# Semi-analytical modeling of cutouts in rectangular plates with variable thickness – Free vibration analysis

## Semi-analytical modeling of cutouts in rectangular plates with variable thickness – Free vibration analysis

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Semi-analytical modeling of cutouts in rectangular plates with variable thickness – Free vibration analysis Igor Shufrin a,∗, Moshe Eisenberger b a

School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia Faculty of Civil and Environmental Engineering, Technion – Israel Institute of Technology, Rabin Building, Technion City, Haifa, 32000, Israel

b

a r t i c l e

i n f o

Article history: Received 29 April 2014 Revised 11 January 2016 Accepted 18 February 2016 Available online xxx Keywords: Free vibrations Variable thickness plates Rectangular cutouts Multi term extended Kantorovich method

a b s t r a c t This paper presents a new semi-analytical method for modeling rectangular plates with variable thickness and cutouts. The plate thickness is represented as a ﬁnite sum of multiplications of one-dimensional functions. The plate deﬂections are also assumed in the similar separable form and the variational extended Kantorovich method is applied. In order to enhance the accuracy of the solution, a multi-term formulation of the extended Kantorovich method is developed. It is shown that this representation is very general and it allows the description of a complex variation of the thickness including step thickness changes and cutouts. It is demonstrated that this approach avoids singularities at the cutout areas and it does not require assembly of predeﬁned trial functions or computational domains satisfying plate geometry and boundary conditions. The presented method is applied for the free vibration analysis of rectangular plates with various rectangular cutouts and variable thickness. The accuracy and convergence of the solution is studied through comparisons with other semi-analytical methods (where applicable) and the results of ﬁnite element analysis. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Plates are common structural members in a broad range of engineering applications. In many cases, plates with variable thickness and plates with cutouts ﬁnd a practical use to fulﬁll functional requirements of lightweight and dynamic performance. Cutouts are also designed to provide operational access and ventilation or they may occur as damage during service life. A large amount of literature has been devoted to analysis of rectangular plates with cutouts of different shapes ranging from rectangular and polygonal cutouts to round, elliptical, and smoothed polygonal ones. In general, analyses of plates with cutouts have been aiming at assessment of stress concentrations around cutouts and evaluation of dynamic and stability characteristics of a plate (e.g. see [1–4] and references within). In this paper, we are focusing on the linear vibration behavior of rectangular plates with arbitrary thickness variation and multiple rectangular cutouts. The study of free vibrations is important for the assessment of the system’s dynamic response. The natural frequencies and modes characterize the resonance behavior of the structure, characteristics that are of use when the structure is excited by practically any force. Commonly, the behavior of plates with cutouts was analyzed using discrete methods. Among them were the ﬁnite difference technique [5,6] that was employed in the ﬁrst systematic studies of this kind of plate problems, the boundary element ∗

Corresponding author. Tel.: +61 864888518. E-mail addresses: [email protected], [email protected] (I. Shufrin), [email protected] (M. Eisenberger).

Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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deformation theories in [40,42]. An EKM solution for the forced vibrations of orthotropic plates was derived in . Recently, the EKM was also applied to the solution of shell problems [44,45] and 3D elasticity solution of laminated plates . In all these studies, the solutions were utilized using only one term in the EKM expansion. At the same time, this single-term formulation either fails to solve some of the critical problems of the plate analyses, or produces a relatively poor solution in terms of accuracy . Consequently, to overcome these limitations, a more general multi-term formulation of the EKM was developed and applied to the free vibrations of rectangular plates , the buckling analysis of laminated plates , the static solution of shallow shells , the free vibrations of elastic solids , nonlinear large deﬂection analysis of laminated plates , nonlinear elastic stability of rectangular plates , nonlinear analysis of trapezoidal plates , the interlaminar stress analysis of thick laminated plates , bending analysis of thick annular sector plates [55,56], and 3D linear elastic solutions for laminated plates [57,58]. The main objective of this study is to develop a general analytical approach to modeling of cutouts and discontinuous thickness variation in rectangular plates. To achieve this goal, we derive the multi-term extended Kantorovich method (MTEKM) for the free vibration analysis of rectangular plates with rectangular cutouts and variable thickness. We consider plates with multiple central and corner cutouts, general boundary conditions, and smooth and abrupt thickness variation. As such general combination of conditions has never been treated analytically nor the MTEKM has been used in the cases with discontinuous properties, we study the convergence and capability of the proposed model using the simpler examples available in the literature and in more general cases comparing the results with the ﬁnite element analysis. 2. Free vibration of rectangular plates by the multi-term extended Kantorovich method Consider a thin isotropic plate with variable thickness. The plate has no in-plane restraints, while arbitrary out-of-plane boundary conditions are allowed. Limiting the discussion to small displacement linear vibrations, the total energy of the transverse harmonic vibrations of the plate is given by Reddy  in terms of deﬂections of the mid plane of the plate, w0 (x, y), as:

1 = 2



Lx



0

Ly 0

 

∂ 2 w0 D ∂ x2

2



∂ 2 w0 +D ∂ y2

2

  2 2 ∂ 2 w0 ∂ 2 w0 ∂ w0 + 2ν D + 2(1 − ν )D − ω2 ρ h(x, y )w0 2 dxdy, ∂ x∂ y ∂ x2 ∂ y2

(1)

where Lx and Ly are the plate dimensions in the x and y directions, respectively, D = Eh (x, y )3 /(12(1 − ν 2 ) ) is the plate bending rigidity, E is the modulus of elasticity, h(x, y) is the variable thickness, ω is the frequency of harmonic vibrations, ρ is the mass density, and ν is the Poisson’s ratio. This energy functional is based on Kirchhoff’s classical thin plate theory, which does not account for the inﬂuence of the transverse shear strains on the deformations. Note that this simple theory is used here to emphases the ability of the MTEKM to model the cutout problem and can straightforwardly be extended by using the higher order theories [33,46 ]. According to the principle of minimum energy, the ﬁrst variation of the functional should be equal to zero:

