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ARTICLE IN PRESS
0267-7261/$ - se
doi:10.1016/j.so
�CorrespondE-mail addr
Soil Dynamics and Earthquake Engineering 26 (2006) 909–921
www.elsevier.com/locate/soildyn
3-D shell analysis of cylindrical underground structures underseismic shear (S) wave action
George P. Kouretzisa, George D. Bouckovalasa,�, Charis J. Gantesb
aDepartment of Geotechnical Engineering, Faculty of Civil Engineering, National Technical University of Athens, 9,
Iroon Polytehniou Str., GR-15780, Athens, GreecebDepartment of Structural Engineering, Faculty of Civil Engineering, National Technical University of Athens, 9,
Iroon Polytehniou Str., GR-15780, Athens, Greece
Accepted 5 February 2006
Abstract
The 3-D shell theory is employed in order to provide a new perspective to earthquake-induced strains in long cylindrical underground
structures, when soil-structure interaction can be ignored. In this way, it is possible to derive analytical expressions for the distribution
along the cross-section of axial, hoop and shear strains and also proceed to their consistent superposition in order to obtain the
corresponding principal and von Mises strains. The resulting analytical solutions are verified against the results of 3-D dynamic FEM
analyses. Seismic design strains are consequently established after optimization of the analytical solutions against the random angles
which define the direction of wave propagation relative to the longitudinal structure axis, the direction of particle motion and the
location on the structure cross-section. The basic approach is demonstrated herein for harmonic shear (S) waves with plane front,
propagating in a homogeneous half-space or in a two layer profile, where soft soil overlays the bedrock.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Earthquakes; Pipelines; Tunnels; Shell theory; Strain analysis; Design
1. Introduction
It is generally acknowledged that underground struc-tures suffer less from earthquakes than buildings on theground surface. However, recent earthquakes in Kobe(1995) [1,2], Chi-Chi (1999) [3–5] and Duzce (1999) causedextensive failures in buried pipelines and tunnels, revivingthe interest in the associated analysis and design methods.
In summary, most current analytical methodologies arebased on two basic assumptions. The first is that theseismic excitation can be modeled as a train of harmonicwaves with plane front, while the second assumption statesthat inertia and kinematic interaction effects between theunderground structure and the surrounding soil can beignored. Theoretical arguments and numerical simulationsplead for the general validity of the former statementregarding inertia effects [6], while the importance of
e front matter r 2006 Elsevier Ltd. All rights reserved.
ildyn.2006.02.002
ing author. Tel.: +302107723870; fax: +30 2108068393.
ess: [email protected] (G.P. Kouretzis).
kinematic interaction effects [7,8] can be checked on acase-by-case basis via the flexibility index
F ¼2Emð1� n2l Þ D=2
� �3Elð1þ nmÞt3s
, (1)
where Em is Young’s modulus of the surrounding soil, El isYoung’s modulus of the structure material, nm is Poisson’sratio of the surrounding soil, nl is Poisson’s ratio of thestructure material, ts is the thickness of the cross-section,and D is the structure diameter. The flexibility index isrelated to the ability of the lining to resist distortion fromthe ground [7,9]. Values of the flexibility index higher than20 are calculated for most common tunnels and pipelines,indicating that ignoring overall the soil-structure interac-tion is a sound engineering approach [10,11].Using the previous assumptions, Newmark [12] calcu-
lated axial strains due to longitudinal and bendingdeformation provoked by shear (S) and compressional(P) waves propagating parallel to the structure axis. Kuesel[13] and Yeh [14] extended the relations of Newmark to
ARTICLE IN PRESSG.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921910
account for obliquely incident shear and Rayleigh waves,introducing the angle of wave propagation relative to thelongitudinal structure axis as a random problem variable.Hoop and shear strains were addressed with considerabletime lag relative to axial strains. Namely, more than adecade later, St John and Zahrah [15] presented analyticalrelations for these strain components, while Wang [7]underlined their importance relative to axial strains andproposed that the design of the tunnel lining shouldconform to the hoop strains resulting from S wavespropagating transversely to the structure axis.
