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International Journal of Pressure Vessels and Piping 113 (2014) 1e9

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International Journal of Pressure Vessels and Piping

journal homepage: www.elsevier .com/locate/ i jpvp

Geometrically nonlinear dynamic and static analysis of shallowspherical shell resting on two-parameters elastic foundations

Ö. CivalekAkdeniz University, Faculty of Engineering, Civil Engineering Department, Division of Mechanics, Antalya, Turkey

a r t i c l e i n f o

Article history:Received 23 May 2013Received in revised form8 October 2013Accepted 17 October 2013

This paper is dedicated to Professor Yogen-dra Nath.

Keywords:Shallow shellNonlinear analysisDynamic responseFoundation effectComputational modeling

E-mail addresses: civalek@yahoo.com, ocivalek@ak

0308-0161/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.ijpvp.2013.10.014

a b s t r a c t

In the present study nonlinear static and dynamic responses of shallow spherical shells resting onWinklerePasternak elastic foundations are carried out. The formulation of the shells is based on theDonnell theory. The nonlinear governing equations of motion of shallow shells are discretized in spaceand time domains using the discrete singular convolution and the differential quadrature methods,respectively. The validity of the present method is demonstrated by comparing the present results withthose available in the open literature. The effects of the Winkler and Pasternak foundation parameters onnonlinear static and dynamic response of shells are investigated. Some results are also presented forcircular plate as special case. Damping effect on nonlinear dynamic response of shells is studied. It isimportant to state that the increase in damping parameter causes decrease in the dynamic response ofthe shells. It is shown that the shear parameter of the foundation has a significant influence on thedynamic and static response of the shells. Also, the response of the shell is decreased with the increasingvalue of the shear parameter of the foundation. Parametric studies considering different geometricvariables have also been investigated.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Shell components are widely used in many engineering appli-cations such as aero-spaces, aircrafts, automobile, civil, mechanical,marine and submarine structures. Domes, rockets, nuclear reactors,naval vessels, ship hulls, storage tanks, pressure vessels, petro-chemical pipes, and industrial chimneys are some known exam-ples of these applications. These structures are very often subjectedto dynamic loads such as earthquakes, explosions, seismic tests andsonic booms during their active using periods. It is known that plateand shell structures are generally placed on an elastic foundation inmany applications of civil, mechanical, aerospace, and railroadengineering. Thus, modeling and analysis of such structures underlarge deflections are important study area for the relatedresearchers.

Nonlinear governing equations for dynamic response ofshallow spherical shells based on von Karman’s theory werederived by Von Karman and Tsien [1]. The investigations ofnonlinear static and dynamic analyses of shell structures havereceived considerable attention in the literature. Nath and Jain [2e4] and Jain and Nath [5] presented several papers for nonlinearanalysis of plates and shells. Effect of elastic foundation on non-

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linear response of spherical shells is analyzed by Nath and Jain[3]. The effect of the foundation mass on nonlinear response oforthotropic shells is analyzed by Nath and Jain [4]. Nonlinear staticand dynamic analyses of circular plates, rectangular plates andshallow spherical shells are presented via collocation method[6,7]. Nath presented an analytical solution for large amplituderesponse of circular plates on elastic foundations [8]. Nonlinearanalysis of imperfect shallow spherical shells resting on Pasternakelastic foundation is investigated by Nie [9]. Non-linear static anddynamic instability problem of complete spherical shells is solvedby Lee et al. [10] using the mixed finite element formulation.Nonlinear static and dynamic analyses of shells are detailedinvestigated by Shen [11e13]. Nonlinear analysis of thin isotropicplates under different dynamic loads is investigated by presentauthor [14,15]. Alashti and Khorsand [16] is used a coupledapproach for three-dimensional dynamo-thermo-elastic analysisof a functionally graded cylindrical shell with piezoelectric layers.Nonlinear dynamic response of imperfect eccentrically stiffenedFGM double curved shallow shells on elastic foundation is pre-sented by Duc [17] using the Galerkin method. Nonlinear free vi-bration of spherical shell panel using higher order sheardeformation theory is presented by Panda and Singh [18] using thefinite element method. Paliwal et al. [19] gives analytical solutionfor the large deflection problem of shallow shell on elastic foun-dation. The effect of temperature on the non-linear axisymmetric

