Vibrations of an orthotropic shallow spherical shell on a Pasternak foundation

Composire Strucwes 33 (1995) 135-142 0 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8223/95/$9.50

0263-8223(95)00113-l

Vibrati0n.s of an orthotropic shallow spherical shell on a Pasternak foundation D. N. Paliwal, H. Kanagasabapathy & K. M. Gupta Department of Applied Mechanics, Motilal Nehru Regional Engineering College, Allahabad-211 004, India

Large amplitude free vibrations of a clamped polar orthotropic shallow spherical shell on a Pasternak foundation are investigated using Sinharay and Banerjee’s approach. Numerical results are obtained for movable as well as immovable clamped edges. The effects of shell geometry and foundation parameters on frequency-amplitude characteristics are studied and plotted.

NOTATION A DI,

D2

Arbitrary dimensionless amplitude (
El,E2 el

e2

if)

h k R RO f-76 2 T t V W

WCJ

z

Normalized constant of integration Radial and tangential strains in the middle surface of the orthotropic shell Factor depending on Poisson’s ratio of the orthotropic shell material Non-dimensional foundation modulus, kR “IDI Non-dimensional shear modulus,

Elh3 12( I. - Vl v2)

8z

E2h3 12(1-v&

respectively Young’s moduli of elasticity in orthogonal directions First invariant of middle surface strains Second invariant of middle surface strains Integrand Function of time Shear modulus of foundation Thickness of the orthotropic shell Foundation modulus Radius of the base circle of the shallow spherical shell Radius of the curvature of the shallow spherical shell Cylindrical coordinates Kinetic energy Time Potential energy Displacement component in z direction Maximum amplitude at the centre at any time Distance of the middle surface of the shell from tlhe plane containing edge

GBD, i VlV2 Poissons ratios of orthotropic shell material Geometric parameter, R 2/2Roh Density of the shell material A time parameter, t(D/phR 4)1’2 Non-linear and linear frequencies, respectively

INTRODUCTION Shell elements offer efficient and convenient forms of structures and many appropriate and elegant shapes for wide applications. Use of shells in structures results in reduced weight and genuine use of materials. Shells are extensively employed in process and power plants, chemical, petrochemical, nuclear, marine, aerospace and aeronautical industries and civil engineering construction. Shallow spherical shells are presently being used in pressure vessels, process equipment and storage vessels, domes and shell foundations. There are several instances where these shells remain continu135

D. N. Paliwal,H. Kanagasabapathy, K. M. Gupta

136

ously supported by an elastic medium. With the increasing use of composites, the study of orthotropic shells resting on an elastic media is of special relevance nowadays. Herein the elastic medium is represented by a two-parameter Pasternak foundation model. Isotropic spherical and cylindrical shells on a Pasternak foundation have already been investigated by Paliwal et al.7-10 The authors of the present work have studied the large amplitude

free vibrations of an orthotropic shallow spherical shell on a Pasternak foundation. A new approach due to Sinharay and Banerjee”Y’2 is further extended to orthotropic shells. Nondimensional frequency vs non-dimensional amplitude characteristics are drawn. The influences of a geometric parameter of the shell and the foundation parameters of these characteristics are investigated.

PROBLEM FORMULATION The geometry of the clamped shallow spherical shell on a Pastemak foundation is shown in Fig. 1. The distance of the middle surface of the shell from the plane of the edge is given by

The Pasternak foundation model consists of closely spaced springs interconnected through a shear layer made of incompressible vertical elements, which deform only by transverse shear. The response equation of the Pastemak foundation is given by: p(r)=&(v)-GV2w(r)

(2)

where k and G are the foundation modulus and shear modulus, respectively. The first and second strain invariants for a polar orthotropic shelllm3 are el=.cl+kc2=-+-

(3)

and e2=L1c2=-.

ku r

du -+dr

1

dw 2 ku I iu

2 (-)dr

* r

r

dw dz -- dr ’ dr

(4)

k

Fig. 1. Shallow spherical shell on Pasternak foundation.

J4bration.sof a shell on a Pastemak foundation

137

where

Total strain energy due to bending written as

DI

(V2),+

2(k- y2) r

‘/=y

and stretching

dw

d2W -+c ’ dr . dr2 -

of the shell and that of foundation

may be

1

12

rdr de

[e:-2(k-v2)e2]

(5) Now, according to a new approach due to Sinharay and Banerjee,“? l2 the term [l - (Y&)~] (kU/r)2 is replaced by P[1/2(dw/dr) +dw/dr . dzldr12 and Z1 is substituted for the expression du/dr+ Vu/ r+ 112 (dw/dr)2+dw/dr. dzldr in eqn (5) where !=& and /I is a constant of proportionality that is a function of E’oissons ratio. Thus eqn (S), for total strain energy, can be rewritten as (V3+

