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On the dynamic response of buried orthotropic cylindrical shells
Journal of Sound and Vibration (1987) 113(1), 101-115
ON THE DYNAMIC RESPONSE OF BURIED ORTHOTROPIC CYLINDRICAL SHELLS V. P.
SINGH,
P. C.
UPADHYAY AND
B. KISHOR
Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi-221 005, India (Received 14 November 1985, and in revised form 25 March 1986)
This paper deals with the dynamic response of an orthotropic cylindrical shell, a pipe, buried underground and subjected to seismic excitation. A thick-shell model of the pipe including shear deformation and rotatory inertia has been used. The shell is assumed to be perfectly bonded to the surrounding medium (soil), of infinite extent. Only the axisymmetric response due to an incident compressional wave has been investigated. Effects of the shell orthotropy on its response characteristics have been illustrated by changing the non-dimensional orthotropy parameters of the shell over a wide range. Results have been obtained for different soil conditions-hard (rocky), medium and soft. It is found that the orthotropy parameters do not influence the shell response equally. Even the degree of their influence is found to be highly dependent on the rigidity of the surrounding soil. 1. INTRODUCTION
Buried pipelines perform a vital role in conducting and! or distributing energy, water, communication and transportation in today's world. However, nearly until the 1970's, underground utilities and transportation systems remained somewhat neglected in respect to consideration of earthquake effects upon them. It was only after the reporting of heavy damage of buried gas, water and other pipelines in some recent earthquakes that researchers realized the importance of studying their dynamic response. During the last seven to eight years a number of papers have appeared concerned with different aspects of the problem. Most of the published papers on the dynamic response of underground pipelines have been concerned either with assessing the role and development of soil parameters or proposing better analytical models for the pipe or the surrounding soil. A good account of these works can be found in review articles [1-4]. In efforts to propose a better model and to explain the nature of pipe failures, Ariman et al. [2,5] treated the pipe as a buried shell using Fliigge's shell theory. They concluded that a beam model of the pipe, as had been used by several investigators [6-8], cannot predict the buckling failures of the pipes which are evidenced as the most frequent cause of failure under seismic loading. Chonan [9] and Datta and his co-workers [10] have also used the shell theory in their work. All the work reported so far, however, has been concerned with isotropic and homogeneous buried shells only. During the Second U.S. National Conference (1979) on earthquake engineering Ariman and Muleski [11] emphasized the need for analyzing the dynamic behaviour of pipes made of orthotropic materials. He pointed out that new materials were becoming quite popular in pipeline systems, and were slowly replacing cast iron and steel. In fact, reinforced plastic mortar (R.P.M.) has already found acceptance as a substitute for cast iron and steel in a number of applications such as potable water, irrigation and sewer lines. However, other than a paper by Cole et al. [12], who performed WI 0022-460X/87/040101 + 15 $03.00/0
© 1987 Academic Press Inc. (London) Limited
102
V. P. SINGH, P. C. UPADHYAY AND B. KISHOR
a finite element analysis of buried R.P.M. pipes, to the best of the authors' knowledge, no work has been reported on the effect of orthotropy on the dynamic performance of underground cylindrical shells. On orthotropic cylindrical shells as such, of course, a great deal of work has been done. The aim of the work reported in what follows, therefore, was to study the effect of orthotropy on the dynamic behaviour of a buried infinite cylindrical shell when it is disturbed by a longitudinal wave moving along its length. Results have been obtained for different soil conditions around the pipe-hard (rocky), medium and soft (sandy). It has been found that the orthotropy parameters do not equally influence the shell response. It has also been found that, depending upon the soil conditions, the effects of orthotropy may be quite different and vary significantly. Results of this investigation are expected to be helpful in designing composite pipes, tunnels lining and underground storage cylindrical tanks, etc., in that the orthotropic material parameters can be selected within a desired range for the purpose of limiting and controlling the shell displacements under different soil conditions. Only axisymmetric motion of the shell has been considered. We have taken this simplified version of the problem of pipeline motion due to seismic excitation because our main object here is only to look into the relative influence of orthotropic material parameters on the shell response, when it is placed under different soil conditions, other factors remaining invariant. In the formulation of the problem an approach similar to that of reference [10] has been followed, wherein the tractions in the dynamical equations of the embedded shell are expressed completely in terms of the incident and the scattered field displacements in the surrounding infinite medium. 2. BASIC EQUATIONS AND FORMULATION
Consider an infinitely long thick orthotropic cylindrical shell of mean radius Rand thickness h lying in a linearly elastic, homogeneous and isotropic infinite medium. The shell is excited by a longitudinal wave of wavelength A (= 2'Tr1 g) and speed c moving along the axis of the shell in the medium. Let a cylindrical-polar co-ordinate system (r, fJ, x) be defined such that x coincides with the axis of the shell. The stress-strain relations of the shell material is assumed to be of the form (1)
where G"ij and el} are, respectively, the components of the stress and strain, and Ex l , E o I , E p i and G X l are the four independent moduli. The equations governing the axisymmetric motion of such a shell can be written, with some modifications, from reference [13] as,
Do)
2
2
1 ( a , -a ] w+ [ 0 - a ] .1, [ -R-2 E o + -R 2 +o--p xax2 at 2 x ax w» 2
a ] w+
[ R ax
p
a] ax U+P*I =0 '
--
(2)
2
D xax2 --G [-0x-axa]w+ [a E --.2:_
[ER
---=----
i ] .1, + [ D a 2 I' i ] u+p*=O -1'x at2 '1':< R ax R at2 2,
[D---=---a a 2
2
]'
R ax 2 R
2
2
+ [ Ex -a-2 p 'ata-2] u+p*=O at 2 '1':< 3 ax ]
.1,
(3)
(4)
in which t is the time, u and ware the displacement components of the middle surface in the axial and radial directions, respectively, and t/Jx is the angle of rotation in the axial direction of a straight line which is initially normal to the middle surface.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
103
The stiffness parameters and other constants in equations (2)-(4) are given as
I' = Pol,
(5)
where k is the shear correction factor taken as 7T' / v'f2, and Po is the mass density of the shell material. are the traction components per unit area exerted on the shell by the surrounding medium, and are given by
pr
pf = (l + h/2R)u~n
pf =
(h/2)p~,
pf = (1 + h/2R)cr':x,
(6)
in which crt and cr~x are, respectively, the normal and the shear stresses generated at the outer surface of the shell (i.e., at r = R + h/2) due to the motion of the surrounding medium. The displacement d(r, e, x, t) at any point in the surrounding medium satisfies the equation of motion dV(V . d) - c~V x V x d == a2 d/ at 2 ,
(7)
where, cl={(A+21L)/Pm}1/2 and C2=.JIL/Pm are respectively the longitudinal and the shear wave speeds depending upon the values of the Lame constants A and j.L, and the density, Pm, of the medium (surrounding soil). Because there is a scatterer (shell) lying in the medium, the displacement d at any point in the medium consists of two parts: the incident field d(1) created due to the longitudinal wave moving along the shell and the scattered field des). For an axisymmetric motion, the radial and the axial components of del) and d(s) are written as (8)
where do is a constant depending on the incident wave intensity and having the dimension of length, and d~S)
= [A(ctl R)K 1(c\r/ R)+ig· BK\(c2r/ R)]
d~s)
ei€(x-cl),
= [Ai~"Ko(c\ r] R) + B(C2/R)Ko(c2 r/ R)] ei~(X-ct),
(9)
where I" and K" (n = 0, 1) are the modified Bessel functions of the first and the second kind, respectively, and {3 = gR = 27T'R/ A.