Ö. Civalek / International Journal of Pressure Vessels and Piping 113 (2014) 1e92

response of functionally graded shallow spherical shells is inves-tigated by Bich and Tung [20]. Static and dynamic analysis of anFGM doubly curved panel resting on the Pasternak-type elasticfoundation is presented by Kiani et al. [21]. Some exact and nu-merical solutions of linear and nonlinear analyses of plates andshells with or without elastic foundations are also presented in theliterature [22e31].

In this paper, a new coupled numerical model for geometri-cally nonlinear static and dynamic analyses of shallow sphericalshells resting on two-parameter elastic foundation have beenpresented. The governing equations are derived for nonlineardynamic response of shallow spherical shells on elastic founda-tion in Section 2. A general theory of the discrete singularconvolution (DSC) method is presented in Section 3. In Section 4,governing equations and solution procedures are presented.Numerical results for nonlinear dynamic and static responses arepresented and discussed in Section 5. Finally, conclusions aregiven in Section 6.

2. Governing equations

An axisymmetric shallow spherical shell resting on WinklerePasternak elastic foundations with base radius a, thickness h, cur-vature s, and radius of curvature R, shown in Fig. 1 is considered. Aradial coordinate system (r,q) is considered for shell. The strainedisplacement field for shallow shell can be written as [2]

3r ¼ vuvr

� swþ 12

�vwvr

�2

; (1)

3q ¼ ur� kw: (2)

where u, and w are the displacement components of shells. Basedon the von Karman nonlinear theory, which takes into accountmoderately large deflections and small strains, the equilibriumequation in radial direction is written as [2]

Fig. 1. Geometry and coordinate system of the shallow spherical shell on elasticfoundation.

rvNr

vrþ ðNr � NqÞ ¼ 0: (3)

The force resultants can be written in terms of the stress functionas:

Nr ¼ f1r

(4)

Nq ¼ vf

vr(5)

Using Eqs. (1)e(3), the compatibility equation can be written

v

vrðNr þ NqÞ þ

Ehr

"12

�vwvr

�2þ rs

vwvr

#¼ 0 (6)

By using the values given in Eqs. (4) and (5) in Eq. (6), we obtain

v

vr

�f

rþ vf

vr

�þ Eh

r

"12

�vwvr

�2

þ rsvwvr

#¼ 0 (7)

The moment values in radial and circumferential directions aregiven as

Mr ¼ �D

" v2wvr2

!þ y

rvwvr

#(8)

Mq ¼ �D

"y

v2wvr2

!þ 1

rvwvr

#(9)

where bending rigidity of the shell is given as

D ¼ Eh3

12�1� y2

� (10)

The moment equilibrium is written as

v

vrðrMrÞ �Mq � rQr þ rkw�

rg

v2wvr2

þ gvwvr

!¼ 0 (11)

In the above equation, k and g are the Winkler and Pasternakfoundation parameters. Similarly, we can write the equilibrium ofthe transverse force

v

vrðrQr Þ�rhr

v2wvt2

�rhrcvwvt

þ rqðr; t Þ� rkwþ rg

v2wvr2

þgvwvr

!

þ v

vr

�rNr

vwvr

�¼ 0

(12)

By using Eqs. (11) and (12) and Eqs. (4)e(8), the resultingnonlinear dynamic equation of shell is given as

D

v4wvr4

þ 21r

v3wvr3

þ 1r2

v2wvr2

!� 1

rvf

vrvwvr

� 1rfv2wvr2

� 1rkf� vf

vr� q ðr; t Þ þ kw� g

v2wvr2

þ 1r

vwvr

!