Other, more advanced, analytical methodologies simu-late soil-structure interaction effects, by employing thebeam-on-elastic foundation approach [15] or modeling theunderground structure as a cylindrical shell embedded inan elastic half-space [16–19], and account for slippage atthe soil-structure interface [20,21]. Furthermore, Manolisand Beskos [22] provide a comprehensive overview ofnumerical methods employed for detailed dynamic ana-lyses of underground lifeline facilities [23–27], which canalso take into account for the effect of the free groundsurface on wave scattering, or complex soil stratificationand non-linear behavior, at the expense of handiness.Nevertheless, the above analytical and numerical advance-ments are aimed at case-specific analyses, while currentdesign guidelines [6,28] suggest the use of simpler, New-mark and Kuesel-type of analyses.
Concluding this brief state of the art review, it is pointedout that:
(a)
plane ofwave propagation
longitudinal axisof the structure
The basic analytical solutions presented earlier, whichform the basis of current design guidelines, computeessentially free-field strains and consequently transformthem to peak axial, peak hoop and peak shear strain atspecific points of the cross-section. However, thedistribution of the above strains along the cross-sectionis not known and therefore a systematic structuralanalysis is not possible. Most importantly, as thelocations of peak axial, hoop and shear strains do notcoincide, it is not possible to superimpose the corres-ponding peak strains in order to obtain the overallmaximum strain values (e.g. the principal or the vonMises strains), without being over-conservative.
(b)
SV
wave-structureplane
y
β
ϕ
A second simplification is that strains from shearwaves, the most likely threat for underground struc-tures, are computed for the special 2-D case, where theparticle motion is parallel to the plane defined by thelongitudinal axis of the structure and the direction ofwave propagation. In this way, the random problemvariables are reduced to the angle of seismic waveincidence, and the overall complexity of the analyticalcomputations is minimized.
SHxz
(c)z'S
Fig. 1. Propagation of a shear wave in a plane randomly oriented
relatively to the structure axis.
Finally, published solutions concern uniform geologi-cal formations, and consequently it is not explicitlystated what ground parameters should be used for thecomputation of each strain component when a twolayer system (e.g. soil over bedrock) is encountered.
This uncertainty is reflected into current design guide-lines. Namely, the design guidelines for undergroundpipelines [6,28] call on seismological evidence tounivocally suggest the use of an ‘‘apparent seismicwave velocity of the bedrock’’, at least equal to 2000m/s, for the computation of axial strains, regardless oflocal soil conditions. On the other hand, the designguidelines for tunnels in soil provided by Wang [7] givehead to hoop strains due to vertically propagatingshear (S) waves, and suggest the use of the seismic wavevelocity of the soil rather than that of the bedrock.
Seeking a new perspective to the problem, the 3-D shelltheory has been used for a consistent calculation of thenormal (axial and hoop) and the shear strain distributionover the entire cross-section of a cylindrical undergroundstructure, as well as the resulting principal and von Misesstrains. Due to the relative complexity of the mathematics,the resulting relations are verified against results fromdynamic finite element analyses.To preserve the length limits of the presentation, the
basic methodology and the findings of this study aredemonstrated herein for the generic case of shear (S) wavespropagating at a random direction relative to the structureaxis (Fig. 1). It is common practice to analyze such wavesvectorially into two specific components: (a) an SV wavewith particle motion perpendicular to the plane formed bythe direction of wave propagation and the longitudinalstructure axis, and (b) an SH wave with particle motionparallel to the aforementioned plane. Thus, strains fromrandomly oriented SH and SV seismic waves are addressedseparately, and consequently superimposed to give theoverall strains. The detailed derivation is provided for thecase of uniform ground. The case of undergroundstructures in a soil layer overlaying the bedrock isexamined next, as a variation of the above basic case.
ARTICLE IN PRESS
εh
εαγ
γ
Fig. 2. Strains in underground structures modeled as thin-walled
cylindrical shells.
Amax
direction ofwave propagation
L/cosϕ
L/sinϕ
z
x
z'
structure axis
-Amaxsinϕ
Amaxcosϕ
x'
ϕ
Fig. 3. Vectorial analysis of an obliquely impinging wave into two
apparent waves: one propagating along and the other propagating
transversely to the undeformed structure axis.
