vibration of skew plates by theMLS-Ritz method, Int. J. Mech. Sci., 50 (7), 1133–1141,(2008). https://dx.doi.org/10.1016/j.ijmecsci.2008.05.002

15 Singha, M. K. and Ganapathi, M. Large amplitude freeflexural vibrations of laminated composite skew plates,Int. J. Nonlin. Mech., 39 (10), 1709–1720, (2004).https://dx.doi.org/10.1016/j.ijnonlinmec.2004.04.003

16 Eftekhari, S. A. and Jafari, A. A. Modified mixed Ritz-DQ formulation for free vibration of thick rectangu-lar and skew plates with general boundary conditions,Appl. Math Model., 37 (12–13), 7398–7426, (2013).https://dx.doi.org/10.1016/j.apm.2013.02.040

17 Wang, X., Wang, Y., and Yuan, Z. Accurate vibration anal-ysis of skew plates by the new version of the differentialquadrature method, Appl. Math Model., 38 (3), 926–937,(2014). https://dx.doi.org/10.1016/j.apm.2013.07.021

18 Leissa, A. W. and Kang, J. H. Exact solutions for vibra-tion and buckling of an SS-C-SS-C rectangular plate loadedby linearly varying in-plane stresses, Int. J. Mech. Sci., 44(9), 1925–1945, (2002). https://dx.doi.org/10.1016/S0020-7403(02)00069-3

19 Kang, J. H. and Leissa, A. W. Vibration and bucklingof SS-F-SS-F rectangular plates loaded by in-plane mo-ments, Int. J. Struct. Stab. Dy., 1 (4), 527–543, (2001).https://dx.doi.org/10.1142/S0219455401000299

20 Kang, J. H. and Leissa, A. W. Exact solutions forthe buckling of rectangular plates having linearly vary-ing in-plane loading on two opposite simply supportededges, Int. J. Solids Struct., 42 (14), 4220–4238, (2005).https://dx.doi.org/10.1016/j.ijsolstr.2004.12.011

21 Zhong, H. and Gu, C. Buckling of symmetrical cross-ply composite rectangular plates under a linearly vary-ing in-plane load, Compos. Struct., 80 (1), 42–48, (2007).https://dx.doi.org/10.1016/j.compstruct.2006.02.030

22 Lai, S. K. and Xiang, Y. DSC analysis for buck-ling and vibration of rectangular plates with elasti-cally restrained edges and linearly varying in-plane load-ing, Int. J. Struct. Stab. Dy., 9 (3), 511–531, (2009).https://dx.doi.org/doi.org/10.1142/S0219455409003119

23 Lopatin, A. V. and Morozov, E. V. Buckling of the SSCFrectangular orthotropic plate subjected to linearly vary-ing in-plane loading, Compos. Struct., 93 (7), 1900–1909,(2011).https://dx.doi.org/10.1016/j.compstruct.2011.01.024

24 Wang, X., Wang, X., and Shi, X. Differential quadra-ture buckling analyses of rectangular plates sub-jected to non-uniform distributed in-plane load-ings, Thin Wall Struct., 44 (8), 837–843, (2006).https://dx.doi.org/10.1016/j.tws.2006.08.008

25 Wang, X., Wang, X., and Shi, X. Accurate buck-ling loads of thin rectangular plates under parabolicedge compressions by the differential quadraturemethod, Int. J. Mech. Sci., 49 (4), 447–453, (2007).https://dx.doi.org/10.1016/j.ijmecsci.2006.09.004

26 Wang, X., Gan, L., and Wang, Y. A differential quadra-ture analysis of vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-planestresses, J. Sound Vib., 298 (1–2), 420–431, (2006).https://dx.doi.org/doi.org/10.1016/j.jsv.2006.06.003

27 Tang, Y. and Wang, X. Buckling of symmetricallylaminated rectangular plates under parabolic edgecompressions, Int. J. Mech. Sci., 53 (2), 91–97,(2011).https://dx.doi.org/10.1016/j.ijmecsci.2010.11.005

28 Kumar, R., Kumar, A. and Panda, S. K. Parametric res-onance of composite skew plate under non-uniform in-plane loading, Struct. Eng. Mech., 55 (2), 435–459, (2015).https://dx.doi.org/10.12989/sem.2015.55.2.435

29 Reddy, J. N. and Liu, C. F. A higher-order shear defor-mation theory of laminated elastic shells, Int. J. Eng. Sci.,23 (3), 319–330, (1985). https://dx.doi.org/10.1016/0020-7225(85)90051-5

30 Soldatos, K. P. A refined laminated plate and shell theorywith applications, J. Sound Vib., 144 (1), 109–129, (1991).https://dx.doi.org/10.1016/0022-460X(91)90736-4

31 Nair, P. S. and Durvasula, S. Vibration of skewplates, J. Sound Vib., 26 (1), 1–19, (1973).https://dx.doi.org/10.1016/S0022-460X(73)80201-9

32 Babu, C. S. and Kant, T. Two shear deformable finite ele-ment models for buckling analysis of skew fibre-reinforcedcomposite and sandwich panels, Compos. Struct., 46(2), 115–124, (1999). https://dx.doi.org/10.1016/S0263-8223(99)00039-2

33 Durvasula, S. Buckling of clamped skewplates, AIAA. J., 8 (1), 178–181, (1970).https://dx.doi.org/doi.org/10.2514/3.5635

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