þ rhv2wvt2

þ crhvwvt

¼ 0

(13)

Ö. Civalek / International Journal of Pressure Vessels and Piping 113 (2014) 1e9 3

where c is the damping coefficient. The governing equations ofmotion for large deflection in the non-dimensional form via Eq. (7)and Eq. (13) can be given as below

R2 v2j

vR2þ 1R

vj

vR

!� jþ

�1� y2

2

�"R�vWvR

�2þ 2saR2

vWvR

#

(14)

R3 v4WvR4

þ21Rv3WvR3

þ 1R2

v2WvR2

!�12

�ah

�2R2"vj

vRvWvR

þ1Rjv2WvR2

#

�12�a3

h2

�R2�sjþvj

vRR3�QR3þR3KW�R3G

v2WvR2

þ1RvWvR

!

þR3v2WvT2

þCR3vWvT

¼ 0

(15)

By using the new differential operators as Lij the nonlineargoverning equations can be written as following compact form

y1L11ðjÞ þ y2L12 þ y3L13ðWÞ ¼ 0 (16)

y4L21 ðj Þ þ�y51L

j22 þ y52L

W22

�þ y6L23

�W�þ y7L24

�j;W

�¼ 0

(17)

The following non-dimensional quantities are used.

R ¼ r=a; W ¼ w=h; j ¼ f1� y2

Eha; Q ¼ qa3=D;

K ¼ ka4=D; s ¼ 1Rh

a2; G ¼ ga2=D; C ¼ c�rha4=D

�1=2;

T ¼ t�D=rha4

�1=2:

(18)

The differential operators given in the governing equations (Eqs.(16)and (17)) are listed below:

L11 ¼

v2

vR2þ 1R

v

vR

!(19)

L12 ¼ j (20)

L13 ¼��

v

vR

��v

vR

�þ 2saR

v

vR

(21)

L21 ¼ v

vR(22)

Lj22 ¼ sj (23)

LW22 ¼ �Q þ KW (24)

L23 ¼

v4

vR4þ2

1R

v3

vR3þ 1R2

v2

vR2

!�G

v2

vR2þ1R

v

vR

!þ v2

vT2þC

v

vT

(25)

L24 ¼"v

vRðjÞ v

vRðWÞ þ 1

Rj

v2

vR2ðWÞ

#: (26)

Also, the constant coefficients in Eqs. (16) and (17) can bewritten asbelow:

y1 ¼ R2; y2 ¼�1; y3 ¼ R�1�y2

2

�; y4 ¼�12

�a3

h2

�R5;

y51 ¼�12�a3

h2

�R2; y52 ¼ R3; y6 ¼ R3; y7 ¼�12

�ah

�2R2:

(27)

In the present study, two types of immovable boundary condi-tions are considered. Related equations are given as follows:

i) Simply supported edge conditions (at R ¼ 1):

W ¼ 0 (28)

v2WvR2

þ y1R

vWvR

¼ 0 (29)

vj

vR� y

1Rj ¼ 0 (30)

ii) Clamped edge conditions (at R ¼ 1):

W ¼ 0 (31)

vWvR

¼ 0 (32)

vj

vR� y

1Rj ¼ 0 (33)

iii) Symmetry conditions at the centre (at R ¼ 0):

vWvR

¼ 0 (34)

v3WvR3

¼ 0 (35)

j ¼ 0 (36)

3. Discrete singular convolution (DSC)

The discrete singular convolution (DSC) method is a relativelynew numerical technique in applied mechanics. The wavelet andthe distributions theories are two fundamental concepts for themathematical bases of the DSC method. DSC was proposed tosolve linear and nonlinear differential equations by Wei [32], andlater it was introduced to solid and fluid mechanics [33e39]. Plateand shells structures have also been modeling via DSC method[40e51]. In general, delta type singular kernels have been per-formed in the concept of kernels. For example Abel and Hilberttype singular kernels are most suitable in the area of aero-dynamics, heat and mass transfer, and plasma diagnostics. It isalso known that there are a few delta sequences kernels arising inthe theory of partial differential equations, Fourier transforms andsignal analysis. For example, the delta sequence kernels of positive

Ö. Civalek / International Journal of Pressure Vessels and Piping 113 (2014) 1e94

and Dirichlet type kernel and have very distinct mathematicalproperties [35].

