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Non-axisymmetric dynamic response of buried orthotropic cylindrical shells
Journal of Sound and Vibration (1988) 121(1) 149-160
N O N - A X I S Y M M E T R I C D Y N A M I C RESPONSE OF BURIED O R T H O T R O P I C CYLINDRICAL SHELLS P. C. UPADHYAY AND B. K. MISIIRA Department of Mechanical Engineering, Institute of Technology, Banaras Hindu Unit'ersity, Varanasi--221005, India (Receiced 20 February 1987, and in revised form 24 April 1987)
This paper deals with the non-axisymmetric dynamic behaviour of buried orthotropic cylindrical shells excited by a combination of P-, SV- and SH-waves. Numerical results have been presented for the case of an incident plane longitudinal wave (P-wave) only. The flexural mode response has been compared with that of the axisymmetric mode. It has been found that, depending upon soil condition and apparent wave speed, shell deformations in the flexural mode may be even greater than in the axisymmetric mode. Also, the shell response is found to be significantly influenced by variations in the shell orthotropy parameters and by the soil condition around the shell. I. INTRODUCTION A good deal of research interest has been generated in the seismic response of buried pipelines due to the great potential for destruction, damage and disruption of utility services during earthquakes. Initially, buried pipelines were modelled as beams but this model was not very satisfactory in explaining the nature o f pipe failures. Muleski et al. [ 1] treated the pipe as a buried shell, using Fliigge's shell theory. Later on some researchers [2, 3] extended the analysis to thick shells incorporating the effects of shear deformation and rotary inertia. New materials, many of them orthotropic, e.g., reinforced plastic mortar (RPM), are now becoming quite popular in lifeline systems and are slowly replacing conventional materials such as cast iron and concrete. A need for analyzing the dynamic behaviour of pipes made o f orthotropic materials was emphasized by Adman [4] during the Second U.S. National Conference on Earthquake Engineering (1979). However, there is very little work available on orthotropic buried pipes/cylindrical shells. Cole et al. [5] have performed a finite element analysis of buried RPM pipes. Recently, Singh et al. [6] have studied the axisymmetric response of buried orthotropic shells. But, since orthotropic materials have different strength and stiffness in di~erent directions, non-axisymmetric response of these shells becomes quite important. Datta et al. [7] have studied the non.axisymmetric response ofburied isotropic shells. To the best of the authors' knowledge there is no work available giving the non-axisymmetric response of buried orthotropic pipelines. The aim of work reported in this paper, therefore, was to study the non-axisymmetric dynamic response of buried orthotropic cylindrical shells embedded in a linearly elastic, homogeneous and isotropic medium. A thick shell theory formulation, which includes the effect of shear deformation and rotatory inertia, is presented. The decision for using thick shell model was taken on the basis of results presented in an earlier paper [8] wherein a comparison of thick and thin shell predictions have been discussed for this type of problem. The shell is assumed to be excited by a combination of P-, SV- and SH-waves. Results are presented, however, for the case of an incident plane longitudinal 149 0022-460X/88/030149+ 12 $03.00/0 9 1988 Academic Press Limited
150
UPADIIYAY
P. C .
AND
II. K .
MISIIRA
wave (P-wave) only. The flexural mode (n -- I) response has been compared with that of the axisymmetric mode ( n = 0 ) and it has been found that, depending upon the soil condition and apparent wave speed, shell deformation in the flexural mode may be even greater than in the axisymmetric mode. Thus, for orthotropic shells, non-axisymmetric response assumes considerable importance. The shell response is found to be significantly influenced by the variations in the shell orthotropy parameters, as well as by the soil conditions around the shell. 2. BASIC EQUATIONS AND FORMULATION Consider an infinitely long thick orthotropic cylindrical shell of mean radius R and thickness h embedded in a linearly elastic, homogeneous and isotropic infinite medium. The shell is excited by a seismic wave (a combination of P-, SV- and SH-waves) of wavelength A ( = 2,-'r/~') and speed c(= t o / f ) moving along the shell axis. Let a cylindrical-polar co-ordinate system (r, O,x) be defined such that x coincides with the axis o f the shell. In addition, if z is measured normal to the middle surface of the shell,
z=r-R,
-h/2<~z<~h/2.
(l)
Equations governing the non-axisymmetric motion o f a cylindrical shell can be derived by following reference [9]. In a matrix equation form they may be written as
[L](U}+(P*} =0,
(2)
where [L] is a ( 5 x 5 ) matrix operator with elements 9)2
O2
L,, =
[E',,
D'\
a2
R:l(h+ I/ R:)
t.,2 =-t.2, = -
(h+UR2)+kR 2
k~,
L,,=-~.~,=
~
k~
_ _
Lo.~= - L 4 1 -
2
Eppo, a R Ox'
-
( h + U R 2) ~ , i)
Lt5 = Lsn = k.~G.,:h "~x'
a2 [E'p D'\ ,32 : a~ L2z = G,olz 72+0.x ~-~'+ - ~ ) - ~ - (koG:o/ g :)(h + I / R 2) _ ph c~t""
G,,I a2
D' ,')"
(
_.?)
R,.)02+ k2o
L_.~=Lj2=--ff- ax 2 !
(h+l/R2)-pl/R~t2,
a~
L 2.~= L.~2= -~ ( Epvo, + G,o h ) ,'Ix ;JO'
L2s = L~2 = O,
a: D' a', L33=G,ol~x'.-~ R: aO~ k'oG:o(h+ I / R 2 ) - P I., ' L34 = L4~ = 0,
l-3s = I-$3 = a-"
G,JI
.
(Dvo, + G, oh3/12) ~x aO" 1 \
.,
i)"
,,}~
,"
BURIED
ORTHOTROPIC
CYLINDRICAL
D ~2
G,ol ~2
L45 = L54 - R 3x 2
i)2
9
L55=Dc~x24
R 2
R
.~
t3t 2 '
k~G~.h-pl
c~0 2
-
151
pl 02
R 3 ~02
G~ol i)2
SHELLS
..
~.2
i)t 2,
and
{U}=Lw
v
~'o
u
~,,,jT
with w, v and u as the displacement components of the middle surface of the shell in the radial, axial and tangential directions respectively, and ~b~ and ~o being the angles o f rotation of a straight line initially normal to the middle surface in the axial and tangential directions respectively. The elements of [ P * ] are given by
P*=(1
h \
,
e,=hp,, 2
where o'*=, or*0 and or* are the stresses at the outer surface o f the shell (i.e., at z = + h / 2 ) . Different constants appearing in the expressions for the Lis are defined as
E,h 9 , Ep - 1 - V.~oVox I = h3/12
and
,_ Eoh Ep - 1 - v,,ovo~' k~ = ko = r r / x / ~
h2 D = E,-~,
D'=
h2 E
t -
P 12'
(shear correction factors)
where E,, Eo are the Young moduli, v,o, vo, the Poisson ratios, G~o, G~: and G:o the shear moduli and p is the desntiy of the shell material. For the evaluation of [P*], the or* are determined in terms of the incident and scattered fields in the surrounding ground. The total displacement field in the ground is written as u =
u('>+u (~),
(3)
where superscripts i and s represent incident and scattered parts respectively. By solving the wave equation in the surrounding infinite medium, components of the incident and scattered fields can be written as
U(ri)= [{ yl ~,(yr/ R )}dt -{flS l" ( 6r/ g )}d2 + {n( g / r)l.( Sr/ g )}d3] x c o s nO cos (~x-tot)
U~o')= [-{n( R / r)I.( yr/ R )ld, + {n( R / r)fl l.(~Sr/ R )}d2-( 8 IL (,Sr/ R )}d3] xsin nO cos (~:x- tot)
,,!?= [-(131,,(r,/R)}d,
+ {6"-I.(ar/R)}d2] cos nO sin (~x - tot),
(4)
ut/~= [{ y K " ( yr/ R ) } A - { f l S K" ( ar/ R )} B + {n( R / r)K.( Sr/ R )}C ] x c o s nO cos ( ~ x - tot),
u~'~= [-{n( R / r)Kn( yr/ R )}A + {u( R / r)fl K.( 6r/ R )} B - { 6K" ( Sr/ R )}C ] x sin nO cos (~x - tot), u!,:' = [ - { f l K . ( y r / R ) } A + { 6 2 K . ( S r / R ) } B ]
cos nO sin (~x-tot).