δ =



Lx



0

Ly

 

D w0,xx + ν w0,yy

0



   δ w0,xx + D w0,yy + ν w0,xx δ w0,yy + 2(1 − ν )Dw0,xy δ w0,xy −ω2 ρ hw0 δ w0 dxdy = 0. (2)

This variational statement describes a free vibration problem of a rectangular plate of a general thickness conﬁguration. Further application of Green’s theorem together with the localization lemma of the calculus of variations yields a partial differential eigenvalue problem with variable coeﬃcients in terms of eigenfrequencies ω and eigenfunctions w. However, a close-form solution is not available for this kind of a problem. Alternatively, the semi-analytical multi-term extended Kantorovich method, which is directly applied on the variational statement, is developed next. 2.1. The multi-term extended Kantorovich method In the multi-term extended Kantorovich method (MTEKM), the plate deﬂections are represented as:

w0 (x, y ) =

N

wi (x )Wi (y ) = wT W,

(3)

i=1

where wi (x) are functions of x only, Wi (y) are functions of y only and N is the number of terms in the series. The bold symbols w and W designate column vectors of size N × 1 comprising of functions of wi (x) or Wi (y), respectively. The plate thickness is also taken as a sum of multiplications of one-dimensional functions, as follows:

h(x, y ) = h0

MH

hi (x )Hi (y ),

(4)

i=1

where hi (x) are the functions of thickness distribution in the x direction, Hi (y) are the distributions in the y direction, h0 is the thickness of the plate at the origin of the coordinate system, and MH is the number of terms in the thickness expansion. Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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This representation is very general, and we will show that it allows the description of complex thickness variation cases, including step thickness and cutouts. It is important to note that the number of terms that is used in the thickness expansion of Eq. (4) does not affect the number of terms that is used in the solution expansion of Eq. (3). In accordance with Eq. (4), the bending rigidity of the plate, D, becomes:

D=

E h0





3

12 1 − ν 2



MH

3 hi (x )Hi (y )

= d0

MD

i=1

di (x )Di (y ),

(5)

i=1

with d0 = Eh0 3 /(12(1 − ν 2 )) and MD is the number of terms in the corresponding expansion of the bending rigidity given by the formula for the tetrahedral numbers as MD = MH (MH + 1)(MH + 2)/6. In the MTEKM, the functions in the y directions are ﬁrst assumed known. Consequently, substitution of Eqs. (3)–(5) into Eq. (2) yields:

δ=





Lx



Ly

d0 0

d0

MD

0

MD

i=1

2 ( 1 − ν )d 0

MD



δ w,xx



di (x )Di (y )

i=1



di (x )Di (y ) w,xx T WWT + ν wT W,yy WT

 ν w,xx T WW,yy T + wT W,yy W,yy T δ w

di (x )Di (y )w,x T W,y W,y T δ w,x − ω2 ρ h0

i=1

(6) MH

hi (x )Hi (y )wT WWT δ w )dxdy = 0.

i=1

Then, integrating Eq. (6) with respect to y we have:

δ =





Lx

w,xx

0

MD

di (x )S1i + w

i=1

 w,xx T

+

T

MD



di (x )S2i + wT

i=1

T

MD

 di (x )S2i

T

δ w,xx + w,x T

MD

i=1 MD

di (x )S4i δ w,x

i=1

di (x )S3i − ω2

i=1

MH



(7)

 δ w dx = 0,

hi (x )S5i

(7)

i=1

where the coeﬃcients S1 i through S5 i are the matrices of size N × N given as:

 S 1i = d0

Ly

Di (y )WWT dy,

0



S 2i = ν d0

Di (y )WW,yy T dy,

(8b)

Di (y )W,yy W,yy T dy,

(8c)

0

 S 3i = d0

Ly

Ly

0

S4i = 2(1 − ν )d0

S 5i = ρ h0

 0

(8a)

Ly



Ly

0

Di (y )W,y W,y T dy,

(8d)

Hi (y )WWT dy.

(8e)

Note that the terms of the S-matrices are scalars resulted from the integration over the y direction. Integrating Eq. (7) by parts, we obtain a system of homogeneous ordinary differential equations with variable coeﬃcients: MD

di (x )S1i w,xxxx + 2

i=1

+

MD

di,x (x )S1i w,xxx +

i=1 MD i=1

di,x (x )(2S2i − S4i )w,x +



MD





di,xx (x )S1i + di (x ) S2i + S2i T − S4i

i=1 MD i=1

di,xx (x )S2i T + di (x )S3i − ω2

MH



w,xx

 hi (x )S5i w = 0,

(9a)

i=1

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and the associated boundary conditions:



Q=

MD

di (x )S1i w,xxx +

MD

di,x (x )S1i w,xx

x=Lx

x=Lx

 x = L x ˆ

or w

=w ,

M M D D x=0 x=0

+ di (x )(S2i − S4i )w,x + di ,x (x )S2i w

=0

i=1 i=1 i=1

i=1

(9b)

x=0

x=Lx

MD

M= = 0 or w,x (−di (x )S1i w,xx − di (x )S2i w )

i=1

x=Lx

ˆ ,x =w

x=0

x=0

x=Lx

,

(9c)

x=0

ˆ and w ˆ ,x are prescribed displacements and rotations at the plate edges. Eq. (9) deﬁne an ordinary differential where w eigenvalue problem in terms of the unknown natural frequency ω and eigenfunctions, wi . This problem is solved using the exact element method for multi-term eigenvalue problems [47,60], which is outlined next. 2.2. The exact element method The exact element method solution is based on derivation of a dynamic stiffness matrix for variable cross-section elements . In this method, the exact stiffness matrix of an element is constructed using the shape functions, which are obtained as a solution of governing equations with special geometric boundary conditions. The terms of this stiffness matrix are holding actions at the ends of the element, which are deﬁned based on the natural boundary conditions. The main advantage of the exact element method over a direct solution of a set of differential equations is that it enables assembling of complex thickness variations from sections of different forms. First, we transform Eq. (9) into a dimensionless form using the normalized coordinate ξ = x/Lx . Thus, the equation reads:

