AJSR_46_08.pdf

American Journal of Scientific Research

ISSN 1450-223X Issue 46 (2012), pp. 67-78

© EuroJournals Publishing, Inc. 2012

http://www.eurojournals.com/ajsr.htm

Bending and Deflection Analysis of Thin FGM Skew

Plates using Extended Kantorovich Method

Amin Joodaky

Corresponding Author, Graduate Student.Young Researchers Club

Arak Branch, Islamic Azad University, Arak P. O. Box: 38135, I.R. Iran

E-mail: [email protected]

Mohammad Hossein Kargarnovin

Department of Mechanical Engineering Sharif University of Technology

Tehran P. O. Box: 14588-89694, I.R. Iran


Fax: +9821-6600-0021

Saeid Jafari Mehrabadi

Department of Mechanical Engineering, Islamic Azad University

Arak Branch, Arak P. O. Box: 38135, I.R. Iran


Amir Hossein Nasrollah Barati

Young Researchers Club, Aligudarz Branch, Islamic Azad University

Aligudarz, I.R. Iran


Abstract

An accurate approximate closed-form solution is presented for bending of a FGM

skew plate with clamped edges, loaded uniformly by using the extended Kantorovich

method. Successive application of EKM together with the idea of weighted residual

technique converts the governing partial differential equation into two ordinary differential

equations. The obtained ODE’s are solved iteratively and the results are compared with

other existing results. It is shown that some factors such as angle of skew plate and power

law index of FGM have important effects on the obtained results.

Keywords: Extended Kantorovich Method; Bending analysis; Von Karman plate theory;

Skew plate; Uniform loading; Galerkin method

1. Introduction Extended Kantorovich Method (EKM) introduced by Kerr [1] using the idea of the well-known

Kantorovich method [2] to obtain highly accurate approximate closed-form solution for torsion of

prismatic bars with rectangular cross-section. The method employs the novel idea of Kantorovich to

reduce the governing partial differential equation of a two-dimensional (2D) elasticity problem to a

double set of ordinary differential equations. Since then, EKM has been extensively used for various


Plates using Extended Kantorovich Method 68

2D elasticity problems in Cartesian coordinates system. Among these applications, one can refer to

eigenvalue problems [3], buckling [4] and free vibrations [5] of thin rectangular plates, bending of

thick rectangular isotropic [6,7] and orthotropic [8] plates and free-edge strength analysis [9]. Most

recent EKM articles include vibration of variable thickness [10] and buckling of symmetrically

laminated [11] rectangular plates. Accuracy of the results and rapid convergence of the method

together with possibility of obtaining closed-form solutions for ODE systems have been discussed in

these articles and others [12]. Finally, a few articles consider polar coordinates and such as using the

EKM for sector plates [13]. All these applications of the EKM, are devoted and restricted to the

problems in the Cartesian and Polar coordinate systems. Among the open literatures, except previous

studies of the authors of the present article [14], no researcher has employed EKM in terms of Oblique

coordinates system. Based on other solution methods, several articles have studied bending, buckling,

vibration, stress and other analysis for skew plates in term of oblique coordinate system [15-20].

Functionally graded materials (FGMs) are composite materials, which are microscopically

inhomogeneous and their mechanical properties vary continuously in one (or more) direction(s). This is

achieved by gradually changing the composition of the constituent materials along one direction,

usually in the thickness direction only, to obtain smooth variation of material properties and optimum

response to externally applied loading. The concept of FGMs, was first introduced in Japan in 1984.

Since then it has gained considerable attentions [21, 22]. Many different applications of FGMs, could

be found in references [23].

This study aims to examine the applicability of the EKM to obtain highly accurate approximate

closed-form solutions for 2D elasticity problems in oblique coordinate system. Bending and

consequently, deflection of thin FGM skew plates with clamped edges subjected to uniform loading

conditions, is considered. Application of the EKM together with the idea of weighted residual

technique converts the forth-order governing equation into two ODEs in terms of X and Y in oblique

coordinates. Both resulted ODEs, are then solved iteratively in a closed-form manner and a very fast

convergence is achieved. A close study on the obtained results revealed that the method provides

sufficiently accurate answers for FGM skew plates. Comparison of the results of various points of the

FGM skew plate show very good agreement with results out of FEM software and other valid

literatures.

2. Analysis and Solution Method 2.1. FGM Plate

Figure 1: Rectangular FGM plate in Cartesian system of coordinates.