(5)
152
p.c.
UPADHYAY A N D B. K. MISHRA
where I. and K . are the modified Bessel functions o f the first and second kind respectively, and T = f l ~ / 1 - ( c / c , ) 2, 8 = f l , / l - ( c / c z ) 2, ~ = ~ R = 2 z r R / A , and c=oJ/r c, and c2 are, respectively, the speeds o f p r o p a g a t i o n o f longitudinal and shear waves in the surrounding m e d i u m , d,, dz a n d d 3 d e p e n d on the intensities o f the P-, SV- and SH-parts o f the incident wave, respectively, and have the dimension of length. A, B and C are arbitrary constants. (') denotes differentiation with respect to the a r g u m e n t s of the Bessel functions. N o w u (i~ and u (~) being known, by using the s t r e s s - d i s p l a c e m e n t relation for the surrounding m e d i u m o',., o-,o and o',x at r - - R + h / 2 can be obtained. Thus [ P * ] is determined in the form {P*} = (1 + 17/2)(tz/R)[g][l]{m}~
(6)
where [g], [I] and {m} are (5 x 5), (5 x 6) and (6 x 1) matrices respectively. Terms of these matrices are as follows: g,, = cos nO cos (~:x - tot),
gz2 = sin nO cos (~x - tot),
g44 = cos nO sin ( ~ x - tot),
g55 = g44,
1,, = ( 2 - r2)E2I.(cq) + 2V2I~(a,),
g . = g22, when i # j ,
go = 0
1,2= -2/382I ~(a2),
1,3 = [ 2 n / ( 1 + IT/2)]8 I ' ( a 2 ) - [2n/(1 + IT/2)2]In (a2),
i,4=(2_r2)e2K.(ot,)+ 2T2K,~(ot,),
!,5= -2/38 2K.(ct2) .,, ,
1,6 = [2n/(1 + IT/2)]8 K:,(a2) - [ 2 n / ( 1 + I7/2)2]K.(a2), 12, = [2n/(1 + 17/2)"] { I . ( a , ) - a , I ' ( a , ) } , 122 = [ 2 n / ( l + 1]/2)z]/3 {o~21"(o~2) - I . (a2)}, 123 = -c52I ~(a2) + [8/(1 + 17/2)]I'(a2) - [ n 2 / ( 1 + 17/2)2] I.(a2), 124 = [2n/(1 + 17/2) 2] { K . ( a , ) - a , K ' ( a , ) } . 12s = [2n/(1 + 1712)2]/3{a2K'(a2)" K . ( a 2 ) } . 126 = - B2K~(ot2) + [81(1 + IT/2)] K'~ (a2) - [n2/( 1 q- t7/2)2] K. (a2).
13s= (h/2)lzs 14, = -2/3711.(a,),
f o r j = 1 to 6, 142 = ~(2/32- r 2 E 2 ) I ~(Cr
l.~3=[n/(l+l-~/2)]~I.(ot2), I,~ = ~(2/3 ~ - r % 2 ) K ' ( a 2 ) ,
15~=(h/2)14s
forj=lto6,
144 = .2/3TK:,(ot,),
!,~ = [,,/(1 + 17/2)]13 K o ( . : ) , and{m}=[d,
de
d3
A
B
Clr.
In these expressions !~= h / R and p. is the shear m o d u l u s o f the surrounding m e d i u m , ~.2= c2/c~=2(1 _ v m ) / ( 1 - 2 u . , ) , v,. is the Poisson ratio o f the surrounding m e d i u m ,
e=~c/c,,
a , = (1 + i~/2)3,,
a2 = (1 + 1T/2)8,
l'~(cr;) = ( n / a , ) I . ( c ~ / ) + I . + , ( a i ) . I ~(a,) = [1 + (n2/a~) - (n/ct ~)]I.(ai) - (1 / c~,)I.+,(a,). K~(a;) = (nla,)K~(a,) - K . + , ( a , ) , K~(c~,) = [ 1 + ( n 2 / a 2) - ( n / a ~ ) ] K . (al) + ( 1 / a , ) K . + t ( a , ) . If the shell displacements are now a s s u m e d to be of the form
W=WoCOSnOcos(~x-wt),
U=vosinnOcos(~x-tot),
Sx = $xo cos nO sin (~x - wt),
U=UocosnOsin(~x-tot),
~o = ~oo sin nO cos (_Ex- tot)
(7)
BURIED ORTIIOTROPIC C Y L I N D R I C A L SttELLS
153
and these along with [ P * ] from equation (6) are substituted into equations (2), a set of five simultaneous equations is obtained. Three more equations are obtained by enforcing the b o u n d a r y condition that the displacement be c o n t i n u o u s at the outer surface of the shell: i.e.,
o+(h/2)~,o=(z~o'~+i,';b.=,,+h/2,
w = (u'." + .<.'~).=,~+h/2,
t, + (h/2)e,x = (u~'l + u?l),= R+h/2.
