j p dwivedi
TRANSCRIPT
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NON-AXISYMMETRIC DYNAMIC
RESPONSE OF IMPERFECTLY
BONDED BURIED ORTHOTROPIC
THIN EMPTY CYLINDRICAL SHELLDUE TO INCIDENT SHEAR WAVE (SV
WAVE)
RAKESH SINGH RAJPUT*
Reader, Mechanical Engineering, Directorate of Technical Education,
Bhopal, M. P., India
SUNIL KUMARProfessor, Mechanical Engineering, Rajeev Gandhi Technical University
Bhopal, M. P., India
ALOK CHAUBEY
Professor, Mechanical Engineering, Rajeev Gandhi Technical University
Bhopal, M. P., [email protected]
J. P. DWIVEDI
Professor, Mechanical Engineering, IT-BHU
Varanasi, U. P., India
Abstract
This paper is deals with the non-axisymmetric dynamic response of imperfectly bonded buried orthotropic thin
empty pipelines subjected to incident shear wave (SV-wave). In the thin shell theory the effect of sheardeformation and rotary inertia is not considered. The pipeline has been modeled as an infinite thin cylindrical
shell imperfectly bonded to surrounding. A thin layer is assumed between the shell and the surrounding medium
(soil) such that this layer possesses the properties of stiffness and damping both. The degree of imperfection of
the bond is varied by changing the stiffness and the damping parameters of this layer. The non axisymmetric
results are comparing with axisymmetric.
Key words: - Shear wave, Orthotropic, Imperfect bond, Seismic Wave, Non-Axisymmetric, DynamicResponse, Buried Pipelines, and Thin Shell.
1. IntroductionDuring past few years a number of studies like Cole Ritter and Jordon (1979) and Singh et al (1987) on the
axisymmetric dynamic response of buried orthotropic pipe/shells have been reported. Later Chonan (1981);Dwivedi and Upadhyay (1989; 1990; 1991); and Dwivedi et al (1991) have analyzed the axisymmetric problems
of imperfectly bonded shell for the pipesmade of orthotropic materials.
*Corresponding Author
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Upadhyay and Mishra (1988) have presented a good account of work on non-axisymmetric response of buried
thick orthotropic pipelines under seismic excitation. Again Dwivedi et al (1992a; 1992b); Dwivedi et al(1993a; 1993b; 1996); and Dwivedi et al (1998) have analyzed the non-axisymmetric problems of imperfectly
bonded buried thick orthotropic cylindrical shells. Kauretzis et al (2007) have presented analytical calculations
of blast induced strains on buried pipe lines. Hasheninajad and Kazemirad (2008) have reported dynamicresponse of eccentric tunnel in poro-elastic soil under sesmic excitation. Lee et al (2009) have done the risk
analysis of buried pipelines using probabilistic method. But in all these analyses pipelines have been modeled
as thick shell. As far as the non-axisymmetric dynamic response of thin shell is concerned, no work has beenreported so far. There is no work available discussing the effect of bond imperfection on the non axisymmetric
response of buried thin pipes made of orthotropic materials. Therefore, in present paper, the effect of imperfect
bond on the non- axisymmetric dynamic response of buried orthotropic thin pipelines has been analyzed.
2. Basic Equations and FormulationThe cylindrical pipeline has been modeled as an infinitely long cylindrical shell of mean radius R and
thickness h. It is considered to be buried in a linearly elastic, homogeneous and isotropic medium of infinite
extent. Basic approach of the formulation is to obtain the mid plane displacements of the shell by solving theequations of motion of the orthotropic shell. Traction terms in the equations of motion are obtained by solving
the three-dimensional wave equation in the surrounding medium. Appropriate boundary conditions are applied
at the shell surfaces. Equations arising out of boundary conditions along with the equations of motion of the
shell are simplified to yield a response equation in matrix form. Equation governing the non axis-symmetric
motion of an infinitely long orthotropic cylinder has been derived following the approach of Herrmann andMirsky (1957).