The method of DSC is an effective and simple approach for thenumerical verification of singular convolutions, which occurcommonly in mathematical physics and engineering. The discretesingular convolutionmethod has been extensively used in scientificcomputations in past ten years. For more details of the mathe-matical background and application readers may refer to Refs. [31e36]. We used same parameters and notation given by Wei [32e35]for formulation of DSC method. In the context of distribution the-ory, a singular convolution can be defined by Ref. [37]

F�t� ¼ �

T*h��t� ¼

ZN�N

Tðt � xÞhðxÞdx (37)

where T is a kind of singular kernel such as Hilbert, Abel and deltatype, and hðtÞ is an element of the space of the given test functions.In the present approach, only singular kernels of delta type arechosen. This type of kernel is defined by Ref. [38]

T ðx Þ ¼ dðrÞ ðx Þ; ðr ¼ 0;1;2;.; Þ: (38)

where subscript r denotes the rth-order derivative of distributionwith respect to parameter x.

In order to illustrate the DSC approximation, consider a functionF(x). In the method of DSC, numerical approximations of a functionand its derivatives can be treated as convolutions with some ker-nels. According to DSC method, the rth derivative of a function F(x)can be approximated as [34]

FðrÞ�x�z

XMk¼�M

dðrÞD;sðxi � xkÞf ðxkÞ; ðr ¼ 0;1;2;.; Þ: (39)

where D is the grid spacing, xk are the set of discrete grid pointswhich are centered around x, and 2M þ 1 is the effective kernel, orcomputational bandwidth. It is also known, the regularized Shan-non kernel (RSK) delivers very small truncation errors when it usethe above convolution algorithm. The regularized Shannon kernel(RSK) is given by Ref. [39]

dD;sðx� xkÞ ¼ sin½ðp=DÞðx� xkÞ�ðp=DÞðx� xkÞ

exp

"� ðx� xkÞ2

2s2

#; s > 0

(40)

The researchers is generally used the regularized delta Shannonkernel by this time. The required derivatives of the DSC kernels canbe easily obtained using the below formulation [40]

dðrÞD;s

�x� xj

� ¼ dr

dxrhdD;s

�x� xj

�i x¼xi

; (41)

For example, the second order derivative at x ¼ xi of the DSCkernels for directly given [31]

dð2ÞD;s

�x� xj

� ¼ d2

dx2

hdD;s

�x� xj

�i x¼xi

; (42)

The discretized forms of Eq. (42) can then be expressed as [40]

f ð2Þ ðx Þ ¼ d2fdx2

x¼xi

zXM

k¼�M

dð2ÞD;sðkDxNÞfiþk;j: (43)

The kernels for second-order derivative at xsxk, can be given asfollows [36]

dð2Þs;Dðx� xkÞ ¼ �ðp=DÞsinðp=DÞðx� xkÞ exp

h� ðx� xkÞ2=2s2

i

ðx� xkÞ

� 2cosðp=DÞðx� xkÞ

ðx� xkÞ2exp

h� ðx� xkÞ2=2s2

i

� 2cosðp=DÞðx� xkÞ

s2exp

h� ðx� xkÞ2=2s2

i

þ 2sinðp=DÞðx� xkÞpðx� xkÞ3=D

exph� ðx� xkÞ2=2s2

i

þ sinðp=DÞðx� xkÞpðx� xkÞs2=D

exph� ðx� xkÞ2=2s2

i

þ sinðp=DÞðx� xkÞps4=D

ðx� xkÞexph� ðx� xkÞ2=2s2

i(44)

For x ¼ xk, this derivative is given by Ref. [36]

dð2Þs;D ð0 Þ ¼ �

3þ�p2=D2

�s2

3s2¼ � 1

s2� p2

3D2 (45)

4. Method of solution

Using DSC method to discretized the spatial derivatives in Eqs.(16) and (17), the derivatives of the displacement components canbe given by

y1D11ðjÞ þ y2D12 þ y3D13ðWÞ ¼ 0 (46)

y4D21

�j�þ�y51D

j22 þ y52D

W22

�þ y6D23

�W�þ y7D24

�j;W

�¼ 0

(47)