(8)
Thus a total o f eight equations is obtained. These eight equations, when simplified, give the final response equation, which may be put into the form [ Q] { Uo} = d, { F'} + d2{ F 2} + dj{ F3},
(9)
where [ Q] is an (8 x 8) matrix and {F~}, {F 2} and {F j} are (8 x 1) matrices, the elements being as follows:
172112) + ,f'/2,
O,, = - k~13217- k 20n 2(r/.,/,72) #7( l + 172112) - IT(n,/r/2) N ( 1 +
Q,2 = h i - k~(r/.d r/glT(i + 172/12) - (r/,/r/D NIT(1 + IT2/12), 0,3 = n[(71,/~,) N(I7 ~/6) + kl(o.,/~)(2 + 17~/6)]>
0,4 = -(vo
Q,5= 2k~O,
Q~,=(l+lT]2)fdts,
Qt6=(l +lT/2)fil,,,
Q,8=(1+17]2)~1,6,
02,=0,2,
022 = - ( r / 3 / r / : ) IT/32- (r/,/r/z) Nn 217( 1 + IT2/12) - k~(r/,/n2) + .0 2, 0-
=
- ( r/3/n2)(17"/6)13 2 + ( 71,/r/9 Nn2(lTZ/6) + k2o(r/J 779(2 + h2/6) + (/T/6)-O 2,
O,,=-n[(Z,o
Q26=(l+#T/2)4h,,
02~=0,
Q28=(1+i'1]2)ft!26,
Q3,=Q,~,
Q32=Q23,
Q33 = -(rh/r/t)(#]'/3213) - (r/,/r/2) Nn 2(1713)- k:o(r/4/r/2) [ (41 #7) + (/~13)] + ( a 213),
Q.= -n[(~'o,N/r/9+(r/dr/9](g/3)B,
Qj, = 0, QJ7 = Q27,
QJ8 = Q2,,
Q., = Q , . ,
Qj6 = Q ~ ,
Q.~ = Q ~ ,
Q.J = QJ,,
Q~4 = - ( N/r/:)l~fl 2 _ n2( r/j/r/~)l~( 1 + / ~ : / 1 2 ) + / 2 ~, Q~s= -(N/r/~)(l~2/6)fl2+(r/j/r/~)n2(l~2/6)+(h/6)12 ~,
Q5,
=
Q~6=(l+h/2)~i~,
Q,,=(l+lT/2)~tl~,
Q,s=(l+g/2)l~l,6,
Qis,
Q~j QJ~,
Q~2 Q2s, =
=
Q5. Q,~, =
Qss = - ( N/r/~)(l;/a)fl ~- n:( Oj/ r/~)(/;/3) - (4k]/ I~) + (/2 ~/3), Q~6 = Q~6,
Qs7 = Q~7,
Q61 = 1,
Q6~ = fl6K'(a.),
Q66 = - y K ' ( a , ) , Q7~ = 0,
Q~s = Q.s,
Q7= = QTa = l,
Q68 = - [ n / ( 1 + 1;/2)]K.(a,),
Q8~ = Qss = l,
QTs = 8K'(a~),
Q86 = f l K . ( a i ) ,
F~ = - ( l + l~/2)fil,~, F~ = F~,
Q76 = [ n / ( 1 + 1~/2)]K.(at),
Q74 = Q7~ = 0,
Q77 = - [ n / ( 1 + l;/2)]flK.(a=), Qsi = Q~2 = Q~3 = 0,
Q6, = Q63 = Q6~ = Q~5 = 0,
Q87 = - 6 Z K . ( a ~ ) ,
F~2=- ( l + i~/2)fil..,,
F~ = - ( 1 + 1;/2)fil~,,
F~ = F~,
Qs8 = 0,
154
P. C . U P A D H Y A Y
F~-- yl'(aq),
AND
B. K. M I S H R A
F~= " [ . / ( 1 + #7/2)]I.(.,),
F2t = - ( l + lT/2)~l,2,
F~= - ( l + l-i/2)fd,,_,
FJ = - ( 1 + 17/2)fil,2,
F~ = F~,
F~ = -t3 I.(,~,).
F~ = F~,
F~ = [n/(1 + 17/2)],8I.(0t2),
F~] = -/36I'(0t2),
F; = - { l + & 2 ) ~ t , , ,
F~' = - 0 + tZ/2)~t:,,
F l = -(1 + h/2)fa,,,
F33 = F 3,
F63 = In/(1 +/T/2)]I. (c,2),
F~ = a2l.(cr2),
F~
, q = FI,
-6I'(ae2) ,
=
F ] = 0.
Here 77, = Eo/ E~, ~2 = G . , / Ex, r/3= G.,o/ E,, 71~= G:o/ E,, N = 1 / ( 1 - V.,oVo.~), 12 = /, / G~.., f22 = 17fir2f12e'-]13, e = c / c t and 13= p,,/p, p,, being the density o f t h e surrounding medium. The response vector { Uo} in equation (9) represents {Uo}=[Wo
/3o
(~oo)
Uo ( h ~ o )
a
B
C
JT.
Results are presented for the case d2 = da = 0 and thus taking the incident wave as totally composed o f P-waves. For this particular case equation (9) becomes [o]{O} ={F'},
(10)
where h
h
A
B
. e l T, "* I .I
W, V and 0 are the non-dimensional amplitudes of the middle surface of the shell in the radial, tangential and axial directions respectively.
3. RESULTS AND DISCUSSION Results are presented for a transversely isotropic shell with r-O as the plane ofisotropy. Thus Eo = E:, G.~: = Gxo, V~o = v:s, V~o = v~:, and G:o = Eo/2(1 + Vo:), and consequently 73 = r/2 and r/4 = G:o/E~ = "qJ2(1 + Vo:). Further, vs_.= V:,o= 0.3 has been assumed, vo~ is given by the relation E o / E , = poJV~o or v0, = ~tVxo. Three different values of 7, and r/2 were used in the calculations rtt =0.05, 0.1 and 0.5; r/z=0.02, 0"05 and 0.1. These are expected to cover a wide range of orthotropic materials. Results are presented for two soil conditions--very soft (/i = 0.001, v,,, = 0.45) and hard ( f i = l . 0 , v,,,=0-25). A constant value of I]=0.05 and 13=0.3 has been assumed throughout. The flexural mode 01 = 1) and the axisymmetric mode (n = 0 ) responses are compared. Figures 1-6 show the variations of the radial displacement (1~r) with the nondimensional wavelength parameter / 3 ( = 2 r r R / A ) . Figures I-5 correspond to soft soil (fi =0.001) whereas Figure 6 is given for the hard and rocky ground condition. The first two figures, 1 and 2, drawn for a very low apparent wave speed (E= 1.001), indicate that the variations in orthotropy parameters r/, and 77., affect the radial displacement o f the shell quite significantly under the soft soil. They also indicate that, excepting for r/t = 0"5 in Figure 1 and r/2 = 0.02 in Figure 2, the axisymmetric mode (n = 0) is dominant in comparison to the flexural mode (n = 1). When the wave speed is increased to C'=2.0, I~z vs. fl plots for the soft soil are shown in Figures 3 and 4 with, respectively, 77, and rt2 as parameters. The orthotropy parameters rh and ~72 are still found to have a significant
B U R I E D O R T H O T R O P I C C Y L I N D R I C A L SIIELLS I
I
[
]55
i
o~
oO
It
On
oo
oz
04
06
08
nO
Figure 1. Radial displacement (l~') versus wavelength parameler (13) for .6. = 0.001 and ~ = 1.001 with rl 0 as parameter,
v,,, = 0 . 4 5 ,
r/~=O.05.
--,
, =0;
--
'
- , n = I.
i
'
u
,
I
'
I
'
005
9
Ol
io
oo
B
Figure 2. Radial displacement (IF') xersus xva~elenglh parameler (/3) for ~ =0-001 and E-- 1.001 with ~z as parameter. ,,,, =0.45. ~, = 0 . 1 . Key as Figure I.
02
'
I
'
i
'
i
'
I
'
|
~,, 0 - 0 ~
oo
f/~/-"
013
J f ,,~,~ ~//j/.. ~ 05
04 # 02 00OO
p .#~+S 02
~
04
06
08
nO
B
Figure 3. Radial displacement (l~') ',ersus wavelength parameter (3) for ,6. =0.001 and ,3= 2.0 with rh as parameter. ~,,,, = 0.45, rl_,= 0"05. Key as Figure I.
156
P. C. U P A D t I Y A Y A N D B. K. M I S I I R A
IZ
I
I
I
"]
i
'
tO
~
05
04
OZ O0
/002
..////~
1~06
O0
-
B
Figure 4. Radial displacement ( I t ' ) versus wavelength parameter (/3) for/.i = 0.001 and E = 2.0 with rlz as parameter, v,, =0-45, r/, = 0 - I . Key as Figure I.
I-4 'Z
005
/fo
I t os
o6
9
OI
r
//
o~
oo
oz
o4
/9
OI6
08
io
Figure 5. Radial displacement (if,') versus wavelength parameter (/3) f o r / i =0.001 and E = 5 - 0 with Tit as parameter, v,,, = 0.45, 02 = 0.05. Key as Figure I.
.y-
o,
~, 9 0 1 . 0 - 0 5
o~
#
J~ C4
T/,'0%OI.O0~ / /
/
I" /
/ / ~
,'
o
O,o, 0000
'Z_-..,_"i ,
1
02
,
l
04
I
06
l
08
i
o
Figure 6. Radial displacement (t,~,) versus v.'avelength parameter (/3) for /7. = 1.0 and ~?= 2.0 with r/i as parameter, v,, = 0.25, rl2 = 0.05. Key as Figure !.