A4 w,ξ ξ ξ ξ + A3 w,ξ ξ ξ + A2 w,ξ ξ + A1 w,ξ + A0 w + ω2 Ah w = 0,

(10)

and the boundary conditions at both ends are:

ξ =0 or w ξ =0 = w ˆ , ξ =0 + A7 L w,ξ ξ =0 + A8 L w

=0 ξ =0

ˆ ,ξ

ML = A9 L w,ξ ξ ξ =0 + A10 L w

= 0 or w,ξ ξ =0 = w Q L = A5 L w,ξ ξ ξ ξ =0 + A6 L w,ξ ξ

ξ =0

(11a)

ξ =0

ξ =1 or w ξ =1 = w ˆ , ξ =1 + A7 R w,ξ ξ =1 + A8 R w

=0 ξ =1

ˆ ,ξ

MR = A9 R w,ξ ξ ξ =1 + A10 R w

= 0 or w,ξ ξ =1 = w

,

QR = A5 R w,ξ ξ ξ ξ =1 + A6 R w,ξ ξ

ξ =1

ξ =1

(11b)

(11c)

,

(11d)

where QL, ML, QR and MR are the N × 1 column vectors. The coeﬃcients A0 …A4 and Ah are the N × N matrices comprising of the functions of ξ , which are deﬁned as:

Ah = −

MH

Lx 4 hi (ξ )S5i ,

(12a)

i=1

A0 =

MD





Lx 2 di,ξ ξ (ξ )S2i T + Lx 4 di (ξ )S3i ,

(12b)

i=1

A1 =

MD

Lx 2 di,ξ (ξ )(2S2i − S4i ),

(12c)

i=1

A2 =

MD





di,ξ ξ (ξ )S1i + Lx 2 di (ξ ) S2i + S2i T − S4i



,

(12d)

i=1

A3 = 2

MD

di,ξ (ξ )S1i ,

(12e)

i=1

A4 =

MD

di (ξ )S1i .

(12f)

i=1

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The coeﬃcients A5 L ...A10 L and A5 R ...A10 R are the N × N matrices of scalar coeﬃcients computed at both ends of the element as:

A5 L = −

A6 L = −

A7 L = −

1

MD

Lx

i=1

S 3 1i

1

MD

Lx

i=1

S 3 1i

di (0 ); di,ξ (0 );

A5 R =

1

MD

Lx

i=1

A6R =

MD 1 di (0 ); ( S 2i − S4i ) Lx

S 3 1i 1

MD

Lx

i=1

S 3 1i

di,ξ (1 ),

(13b)

(13c)

i=1

MD MD 1 1 S 2i di ,ξ (0 ); A80 R = S2i di ,ξ (1 ), Lx Lx i=1

A9L =

(13a)

MD 1 di (1 ), ( S 2i − S4i ) Lx

A7 R =

i=1

A8 L = −

di (1 ),

1

MD

Lx

i=1

S 2 1i

A10 L = S2i

MD

(13d)

i=1

di (0 ); A9R = −

di (0 );

1

MD

Lx

i=1

S 2 1i

2 [A10 ]R = − Si ( )

di (1 ),

MD

i=1

(13e)

di (1 ).

(13f)

i=1

Note that the function hi (x) and di (x) describing the variation of plate’s thickness and rigidity are also translated into functions of ξ . In the exact element method, the solution is assumed as inﬁnite power series of the following form :



w1 ( ξ ) ∗ wN ( ξ )

w=

=

⎧∞ ⎫ ⎪ ⎪ ⎪ X1,i ξ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ i=0 ⎬ ∗

∞ ⎪ ⎪ ⎪ ⎪ ⎪ i⎪ ⎪ ⎩ XN,i ξ ⎪ ⎭

=

X1,i ∗ XN,i

i=0

 ξi =

Xi ξ i ,

(14)

i=0

i=0

where the Xi are unknown N × 1 coeﬃcient vectors. We also assume that the plate thickness varies as polynomial, so the variable A-coeﬃcients can be written as ﬁnite series as well:

Aj =

Kj

a j,k ξ k , j = 0 . . . 4,

(15a)

ah,k ξ k ,

(15b)

k=0

KH

Ah =

k=0

where aj,k and ah,k are the N × N matrices corresponding to the polynomial variation of di (ξ ) and hi (ξ ) and Kj are the number of terms in the jth matrix coeﬃcient. By substituting the assumed functions (14) and their corresponding derivatives together with the variable coeﬃcients (15), back into Eq. (10), we obtain: ∞



i=0

min (K4 ,i )

min (K3 ,i )

+

a4,k (i − k + 1 )(i − k + 2 )(i − k + 3 )(i − k + 4 )Xi−k+4

k=0

a3,k (i − k + 1 )(i − k + 2 )(i − k + 3 )Xi−k+3 +

k=0 min (K1 ,i )

+

k=0

min (K2 ,i )

a2,k (i − k + 1 )(i − k + 2 )Xi−k+2

k=0

a1,k (i − k + 1 )Xi−k+1 + ω

min (Kh ,i ) 2

k=0



min (K0 ,i )

ah,k Xi−k +

a0,k Xi−k

ξ i = 0.