The cylindrical shell buried in a linearly elastic, homogeneous and isotropic medium of infinite medium, a thin
layer is assumed between the shell and the surrounding medium (soil). The degree of imperfection of the bond is
varied by changing the stiffness and the damping parameters of this layer. The shell is excited by a shear
vertical wave (SV- wave). A wavelength (=2/) is considered which strikes the shell at an angle withthe axis of the shell. Let a cylindrical polar co-ordinate system (r,, x) which is defined in such a way that xcoincides with the axis of the shell and, in addition, z is measured normal to the shell middle surface, which is
given as
h/2zh/2-, Rrz (1)The equation of motion of shell in the matrix form is given as Rajput et.al (2010)
[{L} {U}] + {P*} = 0 (2)
Where [L] is 3 3 a matrix and {U} = [w v u] T
With w, v and u as the displacement components of the middle surface of the shell in the radial, tangential andaxial directions respectively, the elements of {P*} are given by Herrman and Mirsky (1957) as:
,1P,1
h/2
h/2-
*
2
2/
_h/2
*
1 zh
zz
R
z
R
zP
,1P,1
h/2
h/2-
*
4
2/
2/
*
3 zx
h
hzR
z
R
zzP
2/
2/
*
5 1h
hzxR
zzP
For the evaluation of {P*}, ij at z = (h/2) must be determined in the terms of incident and scattered field in thesurrounding ground. The total displacement field in the ground is written as
d=d(i) + d(s)
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Where i and s represents the incident and scattered parts respectively. By solving the wave equation in the
surrounding infinite medium the components of incident and scattered fields can be written as (Chonan, S.,1981):
)(expcos
1
5
3
'
11
'
)( ctxin
BR
r
Ir
R
n
BR
rIB
R
rI
d
n
nn
i
r
)(expsin
)(
5
'
311
)( ctxin
BR
rI
BR
rI
r
RinB
R
rI
r
Rrn
d
n
nn
i
)(expcos3211)( ctxinxBR
rIB
R
rIid
nn
i
x
(3)
Where B1 ='1B /R, B3 =
'3B /R
2 and B5 ='5B /R. (
) denotes differentiation with respect to the argument of
the Bessel functions. The constants B1, B3 and B5 depend on the parameters of the incident wave and may be
expressed as:
3n52
2n3
1
11n1
A)1(B,
Ai)1(B,
Ai)1(B (4)
)ctx(iexpncosBR
rK
r
RnB
R
rKiB
R
rKd 6n4
'n12
'n
)s(r
)ctx(iexpnsinB
R
rK
BR
rK
r
RinB
R
rK
r
Rn
d
6'n
4n12n
)s(
)ctx(iexpncos
BR
rKBR
rKid 4n
22n1)s(
x
(5)
Where dr, d, dx are components of displacement vector,
321 ;; AAA
are amplitudes of P, SV and SH wave
respectively.
Here ./Band/,/ '662'
44
'
22 RBRBBRBB are constants.Stress fields due to the incident wave can be obtained by plugging above equations into the stress-displacement
relations of the medium, and are given by:
)ctx(iexpncos
BR
rI
r
R
R
rI
r
Rn2
BR
rIi2
BR
rI2
R
rI2
R
5n'n
3
"
n
2
1
1"n
2n
22
21
)i(
rr
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)(expsin2
2
5
2
'"2
3
'
1
1
'
)(ctxin
B
R
rI
r
nR
R
rI
r
R
R
rI
BR
rI
r
R
R
rI
r
Rin
BR
rI
R
rI
r
R
r
Rn
R
nnn
nn
nn
i
r
)(expcos
22
51
'2
2
2
11
'
1
)(ctxin
BR
rI
r
Rin
R
rIB
R
rIi
Rn
nn
i
rx
)(expcos
2
2
22
6
'
4
"2
1
2
"22
2
2
1
)(ctxin
BR
rK
r
R
R
rK
r
Rn
BR
rKi
BR
rK
R
rK
R
nn
n
nn
s
r
)(expsin2
2
6
2
'"2
4
'
1
2
'
)(ctxin
BR
rK
r
nR
R
rK
r
R
R
rK
BR
rK
r
R
R
rK
r
Rin
BR
rK
R
rK
r
R
r
Rn
R
nnn
nn
nn
s
r
)(expcos
22
61
4
'2
2
2
12
'
1
)(ctxin
BR
rK
r
Rin
BR
rKB
R
rKi
Rn
nn
s
rx
(6)
WhereIn ( ) Modified Bessel function of first kind
Jn ( ) Bessel function of first kind
Kn ( ) Modified Bessel function of second kindWith the help of above equations the stresses at the outer surface of the shell (z = h/2 or r = R + h/2) can be
obtained. Thus {P*} in Eq. (2) can be determined.