G.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921 911
2. Strains from SH-waves in uniform ground
To aid the following presentation, Fig. 2 defines thestrains that are considered in the analysis of thin cylindricalshells (membranes), namely,
ea ¼ ez ¼@uz
@zðaxial strainÞ, (2)
eh ¼ eyy ¼1
r
@uy
@yþ
ur
rðhoop strainÞ, (3)
g ¼ gyz ¼1
r
@uz
@yþ@uy
@zðshear strainÞ, (4)
where uz, ur and uy are the displacement componentsimposed by the shear wave, in a cylindrical coordinatesystem fitted to the longitudinal axis of the structure. Dueto the small thickness of the cross-section, the remainingthree strain components (one radial and two shear), arecustomarily neglected as they are fairly insignificant.
For an SH wave propagating in a direction z0 whichforms an angle j with the z axis of the structure (Fig. 3),the displacement field is described as
ux0 ¼ Amax cos b sin2pL
z0 � Ctð Þ
� �, (5)
uy0 ¼ 0, (6)
uz0 ¼ 0, (7)
where b is the angle between the particle velocity vectorand the propagation plane, Amax is the peak particledisplacement of the seismic motion, C is the shear wavepropagation velocity, L is the wavelength and t stands fortime. As initially introduced by Kuesel [13], propagation ofan SH wave at an angle j relative to the structure axis isequivalent in terms of strains to the following apparentwaves:
�
An SH wave propagating along the structure axis, withwavelength L/cosj, propagation velocity C/cosj andmaximum amplitude Amax cos b cosj. � A P wave propagating along the structure axis, withwavelength L/cosj, propagation velocity C/cosj andmaximum amplitude �Amax cos b sinj.
�
A P wave propagating transversely to the structure axis,with wavelength L/sinj, propagation velocity C/sinjand maximum amplitude Amax cos b cosj. � An SH wave propagating transversely to the structureaxis, with wavelength L/sinj, propagation velocity C/sinj and maximum amplitude Amax cos b sinj.
Thus, strains may be computed separately for each oneof these apparent waves and consequently superimposed.
2.1. Apparent SH wave propagating along the axis of the
structure
Let us consider the Cartesian coordinate system of Fig.3, and an axially propagating SH wave inducing motion inthe xz plane. For harmonic waves with plane front, groundmotion can be described as
ux ¼ Amax cos b cos f sin2p
L= cos fz�
C
cos ft
� �� �. (8)
In a cylindrical coordinate system fitted to the longitudinalaxis of the structure (Fig. 4a), ground displacement can bedecomposed into the following radial and tangentialcomponents:
ur ¼ Amax cos b cosf sin y sin2p
L= cosfz�
C
cosft
� �� �,
(9)
uy ¼ Amax cos b cosf cos y sin2p
L= cosfz�
C
cosft
� �� �.
(10)
According to the ‘‘thin shell’’ theory adopted herein, aswell as Eqs. (2)–(4), the above displacement field results in
ARTICLE IN PRESS
x
y
ux
ur=uxsinθ
uθ=uxcosθr
x=rsinθ
r
x=rsinθ
y
x
uy
uθ=uysinθur=uycosθ
(a) (b)
θ
θ
θ
θ
Fig. 4. Definition of the coordinate system for the calculation of strains due the propagation of (a) an oblique SH wave and (b) an oblique SV wave.
(a) (b)
x
z
x
z
Fig. 5. Effect of interface friction on structure displacements: (a) rough interface, (b) smooth interface model.
G.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921912
pure shear strain (i.e. ea ¼ eh ¼ 0) with amplitude:
g ¼Vmax cos b
C� cos2f � cos y, (11)
where Vmax ¼ ð2pAmax=LÞC is the peak particle velocity ofthe harmonic shear wave.