(16)

k=0

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To satisfy this equation for every ξ , we must have: min (K4 ,i )

a4,k (i − k + 1 )(i − k + 2 )(i − k + 3 )(i − k + 4 )Xi−k+4

k=0 min (K3 ,i )

+

a3,k (i − k + 1 )(i − k + 2 )(i − k + 3 )Xi−k+3 +

k=0 min (K1 ,i )

+

a1,k (i − k + 1 )Xi−k+1 + ω2

min (Kh ,i )

 i! =− (a4,0 )−1 (i + 4 )! min (K3 ,i )

min (K4 ,i )

+

a0,k Xi−k = 0,

(17)

k=0

a4,k (i − k + 1 )(i − k + 2 )(i − k + 3 )(i − k + 4 )Xi−k+4

k=1

a3,k (i − k + 1 )(i − k + 2 )(i − k + 3 )Xi−k+3 +

k=0 min (K1 ,i )

a2,k (i − k + 1 )(i − k + 2 )Xi−k+2

min (K0 ,i )

ah,k Xi−k +

k=0

or

+

k=0

k=0

Xi+4

min (K2 ,i )

min (K2 ,i )

a2,k (i − k + 1 )(i − k + 2 )Xi−k+2

k=0

a1,k (i − k + 1 )Xi−k+1 + ω

min (Kh ,i ) 2

k=0



min (K0 ,i )

ah,k Xi−k +

k=0

a0,k Xi−k .

(18)

k=0

Thus in Eq. (18), we have a recurrence formula for calculating all vector terms in Eq. (14), except the ﬁrst four, which should be determined using the geometric boundary conditions . In the current classical thin plate formulation, there are 2 N degrees of freedom at each end of the element resulting in the 4N × 4 N element stiffness matrix. Thus the 4 N basic shapes for derivation of the stiffness matrix are found using the following geometric boundary conditions:

w j=i (0 ) = 1, w j=i (0 ) = w j,ξ (0 ) = w j (1 ) = w j,ξ (1 ) = 0, for i = 1 . . . N,

(19a)

w j=i−N,ξ (0 ) = L, w j (0 ) = w j=i−N,ξ (0 ) = w j (1 ) = w j,ξ (1 ) = 0, for i = N + 1 . . . 2N,

(19b)

w j=i−2N,ξ (1 ) = 1, w j (0 ) = w j,ξ (0 ) = w j=i−2N (1 ) = w j,ξ (1 ) = 0, for i = 2N + 1 . . . 3N,

(19c)

w j=i−3N,ξ (1 ) = L, w j (0 ) = w j,ξ (0 ) = w j (1 ) = w j=i−3N,ξ (1 ) = 0, for i = 3N + 1 . . . 4N.

(19d)

It is seen from Eq. (18) that the terms Xi +4 tend to 0 as i→∞ and the assumed solution of Eq. (14) converges to the exact solution of the system, Eq. (10). Thus, by setting the criterion for the recurrence calculations of Eq. (18) such that the last 4 N terms are less than an arbitrary small tolerance value, we obtain a very accurate solution for the shape functions [47,48]. In the numerical study presented in Section 3, this criterion was taken as 10−14 . A column of the stiffness matrix represents the holding actions at the ends of the element required to maintain the shape function. These actions are calculated by means of the natural boundary conditions of Eq. (11) as:

SM (1..N, i ) = A5 L wi,ξ ξ ξ |ξ =0 + A6 L wi,ξ ξ |ξ =0 + A7 L wi,ξ |ξ =0 + A8 L w i |ξ =0 ,

(20a)

SM (N + 1..2N, i) = A9 L wi,ξ ξ |ξ =0 + A10 L wi |ξ =0 ,

(20b)

SM (2N + 1..3N, i) = A5 R wi,ξ ξ ξ |ξ =1 + A6 R wi,ξ ξ |ξ =1 + A7 R wi,ξ |ξ =1 + A8 R wi |ξ =1 ,

(20c)

SM (3N + 1..4N, i) = A9 R wi,ξ ξ |ξ =1 + A10 R wi |ξ =1 ,

(20d)

where wi is the ith shape vector calculated using the boundary conditions (19) and a prescribed value of ω. The element stiffness matrices are populated through the sequential calculation of the shapes corresponding to each element. The global stiffness matrix is then constructed by assembling of all elements and imposing of the particular boundary conditions. Since, the shape functions depend on the angular frequency, ω (see Eq. (18)), this matrix is the dynamic stiffness matrix and the natural frequencies are those causing this matrix to become singular. In this study, we employed a simple bisection procedure to search for the frequencies leading to the zero determinant of the dynamic stiffness matrix. The eigenfunctions are obtained by multiplying the shape functions by the corresponding nodal displacements [40,60]. 2.3. The iterative extended Kantorovich procedure The procedure presented in Sections 2.1 and 2.2 is reiterated using the calculated eigenfunctions in one direction as the known functions for the solution in the second direction as follows. Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Fig. 1. Treatment of cutouts (a) a central cutout within a plate with variable thickness, (b) a corner cutout within a constant thickness plate.