(10)
When the arguments of I" and K" are imaginary (for c] Cl > 1·0), these functions become J and Y respectively; K" can, of course, be alternatively expressed in terms of Hankel functions. It may be noted that d'", as given by equations (8), satisfies equation (7) and represents a travelling wave with wavelength A = 27T'/ g moving along the axis of the pipe with speed C = w / g. In the case of a plane longitudinal wave moving at an angle rp with the axis of the shell, equations (8) will represent the axisymmetric components of that wave and the apparent wave speed of such a wave along the shell axis will be (ctlcos ep). The constants A and B in equations (9) are determined by applying the condition of continuity of displacements at the outer surface of the shell: i.e., /I
/I
d(s» ( W ) r:R+h/2 -- (d(i)+ r r
.
r-R+h/2,
A perfect bond between the shell and the surrounding ground has been assumed because our main aim, here, is to study the effect of orthotropy. Moreover, it has been reported
104
v.
P. SINGH, P. C. UPADHYAY AND B. KISHOR
[8] that, excepting under very specific soil conditions, the slip does not have much influence on the stress, etc., in the pipe. Now, upon assuming the shell displacement in the form
u = u(r) eieCx-cl) = {uo + (r - R)r/Jx} eiecx-cll, (12) and using the two boundary conditions (11), the constants A and B in equations (8) can be determined in terms of Uo and wo, which are the displacements at the middle surface of the shell. Finally, the scattered field displacement comes out as d~S) = (11 D)[{(cl cZ):K.O( a 2)K I (ret! R) - f:lzKo(at)K] (rczI R)}wo,
+ i{K](al)K1(rc2/ R) - K I(a2)KI ( rcll R)}(c1f:l)uo] eieCx-cl), d~s) = (1/ D)[ {(C1C2)K1 ( al)Ko(rc21 R) - f:l 2KI(a2) Ko(rc11 R)}uo
+ iC2f:l{:K.o(al):K.o(rc21 R) - :K.o(a2) Ko(CI r] R)}wo] eiL'Cx-ct), D = C1c2Kl(a]):K.o(a2) - f:l 2:K.o(at)Kt(a2), at
= c1(l + hI2),
a2 = cil + h/2),
h = hi R,
Wo = wo- doctIt(al)'
Uo = [uo+(h/2)~x -i,BdoIo(a])],
(13)
Now, with d(i) and dCs) fully known from equations (8) and (13), the tractions given by equation (6), can be determined by calculating
(14)
rt,
as
* _ Ci) Cs» r-R+h/2-_ [ A {(ad; + d; + ad;)} +2JL{ad;}] O"rr-(lTrr+O"rr , ar r ax ar r=R+h/2 >I<
IT rx
= (O"~'I) +O"rx(s) )r-R+h/2 = [
JL
{(ad; ad;)}] -a-+-ax
r
r=R+I./2
,
(15)
where, d; = (d~l) + d~s» and d; ::::: (d~)+ d~s» are the total radial and axial displacement fields, respectively. Hence, after a few steps of simplification, one gets
pt =;; (JL/ R)[PlI wo+ (h/2)Pt2~x + Pt3UO] ei,Hx-cl) - (1 + h/2)(do/ R)JL[(x 2,/ - 2f:l2)1o(at) +2CiI I(al)/ all eUCX-CI), pt::::: (JLIR)[P31 Wo + (h/2)P32~X + P33UO]
eif(x-cl)
+ 2(1 + h(2)(JLI R)[ido(ct,B)I1 (reV R)] ei
(1
[c
2 ]
.( ii\
PI2=-P21"",PI3=-Pn=-11+"2}f:l
( ii\[-
P22=P33=P23=P32= 1+"2} ci
[-y2x2Kt(a2):K.o(at). ] D -r-2, y2x2KI(al)K](a2)] D .
(17)
105
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
pr
When these values of and the displacements u and w from equations (12) are substituted into the shell equations (2)-(4), after some lengthy algebra, the simplified final equations for the determination of {Z} = {wol do, (hI2)rjixl do, aol do} are put into the matrix form [[ Q] - ~[P] -
n 2 [ N ]]{Z } = -,l[P]{F} + MH}(1 + h12),
(18)
where n 2 = (PohRI Gxl)a? = X 2 f3 2 ciifLii is the non-dimensional frequency, in which ~ = JLIOxl and ii = PolPm are, respectively, the modulus of rigidity and the density of the surrounding soil (medium) measured in terms of the corresponding values for the shell material, and c = c] C 1 is the apparent wave speed. The matrix [P] is as given in equations (17), and the components of other matrices [Q], [N], {Z}, {F} and {H} are given by
(-2if3e) (ih13 ~:) ~ (f2 JI2 f32 + k 2) (!h2~) h 'T/3 Q23 (ii~:)
(h~: (1 + ~;) + iie(32) [Q]=
-Q12 -Q13
{Z} =
W} {UV
,
O
=
{WOld } ,Jix_hI2do
(19)
uoldo
and (20) are the orthotropy parameters. It should be noted that the right side of equation (18) represents the excitation in which the matrix [P] depends on the geometry of the pipe, the material properties of the surrounding medium, and the wavelength and wave speed of the excitation.
3. RESULTS AND DISCUSSION
The results to be presented here were obtained mainly with the aim of bringing out the effect of varying the orthotropy parameters '1/1, 'T/2 and 'T/3 on the radial and the axial displacements of the shell under different soil conditions and also at different wave speeds c( = c] c1 ) . Ranges of orthotropy parameters have been selected on the basis of orthotropic constants given for different kinds of composites in reference [14]. For each 1]1' 'T/2 and 1]3, three values have been taken: 'T/I=0'05, 0·50 and 1·0; 'T/2=0'005, 0·05 and 0'10;
106
V. P. SINGH, P. C. UPADHYAY AND B. KISHOR
113 = 0'01,0'02 and 0·20. This is expected to cover a wide range of orthotropic (composite) materials. In each study of the effect of a variation in one orthotropy parameter, the other two have been kept constant. To have a comparative picture and visualize the effect of orthotropy, results for the equivalent isotropic cases have also been plotted, denoted by the label (ISO). Since the modulus of rigidity of a soft clay is nearly 1000 times less than that of hard rocks, to simulate all kinds of combinations of ground to pipe rigidity ratio ji has been varied from 0·01 to 10·0. The thickness to radius ratio of the shell, h, the ratio of the shell and the soil densities, p, and the Poisson's ratio of the soil, m, have been kept constant throughout: ii = 0'05, P= 0·30 and m = 0·25. Before proceeding to compute the response, critical (resonating) wave speeds were determined from the equation
[[ Q] - ji(P] - {J2[N]] = O.
(21)
Figure 1 shows the plots of such critical wave speeds versus wavelength (f3), with 111, 112 and 113 as parameters. It is found that real values for the critical wave speeds exist only under a very hard (rocky) soil condition (ji = 10·0). In soft (fl = 0·01) or medium hard (jl = 0·1 or 1'0) soil no real resonating wave speeds exist. This is why, except for one, all the curves drawn in Figure 1 are for jl = 10·0 only. For jl = 1·0, only for the combination of 111 = O'5, ""2 = 0·05 and 113 = 0·02 were real critical wave speeds found to exist, and then only over a limited range of f3 dotted line on top.
= 4 to
8, as shown by the double chain
0·58 r----,--,---,---....,....--,----,---.------,
---
\'0-<""""'P:=1-0 7'J "'0·50 ~ <, '1
t
/~ ....... ',0 ""'l!!
0·56
\\
0'5
\
l\..l
0'54
0·52
\
--
TJ2 =0,05'7'3 J "'0,02
.... ....
1) "'0-05 1
0'02
-_
_ - - .......
\\
"'
--
--_<::,,""0'01
.......
-........
-,
...... ........