Now the mid plane displacement and slopes are assumed to be of the form;
0ww cosn exp[i(x-ct)]
0vv sinn exp[i(x-ct)]
0uu cosnexp[i(x-ct)] (7)
Plugging Eq. (7) in Eq. (2) and (6) along with the expression for {P*}, a set of six simultaneous algebraicequations are obtained.Three more equations are obtained by imposing the boundary conditions at the inner and
outer surfaces of the shell: i.e.,
2/
)()()( hRr
s
r
i
r ddw
2/
)()()()2/( hRr
siddhv
2/
)()( )()2/(hRr
s
x
i
xx ddhu (8)Boundary conditions at the outer surface of the shell (r = R + h/2) are obtained by assuming that the shell and
the continuum are joined together by a bond which is thin, elastic and inertia less. This implies that the stress at
the shell-soil interface is continuous. To take the elasticity of the bond into account, the stresses in the bond are
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assumed proportional to relative displacements between the shell and continuum. shear modulus of medium
and density of shell material.
2/2/ ])()([()( hRrxs
xi
xxxhRrrx Rrut
ZS
2/2/ )])([()( hRrsr
irrrhRrrr w
tZS
2/2/ ])()([()( hRrsi
hRrr Rrut
ZS
(9)
,.RSr
R
RS .
and RSxx
.
are the non-dimensionalized stiffness coefficient of the bond in
radial , tangential and axial direction respectively.
1cZrr
1cZ
and1cZx
x
are the non-dimensionalized damping coefficient of the bond in
radial, tangential and axial direction respectively.
Thus a total of six algebraic equations are obtained. These six equations when simplified give the final response
equation, which may be put into the form
3523110 BBFB}}{{ FFUQ (10)
;6423
6
3
4
3
2
3
0
3
0
3
0 T
T
BBBUVWAB
AB
AB
Au
Av
AwU
Where [Q] is a (6x6)) matrix and {F1}, {F2} and {F3} are (61) matrices. But for the response of shear verticalwave the amplitudes due to shear waves
1B and 5B would be zero so the effect of {F1} and {F3} matrices
would be eliminated. Putting values of 1B = 5B = 0 and substituting values of 3B Eq. (6) becomes as
22
20 F)1(}}{{
AiUQ
n
(11)Elements of Matrix F
;222
1 12
122
21
31 nn II
hF
;
21
2
21
2
11132
h
IInhF nn
;22
1 113
3 nIih
F
;
21
, 13513
4
h
InFIF n
n
)](2)(){2[)({ 1"2
1
2
2
2
1
1
1
'3
6
nn
rr
rrn II
iIF
Elements of matrix Q in Eq. (11) and matrix L in Eq. (2) are same as provided in Rajput et.al (2010).
3. Results and DiscussionsResults are presented for a transversely isotropic shell with r- as the plane of isotropy.Consequently zEE , xxz GG , xzx , zz , )1(2/EG zz . Thus wehave 23 , and )1(2/E/G z1xz4 . In addition 3.0xz has beentaken in the numerical calculations. Different values of 1 and 2 used are as follows 1 = 0.5, 0.1, 0.05 and
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2 = 0.1,05,.02, has been varied between 0.1 to 10.0 to take into account different soil conditions around thepipe, = 0.1 corresponds to the soft soil , =1 for medium soil and for hard rocky soil = 10.0 For all the
values of , m = 0.25 has been assumed. Thickness to radius ratio of the shell (h ) has been taken as 0.01
and the density ratio of the surrounding medium to the shell ( ) has been taken as 0.75. Non-dimensional
amplitude of the middle surface of the shell in the radial and axial directions (W andU) have been plottedagainst the non-dimensional wave number of the incident shear vertical wave (SV-wave) ( = 2R/). The shellresponse has been shown non-axisymmetric (n = 1) flexural mode and axisymmetric mode taking stiffness
coefficient (x
r ), damping coefficient ( x r ) as bond parameter and 1 , 2 as the shell
orthotropic parameters.