The differential operators in the governing equations (46) and (47)are given as:

D11 ¼ J2R þ

1RJR (48)

D12 ¼ J0jj (49)

D13 ¼ J1RJ

1R þ 2saRJR (50)

D21 ¼ J1R (51)

Dj22 ¼ sJ0

jj (52)

DW22 ¼ �Q þ KJ0

WW (53)

D23 ¼ J4R þ 2

1RJ3R þ

1R2

J2R � G

�J2R þ

1RJ1R

�þ J2

T þ CJ1T (54)

D24 ¼ J1RðjÞJ1

RðWÞ þ 1RjJ2

RðWÞ (55)

and the elements of the DSC operators are defined below

JnR ðW Þ ¼ vðnÞðW Þ

vRðnÞ¼

XMk¼�M

dðnÞD;s

�kDR

�ðW Þiþk;j (56)

Ö. Civalek / International Journal of Pressure Vessels and Piping 113 (2014) 1e9 5

JnR ðj Þ ¼ vðnÞðj Þ

ðnÞ ¼XM

dðnÞD;s

�kDR

�ðj Þiþk;j (57)

vRk¼�M

The resulting equation can be given as

f½LD� þ ½NLD�gfXg ¼ fFg (58)

Where the LD and NLD are the linear and nonlinear operators,respectively. The resulting nonlinear equation has been solvedusing the NewtoneRaphson method [52e54]. The procedure isbased on an incremental iterative method. For temporal dis-cretization of the displacement term, appearing in the givengoverning equations (Eqs. (16)e(17)), differential quadrature (DQ)time integration method is used. The related formulations aregiven below.

4.1. Differential quadrature (DQ)

The method of differential quadrature is an effective numericaldiscretization technique for the approximation of time and spacederivatives. Recently, the method of differential quadrature (DQ)has been extended to solve initial value problem in time domain[50e54]. In these studies highly accurate numerical results areobtained for some initial value problems. By this time, nonlinearanalysis of plates and shells has been performed using the differentmethods [55e67]. We used a couple approach in this study basedon the differential quadrature time integration.

The weighting coefficients of the first order derivative aredetermined by a simple algebraic formulation without any re-striction on the choice of grid points proposed by Shu and Richards[64]. The weighting coefficients of the second and higher orderderivatives are determined by a recurrence relationship in this newapproach called generalized differential quadrature. Then, Shu andChew [56] and Shu and Xue [57] have developed a different alge-braic formulations to compute theweighting coefficients of the firstand second order derivatives in the differential quadrature methodbased on a trigonometric trial function. Namely, unlike the differ-ential quadrature that uses the polynomial functions, such asLagrange interpolated, and Legendre polynomials as the test func-tions, harmonic differential quadrature (HDQ) uses harmonic ortrigonometric functions as the test functions. As the name of thetest function suggested, this method is called the HDQmethod. Theharmonic test function hk(x) used in the HDQ method is defined as[57];

hkðxÞ ¼ sin ðx�x0Þp2 ,,,sin ðx�xk�1Þp

2 sin ðx�xkþ1Þp2 ,,,sin ðx�xNÞp

2

sin ðxk�x0Þp2 ,,,sin ðxk�xk�1Þp

2 sin ðxk�xkþ1Þp2 ,,,sin ðxk�xNÞp

2

(59)

According to the HDQ, the weighting coefficients of the first-order derivatives Aij for i s j can be obtained by using thefollowing formula:

Aij ¼ðp=2ÞPðxiÞ

P�xj�sin��xi � xj

��2 p; i; j ¼ 1;2;3;.;N (60)

where

PðxiÞ ¼YN

j¼1;jsi

sin�xi � xj

2p

�; for j ¼ 1;2;3;.;N (61)

The weighting coefficients of the second-order derivatives Bij fori s j can be obtained using following formula:

Bij ¼ Aij

�2Að1Þ

ii � pctg�xi � xj

2

�p

; i; j ¼ 1;2;3;.;N (62)