In a FGM plate either of material properties namely P is primarily a function of volume

fractions, V, of all comprising materials which can be expressed for two different constituents as [25],

1 1 2 2P PV PV= + (1)

69 Amin Joodaky, Mohammad Hossein Kargarnovin

Saeid Jafari Mehrabadi and Amir Hossein Nasrollah Barati

where indices 1 and 2 address the constituting materials one and two respectively. Moreover, referred

to Figure (1), for example Vi the volume fraction of the constituting materials, can be expressed

according to the power law distribution as [25];

1

12

2( )

2

( ) 1 ( )

,n

h zV z

h

V z V z

−=

= −

(2)

where n denotes the power law index which takes values greater or equal than zero and h is the plate

thickness. Using Eqs. (1) and (2), the material properties P such as Young’s modulus E for a FGM

plate made of metal at the top surface and ceramic at the lower surface can be written as follow:

1( )

2m cm

cm c m

nz

E z E Eh

E E E

= + ⋅ −

= −

(3)

That cE and mE

are Young’s modulus of elasticity coefficient for Ceramic and Metal

respectively. n is the non-negative exponent of power, h is the plate thickness and z is the coordinate in

the thickness direction. Generally, the value of Poisson’s ratio varies in a small range, for simplicity, it

is assumed a constant. In this study we consider a FGM plate in which the upper surface of plate (z =

h/2) is purely metallic and the lower surface (z = -h/2) is purely ceramic.

Classical nonlinear laminated plate theory and the concept of physical neutral surface are

employed to formulate the basic equations of the FGM thin plate [25]. According to the classical

theory of elasticity, in a pure bending of a FGM plate a surface can be found (neutral plane) whose

strains and stresses are zero. The distance of this plane to the mid-surface plane (z = 0z) is defined as

Figure (1) and [25];

2

2

2

2

0

( )

( )

h

h

h

h

zE z dz

E z dz

z−

−

=

∫

∫ (4)

Using the physical neutral surface concept and classical nonlinear Von Karman plate theory

[25], the displacements take the following forms:

( )0 0

wu u z z

x

∂= − −

∂

( )0 0

wv v z z

y

∂= − −

∂

( ),w w x y= (5)

where u and v stand for the in plane displacement and w for the z direction deflection of an arbitrary

point in the plate. Moreover, u0 and v0 represent the in plane displacement of any point on the neutral

surface. If small displacement theory is considered then, 0u and 0v

are very small compared to other

terms therefore they can be neglected. Therefore, displacements field take the following form:

( )0

wu z z

x

∂= − −

∂

( )0

wv z z

y

∂= − −

∂

( ),w w x y= (6)



The strain field (kinematic relation) is [25],

{ } ( ){ }(1)

0z zε ε= −

where

{ } { }

{ }(1)

2 2 2

2 2 2 2

, ,

2,,

x y xy

T

Tw w w

x y x y

ε ε ε ε

ε

=

∂ ∂ ∂= − − −

∂ ∂ ∂ ∂

(7)

and stress field become [25],

{ } [ ]{ }Qσ ε= (8)

where

[ ]2

2

1

2

1 0( )

1 01

0 0

E zQ

υ

υ

υυ

−

=−

(9)

In a plate-bending problem, moments are defined as [25],

{ } [ ]{ }(1)M D ε= (10)

where

[ ]2

1

2

1 0

1 0

0 0

D D

υ

υ

υ

−

=

(11)

and

( ) ( )02

2

2

21

h

h

z z E zD

υ−

−

−= ∫ (12)

If no axial force existences differential equation of motion based on the physical neutral

surface, can be expressed as [25] 2 22

2 22 0

xy yxM MM

qx x y y

∂ ∂∂+ + + =

∂ ∂ ∂ ∂ (13)

Or in terms of w [26-29],

4 ( , ) /w x y q D∇ = (14)

The last equation is the well-known governing equation for a thin isotropic plate, in which w(x,

y) is the deflection, q is the applied distributed load, and D (flexural rigidity) for a FGM plate. Now,

consider a thin skew FGM plate with dimensions of 2a×2b and tilting angle of ϕ along with reference

rectangular (x, y) and local oblique (X, Y) system of coordinates as depicted in Figure (2).