The bond parameters have been varied between zero and infinity,x =
= r = x = = r = 0
corresponds to perfect bonding between the shell and surround shell.
Figure 1 to figure 3 shows the effect of stiffness coefficient on the axial displacement of the shell at different
angle of incidence of wave, at different condition of the soil with increasing wave number
Figure1. Axial displacement )(U vs. wave number ( ) withx = 0.1, 10,100 as
Parameter
10x 100x
1.0x
1.0x
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Figure2. Axial displacement )(U vs. wave number ( ) withx = 0.1, 10,100 as
Parameter
Figure3. Axial displacement )(U vs. wave number ( ) withx = 0.1, 10,100 as
Parameter
100,10,1.0x
1.0x
100x
1.0x
100x
1.0x
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Figure 4to figure 6 shows the effect of damping coefficient on the axial displacement of the shell at different
angle of incidence of wave, at different condition of the soil with increasing wave number
Figure4. Axial displacement )(U vs. ( ) withx = 0.1, 10,100 as Parameter
Figure5. Axial displacement )(U vs. wave number ( ) withx = 0.1, 10,100 as
Parameter
100,10,1.0x
100x
1.0x
100,10,1.0x
100,10,1.0x
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Figure6. Axial displacement )(U vs. wave number ( ) withx = 0.1, 10,100 as
Parameter
Figure 7 to figure 10 shows the effect of stiffness coefficient on the radial displacement of the shell at different
angle of incidence of wave, at different condition of the soil with increasing wave number
Figure7. Radial displacement )W( vs. wave number ( ) withr = 0.1, 10,100 as
Parameter
100,10,1.0x
100x
100,10,1.0x
1.0r
100r
100,10,1,0r
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Figure8 Radial displacement )(W vs. wave number ( ) withr = 0.1, 10,100 as
Parameter
Figure9 Radial displacement )(W vs. wave number ( ) withr = 0.1, 10,100 as
Parameter
100r1.0r
100,10,1,0r
100,10,1.0r
100,10,1.0r 1.0r
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Figure10. Radial displacement )(W vs. wave number ( ) with r = 0.1, 10,100 asParameter
Figure11. Radial displacement )(W vs. wave number ( ) withr = 0.1, 10,100 as
Parameter
100,10,1.0r
100,10,1.0r
100,10,1.0r
100,10,1.0r
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Figure12. Radial displacement )(W vs. ( ) with r = 0.1, 10,100 as Parameter.
Conclusions
Based on the results presented following general conclusions could be drawn-.
Unlike the behavior observed in the non-axisymmetric mode a loose contact between the shell andthe surrounding soil does not always give more shell displacement as compared to those for a
perfectly bonded shell. Therefore assuming a perfect bond may not always lead to a safe and
conservative estimate of displacements.
Effects of bond parameters depend upon the soil conditions and the incidence angle andwavelength of the incident wave. In hard and rocky surroundings bond imperfections show more
prominent effects on the shell response.
The flexural mode response assumes considerable importance in soft soil condition and at higher
apparent wave speed. Flexural mode response due to incident shear vertical wave is significant only at large angle of
incidence.
For large angle of incidence radial deflection is higher in flexural modes at larger wavelength but atthis value it is the axial deflection in axisymmetric mode, which is higher. Thus for larger wavelength
axisymmetric mode is more important because the most common cause of pipeline failure is excessive
axial deformation, while at smaller wavelength the flexural mode has much importance for axialdisplacement.
Both the shell orthotropic parameters influence the radial displacement equally well but 2 has astronger influence on the axial displacement than 1 . As the Stiffness and the damping coefficient isincrease the axial and radial response increase.