The weighting coefficients of the first-order and second-order de-rivatives Aij

(p) for i ¼ j are given as

AðpÞii ¼ �

XNj¼1;jsi

AðpÞij p ¼ 1 or 2; and for i ¼ 1;2;.;N (63)

A natural, an often convenient, choice for sampling points is thatof equally spaced point. The equally sampling gird (E-SG) points aregiven for temporal discretization as;

tj ¼j� 1N � 1

for j ¼ 1;2;.;N: (64)

Once the sampling points are selected, the weighting co-efficients can be obtained. The first and second order derivatives(velocities and accelerations) can be obtained as [52];8>>>>><>>>>>:

_u0ðtÞ_u1ðtÞ$$$

_uNðtÞ

9>>>>>=>>>>>;

¼ ðTÞXNj¼0

Aiju�sj�

¼ ðTÞ

2666664

A10 A11 $ $ A1NA20 A21 $ $ A2N$$

$$

$$

$$

$$

$ $ $ $ $AN0 AN1 $ $ ANN

3777775

8>>>>><>>>>>:

u0ðs0TÞu1ðs1TÞ

$$$

uNðsNTÞ

9>>>>>=>>>>>;

(65)

8>>>>><>>>>>:

€u0ðtÞ€u1ðtÞ$$$

€uNðtÞ

9>>>>>=>>>>>;

¼ ðTÞ2XNj¼0

Biju�sj�

¼ ðTÞ2

2666664

B10 B11 $ $ B1NB20 B21 $ $ B2N$$

$$

$$

$$

$$

$ $ $ $ $BN0 BN1 $ $ BNN

3777775

8>>>>><>>>>>:

u0ðs0TÞu1ðs1TÞ

$$$

uNðsNTÞ

9>>>>>=>>>>>;

(66)

Details on the development of the DQ methods and on its ap-plications to the structural and appliedmechanics problemsmay befound in the open literature. It was also reported from the previousstudies that the ChebysheveGausseLobatto or non-equally sam-pling grid (NE-SG) points for temporal discretization is better thatthe equally spaced as;

tj ¼12

�1� cos

�j� 1N � 1

�p

; (67)

It is known that the choosing the type of sampling points andnumber of sampling points has an important effect on the accuracyof the results in differential quadrature methods. It is proposed thatthe optimal selection of the sampling points in the eigenvalue anddynamic problems is the normalized ChebysheveGausseLobattopoints.

5. Numerical results and discussions

In this section some results will be presented for nonlinear staticand dynamic response of shallow spherical shells with or without

Table 1Comparisons of central deflection of clamped supported spherical shell (qa4/Eh4 ¼ 5).

Shellparameter(s)

Gridnumbers

Literature results Ref. [30]

PresentDSC results

Kanematsu andNash Ref. [29]

Nath andAlwar Ref. [31]

1 (0.862)a (0.938) (0.856) (0.86)N ¼ 11 0.864N ¼ 13 0.865N ¼ 15 0.866N ¼ 17 0.869N ¼ 19 0.869

2 (0.840)a (0.925) (0.827) (0.84)N ¼ 11 0.839N ¼ 13 0.844N ¼ 15 0.846N ¼ 17 0.846N ¼ 19 0.846

5 (0.402)a (0.352) (0.372) (0.40)N ¼ 11 0.389N ¼ 13 0.395N ¼ 15 0.397N ¼ 17 0.398N ¼ 19 0.398

a Results given in Ref. [6].

Table 3Nonlinear maximum deflection (*0.01 m) of clamped circular plate.

Time Present DSC results Ref. [65] a

N ¼ 13 N ¼ 13 N ¼ 15 N ¼ 17

Dt ¼ 0.1 Dt ¼ 0.05 Dt ¼ 0.05 Dt ¼ 0.05

0.0000 0.8093 0.8020 0.8011 0.8011 0.80000.0012 0.6084 0.5993 0.5955 0.5955 0.59390.0036 �0.4476 �0.4335 �0.4331 �0.4331 �0.43260.0060 �0.7318 �0.7163 �0.7173 �0.7172 �0.71520.0072 �0.3142 �0.3108 �0.3068 �0.3068 �0.30310.0096 0.6933 0.6900 0.6839 0.6838 0.68350.0108 0.7985 0.7886 0.7848 0.7848 0.78530.1200 0.5016 0.4853 0.4843 0.4843 0.48480.1320 �0.0438 �0.0416 �0.0410 �0.0411 �0.04090.0156 �0.8142 �0.7982 �0.7968 �0.7968 �0.79720.0180 �0.1865 �0.1705 �0.1642 �0.1641 �0.1646

a Results taken from Ref. [66].