1. A set of functions Wi (y) is selected and the coeﬃcient matrices of Eqs. (8) and (12) are calculated. 2. The ﬁrst natural frequency, ω1 x and the corresponding polynomial eigenfunctions, wi (x) are found for the x direction using the exact element method outlined in Section 2.2. 3. The obtained functions, wi (x) are used as the known ones and the coeﬃcients of Eqs. (8) and (12) are calculated for the y direction. All functions of y are replaced with functions of x and the integration is performed over the length Lx . 4. The ﬁrst natural frequency, ω1 y and the corresponding polynomial eigenfunctions, Wi (y) are found for the y direction using the exact element method. 5. Steps 2–4 are repeated until the natural frequencies ω1 x and ω1 y converge. In the numerical study presented in Section 3, a relative difference of a natural frequency between two subsequent iterations of 0.001% is taken as the convergence criterion. 6. Steps 1–5 are repeated sequentially for the higher frequencies. The main advantage of this iterative procedure is that it does not depend on the assumption made at the ﬁrst cycle. As a result, neither the boundary conditions nor discontinuous thickness variation needs to be satisﬁed by this initial assumption. The only restriction applied to the starting functions Wi is that they have to be linearly independent. Otherwise, the matrix S1 i of Eq. (8) resulting from the integration over the assumed direction becomes singular, and it will be impossible to calculate the inverse of the ﬁrst polynomial term of the a4 coeﬃcient in the recurrence formula (18). For the numerical calculations presented in Section 3, we used Wi (y) = sin(i∗ y/Ly ). 2.4. Treatment of cutouts The cutouts are realized by proper representation of the plate as composed of several subdomains, while in each of them the thickness is deﬁned by the sum as in Eq. (4). The thickness components are chosen so that in the cutout areas the thickness sum up to zero. Fig. 1 illustrates two general representations of the thickness components that result in areas with zero thickness. Case “a” presents the treatment of a central cutout, while in case “b” corner cutouts are realized. In general, the simultaneous description of a number of cutouts will require additional terms in the series that form the thickness variation. In order to further demonstrate the proposed modeling of cutouts, the case “b” of corner cutouts is presented in details. To model two corner cutouts as shown in Fig. (1b), the plate is divided into two sections in the x direction and three sections in the y direction and the thickness is represented using two terms:

h(x, y ) = h0 h1 H1 + h0 h2 H2 ,

(21)

where the thickness functions are deﬁned as:



h1 =

H1 =

1 1 h1( ) , 0 ≤ x ≤ Lx( )

; (2 )

2 1 1 h1( ) , Lx( ) < x ≤ Lx( ) + Lx

h2 =

⎧ 1 ( ) (1 ) ⎪ ⎨H1 , 0 ≤ x ≤ Lx



1 1 h2( ) , 0 ≤ x ≤ Lx( )

, 2 1 1 2 h2( ) , Lx( ) < x ≤ Lx( ) + Lx( )

2 1 1 2 H1( ) , Lx( ) < x ≤ Lx( ) + Lx( )

⎪ ⎩

(3 )

(1 )

(2 )

H1 , Lx + Lx

(1 )

; (2 )

(3 )

< x ≤ Lx + Lx + Lx

H1 =

(22a)

⎧ 1 ( ) (1 ) ⎪ ⎨H2 , 0 ≤ x ≤ Lx

2 1 1 2 H2( ) , Lx( ) < x ≤ Lx( ) + Lx( )

⎪ ⎩

(3 )

(1 )

(2 )

H2 , Lx + Lx

(1 )

, (2 )

(22b)

(3 )

< x ≤ Lx + Lx + Lx

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where the superscript indexes designate the thickness sections and Lx(1 ) , Lx(2 ) , Ly(1 ) , Ly(2 ) and Ly(3 ) are the lengths of the sections in the x and y directions respectively. Accordingly, the bending rigidity is described by the four term series as:

D(x, y ) = d0 (d1 D1 + d2 D2 + d3 D3 + d4 D4 ).

(23)

Substituting the thickness values given in Fig. 1b, the terms in Eq. (23) are deﬁned for all sections as follows: 1 1 1 1 d1( ) = 1, d2( ) = d3( ) = d4( ) = 0,

(24a)

√ √ 2 2 2 2 d1( ) = 1, d2( ) = − 3, d3( ) = 3, d4( ) = −1,

(24b)

√ √ 1 1 1 1 D1( ) = 1, D2( ) = 3, D3( ) = 3, D4( ) = 1,

(24c)

2 2 2 2 D1( ) = 1, D2( ) = D3( ) = D4( ) = 0,

(24d)

√ √ 3 3 3 3 D1( ) = 1, D2( ) = 3, D3( ) = 3, D4( ) = 1.

(24e)

As it was mentioned above, the existence of the solution depends on the uniqueness of the matrix A4 in Eq. (12e). Thus, to demonstrate the ability of this method to overcome the singularity at the cutout area, we only need to examine the terms, which are involved in deﬁnition of this coeﬃcient. Consequently, assuming the ﬁrst known solution in the y direction, S1 i integration coeﬃcients become (see Eq. (8)):



S11 = d0 ⎝

1 Ly( )

1 D ( ) WWT dy +

1 

0



1 Ly( )

S12 = d0 ⎝

2 D ( ) WWT dy +



1 D ( ) WWT dy +

1 Ly( )

1 2 Ly( ) +Ly( )



Ly

D

(1 )

4 

0



Ly 1 2 Ly( ) +Ly( )

0

1 D ( ) WWT dy +

(1 )

3 D( ) WWT dy⎠,

1  1

2 D ( ) WWT dy +



1 2 Ly( ) +Ly( )

2 D ( ) WWT dy +



Ly 1 2 Ly( ) +Ly( )

0

 WWT dy +

(1 )

(2 )

Ly + Ly

D

(2 )

4 

1 Ly( )

1

⎟ 3 D( ) WWT dy⎠,

2  √

⎟ 3 D( ) WWT dy⎠,

3  √

WWT dy +

Ly

1 2 Ly( ) +Ly( )

D

(3 )

4 

0

(25b)

3



(25a)

3

3 

1 Ly( )

3

1 2 Ly( ) +Ly( )

2 

1 Ly( )

3  √

0

Ly

1

3





1 

1 Ly( )

2  √

0

S14 = d0 ⎝

1 2 Ly( ) +Ly( )

1

S13 = d0 ⎝



(25c)

⎞ WWT dy⎠.