BURIED
ORTilOTROPIC
CYLINDRICAL
SliELLS
157
influence on the radial displacement. However, unlike Figures l and 2, here flexural mode displacements are always greater than their corresponding values in the axisymmetric mode. When ~ is further increased to 5.0, as shown in Figure 5, no such general trend is observed. While for .01 = 0.5 the radial displacement is always higher in the flexural mode, for rh = O"1 and 0.05 it depends on the value of/3 as to which mode will give the larger displacement. A common feature o f all the figures (I-5) is that, in general, variations in "01 and "02 produce more changes in the axisymmetric mode than in the flexural mode. In the flexural mode changes are, in fact, negligible up to 13 -~0.6 or so. For bringing out the effect of the surrounding soil condition, fie' for hard and rocky ground (/.7.= !.0) has been plotted in Figure 6, with "01 as parameter. A comparison of Figures 6 and 3 shows that changes in the orthotropy parameters make very little difference in )7V in the hard soil as compared to the differences observed in the soft soil. In fact, in the axisymmetric mode absolutely no difference is realized; only in the flexurai mode some changes are observed. It may be noted that a different trend was obtained in the soft soil (see Figure 3) where changes were more prominent in the axisymmetric mode than in the flexural mode. Further, under the hard ground condition, the dominance of the n = 0 or n = ! mode becomes wavelength dependent. Up to/3 =0.5-0.6 the axisymmetric mode prevails, and beyond that the flexural mode becomes dominant. Axial displacement ( O ) results have been presented through Figures 7-11. Figures 7-10 correspond to soft soil and Figure 11 is given for the hard soil. As seen from Figures 7 and 8, at low apparent wave speed ( ~ = 1.001) axisymmetric mode is found to be dominant. However, at an increased wave speed (? = 2.0), excepting for very small /3 (large wavelengths), the ftexural mode (n = 1) gives higher values of /.St as shown in Figures 9 and 10. In all the four figures (7-10) the maximum value of O in the axisymmetric mode (n = 0) is found to occur at very large wavelengths (i.e., low fl), while in the flexural mode (n = 1) the peak value of O is observed a r o u n d / 3 = 0 . 6 . At still higher wave speeds (say, ,~= 5.0), qualitatively similar results were obtained. Figure 11 shows the axial displacements of the shell in hard soil. It is seen that the variations in orthotropic parameters do not produce significant changes in the axial
I
I
I
//"
o,3 <--.
J oo
oz
I
o4
B
016
Figure 7. A x i a l displacement ( U ) versus wavelength parameter (~)) f o r / ~ = 0.00t and ~ = 1.001 w i t h "~t as p a r a m e t e r , v,,, = 0 . 4 5 , r/2 = 0 . 0 5 . K e y as F i g u r e I.
158
P.c.
UPAI)IIYAY A N D II. K. MISHRA '
i
'
I
'
l
'
~ ' ~ /
005
~
,
"" ~,[
002
,
I
oz
\
005
.I
~"I
'''
o /"
oo
I
04
I
-.
,
o$
I 08
, ,0
Figure 8. Axial displacement ( U ) versus wavelength parameter (/3) for/2 =0.001 and t;-- 1.001 with ~Tz as parameter. ~,,,, =0.45, rh =0-1. Key as Figure I.
}6
'
'
"
'
>S -~-7\
'
/
v~.C- , . ",2"\ 1
1" ~,,o,2X">, ,,1
/I
e*
.o20;
I
-
"1
I
/
t~,
/ / ' I-
col
,
I
i
oo
I
I 02
,
I
if4
oo?--,
,
B
I
06
,
I 08
io
Figure 9. Axial displacement ( U ) versus ~avelcngth parameter (/3) for /2 =0.001 and t~=2-0 with rh as parameter, v,,, = 0-45, 7: = 0.05. Key as Figure 1.
displacement (U) of the shell in the axisymmetric mode. Changes are prominent in the flexural mode only. Further, in hard soil, again a mixed trend is seen where the axisymmetric mode remains dominant only up to /3---0.6, and f o r / 3 ~ 0 . 6 the flexural mode displacement becomes larger. These observations are similar to those for the radial displacement in Figure 6. Although rh and 7-/_,are equally effective in producing changes in the radial displacement, for controlling the axial displacement r/_, is found to be more effective than r/t. Though for smaller values of/:3 the axial displacement in the axisymmetric mode is always higher than that in the flexural mode, in soft soil and at low wave speed (Figures 7 and 8) it becomes many times larger. From the above results it is evident that, unlike the situation for isotropic shells, the non-axisymmetric response of orthotropic shells is of considerable importance. This is
BURIED
ORTHOiROPIC
64
CYLINDRICAL
I
I
I
40
/
'
SHELLS
159
I
\
/
x
/ ao
/
002
/ /
,/
/
2,
.
/I
/ //
OI /
[ ~;C._~ %o'
"~
oo~ '
o"
'
&
'
o"
'
,o
B
Figure I0. Axial displacement (O) versus wavelength parameter (,/3) for/2 =0.001 and ~=2.0 with 172 as parameter. ~,,,,= 0.45, rh = 0-1. Key as Figure I.
~2
i
I
I
o,
9
i
'
|
,,//So~ ! / /,
O-02 . ~ . .
o
oo
(>2
04
o6
o8
io
B
Figure II. Axial displacement (U) versus wavelength parameter (/3) for /i = 1.0 and ~=2-0 with r/2 as parameter, v,, =0-25, r/2 = 0'!. Key as Figure I.
because o f the fact that the deformations in the flexural m o d e may be even greater than those in the axisymmetric mode. By suitably adjusting the different elastic moduli o f the orthotropic shell, it s h o u l d be possible to control the shell displacements appreciably, particularly in the soft soil. When the incident wave is o f larger wavelength (smaller/3), the axial displacement in the axisymmetrie m o d e is always higher than in the flexural mode. Therefore for the larger wavelengths the axisymmetric mode is more relevant, as the most c o m m o n cause o f pipeline failure in seismic environment is the excessive axial deformation [ 1]. However, when the incident wave is o f smaller wavelengths (less than twice the circumference o f buried pipeline, i.e.,/3 > 0.5), the non-axisymmetric response b e c o m e s quite important.
4. CONCLUSION On the basis o f the results presented, the following conclusions can be made. (1) The flexural m o d e response assumes considerable importance in soft soil conditions and at higher a p p a r e n t wave speeds. (2) Variations in the shell orthotropy parameters are more effective in influencing the shell displacements in soft soil than in hard soil.
160
P. C. U P A D t I Y A Y A N D B. K. M I S H R A
(3) The o r t h o t r o p y p a r a m e t e r s r/, a n d r/2 are e q u a l l y effective in c o n t r o l l i n g the r a d i a l d i s p l a c e m e n t , but ~/z affects the axial d i s p l a c e m e n t m u c h m o r e than r/,. (4) F o r i n c i d e n t waves o f larger wavelengths, the axial d e f o r m a t i o n in the a x i s y m m e t r i c m o d e is a l w a y s h i g h e r t h a n in the h i g h e r m o d e s .