Axial deflection and radial deflection both are increase when the value of bonding parameter stiffnesscoefficient (
x r), damping coefficient (
x r) increase from zero to infinity (perfect
to imperfect bonding) as parameters.
Nomenclature
A Amplitude of the plane wave
321 ;; AAA
Amplitudes of P-SV-SH wave respectively
B1 .,B6 Arbitrary constants
'6
'1 B.......B Arbitrary constants
100,10,1.0r
100r
1.0r
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c Apparent wave speed along the axis of the shell
dr, d, dx Components of displacement vector
ER, E, EX Young modulus of the shell.
xr eee ;;
Unit vector in co-ordinate direction
{F1}, {F2}, {F3} Column vector
Gx, Gxz, Gz Shear moduli of the shell
H Vector displacement in the medium
H'
Displacement potential corresponding to SV wave
Hr, H, HX Components of vector potential(Hx corresponding to SH wave)
h Thickness of the shell
h (=h/R) Non dimensional thickness of the shellIn ( ) Modified Bessel function of first kind
Jn ( ) Bessel function of first kindKn ( ) Modified Bessel function of second kind
kx, k Shear correction factor
{L} Matrix operator
xxxx MMMM ;;;
Stress resultant moments
xxxx NNNN ;;; Stress resultantsn Mode shape number in the tangential direction{P*} Column matrix
R Mean radius of the shell
r Radial coordinate
t Time
U Non-dimensional amplitude of the shell in axial directionu Displacement of the shell middle surface in the axial directionu0 Displacement amplitude of the shell middle surface in the axial direction
uz, u, ux Displacement component of a point in the shell
V Non-dimensional amplitude of the shell in the tangential directionv Displacement of the shell middle surface in the tangential direction
v0 Displacement amplitude of shell middle surface in tangential direction
W Non-dimensional amplitude of the shell in the radial directionw Displacement of the shell middle surface in the radial directionw0 Displacement amplitude of the shell middle in the radial directionx Coordinate along the shell axis
z Coordinate normal to middle surface of the shell
Angle of incidence of the wave(=2R/) Non-dimensional wave number of incident wave1, 2, 3, 4 Non-dimensional shell orthotropic parameters of the shellz Normal to middle surface of the shell Tangential direction Wave length of the incident wave Lames constant Modulus of rigidity
xzGNon-dimensional modulus of rigidity of medium
m Poisson ratio the mediumx, x, z, zx, xz Poisson ratios of the shell
(= 2 cos / ) Apparent wave number Density of the shell materialm Density of the medium
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m Non-dimensional density of the medium
ij Components of stress tensor Scalar displacement potential in the mediumx Angle of rotation in r-x plane Symmetry constant =1for n=0, =2 for n=1xo Amplitude ofx Angle of rotation in r- plane
Subscripts
m Medium
r Radial directionx Axial direction
z Normal to middle surface of the shell
Tangential direction
Superscripts
i Incident wave
s Scattered wave
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Biographical notes
Rakesh Singh Rajput received M. Tech. from NIT Allahabad, India and pursuing Ph.D. from Rajeev Gandhi Technological University,
Bhopal, India. He is a Reader in Mechanical Engineering, and working with Department of Technical Education (DTE), Govt. of M. P.,
India. His research interests include machine design and mechanics. He has written book on Engineering Mechanics.Dr. Sunil Kumar is a Professor of Mechanical Engineering, Rajeev Gandhi Technological University, Bhopal, India. He has received M.
Tech from IT-BHU, Varanasi, India and Ph. D. from IIT Delhi, India. He has more than 20 years of experience in teaching and research. His
current area of research includes Combustion mechanics and System Dynamics.
Dr. Alok Chaube is a Professor of Mechanical Engineering, Rajeev Gandhi Technological University, Bhopal, India. He has received M.Tech from IIT Delhi, India and Ph. D. from IIT Roorkee, India. He has more than 20 years of experience in teaching and research. His
current area of research includes Fluid mechanics and System Dynamics.