(25d)

1

Application of the exact element method for the x direction requires two elements for the continuous and cutout areas. Then, substituting Eqs. (24a), (24b) and (25) into Eq. (12f) we obtain A4 matrix coeﬃcients for both elements in the x direction as follows: (1 )



(1 )

A4 = d1 S11 = d0

0

1 Ly( )

 T

WW dy +

1 2 Ly( ) +Ly( ) 1 Ly( )

 T

WW dy +

2 2 2 2 2 A4( ) = d1( ) S11 + d2( ) S12 + d3( ) S13 + d4( ) S14 = d0



1 2 Ly( ) +Ly( ) 1 Ly( )

Ly

1 2 Ly( ) +Ly( )



T

WW dy ,

WWT dy.

(26a)

(26b)

Eq. (26) clearly show that both A4 matrices become singular only when the functions Wi are linearly dependent. 3. Numerical study and discussion The semi-analytical method developed in Section 2 is applied to the free vibration analysis of isotropic plates with variable thickness and rectangular cutouts. For brevity of notation, the plate boundary conditions are denoted starting from the edge x = 0 to x = Lx ; and from y = 0 to y = Ly consequently. For example, SSCC denotes a plate with simply supported (S) edges at x = 0 and x = Lx , and clamped (C) at y = 0 and y = Ly . The accuracy and convergence of the results are examined by comparisons with other numerical and analytical methods. In all cases, the natural frequencies are expressed in terms of the dimensionless factor that is deﬁned by:

!

= ω Ly

2

ρ h0 d0

.

(27)

Also Poisson’s ratio ν is taken as 0.3 in all examples. Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Fig. 2. A square simply supported (SSSS) plate with a central cutout.

Table 1 Normalized natural frequencies of a square SSSS plate with a central cutout, ν = 0.3, a = 0.4, (Fig. 2). MTEKM

Ref. 

Mode N

1

2

3

4

5

1 2 3 20.724

21.508 20.807 20.763 41.907

46.232 41.442 41.131 71.499

75.416 71.647 71.410 85.418

107.118 85.129 82.923 118.720

124.860 118.633 117.971

3.1. Square plates with central cutouts and constant thickness A square SSSS plate with constant thickness and a centrally located square cutout is studied ﬁrst. The geometry of the plate is shown in Fig. 2. In order to compare the obtained results with those available in the literature, two sizes of the cutout were examined. Table 1 presents the ﬁrst ﬁve natural frequencies of the plate with cutout size of 0.4Ly . The results of the MTEKM were obtained using one to three terms in the series expansion are compared with those calculated using the Rayleigh–Ritz method . It was found that the number of MTEKM iterations (the one-dimensional solution in either x or y direction, see Section 2.3) required to achieve the convergence in terms of the natural frequencies varies for different frequencies. For instance, the single-term solution required between 4 to 10 iterations, while 2 and 3 term solutions were converging at 8–12 iterations. This performance is attributed to the more complex combination of the series functions in the multi-term expansions. The results in Table 1 demonstrate that although the single-term MTEKM solution provides a reasonable approximation of the plate frequencies, the additional terms in the series are required to improve the accuracy. This is particularly prominent for the 4th mode, in which an including of the second term into the solution enhances the accuracy by 20.5%. At the same time, with the increase in the number of terms the frequencies converge very fast yielding almost the same results for 2 and 3 term solutions. As both methods (the Rayleigh–Ritz and MTEKM) give the upper bounds for the natural frequencies, it is also seen that the MTEKM provides similar accuracy as the Rayleigh–Ritz method with as little as 2 terms, while the 3 term solution yields the better approximation for all modes. The mode shapes for the ﬁrst three symmetric-symmetric (SS), anti-symmetric–symmetric (AS), and anti-symmetric–anti-symmetric (AA) modes are also shown in Fig. 3. It is apparent from these plots that the most of these shapes cannot be accurately approximated using the single term formulation (e.g. [39–41, 44,45]), which is only able to describe shapes with nodal lines that are parallel to the plate edges. For example, the second SS mode (the 4th mode in Table 1 mentioned above) is a diagonal mode that requires a multi-term description. It was shown in the previous studies that the accuracy of the MTEKM procedure largely depends upon the ability of a speciﬁc number of functions in the assumed solution of Eq.(3) to represent the true vibration mode of the plate . In general, a 2-term solution yields a reasonable approximation for the frequencies corresponding to the modes with nodal lines that are parallel to the plate edges, while the 3rd term only introduces small corrections. For the diagonal modes however, the contribution of 3rd term is more substantial, while the 4th one is only served for the small tuning . Thus in order to focus on the modeling ability of the MTEKM in implementation of the cutouts and abrupt thickness variation, all calculations in the present study are carried out using up to 3 terms in the series expansion, while the accuracy is demonstrated by comparisons with other methods. The next example is a SSSS plate with a larger cutout of 0.5Ly presented in Table 2. The ﬁrst ten normalized natural frequencies calculated using the MTEKM are given together with those obtained by the mesh-free conforming radial point interpolation method , Rayleigh–Ritz method , the equivalent variable thickness plate model , and the ﬁnite element method (FEM), which were obtained using ANSYS software package. In the FEA, the plate was modeled using 4-node thin shell elements SHELL63 with overall 16,160 degrees of freedom. Good agreement between the results is observed along the whole range of the values. Comparisons of the upper bound frequencies obtained by the MTEKM, the FEM, and the Rayleigh–Ritz method conﬁrm very good accuracy of the MTEKM solution. It is seen that the solutions obtained with as little as three terms are with up to 1% discrepancy with the results obtained by the FEM with a very ﬁne mesh. Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Fig. 3. Mode shapes and normalized natural frequencies of a square simply supported plate with a central cutout, ν = 0.3, a = 0.4, (Fig. 2).

Ref. .

Table 2 Normalized natural frequencies of a square simply supported plate (SSSS) with a central cutout, ν = 0.3, a = 0.5 (Fig. 2). Mode

1 2 3 4 5 6 7 8 9 10

MTEKM

Liu et al. 