1)
=0,2
. . . . . . """y
---.... .... -.... -----
--
0·50 I.-..._---'-_ _..L...._ _L-_....,.L_ _~_---l_ _-'-_----" 20 10 30 o ~ {3
Figure 1. Critical wave speeds (c) vs. wavelength (fJ) with 1/1,1/2 and 1/3 as parameters. ii=0'05, p=0'30, m = 0'25; fl = 10·0 except as indicated. - , 1/2 = 0'05, 1/3 = 0·02; - . - , 1/, = 0'50, 113 = 0'02; ---, 1/, '" 0,50, 1/2 = 0'05.
From Figure 1 it is seen that the orthotropy parameters '71 and '72 have practically no effect on the critical wave speeds. However, as shown by the dotted lines, 113 is found to have a significant effect in altering the critical values of c. As "TJ3 increases the critical c values decrease.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
107
Further, looking at the values of the critical wave speeds indicates that even under very hard soil conditions the shell cannot resonate if it is excited by a plane longitudinal wave. Comparing the plots for fi = I- 0 (chain dotted) with those for fi = 10·0 at the same respective values of 1/1> 1J2 and 7'/3, shows that the increase in {i decreases the critical wave speed. This fact has been reported earlier also [10]. The radial response of the shell is shown in Figures 2-7, which give displacement W( = woldo) versus f3 (=21TRI A) plots with 1/1' 1J2, 7'/3 and {i as parameters. Results were obtained for values of f3 up to one only, because going beyond one would mean shorter wavelengths of the order of shell radius, which is not of much relevance when studying the response due to seismic excitations.
20
15
S $l10
5
Figure 2. Radial displacement ( W) vs. wavelength ((3) under different soil conditions (fL) with 111 as parameter and c= 1·005. h = 0-05, P = 0-30, 112 = 0-05, 113 = 0,02, m =0,25. - , fL = 10; - '-, fL = 0'1; ---, JI =0-01.
The effects of varying 7]1, 1)2 and 7]3 on the radial displacement are shown, respectively, in Figures 2, 3 and 4 for the wave speed c = 1·005. From Figure 3 it is evident that the changes in 7/2 do not have any practical influence on the radial displacement, irrespective of whether the surrounding soil is soft Or hard. On the other hand, Figures 2 and 4 indicate that variations in the values of 1JI and 1J3 have a strong influence on the radial displacement, especially when there is a soft soil (fi = 0·01 and 0'10) around the shell. However, the nature of the effects due to the changes in 7/1 is opposite to that for 'TJ3' Whereas an increase in 'TJ\ decreases the magnitude of the radial displacement, with increase in 7/3 it increases. Further, upon comparing the results for different {i, it appears that if the shell is buried under a very hard and rocky (fJ- = 10· 0) surrounding, even with appreciable changes in 7/1 and 1J3 the radial displacement of the shell is not going to be much affected. Therefore, it is only under soft soil conditions that some control over the radial displacement of the shell may be achieved by altering the orthotropy parameters 7/1 and 'TJ3' To visualize the nature of changes in the above results with increase in the wave speed, Figures 5-7 show the radial displacements at c = 5 ,0. Results were obtained for c = 10·0 also. It is found that even at the higher wave speeds effects of changing "71, 7/2 and 7/3
108
V. P. SINGH, P. C. UPADHYA Y AND B. KISHOR
Figure 3. Radial displacement ( W) vs, wavelength ({3) under different soil conditions (fi) with and c = 1·005. .", = O· 50, 173 = 0·02. Other parameters, except "'2, as Figure 2. Key as Figure 2.
"'2 as parameter
'2.0
15
'?:
tv
o
~
10
5
Figure 4. Radial displacement (W) vs. wavelength (f3) under different soil conditions (fi) with n, as parameter and c = 1·005. "72 = 0,05; other parameters, except 113' and key as Figure 3.
remain qualitatively the same as discussed in the preceding paragraph for c == 1·005. Of course with the increasing C, the amplitude of the radial displacement also increases. By comparing the plots of the orthotropic cases with their corresponding isotropic counterparts, it can be seen that, in general, the effect of orthotropy introduced through 'TJ2 and 'TJ3 is to bring down the radial displacement of the shell in all the soil conditions. In hard soil, of course, this effect is not so strong as in soft soil.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
109
150
~
100
'"o
50
0·5o__ ....
_---- -.: -_:..-SQ--0·5
1.0
/3 Figure 5. Radial displacement ( W) vs. wavelength (13) under different soil conditions (Ii) with '11 as parameter and c= 5·0. Parameters and key as Figure 2, except c= 5'0.
150 I
/1 II 0'005,0,05,0'10 $:
,//
----/'>/
100
N
, ",.-....
52
..
~/
"
/;/ I
-
--- .......150
/
/
}/ /I-- 0'005,0,05,0'10
50
If/ 7
// /1 //
'£ -
----
0'005,0'05,0,10 _ _ .... ..----
-----
....
0·5
/3
Figure 6. Radial displacement (W) vs. wavelength ((3) under different soil conditions (Ii) with '12 as parameter and c= 5·0. Parameters and key as Figure 3, except c= 5'0, and - - - - jL = 1·0.
Effects of changing 'TIl, 'TI2 and 'TI3 on the axial component of the displacement (U), are shown in Figures 8-14. Figures 8 and 9 show the changes in U due to variations in 'TIl and 'TI2, respectively, for if = 1·005. From these figures it appears that, for if = 1,005, changes in "'It and '172 do not have any significant influence on the axial displacement of the shell, irrespective of the soil condition (i.e., {i). In fact, though it is not shown in the figures, for {i = 10·0 these changes in 'TIl and 'TI2 were found to have absolutely no effect on the magnitude of
110
V. P. SINGH, P. C. UPADHYAY AND B. KISHOR
150
0 ':2 / I
'13 =0'01,0'0:2,0 ':2
--
./-./ /".",..,.-"-
O'OU
,// / 1/
I'
//
50
1/ ' -:
,,;;1
_//150 ,,-
__ ..,./,
/
/-
'/
i' -:0'01
I '/
,
O.:.?~ __--
/.~/
Y
41"
_--------0.01
_
_
0 '5
1·0
/3 Figure 7. Radial displacement ( W ) V5. wavelength (f) under different soil conditions (;L) with and c =: 5 ,0. Parameters and key as Figure 4, except C'" 5.0.
1)3
as parameter
25
20 :::l NO
15
10
I
I
//r-.;;;;:;-__
5
'-
0 .05
~--.:.::::----
I'
0 ·50,1· 0
-
--o:os1
.-...-.,,-.
-If':, 0·501'0 I
Figure 8. Axial displacement (U) vs, wavelength (13) under different soil conditions (;L) with '1), as parameter and if= 1·005. Parameters and key as Figure 2, except ----, jl '" 1·0.
U. However, as seen in Figures 10 and II,. alterations in TJ3 have a very strong influence on the axial displacement under all kinds of soil conditions. Even for fi = 10·0 changes in 713 have quite a significant effect on the magnitude of U, as seen from Figure 11. The only difference is that under the soft soil conditions the effect of 'T} 3 is prominent when f3 < O: 5, whereas for a very hard surrounding soil (fi = 10·0) effects are significant in the range of f3 between O· 5 and 1·0. In both the figures, however, U is found to increase with an increase in the 1')3 value.
111
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
30,----,--,--,----.---,--r---,--.----,----,
25
20 :::>
N
~ 15
10
/
, I
,0---- _---
0'1
0'005,0,05
...- - -
Q:'O.2~~02,.Q:19 0·5
1·0
f3
4
2
0·5 {3
Figure 10. Axial displacement ( U) vs, wavelength ((3) under different soil conditions (iL) with '13 as parameter and c= 1·005. Parameters and key as Figure 4.