If all edges are clamp, then the deflection and its first derivative with respect to the normal

direction to the boundary should vanish at such boundaries, i.e.:

0, / 0Xw dw d= = for X a= ± where b Y b− ≤ ≤

0, / 0Yw dw d= = for Y b= ± where a X a− ≤ ≤



2.2. Iterative Solution by EKM

Figure 2: Skew plate in oblique coordinate (X,Y).

According to the Extended Kantorovich Method (EKM)[1], the function for the deflection of

the plate is assumed as multiplication of different single variable functions as:

( , ) ( ) ( )X Y f X g Yij i jw ≅ ⋅ (15)

where ( )Xfi and ( )j

Yg are unknown functions to be determined and subscripts i and j denote number

of iterations. For the plate deflection function of w(x, y), according to the Galerkin weighted residual

method, we have [13]:

4 0( )a b

a bD X Yw q wd dδ

− −∇ =−∫ ∫ (16)

Now, for a prescribed function of ( )j

Yg , j=0 and referred to the Eq. (15), wδ becomes:

( )j

Yi

g fwδ δ= ⋅ (17)

Substitution of Eq. (15) into Eq. (16) in conjunction with Eq. (17) leads to,

( )40( )i j j

a b

a b if dXD f g q g dY δ

− −= ∇ ⋅ − ∫ ∫ (18)

Based on the existing rules in the variational principle, Eq. (18) is satisfied if the expression in

the bracket is vanished, hence,

( )4 ( ) 0i j j

b

bD f g q g dY

−∇ ⋅ − =∫ (19)

in which operator ∇ is in the Cartesian coordinate, (x, y), and has to be converted to the oblique

coordinate (X, Y), i.e.:

tanX x y ϕ= − and / cosY y ϕ= (20)

Hence, by doing some mathematical manipulations one would get:

2 2 22

2 2

1( 2 sin )

cos X YX Y

ϕϕ

∂ ∂ ∂∇ = − +

∂ ∂∂ ∂

And consequently, 4∇ becomes,

( )44 4 4 4 4

1 22 1 2 sin 4 sin

4 4 2 2 3 3 4cos X X Y X Y X Y Y

ϕ ϕ

ϕ

∂ ∂ ∂ ∂ ∂∇ + + − + +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

=

(21)

Having on hand 4∇ operator in conjunction with Eq.(14), governing equation becomes:



( )4 4 4 4

2

4 2 2 3 3

4

4

4

4cos

( , )

2 1 2sin 4sinD

D

X Y

w w w w

X X Y X Y X Y

wq

Y

wϕ

ϕ ϕ∇∂ ∂ ∂ ∂

+ + − +∂ ∂ ∂ ∂ ∂ ∂ ∂

∂

∂

= +

=

(22)

Eq.(22) along with Eq.(15), and with an initial arbitrary guess of ( )j

Yg , j=0 turns the result

into a forth order ODE as:

( )2 2

42 2

3 3 4

3 3 4

4 ( ) ( )( ) 2( , ) ( )

4 4cos

( ) ( ) ( ) ( ) ( )4 sin ( ) ( , ) 0

( . ) 2 1 2sinY XD f X

D X Y g YX

g Y f X g Y f X g Yf X q X Y

Y XX Y Y

g ff g q

Y Xϕ

ϕ

ϕ

∂∇ +

∂

∂ ∂ ∂ ∂ ∂+ + − =

∂ ∂∂ ∂ ∂

∂ ∂− = + −

∂ ∂

(23)

The integrating with respect to Y of Eq. (19) as,

( ) ( )4

42

4

( )4( )( )

b b

i j j bb

A

d f XdY g Y dY

dXD f g q g

−−+∇ ⋅ − = ∫∫

��

( ) ( )

32

2 2 32

2 2 3

( ) ( ) ( ) ( )2 1 2sin ( ) 4sin ( )

b b

b b

AA

d g Y d f X dg Y d f Xg Y dY g Y dY

dYdY dX dXϕ ϕ

− −+ + − +

∫ ∫

��

( ) ( )1 0

3

3

5

4 4

4

( ) ( ) ( ) cos4sin ( ) ( ) ( ) ( ) 0

bb b

b b b

AA A

dg Y df X d g Y qg Y dY g Y dY f X

dX DdY dYg Y dY

ϕϕ

− − −− +

− =∫ ∫ ∫��

��

(24)

and in concise form,

54 3 2 1 0

4 3 2( ) ( ) ( ) ( )( )