Ö. Civalek / International Journal of Pressure Vessels and Piping 113 (2014) 1e96

an elastic foundation. The effects of the some geometric quantities,boundary conditions and foundation parameters on nonlinearstatic and dynamic responses of circular plates and shallowspherical shells on elastic foundations are investigated.

5.1. Convergence study and accuracy

To illustrate the efficiency and accuracy of the proposed coupledapproach, comparisons study is made for central deflection ofclamped supported spherical shell. The results are presented inTable 1 for different shell parameters. The analytical and numericalresults from the open literature are also presented. It is shown thatfrom Table 1 (without foundation), there is good agreement amongthe results for all type shell parameters [30,31] usingN¼ 17. Secondconvergence study is also made for spherical shell on elasticfoundation (K ¼ 100; G ¼ 50; Q ¼ qa4/Eh4 ¼ 30; s ¼ 3). The resultsare presented in Table 2 and compared with the results presentedby Jain and Nath [5]. Chebyshev series method is used by Jain andNath [5] for computation. There is good agreement with the resultsof Jain and Nath [5]. N ¼ 15 is also gives efficient and accurate re-sults for the nonlinear problem of shells on elastic foundation.

Another convergence study for nonlinear dynamic response ofcircular plates under uniformly distributed load is considered.Circular plate with clamped edge is considered for nonlineardeflection problem. The material and geometrical parameters forplate are as follows: r ¼ 8000 kg/m3, n ¼ 0.33, a ¼ 50 cm; h ¼ 1 cm;E ¼ 150 GPa [65]. The results are presented in Table 3. Present DSC-DQ method provide acceptable results with a maximum discrep-ancy of 1.39% for N ¼ 15 points using Dt ¼ 0.05. The results relatedto convergence of the proposed methodology are also presented inFigs. 2 and 3 for circular plate and spherical shells. For time

Table 2Convergence study of central deflection of spherical shell on elastic foundation(K ¼ 100; G ¼ 50; Q ¼ qa4/Eh4 ¼ 30; s ¼ 3).

Ref. [5] Present DSC-HDQ results

N ¼ 11 N ¼ 13 N ¼ 15 N ¼ 17

0.988 1.015 1.002 0.991 0.991

convergence study, the results related to nonlinear dynamicresponse of clamped spherical shells given by Nath et al. [6] areused. It can be noted from these figures that converged results canbe obtained for a time step size Dt ¼ 0.05 and N ¼ 15.

5.2. Nonlinear response of shallow shells

In this section some parametric results related to nonlineardynamic and static response of shallow spherical shells with orwithout elastic foundations have been presented.

Effect of damping coefficients on nonlinear dynamic response ofclamped spherical shell (s ¼ 1.1; Q ¼ 6) is depicted in Fig. 4. It isobserved that as the damping coefficient increases the nonlineardeflection and as well as the peak amplitude decrease. The effectsof curvature on the nonlinear transient response of the shallowshell under dynamic step load are examined and results depicted inFig. 5. As the curvature of the shallow spherical shell increases, thenonlinear deflection of the shell decreases. It can be also seen thatnot only the amplitude but also period is affected with the curva-ture. Fig. 6 displays the nonlinear dynamic response of clampedspherical shell with different Winkler parameters under dynamicstep load (qa4/Eh4 ¼ 5.5; s¼ 2.0). It is clearly shown from this figure

Fig. 2. Convergence of nonlinear static response of simply supported circular plate.

Fig. 5. Nonlinear dynamic response of clamped spherical shell with different curvatureunder dynamic step load (qa4/Eh4 ¼ 5.5).