In the above derivation, the relative displacement at thesoil-structure interface uz is considered to be zero, implyingthat the shell is ‘‘rough’’ and does not permit slippage tooccur (Fig. 5a). If the shell is ‘‘smooth’’, i.e. it is free to slipover the surrounding soil, then g ¼ 0 and according to Eq.(2) an axial displacement will emerge (Fig. 5b), which isequal to
uz ¼ �x@ux
@z(12)
or
uz ¼ � r sin y �2pAmax cos bcos2f
C
� cos2p
L= cosfz�
C
cosft
� �� �. ð13Þ
Hence, shear strain g will now give its place to an axialstrain with amplitude equal to
ea ¼ r sin y �amax � cos b
C2cos3f, (14)
where amax ¼2pL
C� �2
Amax is the peak ground acceleration.Note first that the two interface models examined
previously, which are shown schematically in Fig. 5,cannot physically co-exist and consequently the aboveaxial and shear strains should not be superimposed. Inaddition, the axial strain corresponding to the ‘‘smooth’’interface model is roughly one order of magnitude less thanthe shear and principal strains ðe1;3 ¼ �g=2Þ correspondingto the ‘‘rough’’ interface model. Thus, in extent of theseobservations, the ‘‘rough’’ interface assumption will beadopted hereafter as more conservative.
2.2. Apparent P wave propagating along the axis of the
structure
In the coordinate system of Fig. 4a, the harmonicdisplacement induced to the shell by an axially propagating
ARTICLE IN PRESSG.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921 913
P wave can be expressed as
uz ¼ �Amax cos b sinf sin2p
L= cosfz�
C
cosft
� �� �. (15)
According to Eqs. (4)–(6), this displacement will merelycause axial strain (i.e. eh ¼ g ¼ 0), with amplitude
ea ¼ �Vmax cos b
C� sinf � cosf. (16)
2.3. Apparent P wave propagating transversely relatively to
the axis of the structure
This apparent wave will enforce a harmonic displace-ment on the shell that is expressed as
ux ¼ Amax cos b cosf sin2p
L= sinfx�
C
sinft
� �� �. (17)
The vectorial analysis of ux to the cylindrical coordinatesystem of Fig. 4a yields the following displacementcomponents in the radial and tangential directions:
ur ¼ Amax cos b cosf sin y sin2p
L= sinfx�
C
sinft
� �� �,
(18)
uy ¼ Amax cos b cosf cos y sin2p
L= sinfx�
C
sinft
� �� �.
(19)
In this case, only hoop strain will develop, withamplitude equal to
eh ¼Vmax � cos b
C� sinf cosf � cos2y (20)
as the axial and shear strain components resulting fromEqs. (2) and (4) become zero (i.e. ea ¼ g ¼ 0).
2.4. Apparent SH wave propagating transversely relatively
to the axis of the structure
The propagation of a transverse SH wave induces thefollowing harmonic axial displacement on the structure:
uz ¼ �Amax cos b sinf sin2p
L= sinfx�
C
sinft
� �� �, (21)
which, according to Eqs. (2)–(4), gives shear strain amplitude
g ¼ �Vmax cos b
C� sin2f � cos y (22)
and ea ¼ eh ¼ 0.
3. Strains from SV waves in uniform ground
In Cartesian coordinates, a shear SV wave propagatingalong z0, at an angle j relatively to the axis z of thestructure (Fig. 3), induces the following displacement field:
ux0 ¼ 0, (23)
uy0 ¼ Amax sin b sin2pL
z0 � Ctð Þ
� �, (24)
uz0 ¼ 0. (25)
In terms of displacement, this wave can be decomposedto the following apparent waves:
�
An SV wave propagating along the structure axis z, withwavelength L/cosj, propagation velocity C/cosj andmaximum amplitude Amax sin b. � An SV wave propagating transversely relatively to thestructure axis, with wavelength L/sinj, propagationvelocity C/sinj and maximum amplitude Amax sin b.
3.1. Apparent SV wave propagating along the axis of the
structure
The displacement applied to the structure by itssurrounding medium is equal to
uy ¼ Amax sin b sin2p
L= cosfz�
C
cosft
� �� �, (26)
or, in the cylindrical coordinate system of Fig. 4b,
ur ¼ Amax sin b cos y sin2p
L= cosfz�
C
cosft
� �� �, (27)
uy ¼ Amax sin b sin y sin2p
L= cosfz�
C
cosft
� �� �. (28)
According to the strain definitions provided earlier(Eqs. (2)–(4)), the above displacements result in pure shearstrain on the shell, with amplitude equal to
g ¼Vmax sin b
C� cosf � sin y (29)
while the corresponding axial and hoop strain componentsbecome zero (i.e. ea ¼ eh ¼ 0).