2(kyv2) .$y. $1

rdr de

The kinetic energy of the shell is given by T=ph 2

(7)

in which inertial effect, due to radial displacement U, is neglected. This neglect introduces an error of less than 4% according to Mottershead.4 Forming the Lagrangian function L= T- V and applying Hamilton’s principle one gets 8

fiLdt=O. s t1

(8)

If one sets B=

(9

where 2(k-v2) -(v2w2)2+

_-

r

12

dw

d2W

12

.-g.---g=p:

rdr de dt

h2

or F(r,u,u,,w,w,w,,wt)

dr d0 dt

(10)

138

D.N.Paliwal, H. Kanagasabapathy, K. M. Gupta

in which u,=-

au ar

PW

aW

w,=-

;

;

ar

i3W

wrr=-

*

’

wt=-

at

2(k-

v2)

dw -

ar2

and -(V’w)‘+

r

’

d=w -

&.

12 _=

’ dr2 -Fe1

(11)

+ Euler’s equations

from the calculus of variations

are as follows:

(- )

aF a aF =() -_au i3r au,

(12)

g-p,)-$(g)+$(E)=o.

(13)

Application +-5

*Cl=-

of eqn (12) to eqn (11) yields

“z;= f(t)*

(14)

Similarly, applying eqn (13) to eqn (11) results in V4w-cc’f(t)r’-‘.

V2w-cx2f(t)rv-2.

(V-1)

E-k

-sV’w-~

(

=----

ph a2w D1

2a2f(t)r”-1

at2

The following expression, verse displacement. w=Aw,(t)

R,

)

1

aw

1 [ 2r

() ar

(15)

-z’

which satisfies the clamped

boundary

condition,

is assumed

for trans-

r2 ’

( ) 1 -F

3

-

kw

(16)

in which A is a non-dimensional amplitude ( I 1). Now, to solve eqn (14), eqn (16) is substituted area of the shell. Thus one obtains “f;= f(t)=

0

12~

16f~,vR’-~ R,(v+3)(++5)

+

128A=~~iK-~ (V+3)(ii+5)(?+7)

into eqn (14) and integration

is carried over the

.

(17)

V7bration.sof a shell on a Pastemak foundation

139

Galerkin’s error minimizing technique is used to solve eqn (15). Substituting eqn (16) in eqn (15) and making use of eqn (17) and then applying Galerkin’s technique, the equation for the time function wO(t) is obtalined as follows: 15367 (v+3)2(v+5)2

+

36864V +(20)

(v-3)2(?+5)2@+7)

AW2

+ 2568 + 35

1

A2W3=0

(18)

where WZ,

t=--

R2 2R,h ’

z=t

D1 [ phR4

1

‘I2 ,

kR4 GR2 A=and p=DI DI ’

When there is no elastic foundation, 1=~=0. Also, in the case of isotropic materials, v1=v2=v,

v=Jv1v2=v. Thus, eqn (18) reducles to 36864~

16P + 5

+lO

196608 v [ (v+3)2(v+5)2(v+7)2

2568 +

35

(v+3)2(v+5)2)(v+7)

1

A2w3=0

+

488 5

1

(205)Aw2

*

Thus as a special case of the present work, when v1=v2=v, and A=O, G=O, our result is exactly the same as that of Sinha.ray and Banerjee,12 eqn (8). Equation (18) is rewritten as: a2W 7+p1W+p2AW2+p3A2W3=0 in which pl, p2 and pj are the coefficients of W, AW2 and A2 W3 in eqn (18), respectively. If the initial conditions are W=l and awl&=0 at z=O, one obtains the following expression the approximate natural frequency (w*) from the solution of the nonlinear eqn (19).13

(19)

for

where o is the linear frequency.

NUMERICAL

RESULTS

Numerical results are obtained for the shells made of fiber glass-reinforced plastics and the two-way reinforced concrete for the following particular values of various parameters.

140

D. N. Paliwal,H. Kimagasabapathy, K. M. Gupta

5=0.5, 1.25, 2.50, 5.0; 1=20, 50, 100; jJ=20, 50, 100; V=& v,=O*O79 for fiber glass reinforced =0*1386 for two-way reinforced

plastics concrete. -------

Movable Immovable

EVALUATION OF p For a simply supported edge duldr can be neglected, as its contribution to radial strain is small, whereas rotation (dwldr) is large, so term 1/2[ 1/2(dwldr)2 + dwldr . dzldr] is more pronounced. For a clamped edge, du/dr is a minimum at the centre and maximum at the edge; therefore, it is quite reasonable to assume duldr=1/2[1/2(dw/dr)2+dw/dr. dz/dr]=average of the radial strain. Under this assumption, putting dV/d/3=0, for minimum potential energy, p is obtained as /3=2v1vz. In a similar manner p=vlv2 for a simply supported edge.12

DISCUSSION

0.999’

Parametric studies involving non-dimensional frequency ratios with the dimensionless ampli1.02

1.00~

-0..