4 3 2

d f X d f X d f X df XA A A A A f X A

dXdX dX dX+ + + + = (25)

Dividing both sides by A4 , yields: 4 3 2

3 0 52 14 3 2

4 4 4 4 4

( ) ( ) ( ) ( )( )

X X X XX

XX X X

A AA A Ad f d f d f dff

A A A d A Ad d d+ + + + = (26)

In order to obtain the homogeneous solution for the above forth order ODE we assume

f(X)=CemX

then, the corresponding characteristic equation will be:

4 3 23 02 1

4 4 4 4

0A AA A

m m m mA A A A

+ + + + = (27)

Eq. (27) has four roots: 1 1im a i b= ± ± . Therefore, the final answer for the homogeneous and

particular solutions for f(X) becomes as:

1 1 1151 1 2 1 3 1 14

( ) cos( ) sin( ) cos( ) sin( )XX X Xaa a a

X C b X C b X C b X C b X Ce e e ef = + + + + (28)

where 055 /C A A= . The updated forms of boundary conditions for clamped plates in terms of f(X) and

g(Y) would be as:

0, / 0i i

f df dX= = for X a= ±

0, / 0j j

g dg dY= = for Y b= ±

Solving Eq. (25) in conjunction with the first series of new boundary data leads to the first

estimate of the function ( ) 1,i X if = , which we call it ( )f X . Now, having on hand the primary answer



for ( )if X , it is possible to continue the procedure by introducing the obtained function ( )f X to the

Eq.(26). This yields to a new form of wδ and the Galerkin equation as,

( ) jf X giw δδ = ⋅ (29)

and,

( )4( ) 0i j

b a

jb a iD f g q dX dYf gδ− −

∇ ⋅ − =∫ ∫ (30)

Again, in order to satisfy Eq.(30), the bracket should be vanished. The integration with respect

to X of the expression in the bracket leads to the second forth-order ODE in terms of ( )jg Y as,

54 3 2 1 0

4 3 2( ) ( ) ( ) ( )

( )4 3 2

Y Y Y YY

d g d g d g dgB B B B B g B

dYdY dY dY+ + + + = (31)

The corresponding characteristic equation related to Eq.(31) turns to:

03 2 1

4 4 4 4

4 3 2 0B B B B

n n n nB B B B

+ + + + = (32)

Again Eq.(32) has four roots as: 2 2in a b i= ± ± . So g(Y) could be shown as,

2 2 2 22 2 22 51 2 3 4

( ) cos( ) sin( ) cos( ) sin( )Y Y Y Ya a a a

Y D b Y D b Y D b Y D b Y De e e eg = + + + + (33)

where 05 5 /D B B= .

The procedure is continues by solving ODE (31) together with the second series of new

boundary data and obtaining the new prediction for ( ), 1j

Y jg = which we call it ( )Yg . This will finish

the first cycle of our iteration for determination of deflection equation, i.e. 11

( , ) ( ) ( )X Y f X g Yw ≅ ⋅ . Eqs.

(25) and (31) should then be solved iteratively and new updated estimates for functions ( )Xfi and

( )j

Yg are determined. Iterations continue in the same way until a prescribed level of convergence is

achieved. The initial guess for the function of ( )Yg does not need necessarily to satisfy all of the

boundary conditions. Moreover, if we wanted to consider f(X) as an initial guess it would be,

0

2 2 2( ) ( )XX af = − .

It is worth of mentioning that after calculation of deflection, one can determine all other

mechanical parameters, i.e. rotations, forces and moments using well-known expressions presented

elsewhere, see for instance, [26-29].

3. Results and Discussion Consider a FGM skew plate made of Alumina- Aluminum with all sides clamped and subjected to a

uniform loading. The following plate properties and initial guess for0

( )Yg , are considered in the most

cases although some parameters may have other values for examining their effect on the results:

1,0.3 ,15 /12, 2 0.6 , 2 0.8 , 0.008 , 1000 Paa m b m h m qυϕ π == = = = = =�

0 ,

2

2

2min , ( )70 380 , 0.25 ( )Al Alu a

YYE GPa aE Gpa n g == −= = (34)

The convergence of all parameters used in Eqs. (28) to (33) are, tabulated in the Table 1. The

listed values in this table demonstrate that the convergence rate of the method is very high as there are

no major changes after the second iteration.