Effects of '1/1 and '1/2 were found to be practically insignificant (see Figures 8 and 9) for c = 1·005. However, as the apparent wave speed c increases all the three orthotropy parameters '1/1, ?J2 and '1/3 appear to have significant effects on the axial displacement of the shell. This is shown in Figures 12-14, drawn for c= 5·0. Figures 12 and 13 show that changes in 711 and 712 have a significant effect on U, especially under the soft soil conditions (fl = 0·01 and 0'1). But, as ji increases and the rigidity of the surrounding ground becomes comparable to the rigidity of the shell, the effect becomes much less. For ji = 10·0 (not
112
V. P. SINGH, P. C. VPADHYAY AND B. KlSHOR
100
75 ::.
'"o
50
-------ISO
'L-
.. 0·02 __ - - - - _ ;....;;--
25
V~~--------
-"
----------~--0·5
f3 Figure 11. Axial displacement (U) vs. wavelength (f3) under different soil conditions (ji) with 1)3 as parameter and c = I' 005. Parameters and key as Figure 4, except - - - - . ji. = 1·O.
1),=
//
30
V - - .............. J 20 ,
;
r
/ ''/
/--'-
,
1·0
/
~
,,/
»<
'C."7"'-_
_
/ ----.. -,,--,------_
y; I
"~5
0·50 ..._ -
,
I
10
0,05,0,50
/0
40
0·05
~
0·50
J.-A/ 1·0 --------~~~~~== 0·5
1'0
f3 Figure 12. Axial displacement (U) vs, wavelength (f3) under different soil conditions Uil with and C,= 5·0. Parameters and key as Figure 5, except - - --, ji. = 1·0.
1)1
as parameter
shown in the figures) practically no effect was visible. The effect of 1')3, as shown in Figure 14, is again quite considerable but of a lesser degree than observed for c = 1·005. Thus, for c = 5·0 all three parameters ('TIl, "'12 and 'TJ3) are found to influence the axial displacement. However, changes in '1'12 have comparatively greater influence than "'11 and 'TJ3' Also, it may be observed that the maximum.changes in U, due to variations in '171 and 1')2, occur for values of (3 in the range of 0·40 to 0,70, approximately.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
113
Figure 13. Axial displacement (U ) vs. wavelength (13) under different soil conditions (j1.) with "12 as parameter and c= 5·0. Parameters and key as Figure 6.
60
40 ::l
oJ
~
0·2 20 0·02
Figure 14. Axial displacement (U) vs. wavelength ((3) under different soil conditions (j1.) with ''13 as parameter and c= 5·0. Parameters and key as Figure 7.
For still higher wave speeds (c= 10'0), the overall nature of changes in U due to alterations in 'TIl, 'TI2 or 'TI3 were found to be similar to those observed for c= 5·0. But, as the apparent wave speed increases, effects of changes in these parameters are found to be significant only at lower values of {i (=0,01 and 0'10). Therefore, it may be said that at lower wave speeds orthotropy parameters will have a significant effect on the axial displacement of the shell under nearly all soil conditions, excepting only when it is very
114
v. P. SINGH,
P. C. UPADHYAY AND B. KISHOR
hard and rocky. At higher wave speeds these parameters will have a noticeable influence only when the surrounding soil is quite soft and sandy. In a manner similar to that for the effects on radial displacement, introduction of orthotropy through 'TIl, 1/2 and 1/3 is found to lower the axial displacement of the shell quite significantly in comparison to that for the corresponding equivalent isotropic system. This observation is of special interest because most of the failures of buried pipes have been reported as due to excessive axial displacement.
4. CONCLUSIONS
On the basis of the results discussed above, the main conclusions may be listed briefly as follows. (i) Shell resonance can occur only when the shell is buried under a very hard (compared to the shell) and rocky medium, and is excited by waves of smaller wavelengths. In soft soil surroundings no resonance occurs. (ii) Variations in the orthotropy parameters 'TIl and 1/2 do not influence the values of the resonating wave speeds. However, changes in 1/3 alter the resonance wave speeds significantly. (iii) Under soft soil conditions (jj = 0'01-0'1) changes in 1/1 and '1'/3 have a strong influence on the radial displacement of the shell, at all wave speeds, but changes in '1'/2 have no effect, irrespective of the nature of the surrounding soil (hard or soft) and the wave speed c. (iv) Although '1'/2 does not influence the radial displacement of the shell it has a very prominent role in altering the axial displacement component, especially at the higher wave speeds and under the soft ground conditions. (v) The parameter '1'/3 has a strong influence on the axial displacement of the shell at all the wave speeds and under all ground conditions. However, it is greater at the lower wave speeds and with softer surrounding medium. (vi) As the wave speed increases, at lower values of p, (=0'01-0,1) all three orthotropy parameters, 'TJI, 1/2 and 'TJ3, are found to have effective influences on the axial displacement of the shell. Most of the underground pipe failures, due to seismic loading, have been reported to occur because of excessive axial deformation of the shell. Therefore, by making a proper choice of 'TJI, T/2 and 'TJ3, when fabricating the composites, it may be possible to reduce the axial displacement to some extent, depending on the soil conditions. (vii) If the rigidity modulus of the surrounding ground is much greater than that of the shell, say fi = 10,0, alterations in the parameters 7]1, '1'/2 and 'TJ3 do not produce any significant changes in the radial or the axial responses of the shell. (viii) Introduction of orthotropy is helpful in reducing the radial and axial displacements of the shell as compared with those of the corresponding isotropic system.
REFERENCES 1. L. R. L. WANG and M. O'ROURKE 1977 Proceedings of the Current State of Lifeline Earthquake Engineering, ASCE, Los Angeles, California, 252-266. State of the art of buried lifeline
earthquake engineering. 2. T. ARIMAN and G. E. MULESKI 1981 International Journal of Earthquake Engineering and Structural Dynamics 9, 133-151. A review of the response of buried pipelines under seismic excitations. 3. I. NELSON and P. WEIDLINGER 1979 Journal of Pressure Vessel Technology 101, 10-20. Dynamic seismic analysis of long segmented lifelines.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
115
4. K. KUBO, T. KATAYAMA and M. OHASHI 1979 Journal of the Technical Councils of American Society of Civil Engineers, 221-238. Lifeline earthquake engineering in Japan. 5. 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. 6. L. R. L. WANG 1978 Journal of the Teclmical Councils of American Society of Civil Engineers 104, 71-89. Vibration frequencies of buried pipelines. 7. R. A. PARMELLE and C. A. LUDTKE 1975 Proceedings of u.s. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Ann Arbor, Michigan, 406-415. Seismic soil-structure interaction of buried pipelines. 8. M. NOVAK and A. HINDY 1980 Proceedings of Seventh World Conference on Earthquake Engineering 8, 287-294. Seismic analysis of underground tubular structures. 9. S. CHONAN 1981 Journal of Sound and Vibration 78, 257-267. Dynamic response of a circular shell imperfectly bonded to a surrounding continuum of infinite extent. 10. S. K. DATTA, A. H. SHAH and N. EL·AKILY 1982 Journal of Applied Mechanics 49, 141-148. Dynamic behaviour of buried pipe in a seismic environment. 11. T. ARIMAN and G. E. MULESKI 1979 Proceedings of the Second U.S. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Stanford University, California 643-652. Recent developments in seismic analysis of buried pipelines. 12. B. H. COLE, C. J. RITTER and S. JORDAN 1979 Lifeline Earthquake Engineering-Buried Pipelines, Seismic Risk and Instrumentation. (Editors T. Ariman, S. C. Liu and R. E. Nickell). American Society of Mechanical Engineers. Structural analysis of buried reinforced plastic mortar pipe. 13. G. HERRMANN and E. H. BAKER 1967 Journal of Applied Mechanics 34, 81-86. Response of cylindrical sandwich shells to moving loads. 14. R. M. JONES 1975 Mechanics of Composite Materials. Tokyo: McGraw-Hili, first edition.