REFERENCES 1. G. E. MULESKI, T. ARIMAN and C. P. AUMEN 1979 Journal of Pressure Vessel Technology 101, 44-50. A shell model of a buried pipe in a seismic environment. 2. S. K. DATrA, A. H. SHAI-t and N. EL-AKILY 1982 Journal of Applied Mechanics 49, 141-148. Dynamic behaviour of buried pipe in a seismic environment. 3. S. CIIONAN 1981 Journal of Sound and Vibration 78, 257-267. Dynamic response of a circular shell imperfectly bonded to a surrounding continuum of infinite extent. 4. T. AR1MAN and G. E. MULESKI 1979 Proceedings of the 2nd U.S. National Conference on
Earthquake Engineering, Earthquake Engineering Research Institute, Stanford Universit); California, 643-652. Recent development in seismic analysis of buried pipelines. 5. B. W. COLE, C. J. RITTER and S. JORDAN 1979 in Lifeline Earthquake Engineering--Buried Pipelines, Seismic Risk and Instrumentation, ASME (Editors T. Adman, S. C. Liu and R. E. Nickell). Structural analysis of buried reinforced mortar pipe. 6. V. P. SINGII, P. C. UPADtIYAY and B. KISHOR 1987 Journal of Sound and Vibration 113, 101-115. On the dynamic response of buried orthotropic cylindrical shells. 7. S. K. DATTA, A. H. SI-IAH and N. EL-AKILY 1981 in Seismic Risk Analysis and Its Applications to Reliability-based Design of Lifeline Systems (Editors M. Kubo and M. Shinozouka). Tokyo: Gakujutsu Bunken Fukyu-Kai. 8. V. P. S I N G H , P. C. UPADiiYAY and B. KISHOR 1987 Journal of Sound and Vibration 119, 339-345. A comparison o f thick and thin shell theory results for buried orthotropie cylindrical shells. 9. 1. MIRSKY and G. HERRMANN 1957 Journal of the Acoustical Society of America 29, 1116-1123. Nonaxially symmetric motions of cylindrical shells.
N O N - A X I S Y M M E T R I C D Y N A M I C RESPONSE OF BURIED O R T H O T R O P I C CYLINDRICAL SHELLS P. C. UPADHYAY AND B. K. MISIIRA Department of Mechanical Engineering, Institute of Technology, Banaras Hindu Unit'ersity, Varanasi--221005, India (Receiced 20 February 1987, and in revised form 24 April 1987)
This paper deals with the non-axisymmetric dynamic behaviour of buried orthotropic cylindrical shells excited by a combination of P-, SV- and SH-waves. Numerical results have been presented for the case of an incident plane longitudinal wave (P-wave) only. The flexural mode response has been compared with that of the axisymmetric mode. It has been found that, depending upon soil condition and apparent wave speed, shell deformations in the flexural mode may be even greater than in the axisymmetric mode. Also, the shell response is found to be significantly influenced by variations in the shell orthotropy parameters and by the soil condition around the shell. I. INTRODUCTION A good deal of research interest has been generated in the seismic response of buried pipelines due to the great potential for destruction, damage and disruption of utility services during earthquakes. Initially, buried pipelines were modelled as beams but this model was not very satisfactory in explaining the nature o f pipe failures. Muleski et al. [ 1] treated the pipe as a buried shell, using Fliigge's shell theory. Later on some researchers [2, 3] extended the analysis to thick shells incorporating the effects of shear deformation and rotary inertia. New materials, many of them orthotropic, e.g., reinforced plastic mortar (RPM), are now becoming quite popular in lifeline systems and are slowly replacing conventional materials such as cast iron and concrete. A need for analyzing the dynamic behaviour of pipes made o f orthotropic materials was emphasized by Adman [4] during the Second U.S. National Conference on Earthquake Engineering (1979). However, there is very little work available on orthotropic buried pipes/cylindrical shells. Cole et al. [5] have performed a finite element analysis of buried RPM pipes. Recently, Singh et al. [6] have studied the axisymmetric response of buried orthotropic shells. But, since orthotropic materials have different strength and stiffness in di~erent directions, non-axisymmetric response of these shells becomes quite important. Datta et al. [7] have studied the non.axisymmetric response ofburied isotropic shells. To the best of the authors' knowledge there is no work available giving the non-axisymmetric response of buried orthotropic pipelines. The aim of work reported in this paper, therefore, was to study the non-axisymmetric dynamic response of buried orthotropic cylindrical shells embedded in a linearly elastic, homogeneous and isotropic medium. A thick shell theory formulation, which includes the effect of shear deformation and rotatory inertia, is presented. The decision for using thick shell model was taken on the basis of results presented in an earlier paper [8] wherein a comparison of thick and thin shell predictions have been discussed for this type of problem. The shell is assumed to be excited by a combination of P-, SV- and SH-waves. Results are presented, however, for the case of an incident plane longitudinal 149 0022-460X/88/030149+ 12 $03.00/0 9 1988 Academic Press Limited
150
UPADIIYAY
P. C .
AND
II. K .
MISIIRA
wave (P-wave) only. The flexural mode (n -- I) response has been compared with that of the axisymmetric mode ( n = 0 ) and it has been found that, depending upon the soil condition and apparent wave speed, shell deformation in the flexural mode may be even greater than in the axisymmetric mode. Thus, for orthotropic shells, non-axisymmetric response assumes considerable importance. The shell response is found to be significantly influenced by the variations in the shell orthotropy parameters, as well as by the soil conditions around the shell. 2. BASIC EQUATIONS AND FORMULATION Consider an infinitely long thick orthotropic cylindrical shell of mean radius R and thickness h embedded in a linearly elastic, homogeneous and isotropic infinite medium. The shell is excited by a seismic wave (a combination of P-, SV- and SH-waves) of wavelength A ( = 2,-'r/~') and speed c(= t o / f ) moving along the shell axis. Let a cylindrical-polar co-ordinate system (r, O,x) be defined such that x coincides with the axis o f the shell. In addition, if z is measured normal to the middle surface of the shell,
z=r-R,
-h/2<~z<~h/2.
(l)
Equations governing the non-axisymmetric motion o f a cylindrical shell can be derived by following reference [9]. In a matrix equation form they may be written as
[L](U}+(P*} =0,
(2)
where [L] is a ( 5 x 5 ) matrix operator with elements 9)2
O2
L,, =
[E',,
D'\
a2
R:l(h+ I/ R:)
t.,2 =-t.2, = -
(h+UR2)+kR 2
k~,
L,,=-~.~,=
~
k~
_ _
Lo.~= - L 4 1 -
2
Eppo, a R Ox'
-
( h + U R 2) ~ , i)
Lt5 = Lsn = k.~G.,:h "~x'
a2 [E'p D'\ ,32 : a~ L2z = G,olz 72+0.x ~-~'+ - ~ ) - ~ - (koG:o/ g :)(h + I / R 2) _ ph c~t""
G,,I a2
D' ,')"
(
_.?)
R,.)02+ k2o
L_.~=Lj2=--ff- ax 2 !
(h+l/R2)-pl/R~t2,
a~
L 2.~= L.~2= -~ ( Epvo, + G,o h ) ,'Ix ;JO'
L2s = L~2 = O,
a: D' a', L33=G,ol~x'.-~ R: aO~ k'oG:o(h+ I / R 2 ) - P I., ' L34 = L4~ = 0,
l-3s = I-$3 = a-"
G,JI
.
(Dvo, + G, oh3/12) ~x aO" 1 \
.,
i)"
,,}~
,"
BURIED
ORTHOTROPIC
CYLINDRICAL
D ~2
G,ol ~2
L45 = L54 - R 3x 2
i)2
9
L55=Dc~x24
R 2
R
.~
t3t 2 '
k~G~.h-pl
c~0 2
-
151
pl 02
R 3 ~02
G~ol i)2
SHELLS
..