1 term

2 terms

3 terms

24.029 44.211 44.211 75.553 105.854 123.105 123.105 163.856 188.683 199.159

23.527 40.490 40.490 71.534 75.097 112.505 112.505 146.684 176.785 200.663

23.516 40.249 40.249 71.341 73.690 111.815 111.815 144.900 172.777 200.486

Huang and Sakiyama 

Ali and Atwal 

FEM (ANSYS,16160 D.O.F)

MTEKM to FEM % difference

23.107 40.069 40.082 69.750 74.979 109.625 110.589 138.399 170.523 191.811

22.337 39.502 39.563 68.792 75.138 111.371 111.907 144.110 174.732 189.868

23.242 40.329 40.329 69.328 74.091

23.418 39.994 39.994 71.218 72.945 111.572 111.572 144.945 172.870 201.263

−0.42% −0.64% −0.64% −0.17% −1.00% −0.22% −0.22% 0.03% 0.05% 0.39%

The other two methods, the conforming radial point interpolation  and the equivalent plate , seem to offer better accuracy for most frequencies. However, they showed a general inconsistence providing dissimilar values for the modes 2–3 and 6–7, which should be identical due to the plate symmetry. On the other hand, the MTEKM results demonstrate consistent behavior for all the modes. Also here the difference between the solutions obtained using one and two terms is considerable for all frequencies, but the 10th mode, in which the single term formulation yields the highly accurate solution. It is also seen that the convergence of the MTEKM frequencies for this mode is not monotonic, yet the deviations of the results are very small. 3.2. Rectangular plates with corner cutouts and constant thickness The next case demonstrates the ability of the proposed approach to predict the natural frequencies of rectangular plates with corner cutouts. Table 3 presents geometry, boundary conditions, vibration modes and frequencies for three examples of rectangular plates with symmetric and asymmetric corner cutouts. The MTEKM solutions are compared with FEM results obtained using ANSYS with 11,200 D.O.F for the symmetric cases and 12,0 0 0 D.O.F for the asymmetric Z shape. In the FE analysis, the plates were modeled using the thin shell elements as in the previous case. These results reveal the fast convergence and good accuracy of the MTEKM calculations. It is also seen that three terms are suﬃcient to produce very accurate results. A number of interesting observations can be made in the third case of the Z-shaped plate. Firstly, note that this shape requires three terms in the thickness representation. Secondly, it is observed that the single term solution yields poor accuracy for the ﬁrst two modes and completely fails to predict a frequency of the third diagonal mode. At the same Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Table 3 Natural frequencies of rectangular plates with corner cutouts and constant thickness. Boundary conditions

CCCC

SSSS

CCCC

0.5 Ly

0.5 Ly

0.5 Ly

0.5 Ly

0.5 Ly

2/3 Ly

1/3 Ly

1/3 Ly

Ly

0.5 Ly

Ly

0.5 Ly

1/3 Ly

1/3 Ly

0.5 Ly

1/3 Ly

Geometry

Mode 1

MTEKM

25.582 24.656 24.603 24.614

12.879 11.982 11.965 11.940

17.381 15.734 15.381 15.285

Mode 2

1 term 2 terms 3 terms FEM MTEKM

40.071 38.678 38.603 38.777

26.771 25.855 25.840 25.803

40.658 34.610 34.561 34.457

Mode 3

1 term 2 terms 3 terms FEM MTEKM

1 term 2 terms 3 terms FEM

64.970 60.997 60.014 60.184

42736 39.532 39.054 38.847

– 47.029 45.685 44.473

time, it is seen that even for this complex plate conﬁguration, the obtained results are almost converged on the FEM results with as little as three terms. The MTEKM performance in terms of the number of iterations required was found similar to the previous case: the single-term solution converged at 6–10 iterations and the multi-term ones after 8–12 iterations. 3.3. Square plate with smooth thickness variation and cutouts A solution for a plate with smooth thickness variation and cutouts is illustrated in this example. A clamped plate tapered in the x direction with two corner cutouts as shown in Fig. 4 is studied. Table 4 presents the ﬁrst ten normalized natural frequencies that are calculated using one to three terms in the series expansion. Caused by the variable thickness, this case required the higher number of the one-dimensional solutions (MTEKM iterations): the single-term solution converged after 6–16 iterations, while 2 and 3 term solutions needed 12–22 iterations. The MTEKM solution is compared with the ﬁnite element analysis done by ANSYS using the thin shell elements. It is seen that the results are very close to those from the very ﬁne mesh ﬁnite element analysis and the convergence of the MTEKM is relatively rapid. Interestingly, the convergence of the 6th, 8th and 9th frequencies are not monotonic in this case. To further study this trend, Fig. 5 illustrates these frequencies normalized with respect to the FEM results for one to four terms in the MTEKM solution. It can be observed that the single term solution provides the best result for the 6th mode, which is more accurate than the FEM one. The accuracy then decreases for two terms and improves for the three and four terms. The frequencies of the modes 8 and 9 Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Fig. 4. A variable thickness plate with two corner cutouts – smooth thickness variation. Table 4 Normalized natural frequencies of a square CCCC plate with corner cutouts and smooth thickness variation, ν = 0.3. Mode

1 2 3 4 5 6 7 8 9 10

FEM 21510 D.O.F

22.637 44.862 49.254 73.840 92.162 104.879 119.858 132.864 146.467 157.337

MTEKM 1 term

2 terms

3 terms

24.102 48.611 51.484 78.232 94.152 103.881 122.448 134.914 151.112 177.678

22.770 45.086 49.491 74.104 92.317 104.875 120.055 133.390 147.025 160.543

22.662 44.902 49.334 73.934 92.233 104.643 119.916 134.241 147.281 158.031

Fig. 5. MTEKM convergence for 6th, 8th and 9th free vibration modes of a square plate with corner cutouts and variable thickness.

improve non-monotonically from the single term solution reaching the accuracy obtained by the FEM with the four term expansion. Note that the discrepancy between these results is very small: the 8th and 9th frequencies are enhanced by 1.5% and 2.5% from the one to two term solutions respectively and the 6th frequency goes up by 1%, while deviations in the results from two to four terms are under 1% for all modes. The corresponding vibration modes are also plotted out in Fig. 6. These plots verify the ability of the developed approach to accurately describe complex mode shapes. Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Fig. 6. Free vibration modes of a square plate with corner cutouts and variable thickness, N = 3.