Table 1: Convergence of parameters in closed-form solutions of a FGM skew plate listed in Eqs. (28), (33).

Parameters Iterations, i, j

Parameters Iterations, i, j

1 2 3 1 2 3

1ia

5.3075 -5.3837 -5.3837 2ia

-7.0697 -7.0697 -7.0697

1ib

2.6283 3.0633 3.0633 2ib

3.6269 3.6260 3.6258

1iC

0.981e-3 0.790e-3 0.790e-3 1iD

-0.391e-2 -0.392e-2 -0.392e-2

2iC

0.283e-3 -0.767e-4 -0.767e-4 2iD

-0.15e-2 -0.145e-2 -0.145e-2

3iC

0.981e-3 0.790e-3 0.790e-3 3iD

-0.391e-2 -0.392e-2 -0.392e-2

4iC

-0.28e-3 0.767e-4 0.767e-4 4iD

0.1453e-2 0.1454e-2 0.1454e-2

5iC

-0.26e-2 -0.221e-2 -0.221e-2 5iD

0.322e-1 0.322e-1 0.322e-1

The FGM skew plate, is modeled in commercial version of ANSYS software using Solid45

element, similar geometrical and material properties listed in (34) and with all four sides clamped. For

considering FGM properties, twenty layers that elasticity modulus of each is obtained from FGM

function of (3) are considered in z direction and then they are glued. Then a normal uniformly

distributed load clamp condition for supports are applied. Finally, a regular meshing by dividing each

side by fifty parts is considered and then the model was analyzed. Tables 2 shows the maximum

deflection (w) which occurs in the center of the skew plate with corresponding values obtained out of

ANSYS, FEM package.

Table 2: Maximum deflection (w), in center of skew FGM plate with angle of 15�, compared to ANSYS

results.

Deflection Iterations, i, j

FEM 1 2 3 4

w (meter) -0.1616e-4 -0.1526e-4 -0.15428e-4 -0.15428e-4 -0.1535e-4

The stresses and moments functions are defined in terms of lateral deflection of w [26-29].

Therefore, after obtaining these functions in Oblique coordinates system, one can let in them,

deflection amounts of every point of the FGM skew plate and achieve stress components for those

points. Figure 3 shows amounts of the stress components of , ,X Y XYσ σ σ as Sx, Sy and Sxy

respectively, for the clamp FGM skew plate.

Figure 3: Stress components diagrams of clamp FGM skew plate for various points of X-axis along Y=0 axis.



Angle of phi (ω) which is shown in Figure 2, is the factor that changes deflection function of

the skew plate conspicuously. Figure 4 shows the effect of phi (ω) in term of degree, on deflection

function. When phi (ω) is increasing, the non-opposite edges of the skew plate are closer to each other,

so deflection amounts decrease.

Figure 4: Comparison of deflection diagrams of the clamp FGM skew plate vs. X and Y axes in Oblique

coordinates system for different angle of phi (ω) of the skew plate in term of degree.

Figure 5: Comparison of deflection diagrams of the clamp FGM skew plate vs. X and Y axes in Oblique

coordinates system for different amounts of n in FGM function of (3).

Power index of n in FGM function (3), has its effect on deflection diagrams because it changes

elasticity modulus. Figure 5 compares various diagrams for different amounts of n. When n increases,

elasticity decreases, so deflections increase too.

Table 3: Deflection in the center of the FGM skew plate for various n, ϕ and plate dimensions.

Various n and

ϕ Deflection in the center of the FGM skew plate (a, b)

2a=0.5, 2b=1 2a=1, 2b=1 2a=1, 2b=1.5 2a=1, 2b=2 2a=1, 2b=3

n=0

0

15

30

45

60

ϕ

ϕ

ϕ

ϕ

ϕ

=

=

=

=

=

�

�

�

�

�

-0.8859e-5 -0.7093e-4 -0.1228e-3 -0.1417e-3 -0.1464e-3

-0.7600e-5 -0.5985e-4 -0.1047e-3 -0.1216e-3 -0.1271e-3

-0.4740e-5 -0.3578e-4 -0.6361e-4 -0.7585e-4 -0.8162e-4

-0.2007e-5 -0.1441e-4 -0.2618e-4 -0.3211e-4 -0.3540e-4

-0.4788e-6 -0.3296e-5 -0.6094e-5 -0.7664e-5 -0.8744e-5



Table 3: Deflection in the center of the FGM skew plate for various n, ϕ and plate dimensions. - continued