~.2
i)t 2,
and
{U}=Lw
v
~'o
u
~,,,jT
with w, v and u as the displacement components of the middle surface of the shell in the radial, axial and tangential directions respectively, and ~b~ and ~o being the angles o f rotation of a straight line initially normal to the middle surface in the axial and tangential directions respectively. The elements of [ P * ] are given by
P*=(1
h \
,
e,=hp,, 2
where o'*=, or*0 and or* are the stresses at the outer surface o f the shell (i.e., at z = + h / 2 ) . Different constants appearing in the expressions for the Lis are defined as
E,h 9 , Ep - 1 - V.~oVox I = h3/12
and
,_ Eoh Ep - 1 - v,,ovo~' k~ = ko = r r / x / ~
h2 D = E,-~,
D'=
h2 E
t -
P 12'
(shear correction factors)
where E,, Eo are the Young moduli, v,o, vo, the Poisson ratios, G~o, G~: and G:o the shear moduli and p is the desntiy of the shell material. For the evaluation of [P*], the or* are determined in terms of the incident and scattered fields in the surrounding ground. The total displacement field in the ground is written as u =
u('>+u (~),
(3)
where superscripts i and s represent incident and scattered parts respectively. By solving the wave equation in the surrounding infinite medium, components of the incident and scattered fields can be written as
U(ri)= [{ yl ~,(yr/ R )}dt -{flS l" ( 6r/ g )}d2 + {n( g / r)l.( Sr/ g )}d3] x c o s nO cos (~x-tot)
U~o')= [-{n( R / r)I.( yr/ R )ld, + {n( R / r)fl l.(~Sr/ R )}d2-( 8 IL (,Sr/ R )}d3] xsin nO cos (~:x- tot)
,,!?= [-(131,,(r,/R)}d,
+ {6"-I.(ar/R)}d2] cos nO sin (~x - tot),
(4)
ut/~= [{ y K " ( yr/ R ) } A - { f l S K" ( ar/ R )} B + {n( R / r)K.( Sr/ R )}C ] x c o s nO cos ( ~ x - tot),
u~'~= [-{n( R / r)Kn( yr/ R )}A + {u( R / r)fl K.( 6r/ R )} B - { 6K" ( Sr/ R )}C ] x sin nO cos (~x - tot), u!,:' = [ - { f l K . ( y r / R ) } A + { 6 2 K . ( S r / R ) } B ]
cos nO sin (~x-tot).
(5)
152
p.c.
UPADHYAY A N D B. K. MISHRA
where I. and K . are the modified Bessel functions o f the first and second kind respectively, and T = f l ~ / 1 - ( c / c , ) 2, 8 = f l , / l - ( c / c z ) 2, ~ = ~ R = 2 z r R / A , and c=oJ/r c, and c2 are, respectively, the speeds o f p r o p a g a t i o n o f longitudinal and shear waves in the surrounding m e d i u m , d,, dz a n d d 3 d e p e n d on the intensities o f the P-, SV- and SH-parts o f the incident wave, respectively, and have the dimension of length. A, B and C are arbitrary constants. (') denotes differentiation with respect to the a r g u m e n t s of the Bessel functions. N o w u (i~ and u (~) being known, by using the s t r e s s - d i s p l a c e m e n t relation for the surrounding m e d i u m o',., o-,o and o',x at r - - R + h / 2 can be obtained. Thus [ P * ] is determined in the form {P*} = (1 + 17/2)(tz/R)[g][l]{m}~
(6)
where [g], [I] and {m} are (5 x 5), (5 x 6) and (6 x 1) matrices respectively. Terms of these matrices are as follows: g,, = cos nO cos (~:x - tot),
gz2 = sin nO cos (~x - tot),
g44 = cos nO sin ( ~ x - tot),
g55 = g44,
1,, = ( 2 - r2)E2I.(cq) + 2V2I~(a,),
g . = g22, when i # j ,
go = 0
1,2= -2/382I ~(a2),
1,3 = [ 2 n / ( 1 + IT/2)]8 I ' ( a 2 ) - [2n/(1 + IT/2)2]In (a2),
i,4=(2_r2)e2K.(ot,)+ 2T2K,~(ot,),
!,5= -2/38 2K.(ct2) .,, ,
1,6 = [2n/(1 + IT/2)]8 K:,(a2) - [ 2 n / ( 1 + I7/2)2]K.(a2), 12, = [2n/(1 + 17/2)"] { I . ( a , ) - a , I ' ( a , ) } , 122 = [ 2 n / ( l + 1]/2)z]/3 {o~21"(o~2) - I . (a2)}, 123 = -c52I ~(a2) + [8/(1 + 17/2)]I'(a2) - [ n 2 / ( 1 + 17/2)2] I.(a2), 124 = [2n/(1 + 17/2) 2] { K . ( a , ) - a , K ' ( a , ) } . 12s = [2n/(1 + 1712)2]/3{a2K'(a2)" K . ( a 2 ) } . 126 = - B2K~(ot2) + [81(1 + IT/2)] K'~ (a2) - [n2/( 1 q- t7/2)2] K. (a2).
13s= (h/2)lzs 14, = -2/3711.(a,),
f o r j = 1 to 6, 142 = ~(2/32- r 2 E 2 ) I ~(Cr
l.~3=[n/(l+l-~/2)]~I.(ot2), I,~ = ~(2/3 ~ - r % 2 ) K ' ( a 2 ) ,
15~=(h/2)14s
forj=lto6,
144 = .2/3TK:,(ot,),
!,~ = [,,/(1 + 17/2)]13 K o ( . : ) , and{m}=[d,
de
d3
A
B
Clr.
In these expressions !~= h / R and p. is the shear m o d u l u s o f the surrounding m e d i u m , ~.2= c2/c~=2(1 _ v m ) / ( 1 - 2 u . , ) , v,. is the Poisson ratio o f the surrounding m e d i u m ,
e=~c/c,,
a , = (1 + i~/2)3,,
a2 = (1 + 1T/2)8,
l'~(cr;) = ( n / a , ) I . ( c ~ / ) + I . + , ( a i ) . I ~(a,) = [1 + (n2/a~) - (n/ct ~)]I.(ai) - (1 / c~,)I.+,(a,). K~(a;) = (nla,)K~(a,) - K . + , ( a , ) , K~(c~,) = [ 1 + ( n 2 / a 2) - ( n / a ~ ) ] K . (al) + ( 1 / a , ) K . + t ( a , ) . If the shell displacements are now a s s u m e d to be of the form
W=WoCOSnOcos(~x-wt),
U=vosinnOcos(~x-tot),
Sx = $xo cos nO sin (~x - wt),
U=UocosnOsin(~x-tot),
~o = ~oo sin nO cos (_Ex- tot)
(7)
BURIED ORTIIOTROPIC C Y L I N D R I C A L SttELLS
153
and these along with [ P * ] from equation (6) are substituted into equations (2), a set of five simultaneous equations is obtained. Three more equations are obtained by enforcing the b o u n d a r y condition that the displacement be c o n t i n u o u s at the outer surface of the shell: i.e.,
o+(h/2)~,o=(z~o'~+i,';b.=,,+h/2,
w = (u'." + .<.'~).=,~+h/2,
t, + (h/2)e,x = (u~'l + u?l),= R+h/2.