ON THE DYNAMIC RESPONSE OF BURIED ORTHOTROPIC CYLINDRICAL SHELLS V. P.
SINGH,
P. C.
UPADHYAY AND
B. KISHOR
Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University, Varanasi-221 005, India (Received 14 November 1985, and in revised form 25 March 1986)
This paper deals with the dynamic response of an orthotropic cylindrical shell, a pipe, buried underground and subjected to seismic excitation. A thick-shell model of the pipe including shear deformation and rotatory inertia has been used. The shell is assumed to be perfectly bonded to the surrounding medium (soil), of infinite extent. Only the axisymmetric response due to an incident compressional wave has been investigated. Effects of the shell orthotropy on its response characteristics have been illustrated by changing the non-dimensional orthotropy parameters of the shell over a wide range. Results have been obtained for different soil conditions-hard (rocky), medium and soft. It is found that the orthotropy parameters do not influence the shell response equally. Even the degree of their influence is found to be highly dependent on the rigidity of the surrounding soil. 1. INTRODUCTION
Buried pipelines perform a vital role in conducting and! or distributing energy, water, communication and transportation in today's world. However, nearly until the 1970's, underground utilities and transportation systems remained somewhat neglected in respect to consideration of earthquake effects upon them. It was only after the reporting of heavy damage of buried gas, water and other pipelines in some recent earthquakes that researchers realized the importance of studying their dynamic response. During the last seven to eight years a number of papers have appeared concerned with different aspects of the problem. Most of the published papers on the dynamic response of underground pipelines have been concerned either with assessing the role and development of soil parameters or proposing better analytical models for the pipe or the surrounding soil. A good account of these works can be found in review articles [1-4]. In efforts to propose a better model and to explain the nature of pipe failures, Ariman et al. [2,5] treated the pipe as a buried shell using Fliigge's shell theory. They concluded that a beam model of the pipe, as had been used by several investigators [6-8], cannot predict the buckling failures of the pipes which are evidenced as the most frequent cause of failure under seismic loading. Chonan [9] and Datta and his co-workers [10] have also used the shell theory in their work. All the work reported so far, however, has been concerned with isotropic and homogeneous buried shells only. During the Second U.S. National Conference (1979) on earthquake engineering Ariman and Muleski [11] emphasized the need for analyzing the dynamic behaviour of pipes made of orthotropic materials. He pointed out that new materials were becoming quite popular in pipeline systems, and were slowly replacing cast iron and steel. In fact, reinforced plastic mortar (R.P.M.) has already found acceptance as a substitute for cast iron and steel in a number of applications such as potable water, irrigation and sewer lines. However, other than a paper by Cole et al. [12], who performed WI 0022-460X/87/040101 + 15 $03.00/0
© 1987 Academic Press Inc. (London) Limited
102
V. P. SINGH, P. C. UPADHYAY AND B. KISHOR
a finite element analysis of buried R.P.M. pipes, to the best of the authors' knowledge, no work has been reported on the effect of orthotropy on the dynamic performance of underground cylindrical shells. On orthotropic cylindrical shells as such, of course, a great deal of work has been done. The aim of the work reported in what follows, therefore, was to study the effect of orthotropy on the dynamic behaviour of a buried infinite cylindrical shell when it is disturbed by a longitudinal wave moving along its length. Results have been obtained for different soil conditions around the pipe-hard (rocky), medium and soft (sandy). It has been found that the orthotropy parameters do not equally influence the shell response. It has also been found that, depending upon the soil conditions, the effects of orthotropy may be quite different and vary significantly. Results of this investigation are expected to be helpful in designing composite pipes, tunnels lining and underground storage cylindrical tanks, etc., in that the orthotropic material parameters can be selected within a desired range for the purpose of limiting and controlling the shell displacements under different soil conditions. Only axisymmetric motion of the shell has been considered. We have taken this simplified version of the problem of pipeline motion due to seismic excitation because our main object here is only to look into the relative influence of orthotropic material parameters on the shell response, when it is placed under different soil conditions, other factors remaining invariant. In the formulation of the problem an approach similar to that of reference [10] has been followed, wherein the tractions in the dynamical equations of the embedded shell are expressed completely in terms of the incident and the scattered field displacements in the surrounding infinite medium. 2. BASIC EQUATIONS AND FORMULATION
Consider an infinitely long thick orthotropic cylindrical shell of mean radius Rand thickness h lying in a linearly elastic, homogeneous and isotropic infinite medium. The shell is excited by a longitudinal wave of wavelength A (= 2'Tr1 g) and speed c moving along the axis of the shell in the medium. Let a cylindrical-polar co-ordinate system (r, fJ, x) be defined such that x coincides with the axis of the shell. The stress-strain relations of the shell material is assumed to be of the form (1)
where G"ij and el} are, respectively, the components of the stress and strain, and Ex l , E o I , E p i and G X l are the four independent moduli. The equations governing the axisymmetric motion of such a shell can be written, with some modifications, from reference [13] as,
Do)
2
2
1 ( a , -a ] w+ [ 0 - a ] .1, [ -R-2 E o + -R 2 +o--p xax2 at 2 x ax w» 2
a ] w+
[ R ax
p
a] ax U+P*I =0 '
--
(2)
2
D xax2 --G [-0x-axa]w+ [a E --.2:_
[ER
---=----
i ] .1, + [ D a 2 I' i ] u+p*=O -1'x at2 '1':< R ax R at2 2,
[D---=---a a 2
2
]'
R ax 2 R
2
2
+ [ Ex -a-2 p 'ata-2] u+p*=O at 2 '1':< 3 ax ]
.1,
(3)
(4)
in which t is the time, u and ware the displacement components of the middle surface in the axial and radial directions, respectively, and t/Jx is the angle of rotation in the axial direction of a straight line which is initially normal to the middle surface.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
103
The stiffness parameters and other constants in equations (2)-(4) are given as
I' = Pol,
(5)
where k is the shear correction factor taken as 7T' / v'f2, and Po is the mass density of the shell material. are the traction components per unit area exerted on the shell by the surrounding medium, and are given by
pr
pf = (l + h/2R)u~n
pf =
(h/2)p~,
pf = (1 + h/2R)cr':x,
(6)
in which crt and cr~x are, respectively, the normal and the shear stresses generated at the outer surface of the shell (i.e., at r = R + h/2) due to the motion of the surrounding medium. The displacement d(r, e, x, t) at any point in the surrounding medium satisfies the equation of motion dV(V . d) - c~V x V x d == a2 d/ at 2 ,
(7)
where, cl={(A+21L)/Pm}1/2 and C2=.JIL/Pm are respectively the longitudinal and the shear wave speeds depending upon the values of the Lame constants A and j.L, and the density, Pm, of the medium (surrounding soil). Because there is a scatterer (shell) lying in the medium, the displacement d at any point in the medium consists of two parts: the incident field d(1) created due to the longitudinal wave moving along the shell and the scattered field des). For an axisymmetric motion, the radial and the axial components of del) and d(s) are written as (8)
where do is a constant depending on the incident wave intensity and having the dimension of length, and d~S)
= [A(ctl R)K 1(c\r/ R)+ig· BK\(c2r/ R)]
d~s)
ei€(x-cl),
= [Ai~"Ko(c\ r] R) + B(C2/R)Ko(c2 r/ R)] ei~(X-ct),
(9)
where I" and K" (n = 0, 1) are the modified Bessel functions of the first and the second kind, respectively, and {3 = gR = 27T'R/ A.