Fig. 7. A square plate with a central cutout and abrupt thickness variation.

3.4. Square plate with abrupt thickness variation and cutouts The next example is of a SSSS plate with an abrupt change in the thickness, such as to reinforce the opening, as shown in Fig. 7. In Table 5 the comparison of the MTEKM results is given with the results obtained using a FEM thin plate model with 25,790 D.O.F. It is seen that practical convergence is obtained for the three term solution. Note that these frequencies are compared based on similarity of the corresponding shape modes obtained by both methods. For example, Fig. 8 shows the vibration modes corresponding to the ﬁrst six (not repeated) frequencies. It is apparent from these ﬁgures that a multiterm expansion is required to obtain realistic nodal lines in the mode shapes, in the sense that they are not parallel to the boundaries of the plate. For instance, the 5th and 8th mode shapes require a minimum composition of two single term solutions as shown in Fig. 9, which is only possible due to symmetry observed in this case. Moreover, this mode composition is responsible for the non-monotonic convergence of the 8th frequency, Table 5, (see Ref.  for the detailed discussion on the vibration mode compositions in the MTEKM). The MTEKM performance in terms of the number of iterations was found following the trends observed in the previous examples: the single-term solution needed 6–12 iterations, 2 and 3 term solutions converged after 10–16 iterations. Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Table 5 Normalized natural frequencies of a plate with a central cutout and abrupt thickness variation, ν = 0.3. Mode

FEM 25790 D.O.F.

1 2 3 4 5 6 7 8 9 10

23.299 49.561 49.562 80.500 98.432 134.887 134.887 143.477 192.624 192.624

MTEKM 1 term

2 terms

3 terms

24.106 54.832 54.832 84.852 118.042 141.328 141.328 118.042 194.028 194.028

23.346 49.910 49.910 81.098 100.185 135.815 135.815 145.056 194.017 194.017

23.320 49.640 49.640 80.628 98.567 135.230 135.230 143.606 192.969 192.969

Fig. 8. Free vibration modes of a SSSS plate with a central reinforced cutout.

Fig. 9. Composition of 5th and 8th vibration modes of a SSSS square plate with a central reinforced cutout. Table 6 Normalized natural frequencies of the CCSS rectangular plate with two cutouts, ν = 0.3, Ly /Lx = 13/7. Mode

1 2 3 4 5

MTEKM 3 terms

48.822 97.085 127.163 142.931 157.430

FEM 9877 DOF

48.308 106.517 127.349 134.190 166.252

Aksu and Ali  n = 43

n = 191

Experiment

45.990 85.840 112.950 123.230 142.330

47.860 93.620 122.040 136.170 151.950

48.986 95.128 124.967 139.244 153.524

3.5. Rectangular plates with two cutouts Finally, plates with two central cutouts are considered. First, the normalized frequencies of a CCSS rectangular plate shown in Fig. 10 are compared with the results obtained by the ﬁnite difference technique  in Table 6. A good agreement with numerical and experimental results can be observed for all values. Another conﬁguration of two cutouts within a SSSS square plate is considered in the next example, which is illustrated in Fig. 11. Table 7 presents the ﬁrst ten frequencies Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Fig. 10. A rectangular CCSS plate with two cutouts, Ly /Lx = 13/7.

Fig. 11. A SSSS constant thickness square plate with two cutouts. Table 7 Normalized natural frequency of the SSSS constant thickness square plate with two cutouts, ν = 0.3. Mode

1 2 3 4 5 6 7 8 9 10

FEM 22603 D.O.F.

17.597 42.220 45.240 47.581 79.024 85.621 90.440 97.748 129.892 147.982

MTEKM 1 term

2 terms

3 terms

19.460 46.177 48.284 58.982 83.154 95.658 102.989 115.278 131.750 163.512

17.815 42.719 45.856 48.979 79.800 87.875 89.467 99.410 124.443 156.650

17.680 42.332 45.369 47.859 79.184 86.128 91.092 98.145 130.960 148.829

calculated using the MTEKM with one to three terms, and FEM results obtained from a thin plate model with 22,603 degrees of freedom. These results, which are in very good agreement, also demonstrate fast convergence of the MTEKM solution. 4. Conclusions An analytical approach for the vibration analysis of rectangular plates with rectangular cutouts and arbitrary boundary conditions has been developed. The method capitalizes on the multi-term formulation of the extended Kantorovich method and the exact element method. It has been shown that this approach is very general and allows analysis of plates with arbitrary variations of thickness including abrupt variation and rectangular cutouts. The fast convergence of the method has been demonstrated numerically. Despite the non-monotonic convergence observed in some cases, the solution expansion using three functions ensures a reasonable accuracy of the results, which has been demonstrated through comparisons with other numerical and analytical methods. It has been shown that the use of a multi-term expansion overcomes the limitations of the classical single-term Extended Kantorovich method and allows the analysis of complex plate conﬁgurations. The method developed here provides a semi-analytical solution for the free vibration analysis of a broad range of rectangular plates. This method can be further extended to the vibration analysis of plates with various irregularities, either in material properties, boundary conditions or kinematical constraints.

Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020

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Please cite this article as: I. Shufrin, M. Eisenberger, Semi-analytical modeling of cutouts in rectangular plates with variable thickness – free vibration analysis, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.02.020