(8)
Thus a total o f eight equations is obtained. These eight equations, when simplified, give the final response equation, which may be put into the form [ Q] { Uo} = d, { F'} + d2{ F 2} + dj{ F3},
(9)
where [ Q] is an (8 x 8) matrix and {F~}, {F 2} and {F j} are (8 x 1) matrices, the elements being as follows:
172112) + ,f'/2,
O,, = - k~13217- k 20n 2(r/.,/,72) #7( l + 172112) - IT(n,/r/2) N ( 1 +
Q,2 = h i - k~(r/.d r/glT(i + 172/12) - (r/,/r/D NIT(1 + IT2/12), 0,3 = n[(71,/~,) N(I7 ~/6) + kl(o.,/~)(2 + 17~/6)]>
0,4 = -(vo
Q,5= 2k~O,
Q~,=(l+lT]2)fdts,
Qt6=(l +lT/2)fil,,,
Q,8=(1+17]2)~1,6,
02,=0,2,
022 = - ( r / 3 / r / : ) IT/32- (r/,/r/z) Nn 217( 1 + IT2/12) - k~(r/,/n2) + .0 2, 0-
=
- ( r/3/n2)(17"/6)13 2 + ( 71,/r/9 Nn2(lTZ/6) + k2o(r/J 779(2 + h2/6) + (/T/6)-O 2,
O,,=-n[(Z,o
Q26=(l+#T/2)4h,,
02~=0,
Q28=(1+i'1]2)ft!26,
Q3,=Q,~,
Q32=Q23,
Q33 = -(rh/r/t)(#]'/3213) - (r/,/r/2) Nn 2(1713)- k:o(r/4/r/2) [ (41 #7) + (/~13)] + ( a 213),
Q.= -n[(~'o,N/r/9+(r/dr/9](g/3)B,
Qj, = 0, QJ7 = Q27,
QJ8 = Q2,,
Q., = Q , . ,
Qj6 = Q ~ ,
Q.~ = Q ~ ,
Q.J = QJ,,
Q~4 = - ( N/r/:)l~fl 2 _ n2( r/j/r/~)l~( 1 + / ~ : / 1 2 ) + / 2 ~, Q~s= -(N/r/~)(l~2/6)fl2+(r/j/r/~)n2(l~2/6)+(h/6)12 ~,
Q5,
=
Q~6=(l+h/2)~i~,
Q,,=(l+lT/2)~tl~,
Q,s=(l+g/2)l~l,6,
Qis,
Q~j QJ~,
Q~2 Q2s, =
=
Q5. Q,~, =
Qss = - ( N/r/~)(l;/a)fl ~- n:( Oj/ r/~)(/;/3) - (4k]/ I~) + (/2 ~/3), Q~6 = Q~6,
Qs7 = Q~7,
Q61 = 1,
Q6~ = fl6K'(a.),
Q66 = - y K ' ( a , ) , Q7~ = 0,
Q~s = Q.s,
Q7= = QTa = l,
Q68 = - [ n / ( 1 + 1;/2)]K.(a,),
Q8~ = Qss = l,
QTs = 8K'(a~),
Q86 = f l K . ( a i ) ,
F~ = - ( l + l~/2)fil,~, F~ = F~,
Q76 = [ n / ( 1 + 1~/2)]K.(at),
Q74 = Q7~ = 0,
Q77 = - [ n / ( 1 + l;/2)]flK.(a=), Qsi = Q~2 = Q~3 = 0,
Q6, = Q63 = Q6~ = Q~5 = 0,
Q87 = - 6 Z K . ( a ~ ) ,
F~2=- ( l + i~/2)fil..,,
F~ = - ( 1 + 1;/2)fil~,,
F~ = F~,
Qs8 = 0,
154
P. C . U P A D H Y A Y
F~-- yl'(aq),
AND
B. K. M I S H R A
F~= " [ . / ( 1 + #7/2)]I.(.,),
F2t = - ( l + lT/2)~l,2,
F~= - ( l + l-i/2)fd,,_,
FJ = - ( 1 + 17/2)fil,2,
F~ = F~,
F~ = -t3 I.(,~,).
F~ = F~,
F~ = [n/(1 + 17/2)],8I.(0t2),
F~] = -/36I'(0t2),
F; = - { l + & 2 ) ~ t , , ,
F~' = - 0 + tZ/2)~t:,,
F l = -(1 + h/2)fa,,,
F33 = F 3,
F63 = In/(1 +/T/2)]I. (c,2),
F~ = a2l.(cr2),
F~
, q = FI,
-6I'(ae2) ,
=
F ] = 0.
Here 77, = Eo/ E~, ~2 = G . , / Ex, r/3= G.,o/ E,, 71~= G:o/ E,, N = 1 / ( 1 - V.,oVo.~), 12 = /, / G~.., f22 = 17fir2f12e'-]13, e = c / c t and 13= p,,/p, p,, being the density o f t h e surrounding medium. The response vector { Uo} in equation (9) represents {Uo}=[Wo
/3o
(~oo)
Uo ( h ~ o )
a
B
C
JT.
Results are presented for the case d2 = da = 0 and thus taking the incident wave as totally composed o f P-waves. For this particular case equation (9) becomes [o]{O} ={F'},
(10)
where h
h
A
B
. e l T, "* I .I
W, V and 0 are the non-dimensional amplitudes of the middle surface of the shell in the radial, tangential and axial directions respectively.
3. RESULTS AND DISCUSSION Results are presented for a transversely isotropic shell with r-O as the plane ofisotropy. Thus Eo = E:, G.~: = Gxo, V~o = v:s, V~o = v~:, and G:o = Eo/2(1 + Vo:), and consequently 73 = r/2 and r/4 = G:o/E~ = "qJ2(1 + Vo:). Further, vs_.= V:,o= 0.3 has been assumed, vo~ is given by the relation E o / E , = poJV~o or v0, = ~tVxo. Three different values of 7, and r/2 were used in the calculations rtt =0.05, 0.1 and 0.5; r/z=0.02, 0"05 and 0.1. These are expected to cover a wide range of orthotropic materials. Results are presented for two soil conditions--very soft (/i = 0.001, v,,, = 0.45) and hard ( f i = l . 0 , v,,,=0-25). A constant value of I]=0.05 and 13=0.3 has been assumed throughout. The flexural mode 01 = 1) and the axisymmetric mode (n = 0 ) responses are compared. Figures 1-6 show the variations of the radial displacement (1~r) with the nondimensional wavelength parameter / 3 ( = 2 r r R / A ) . Figures I-5 correspond to soft soil (fi =0.001) whereas Figure 6 is given for the hard and rocky ground condition. The first two figures, 1 and 2, drawn for a very low apparent wave speed (E= 1.001), indicate that the variations in orthotropy parameters r/, and 77., affect the radial displacement o f the shell quite significantly under the soft soil. They also indicate that, excepting for r/t = 0"5 in Figure 1 and r/2 = 0.02 in Figure 2, the axisymmetric mode (n = 0) is dominant in comparison to the flexural mode (n = 1). When the wave speed is increased to C'=2.0, I~z vs. fl plots for the soft soil are shown in Figures 3 and 4 with, respectively, 77, and rt2 as parameters. The orthotropy parameters rh and ~72 are still found to have a significant
B U R I E D O R T H O T R O P I C C Y L I N D R I C A L SIIELLS I
I
[
]55
i
o~
oO
It
On
oo
oz
04
06
08
nO
Figure 1. Radial displacement (l~') versus wavelength parameler (13) for .6. = 0.001 and ~ = 1.001 with rl 0 as parameter,
v,,, = 0 . 4 5 ,
r/~=O.05.
--,
, =0;
--
'
- , n = I.
i
'
u
,
I
'
I
'
005
9
Ol
io
oo
B
Figure 2. Radial displacement (IF') xersus xva~elenglh parameler (/3) for ~ =0-001 and E-- 1.001 with ~z as parameter. ,,,, =0.45. ~, = 0 . 1 . Key as Figure I.