n=1

0

15

30

45

60

ϕ

ϕ

ϕ

ϕ

ϕ

=

=

=

=

=

�

�

�

�

�

-0.1783e-4 -0.1424e-3 -0.2467e-3 -0.2852e-3 -0.2937e-3

-0.1523e-4 -0.1201e-3 -0.2102e-3 -0.2437e-3 -0.2550e-3

-0.9504e-5 -0.7172e-4 -0.1275e-3 -0.1521e-3 -0.1637e-3

-0.4031e-5 -0.2892e-4 -0.5246e-4 -0.6449e-4 -0.7168e-4

-0.9587e-6 -0.6628e-5 -0.1224e-4 -0.1535e-4 -0.1762e-4

n=5

0

15

30

45

60

ϕ

ϕ

ϕ

ϕ

ϕ

=

=

=

=

=

�

�

�

�

�

-0.2691e-4 -0.2156e-3 -0.3740e-3 -0.4306e-3 -0.4464e-3

-0.2315e-4 -0.1818e-3 -0.3179e-3 -0.3705e-3 -0.3859e-3

-0.1442e-4 -0.1087e-3 -0.1934e-3 -0.2308e-3 -0.2479e-3

-0.6106e-5 -0.4379e-4 -0.7956e-4 -0.9770e-4 -0.1079e-3

-0.1458e-5 -0.1002e-4 -0.1853e-4 -0.2334e-4 -0.2672e-4

Considering properties of (34), for various n, ϕ and plate dimensions, Table 3 gives center

deflections of the FGM skew plates. Increasing inclination angle of ϕ causes non-opposite edges of

the plate come closer, so deflections decrease. When power index of n in FGM function in (3)

increases, elasticity module decreases, and consequently, deflections increase. It is conspicuous that

enlarging plate dimensions certainly causes larger deflections.

A skew plate, could be compared to, a Rectangular plate when inclination angle equls zero. In

the Table 3, those results for 0ϕ = , could be compared to similar FGM rectangular plate in other

literatures. Table 4 compares the maximum deflection results for a clamped isotropic rectangular plate,

which is obtained, from a skew plate by considering 0ϕ = to a similar rectangular plate in [24] for

different dimensions of the plate.

Table 4: Comparison of maximum deflection (w), in the center of isotropic rectangular plate, which is

obtained from a skew plate by considering, 0ϕ =

with similar results from [24] for different plate

dimension.

Deflection in (a, b) 2a=1 , 2b=1 2a=1 , 2b=1.5 2a=1 , 2b=2

Present [24] Present [24] Present [24] 3

10w−× -0.3905 -0.3894 -0.6775 -0.6800 -0.7806 -0.7851

4. Concluding Remarks Application of EKM based on Galerkin method could successfully obtain a highly accurate

approximate closed-form solution for deflection analysis of FGM skew plates subjected to uniform

loading for clamp boundary conditions. EKM iterative procedure extracts two sets of decoupled

ordinary differential equations in terms of X and Y in Oblique coordinates system from the coupled

forth-order partial differential governing equation of the main problem. The solution procedure then

completes by presenting an exact approximate closed-form solution for two sets of ODE systems in an

iterative scheme. Present study shows that the method provides very fast convergence and highly

accurate predictions. Angle of skew plates has an important role in deflection function. Power index in

FGM function could changes deflection diagram obviously for different presented dimensions of the

skew plate. Comparing to results of other valid literatures and FEM software of ANSYS, there are very



good agreements with the results of the present studies in every case. Finally, by considering

inclination angle of skew plate equals zero, the present studies could be developed and compared to

rectangular plates.

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[2] Kantorovich L.V. and Krylov V.I., 1958. ”Approximate methods of higher Analysis”.

Groningen: Noordhoff.

[3] Kerr A.D., 1969. ” An extended Kantorovich method for the solution of eigenvalue problems”.

Int J Solids Struct 5(6), pp. 559–72.

[4] Yuan S. and Jin Y., 1998. ”Computation of elastic buckling loads of rectangular thin plates

using the extended Kantorovich method”. Comput Struct 66(6), pp. 861–7.

[5] Dalaei M. and Kerr A.D., 1996. ”Natural vibration analysis of clamped rectangular orthotropic

plates”. J Sound Vib 189(3), pp. 399–406.

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