(10)
When the arguments of I" and K" are imaginary (for c] Cl > 1·0), these functions become J and Y respectively; K" can, of course, be alternatively expressed in terms of Hankel functions. It may be noted that d'", as given by equations (8), satisfies equation (7) and represents a travelling wave with wavelength A = 27T'/ g moving along the axis of the pipe with speed C = w / g. In the case of a plane longitudinal wave moving at an angle rp with the axis of the shell, equations (8) will represent the axisymmetric components of that wave and the apparent wave speed of such a wave along the shell axis will be (ctlcos ep). The constants A and B in equations (9) are determined by applying the condition of continuity of displacements at the outer surface of the shell: i.e., /I
/I
d(s» ( W ) r:R+h/2 -- (d(i)+ r r
.
r-R+h/2,
A perfect bond between the shell and the surrounding ground has been assumed because our main aim, here, is to study the effect of orthotropy. Moreover, it has been reported
104
v.
P. SINGH, P. C. UPADHYAY AND B. KISHOR
[8] that, excepting under very specific soil conditions, the slip does not have much influence on the stress, etc., in the pipe. Now, upon assuming the shell displacement in the form
u = u(r) eieCx-cl) = {uo + (r - R)r/Jx} eiecx-cll, (12) and using the two boundary conditions (11), the constants A and B in equations (8) can be determined in terms of Uo and wo, which are the displacements at the middle surface of the shell. Finally, the scattered field displacement comes out as d~S) = (11 D)[{(cl cZ):K.O( a 2)K I (ret! R) - f:lzKo(at)K] (rczI R)}wo,
+ i{K](al)K1(rc2/ R) - K I(a2)KI ( rcll R)}(c1f:l)uo] eieCx-cl), d~s) = (1/ D)[ {(C1C2)K1 ( al)Ko(rc21 R) - f:l 2KI(a2) Ko(rc11 R)}uo
+ iC2f:l{:K.o(al):K.o(rc21 R) - :K.o(a2) Ko(CI r] R)}wo] eiL'Cx-ct), D = C1c2Kl(a]):K.o(a2) - f:l 2:K.o(at)Kt(a2), at
= c1(l + hI2),
a2 = cil + h/2),
h = hi R,
Wo = wo- doctIt(al)'
Uo = [uo+(h/2)~x -i,BdoIo(a])],
(13)
Now, with d(i) and dCs) fully known from equations (8) and (13), the tractions given by equation (6), can be determined by calculating
(14)
rt,
as
* _ Ci) Cs» r-R+h/2-_ [ A {(ad; + d; + ad;)} +2JL{ad;}] O"rr-(lTrr+O"rr , ar r ax ar r=R+h/2 >I<
IT rx
= (O"~'I) +O"rx(s) )r-R+h/2 = [
JL
{(ad; ad;)}] -a-+-ax
r
r=R+I./2
,
(15)
where, d; = (d~l) + d~s» and d; ::::: (d~)+ d~s» are the total radial and axial displacement fields, respectively. Hence, after a few steps of simplification, one gets
pt =;; (JL/ R)[PlI wo+ (h/2)Pt2~x + Pt3UO] ei,Hx-cl) - (1 + h/2)(do/ R)JL[(x 2,/ - 2f:l2)1o(at) +2CiI I(al)/ all eUCX-CI), pt::::: (JLIR)[P31 Wo + (h/2)P32~X + P33UO]
eif(x-cl)
+ 2(1 + h(2)(JLI R)[ido(ct,B)I1 (reV R)] ei
(1
[c
2 ]
.( ii\
PI2=-P21"",PI3=-Pn=-11+"2}f:l
( ii\[-
P22=P33=P23=P32= 1+"2} ci
[-y2x2Kt(a2):K.o(at). ] D -r-2, y2x2KI(al)K](a2)] D .
(17)
105
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
pr
When these values of and the displacements u and w from equations (12) are substituted into the shell equations (2)-(4), after some lengthy algebra, the simplified final equations for the determination of {Z} = {wol do, (hI2)rjixl do, aol do} are put into the matrix form [[ Q] - ~[P] -
n 2 [ N ]]{Z } = -,l[P]{F} + MH}(1 + h12),
(18)
where n 2 = (PohRI Gxl)a? = X 2 f3 2 ciifLii is the non-dimensional frequency, in which ~ = JLIOxl and ii = PolPm are, respectively, the modulus of rigidity and the density of the surrounding soil (medium) measured in terms of the corresponding values for the shell material, and c = c] C 1 is the apparent wave speed. The matrix [P] is as given in equations (17), and the components of other matrices [Q], [N], {Z}, {F} and {H} are given by
(-2if3e) (ih13 ~:) ~ (f2 JI2 f32 + k 2) (!h2~) h 'T/3 Q23 (ii~:)
(h~: (1 + ~;) + iie(32) [Q]=
-Q12 -Q13
{Z} =
W} {UV
,
O
=
{WOld } ,Jix_hI2do
(19)
uoldo
and (20) are the orthotropy parameters. It should be noted that the right side of equation (18) represents the excitation in which the matrix [P] depends on the geometry of the pipe, the material properties of the surrounding medium, and the wavelength and wave speed of the excitation.
3. RESULTS AND DISCUSSION
The results to be presented here were obtained mainly with the aim of bringing out the effect of varying the orthotropy parameters '1/1, 'T/2 and 'T/3 on the radial and the axial displacements of the shell under different soil conditions and also at different wave speeds c( = c] c1 ) . Ranges of orthotropy parameters have been selected on the basis of orthotropic constants given for different kinds of composites in reference [14]. For each 1]1' 'T/2 and 1]3, three values have been taken: 'T/I=0'05, 0·50 and 1·0; 'T/2=0'005, 0·05 and 0'10;
106
V. P. SINGH, P. C. UPADHYAY AND B. KISHOR
113 = 0'01,0'02 and 0·20. This is expected to cover a wide range of orthotropic (composite) materials. In each study of the effect of a variation in one orthotropy parameter, the other two have been kept constant. To have a comparative picture and visualize the effect of orthotropy, results for the equivalent isotropic cases have also been plotted, denoted by the label (ISO). Since the modulus of rigidity of a soft clay is nearly 1000 times less than that of hard rocks, to simulate all kinds of combinations of ground to pipe rigidity ratio ji has been varied from 0·01 to 10·0. The thickness to radius ratio of the shell, h, the ratio of the shell and the soil densities, p, and the Poisson's ratio of the soil, m, have been kept constant throughout: ii = 0'05, P= 0·30 and m = 0·25. Before proceeding to compute the response, critical (resonating) wave speeds were determined from the equation
[[ Q] - ji(P] - {J2[N]] = O.
(21)
Figure 1 shows the plots of such critical wave speeds versus wavelength (f3), with 111, 112 and 113 as parameters. It is found that real values for the critical wave speeds exist only under a very hard (rocky) soil condition (ji = 10·0). In soft (fl = 0·01) or medium hard (jl = 0·1 or 1'0) soil no real resonating wave speeds exist. This is why, except for one, all the curves drawn in Figure 1 are for jl = 10·0 only. For jl = 1·0, only for the combination of 111 = O'5, ""2 = 0·05 and 113 = 0·02 were real critical wave speeds found to exist, and then only over a limited range of f3 dotted line on top.
= 4 to
8, as shown by the double chain
0·58 r----,--,---,---....,....--,----,---.------,
---
\'0-<""""'P:=1-0 7'J "'0·50 ~ <, '1
t
/~ ....... ',0 ""'l!!
0·56
\\
0'5
\
l\..l
0'54
0·52
\
--
TJ2 =0,05'7'3 J "'0,02
.... ....
1) "'0-05 1
0'02
-_
_ - - .......
\\
"'
--
--_<::,,""0'01
.......
-........
-,
...... ........
1)
=0,2
. . . . . . """y
---.... .... -.... -----
--
0·50 I.-..._---'-_ _..L...._ _L-_....,.L_ _~_---l_ _-'-_----" 20 10 30 o ~ {3
Figure 1. Critical wave speeds (c) vs. wavelength (fJ) with 1/1,1/2 and 1/3 as parameters. ii=0'05, p=0'30, m = 0'25; fl = 10·0 except as indicated. - , 1/2 = 0'05, 1/3 = 0·02; - . - , 1/, = 0'50, 113 = 0'02; ---, 1/, '" 0,50, 1/2 = 0'05.