9.

0.98 -

l.

0

;* : p=504

.’ ,* !r

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

A

Fig. 3.

Relation between w*/o and A for the clamped shell of fiber-reinforced plastics.

1.008

5 = 0.5 5= 1.25 \ . ..I --. . . . . . . . . . . . . . . ..___ L

..

p=20+

v, = 0.0891 v, = 0.07 it=20 5 = 1.25

-

Movable

5=2.5’.*... ‘.

‘.

t

-.

‘*.

0.96

‘.

‘*.

v.. =5-o

0.94

513

i

0.92 t

.

0.90

0.88

0.86

‘a*

-

Movable

-------

Immovable

.. \ .. *

VI = 0.0891 v* = 0.07 -

:

A=20

1

jl=20 o.84L 0 Fig.

2.

0.2

.\ .

0.4

0.6

0.8

1.0

0.6

A

Relation between w*/w and A for the clamped shell of fiber-reinforced plastics.

0.8

1.0

A

Eig. 4.

Relation between w*/o and A for the clamped shell of fiber-reinforced plastics.

Vibrations of a shell on a Pastemak foundation

141

-

_______

Movable mv&le

/ -

-Movable _--_-_- nrjnjov&te

0.999

\

Fig. 7.

v1 = 0.1.6 v, = 0.1.2 x=20 p=20

0.93 0

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

A Fie. 5.

Relation between o*/o and A for the clamped shell of two-way reinforced concrete.

c

--

1.007

1.006

1.005

,

Movable

II. I I._ _ ~v&le

:

:

,:

1

a’ d:

: : .’ :

v1 = 13.16 v, = 0.12

u=loo

x=20 c= 1.25

;: L’ :

, 4’

1

1.004-

L----J

0

0.2

0.4

0.6

0.8

1.0

A Relation between o*/o and A for the clamped shell of two-way reinforced concrete.

tudes are presented in Figs 2-4 (for fiber-reinforced plastics) and Figs 5-7 (for two-way reinforced concrete). It is obvious from these characteristics that the effect of changes in material, foundation and geometrical parameters is more pronounced in a frequencyamplitude relationship in the case of immovable edges than movable ones. The frequency ratio first decreases, when geometric parameter ‘5’ is increased in steps from O-5 to 5, shown in Figs 2 and 5. Figures 3 and 6 and 4 and 7 show that foundation parameters A and p have a negligible effect on a frequency-amplitude relation in the case of movable edges, while this relation is considerably influential for immovable edge conditions.

,

813 1.003-

1

1.002-

0.999b0

0.2

0.4

0.6

0.8

1.0

A Fig. 6.

Relation between o*/o and A for the clamped shell of two-way reinforced concrete.

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142

6. 7. 8.

9.

D. N. Paliwal,H. Kanagasabapathy, K. M. Gupta

Stability of Plates and Shells Structures, Ghent University, 1987, pp. 341-6. Nath, Y., Jam, R. K. & Mahrenholtz, O., Orthotropic circular plates and spherical shells on nonlinear elastic subgrade. Arc. Mech. Warszawa,39 (1987) 275-81. Paliwal, D. N., Sinha, S. N. & Choudhary, B. K., Shallow spherical shells on Pasternak foundations. AXE J. Engng Mech., 112 (1986) 175-82. Paliwal, D. N. & Srivastava, R., Non-linear static and dynamic behaviour of shallow spherical shells on a Kerr foundation. Znt. J. Pressure Kssels and Piping, 55 (1993) 481-94. Paliwal, D. N. & Bhalla, V., Large amplitude free vibration of shallow spherical shells on Pasternak foundation. ASME .Z. Rbrations and Acoustics, 115 (1993)70-4.

10. Paliwal, D. N. & Srivastava, R., Vibrations of shallow spherical shells on a Kerr foundation. ASME J. Kbration and Acoustics, 116 (1994) 47-52. 11. Sinharay, G. C. & Banerjee, B., A new approach to large deflection analysis of spherical and cylindrical shells. ASME J. Appl. Mech., 52 (1985) 872-8. 12. Sinharay, G. C. & Banetjee, B., Large amplitude free vibrations of shallow spherical shell and cylindrical shell - new apporoach. Znt. .Z. Nonlinear Mechanics, 20 (1985) 69-78. 13. Stoker, J. J., Non-linear Vibration, Vol. 2, Interscience Publishers, Inc., New York, 1963, pp. 98-103. 14. Timoshenko, S. & Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd edition. McGraw-Hill International Book Co., Tokyo, 1959.