02
'
I
'
i
'
i
'
I
'
|
~,, 0 - 0 ~
oo
f/~/-"
013
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156
P. C. U P A D t I Y A Y A N D B. K. M I S I I R A
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Figure 4. Radial displacement ( I t ' ) versus wavelength parameter (/3) for/.i = 0.001 and E = 2.0 with rlz as parameter, v,, =0-45, r/, = 0 - I . Key as Figure I.
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BURIED
ORTilOTROPIC
CYLINDRICAL
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157
influence on the radial displacement. However, unlike Figures l and 2, here flexural mode displacements are always greater than their corresponding values in the axisymmetric mode. When ~ is further increased to 5.0, as shown in Figure 5, no such general trend is observed. While for .01 = 0.5 the radial displacement is always higher in the flexural mode, for rh = O"1 and 0.05 it depends on the value of/3 as to which mode will give the larger displacement. A common feature o f all the figures (I-5) is that, in general, variations in "01 and "02 produce more changes in the axisymmetric mode than in the flexural mode. In the flexural mode changes are, in fact, negligible up to 13 -~0.6 or so. For bringing out the effect of the surrounding soil condition, fie' for hard and rocky ground (/.7.= !.0) has been plotted in Figure 6, with "01 as parameter. A comparison of Figures 6 and 3 shows that changes in the orthotropy parameters make very little difference in )7V in the hard soil as compared to the differences observed in the soft soil. In fact, in the axisymmetric mode absolutely no difference is realized; only in the flexurai mode some changes are observed. It may be noted that a different trend was obtained in the soft soil (see Figure 3) where changes were more prominent in the axisymmetric mode than in the flexural mode. Further, under the hard ground condition, the dominance of the n = 0 or n = ! mode becomes wavelength dependent. Up to/3 =0.5-0.6 the axisymmetric mode prevails, and beyond that the flexural mode becomes dominant. Axial displacement ( O ) results have been presented through Figures 7-11. Figures 7-10 correspond to soft soil and Figure 11 is given for the hard soil. As seen from Figures 7 and 8, at low apparent wave speed ( ~ = 1.001) axisymmetric mode is found to be dominant. However, at an increased wave speed (? = 2.0), excepting for very small /3 (large wavelengths), the ftexural mode (n = 1) gives higher values of /.St as shown in Figures 9 and 10. In all the four figures (7-10) the maximum value of O in the axisymmetric mode (n = 0) is found to occur at very large wavelengths (i.e., low fl), while in the flexural mode (n = 1) the peak value of O is observed a r o u n d / 3 = 0 . 6 . At still higher wave speeds (say, ,~= 5.0), qualitatively similar results were obtained. Figure 11 shows the axial displacements of the shell in hard soil. It is seen that the variations in orthotropic parameters do not produce significant changes in the axial
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Figure 7. A x i a l displacement ( U ) versus wavelength parameter (~)) f o r / ~ = 0.00t and ~ = 1.001 w i t h "~t as p a r a m e t e r , v,,, = 0 . 4 5 , r/2 = 0 . 0 5 . K e y as F i g u r e I.
158
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displacement (U) of the shell in the axisymmetric mode. Changes are prominent in the flexural mode only. Further, in hard soil, again a mixed trend is seen where the axisymmetric mode remains dominant only up to /3---0.6, and f o r / 3 ~ 0 . 6 the flexural mode displacement becomes larger. These observations are similar to those for the radial displacement in Figure 6. Although rh and 7-/_,are equally effective in producing changes in the radial displacement, for controlling the axial displacement r/_, is found to be more effective than r/t. Though for smaller values of/:3 the axial displacement in the axisymmetric mode is always higher than that in the flexural mode, in soft soil and at low wave speed (Figures 7 and 8) it becomes many times larger. From the above results it is evident that, unlike the situation for isotropic shells, the non-axisymmetric response of orthotropic shells is of considerable importance. This is
BURIED
ORTHOiROPIC
64
CYLINDRICAL
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because o f the fact that the deformations in the flexural m o d e may be even greater than those in the axisymmetric mode. By suitably adjusting the different elastic moduli o f the orthotropic shell, it s h o u l d be possible to control the shell displacements appreciably, particularly in the soft soil. When the incident wave is o f larger wavelength (smaller/3), the axial displacement in the axisymmetrie m o d e is always higher than in the flexural mode. Therefore for the larger wavelengths the axisymmetric mode is more relevant, as the most c o m m o n cause o f pipeline failure in seismic environment is the excessive axial deformation [ 1]. However, when the incident wave is o f smaller wavelengths (less than twice the circumference o f buried pipeline, i.e.,/3 > 0.5), the non-axisymmetric response b e c o m e s quite important.
4. CONCLUSION On the basis o f the results presented, the following conclusions can be made. (1) The flexural m o d e response assumes considerable importance in soft soil conditions and at higher a p p a r e n t wave speeds. (2) Variations in the shell orthotropy parameters are more effective in influencing the shell displacements in soft soil than in hard soil.
160
P. C. U P A D t I Y A Y A N D B. K. M I S H R A
(3) The o r t h o t r o p y p a r a m e t e r s r/, a n d r/2 are e q u a l l y effective in c o n t r o l l i n g the r a d i a l d i s p l a c e m e n t , but ~/z affects the axial d i s p l a c e m e n t m u c h m o r e than r/,. (4) F o r i n c i d e n t waves o f larger wavelengths, the axial d e f o r m a t i o n in the a x i s y m m e t r i c m o d e is a l w a y s h i g h e r t h a n in the h i g h e r m o d e s .
REFERENCES 1. G. E. MULESKI, T. ARIMAN and C. P. AUMEN 1979 Journal of Pressure Vessel Technology 101, 44-50. A shell model of a buried pipe in a seismic environment. 2. S. K. DATrA, A. H. SHAI-t and N. EL-AKILY 1982 Journal of Applied Mechanics 49, 141-148. Dynamic behaviour of buried pipe in a seismic environment. 3. S. CIIONAN 1981 Journal of Sound and Vibration 78, 257-267. Dynamic response of a circular shell imperfectly bonded to a surrounding continuum of infinite extent. 4. T. AR1MAN and G. E. MULESKI 1979 Proceedings of the 2nd U.S. National Conference on
Earthquake Engineering, Earthquake Engineering Research Institute, Stanford Universit); California, 643-652. Recent development in seismic analysis of buried pipelines. 5. B. W. COLE, C. J. RITTER and S. JORDAN 1979 in Lifeline Earthquake Engineering--Buried Pipelines, Seismic Risk and Instrumentation, ASME (Editors T. Adman, S. C. Liu and R. E. Nickell). Structural analysis of buried reinforced mortar pipe. 6. V. P. SINGII, P. C. UPADtIYAY and B. KISHOR 1987 Journal of Sound and Vibration 113, 101-115. On the dynamic response of buried orthotropic cylindrical shells. 7. S. K. DATTA, A. H. SI-IAH and N. EL-AKILY 1981 in Seismic Risk Analysis and Its Applications to Reliability-based Design of Lifeline Systems (Editors M. Kubo and M. Shinozouka). Tokyo: Gakujutsu Bunken Fukyu-Kai. 8. V. P. S I N G H , P. C. UPADiiYAY and B. KISHOR 1987 Journal of Sound and Vibration 119, 339-345. A comparison o f thick and thin shell theory results for buried orthotropie cylindrical shells. 9. 1. MIRSKY and G. HERRMANN 1957 Journal of the Acoustical Society of America 29, 1116-1123. Nonaxially symmetric motions of cylindrical shells.