From Figure 1 it is seen that the orthotropy parameters '71 and '72 have practically no effect on the critical wave speeds. However, as shown by the dotted lines, 113 is found to have a significant effect in altering the critical values of c. As "TJ3 increases the critical c values decrease.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
107
Further, looking at the values of the critical wave speeds indicates that even under very hard soil conditions the shell cannot resonate if it is excited by a plane longitudinal wave. Comparing the plots for fi = I- 0 (chain dotted) with those for fi = 10·0 at the same respective values of 1/1> 1J2 and 7'/3, shows that the increase in {i decreases the critical wave speed. This fact has been reported earlier also [10]. The radial response of the shell is shown in Figures 2-7, which give displacement W( = woldo) versus f3 (=21TRI A) plots with 1/1' 1J2, 7'/3 and {i as parameters. Results were obtained for values of f3 up to one only, because going beyond one would mean shorter wavelengths of the order of shell radius, which is not of much relevance when studying the response due to seismic excitations.
20
15
S $l10
5
Figure 2. Radial displacement ( W) vs. wavelength ((3) under different soil conditions (fL) with 111 as parameter and c= 1·005. h = 0-05, P = 0-30, 112 = 0-05, 113 = 0,02, m =0,25. - , fL = 10; - '-, fL = 0'1; ---, JI =0-01.
The effects of varying 7]1, 1)2 and 7]3 on the radial displacement are shown, respectively, in Figures 2, 3 and 4 for the wave speed c = 1·005. From Figure 3 it is evident that the changes in 7/2 do not have any practical influence on the radial displacement, irrespective of whether the surrounding soil is soft Or hard. On the other hand, Figures 2 and 4 indicate that variations in the values of 1JI and 1J3 have a strong influence on the radial displacement, especially when there is a soft soil (fi = 0·01 and 0'10) around the shell. However, the nature of the effects due to the changes in 7/1 is opposite to that for 'TJ3' Whereas an increase in 'TJ\ decreases the magnitude of the radial displacement, with increase in 7/3 it increases. Further, upon comparing the results for different {i, it appears that if the shell is buried under a very hard and rocky (fJ- = 10· 0) surrounding, even with appreciable changes in 7/1 and 1J3 the radial displacement of the shell is not going to be much affected. Therefore, it is only under soft soil conditions that some control over the radial displacement of the shell may be achieved by altering the orthotropy parameters 7/1 and 'TJ3' To visualize the nature of changes in the above results with increase in the wave speed, Figures 5-7 show the radial displacements at c = 5 ,0. Results were obtained for c = 10·0 also. It is found that even at the higher wave speeds effects of changing "71, 7/2 and 7/3
108
V. P. SINGH, P. C. UPADHYA Y AND B. KISHOR
Figure 3. Radial displacement ( W) vs, wavelength ({3) under different soil conditions (fi) with and c = 1·005. .", = O· 50, 173 = 0·02. Other parameters, except "'2, as Figure 2. Key as Figure 2.
"'2 as parameter
'2.0
15
'?:
tv
o
~
10
5
Figure 4. Radial displacement (W) vs. wavelength (f3) under different soil conditions (fi) with n, as parameter and c = 1·005. "72 = 0,05; other parameters, except 113' and key as Figure 3.
remain qualitatively the same as discussed in the preceding paragraph for c == 1·005. Of course with the increasing C, the amplitude of the radial displacement also increases. By comparing the plots of the orthotropic cases with their corresponding isotropic counterparts, it can be seen that, in general, the effect of orthotropy introduced through 'TJ2 and 'TJ3 is to bring down the radial displacement of the shell in all the soil conditions. In hard soil, of course, this effect is not so strong as in soft soil.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
109
150
~
100
'"o
50
0·5o__ ....
_---- -.: -_:..-SQ--0·5
1.0
/3 Figure 5. Radial displacement ( W) vs. wavelength (13) under different soil conditions (Ii) with '11 as parameter and c= 5·0. Parameters and key as Figure 2, except c= 5'0.
150 I
/1 II 0'005,0,05,0'10 $:
,//
----/'>/
100
N
, ",.-....
52
..
~/
"
/;/ I
-
--- .......150
/
/
}/ /I-- 0'005,0,05,0'10
50
If/ 7
// /1 //
'£ -
----
0'005,0'05,0,10 _ _ .... ..----
-----
....
0·5
/3
Figure 6. Radial displacement (W) vs. wavelength ((3) under different soil conditions (Ii) with '12 as parameter and c= 5·0. Parameters and key as Figure 3, except c= 5'0, and - - - - jL = 1·0.
Effects of changing 'TIl, 'TI2 and 'TI3 on the axial component of the displacement (U), are shown in Figures 8-14. Figures 8 and 9 show the changes in U due to variations in 'TIl and 'TI2, respectively, for if = 1·005. From these figures it appears that, for if = 1,005, changes in "'It and '172 do not have any significant influence on the axial displacement of the shell, irrespective of the soil condition (i.e., {i). In fact, though it is not shown in the figures, for {i = 10·0 these changes in 'TIl and 'TI2 were found to have absolutely no effect on the magnitude of
110
V. P. SINGH, P. C. UPADHYAY AND B. KISHOR
150
0 ':2 / I
'13 =0'01,0'0:2,0 ':2
--
./-./ /".",..,.-"-
O'OU
,// / 1/
I'
//
50
1/ ' -:
,,;;1
_//150 ,,-
__ ..,./,
/
/-
'/
i' -:0'01
I '/
,
O.:.?~ __--
/.~/
Y
41"
_--------0.01
_
_
0 '5
1·0
/3 Figure 7. Radial displacement ( W ) V5. wavelength (f) under different soil conditions (;L) with and c =: 5 ,0. Parameters and key as Figure 4, except C'" 5.0.
1)3
as parameter
25
20 :::l NO
15
10
I
I
//r-.;;;;:;-__
5
'-
0 .05
~--.:.::::----
I'
0 ·50,1· 0
-
--o:os1
.-...-.,,-.
-If':, 0·501'0 I
Figure 8. Axial displacement (U) vs, wavelength (13) under different soil conditions (;L) with '1), as parameter and if= 1·005. Parameters and key as Figure 2, except ----, jl '" 1·0.
U. However, as seen in Figures 10 and II,. alterations in TJ3 have a very strong influence on the axial displacement under all kinds of soil conditions. Even for fi = 10·0 changes in 713 have quite a significant effect on the magnitude of U, as seen from Figure 11. The only difference is that under the soft soil conditions the effect of 'T} 3 is prominent when f3 < O: 5, whereas for a very hard surrounding soil (fi = 10·0) effects are significant in the range of f3 between O· 5 and 1·0. In both the figures, however, U is found to increase with an increase in the 1')3 value.
111
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
30,----,--,--,----.---,--r---,--.----,----,
25
20 :::>
N
~ 15
10
/
, I
,0---- _---
0'1
0'005,0,05
...- - -
Q:'O.2~~02,.Q:19 0·5
1·0
f3
4
2
0·5 {3
Figure 10. Axial displacement ( U) vs, wavelength ((3) under different soil conditions (iL) with '13 as parameter and c= 1·005. Parameters and key as Figure 4.