3.2. Apparent SV wave propagating transversely relatively
to the axis of the structure
The corresponding displacement uy is now
uy ¼ Amax sin b sin2p
L= sinfx�
C
sinft
� �� �, (30)
which can be decomposed in the cylindrical coordinatesystem of Fig. 4b, to
ur ¼ Amax sin b cos y sin2p
L= sinfr sin y�
C
sinft
� �� �, (31)
uy ¼ �Amax sin b sin y sin2p
L= sinfr sin y�
C
sinft
� �� �.
(32)
In this case, the axial and shear strain componentsbecome zero (i.e. ea ¼ g ¼ 0) while the remaining hoop
ARTICLE IN PRESSG.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921914
strain amplitude is:
eh ¼Vmax � sin b
2C� sinf � sin 2y. (33)
The strain amplitudes derived herein for the SV waves arein phase and can be added algebraically to the correspond-ing ones derived earlier for the SH waves. Thus, theexpressions for the amplitude of total structure strains inuniform ground become
ea ¼ �Vmax
2Ccos b � sin 2f, (34)
eh ¼Vmax
2Ccos b � sin 2f � cos2yþ sin b � sinf � sin 2y� �
,
(35)
g ¼Vmax
Ccos b � cos 2f � cos yþ sin b � cosf � sin yð Þ, (36)
where angles j, y and b are defined in Figs. 1, 3 and 4.
4. Soft soil effects
Consider the case of an S wave propagating in a two-layered half-space, consisting of a soft soil layer overlyingthe semi-infinite bedrock (Fig. 6). According to Snell’s law,the angle of wave propagation in soil aS is related to theangle of wave propagation in rock aR as
cos aS ¼ cos aRCS
CR, (37)
where CR is the shear wave propagation velocity in thebedrock and CS is the shear wave propagation in the softsoil layer. In the following presentation we neglect the Pwave resulting from the refraction of the SV component ofthe S wave at the soil–bedrock interface. This assumptiondraws upon the fact that P waves propagate with a velocitythat is substantially larger than that of S waves, and
LS/sinαS
αS SOFT SOIL
LS/cosαS
LR/cosαR
αS
αR
undergroundstructure axis(projection)
BEDROCK
Fig. 6. Analysis of a wave refracted at the soft soil–bedrock interface into
a vertical and a horizontal apparent wave.
therefore both waves do not arrive simultaneously at thestructure.According to the apparent wave concept presented
before, the propagation of a shear wave from the bedrockto the ground surface through the soft soil layer isequivalent in terms of strains with the following apparentwaves (Fig. 6):
�
Fig
CR
An apparent wave propagating vertically, with wave-length LS/sin aS and propagation velocity CS/sin aS.
� An apparent wave propagating horizontally, withwavelength LS/cos aS and propagation velocity CS/cos aS.
We can rewrite the wavelength of the horizontalapparent wave with the aid of Eq. (37), as
LS
cos aS¼
LR CS=CR
� �cos aR CS=CR
� � ¼ LR
cos aR, (38)
i.e. the horizontal apparent wave at the ground surfacepropagates with the same velocity and has the samewavelength as the apparent wave formed due to the timelag of waves impinging at the soft soil–bedrock interface(Fig. 6).Moreover, it is
sin aS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� cos2aS
p¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� cos2aR
CS
CR
� �2s
. (39)
Fig. 7 draws the variation of sin aS with the angle aR, fordifferent values of the CS/CR ratio. Observe that forCS=CR ¼
13we can reasonably assume that sin aSffi1. This
implies that the vertical apparent wave propagates in thesoft soil layer with velocity and wavelength that are moreor less equal to the shear wave propagation velocity andwavelength in soft soil, CS and LS, respectively.
0 10 20 30 40 50 60 70 80 90αR
0.84
0.88
0.92
0.96
1
sinα
S
CS/CR=0.5
0.333
0.25
0.125
. 7. Variation of sin aS with the angle aR, for different values of the CS/
ratio.