Fig. 6. Nonlinear dynamic response of clamped spherical shell with different Winklerparameters under dynamic step load (qa4/Eh4 ¼ 5.5).

Fig. 3. Convergence analysis for time-step for DQ time derivation.

Ö. Civalek / International Journal of Pressure Vessels and Piping 113 (2014) 1e9 7

that the nonlinear dynamic response of the shell decreases with anincrease in the value of Winkler foundation parameters.

Fig. 7 is similar manner as Fig. 6. In this figure (Fig. 7) sinusoidaldynamic load is considered. Nonlinear dynamic response of clam-ped spherical shell with different Winkler parameters under si-nusoidal dynamic load is depicted in Fig. 7. An increase in Winklerparameter K, results in an increase on the nonlinear deflection. Theperiod of the motion is also effected with Winkler foundationparameter.

Nonlinear dynamic response of simply supported spherical shellwith different Pastenak parameters under step load (qa4/Eh4 ¼ 6;s ¼ 3.0; K ¼ 10) is calculated and presented in Fig. 8. It is concludedfrom this figure that increase in shear modulus G of Pasternakfoundation decreases in the nonlinear dynamic response of theshell. The influence of the shear parameter of the foundation on thetransient response of shell is very significant than the Winklerparameter. In Fig. 9, three different foundation combinations areconsidered for clamped spherical shell under step load (qa4/Eh4 ¼ 5; s ¼ 1.0). As similar Figs. 7 and 8, the influence of theWinkler and Pasternak foundations parameters cause decrease inthe nonlinear response of the shells. However, the effect of thePasternak foundation parameters on deflection is clearly shown inthis figure. Namely, the shear parameter of the foundations hasbeen found to have a significant influence than the Winkler springparameter. Variation of the static response of clamped shallowspherical shell (s ¼ 2; K ¼ 75) with load with different shear

Fig. 4. Effect of damping coefficients on nonlinear dynamic response of clampedspherical shell (k ¼ 1.1; Q ¼ 6).

parameters as depicted in Fig. 10. The Pasternak foundation pa-rameters have also been significant effect on nonlinear staticresponse of the shallow spherical shells. From this figure, it may beconcluded that increasing the foundation parameter G will alwaysresult in decreased the deflection.

Fig. 7. Nonlinear dynamic response of clamped spherical shell with different Winklerparameters under sinusoidal dynamic load (qa4/Eh4 ¼ 5.5; s ¼ 3.0).

Fig. 8. Nonlinear dynamic response of simply supported spherical shell with differentPastenak parameters under step load (qa4/Eh4 ¼ 6; s ¼ 3.0; K ¼ 10).

Fig. 9. Nonlinear dynamic response of clamped spherical shell with different foun-dation parameters under step load (qa4/Eh4 ¼ 5; s ¼ 1.0).

Fig. 10. Effect of Pasternak foundation parameters on nonlinear static response ofclamped shallow spherical shell (s ¼ 2; K ¼ 75).

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6. Conclusions

The present study is related to nonlinear dynamic and staticresponses of shallow spherical shells on WinklerePasternak elasticfoundation. A coupled numerical methodology is used for compu-tation. Discrete singular convolution and differential quadraturemethods have been used for the spatial and time wise integrations,respectively. The influence of the Winkler and Pasternak parame-ters of the foundation, curvature of the shell, damping coefficienton the nonlinear dynamic and static response of the shallow shellshave been investigated in detail. It is shown that the shearparameter of the foundation has a significant influence on the dy-namic and static response of the shells. The damping coefficientand curvature of the shell have also been found to have significantinfluence on the results. In general, the static and dynamic responseof the shell is decreased with the increasing value of the foundationparameter. Increase in damping parameter causes decrease in thedynamic response. It may be also concluded that increasing thecurvature of the shell will always result in decreased deflection.Also, the response to a simply supported shell is higher than theresponse to a clamped supported. Consequently, it is possible to saythat the DSC-DQ coupledmethodology presented in this study is aneffective and accurate method for the nonlinear dynamic and staticanalyses of shell structures.

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