Effects of '1/1 and '1/2 were found to be practically insignificant (see Figures 8 and 9) for c = 1·005. However, as the apparent wave speed c increases all the three orthotropy parameters '1/1, ?J2 and '1/3 appear to have significant effects on the axial displacement of the shell. This is shown in Figures 12-14, drawn for c= 5·0. Figures 12 and 13 show that changes in 711 and 712 have a significant effect on U, especially under the soft soil conditions (fl = 0·01 and 0'1). But, as ji increases and the rigidity of the surrounding ground becomes comparable to the rigidity of the shell, the effect becomes much less. For ji = 10·0 (not
112
V. P. SINGH, P. C. VPADHYAY AND B. KlSHOR
100
75 ::.
'"o
50
-------ISO
'L-
.. 0·02 __ - - - - _ ;....;;--
25
V~~--------
-"
----------~--0·5
f3 Figure 11. Axial displacement (U) vs. wavelength (f3) under different soil conditions (ji) with 1)3 as parameter and c = I' 005. Parameters and key as Figure 4, except - - - - . ji. = 1·O.
1),=
//
30
V - - .............. J 20 ,
;
r
/ ''/
/--'-
,
1·0
/
~
,,/
»<
'C."7"'-_
_
/ ----.. -,,--,------_
y; I
"~5
0·50 ..._ -
,
I
10
0,05,0,50
/0
40
0·05
~
0·50
J.-A/ 1·0 --------~~~~~== 0·5
1'0
f3 Figure 12. Axial displacement (U) vs, wavelength (f3) under different soil conditions Uil with and C,= 5·0. Parameters and key as Figure 5, except - - --, ji. = 1·0.
1)1
as parameter
shown in the figures) practically no effect was visible. The effect of 1')3, as shown in Figure 14, is again quite considerable but of a lesser degree than observed for c = 1·005. Thus, for c = 5·0 all three parameters ('TIl, "'12 and 'TJ3) are found to influence the axial displacement. However, changes in '1'12 have comparatively greater influence than "'11 and 'TJ3' Also, it may be observed that the maximum.changes in U, due to variations in '171 and 1')2, occur for values of (3 in the range of 0·40 to 0,70, approximately.
BURIED ORTHOTROPIC CYLINDRICAL SHELLS
113
Figure 13. Axial displacement (U ) vs. wavelength (13) under different soil conditions (j1.) with "12 as parameter and c= 5·0. Parameters and key as Figure 6.
60
40 ::l
oJ
~
0·2 20 0·02
Figure 14. Axial displacement (U) vs. wavelength ((3) under different soil conditions (j1.) with ''13 as parameter and c= 5·0. Parameters and key as Figure 7.
For still higher wave speeds (c= 10'0), the overall nature of changes in U due to alterations in 'TIl, 'TI2 or 'TI3 were found to be similar to those observed for c= 5·0. But, as the apparent wave speed increases, effects of changes in these parameters are found to be significant only at lower values of {i (=0,01 and 0'10). Therefore, it may be said that at lower wave speeds orthotropy parameters will have a significant effect on the axial displacement of the shell under nearly all soil conditions, excepting only when it is very
114
v. P. SINGH,
P. C. UPADHYAY AND B. KISHOR
hard and rocky. At higher wave speeds these parameters will have a noticeable influence only when the surrounding soil is quite soft and sandy. In a manner similar to that for the effects on radial displacement, introduction of orthotropy through 'TIl, 1/2 and 1/3 is found to lower the axial displacement of the shell quite significantly in comparison to that for the corresponding equivalent isotropic system. This observation is of special interest because most of the failures of buried pipes have been reported as due to excessive axial displacement.
4. CONCLUSIONS
On the basis of the results discussed above, the main conclusions may be listed briefly as follows. (i) Shell resonance can occur only when the shell is buried under a very hard (compared to the shell) and rocky medium, and is excited by waves of smaller wavelengths. In soft soil surroundings no resonance occurs. (ii) Variations in the orthotropy parameters 'TIl and 1/2 do not influence the values of the resonating wave speeds. However, changes in 1/3 alter the resonance wave speeds significantly. (iii) Under soft soil conditions (jj = 0'01-0'1) changes in 1/1 and '1'/3 have a strong influence on the radial displacement of the shell, at all wave speeds, but changes in '1'/2 have no effect, irrespective of the nature of the surrounding soil (hard or soft) and the wave speed c. (iv) Although '1'/2 does not influence the radial displacement of the shell it has a very prominent role in altering the axial displacement component, especially at the higher wave speeds and under the soft ground conditions. (v) The parameter '1'/3 has a strong influence on the axial displacement of the shell at all the wave speeds and under all ground conditions. However, it is greater at the lower wave speeds and with softer surrounding medium. (vi) As the wave speed increases, at lower values of p, (=0'01-0,1) all three orthotropy parameters, 'TJI, 1/2 and 'TJ3, are found to have effective influences on the axial displacement of the shell. Most of the underground pipe failures, due to seismic loading, have been reported to occur because of excessive axial deformation of the shell. Therefore, by making a proper choice of 'TJI, T/2 and 'TJ3, when fabricating the composites, it may be possible to reduce the axial displacement to some extent, depending on the soil conditions. (vii) If the rigidity modulus of the surrounding ground is much greater than that of the shell, say fi = 10,0, alterations in the parameters 7]1, '1'/2 and 'TJ3 do not produce any significant changes in the radial or the axial responses of the shell. (viii) Introduction of orthotropy is helpful in reducing the radial and axial displacements of the shell as compared with those of the corresponding isotropic system.
REFERENCES 1. L. R. L. WANG and M. O'ROURKE 1977 Proceedings of the Current State of Lifeline Earthquake Engineering, ASCE, Los Angeles, California, 252-266. State of the art of buried lifeline
earthquake engineering. 2. T. ARIMAN and G. E. MULESKI 1981 International Journal of Earthquake Engineering and Structural Dynamics 9, 133-151. A review of the response of buried pipelines under seismic excitations. 3. I. NELSON and P. WEIDLINGER 1979 Journal of Pressure Vessel Technology 101, 10-20. Dynamic seismic analysis of long segmented lifelines.
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4. K. KUBO, T. KATAYAMA and M. OHASHI 1979 Journal of the Technical Councils of American Society of Civil Engineers, 221-238. Lifeline earthquake engineering in Japan. 5. 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. 6. L. R. L. WANG 1978 Journal of the Teclmical Councils of American Society of Civil Engineers 104, 71-89. Vibration frequencies of buried pipelines. 7. R. A. PARMELLE and C. A. LUDTKE 1975 Proceedings of u.s. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Ann Arbor, Michigan, 406-415. Seismic soil-structure interaction of buried pipelines. 8. M. NOVAK and A. HINDY 1980 Proceedings of Seventh World Conference on Earthquake Engineering 8, 287-294. Seismic analysis of underground tubular structures. 9. S. CHONAN 1981 Journal of Sound and Vibration 78, 257-267. Dynamic response of a circular shell imperfectly bonded to a surrounding continuum of infinite extent. 10. S. K. DATTA, A. H. SHAH and N. EL·AKILY 1982 Journal of Applied Mechanics 49, 141-148. Dynamic behaviour of buried pipe in a seismic environment. 11. T. ARIMAN and G. E. MULESKI 1979 Proceedings of the Second U.S. National Conference on Earthquake Engineering, Earthquake Engineering Research Institute, Stanford University, California 643-652. Recent developments in seismic analysis of buried pipelines. 12. B. H. COLE, C. J. RITTER and S. JORDAN 1979 Lifeline Earthquake Engineering-Buried Pipelines, Seismic Risk and Instrumentation. (Editors T. Ariman, S. C. Liu and R. E. Nickell). American Society of Mechanical Engineers. Structural analysis of buried reinforced plastic mortar pipe. 13. G. HERRMANN and E. H. BAKER 1967 Journal of Applied Mechanics 34, 81-86. Response of cylindrical sandwich shells to moving loads. 14. R. M. JONES 1975 Mechanics of Composite Materials. Tokyo: McGraw-Hili, first edition.