ARTICLE IN PRESS
0 0.4 0.8 1.2 1.6time
-1.2-0.8-0.4
00.40.81.2
u Z, u
Y
30m
D=1.00m
ts=0.002m
1m
zi
CROSS-SECTION
z
x
y
SV
SH
GROUNDDISPLACEMENT
2
Fig. 8. Mesh discretization of the 3-D numerical model, presented
together with the dynamically imposed displacements at a random node,
for the uniform ground analysis (this figure is out of scale).
G.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921 915
In the following, strains due to vertical and horizontalapparent waves are computed separately, using theexpressions for the generic case of uniform ground andconsequently superimposed to provide total strains.
The vertically propagating apparent S wave impinges atthe structure axis with an angle of incidence fV
¼ p2, while
the particle motion forms a random angle bV with thevertical plane passing through the structure axis. Substitut-ing these angles in Eqs. (34)–(36) for uniform ground yields
ea ¼ 0, (40)
eh ¼Vmax
2CSsin bV � sin 2yV, (41)
g ¼ �Vmax
CScos bV � cos yV, (42)
where yV is the polar angle in the cross-section.The horizontally propagating apparent S wave impinges
at the structure axis at a random angle jH, with the particlemotion forming an angle bH ¼ p=2� aS with the horizon-tal plane, while yH ¼ yV � p=2 is the polar angle in thecross-section. In this case, Eqs. (34)–(36) for uniformground give
ea ¼ �Vmax
2CR� cos aR sin 2fH, (43)
eh ¼Vmax
2CR
� cos aR � sin 2fH� cos2yH �
CS
CR� cos2aR � sinf
H� cos 2yH
� �,
ð44Þ
g ¼Vmax
CR
� cos aR � cos 2fH� cos yH þ
CS
CRcos2aR � cosf
H� sin yH
� �.
ð45Þ
Adding of the above (in phase) strain amplitudes, from theapparent vertically and horizontally propagating waves,leads to the following expressions for the total strainamplitudes:
ea ¼ �Vmax
CS
CS
CRcos aR sin 2f�
� �, (46)
eh ¼Vmax
2CS
� sin b� sin 2y� þCS
CR� cos aR � sin 2f
�� sin2y�
�
�CS
CR
� �2
� cos2aR � sinf�� sin 2y�
!, ð47Þ
g ¼Vmax
CS
� � cos b� � cos y� þCS
CR� cos aR � cos 2f
�� sin y�
�
�CS
CR
� �2
cos2aR � cosf�� cos y�
!, ð48Þ
where y� ¼ yV ¼ yH þ p=2, b� ¼ bV, andf� ¼ fH.
5. Numerical verification
Validation of the above analytical relations for uniformground, as well as for soft soil over bedrock, isaccomplished through comparison with numerical resultsfrom elastic 3-D dynamic analyses, performed with the aidof the commercial FEM program ANSYS [29]. It isclarified in advance that the aim of this comparison isnot to check the validity of the assumptions but merely tocheck the complex mathematics that underlay the presentcomputation of shell strains.
5.1. Uniform ground conditions
Fig. 8 illustrates the geometry that was analyzed and theassociated ground motion. Namely, the undergroundstructure is modeled as 3-D hollow cylinder with 30mlength, 1m diameter and 0.002m wall thickness. It isdiscretized into 16 equal shell elements per cross-section,each of 1m length, using the SHELL63 element [29] thathas both membrane and bending capabilities. The structurematerial is considered to be isotropic linear elastic withspecific weight gl ¼ 75KN/m3, Young’s modulusEl ¼ 210GPa and Poisson’s ratio of vlffi0. Note that theexact values of gl, El and vl are of absolutely no importanceto the numerical results, as seismic strain components ea, eh
and g are directly related to imposed displacements alone(Eqs. (2)–(4)). As the underground structure conforms tothe ground motion, the displacements of each node were
ARTICLE IN PRESSG.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921916
set equal to the corresponding ground displacement. Theground motion is harmonic with unit amplitude, featuringa propagation velocity of C ¼ 100m/s, a period ofT ¼ 0.1 s and a propagation direction that forms an anglej ¼ 301 with the structure axis. Thus, to apply the grounddisplacements correctly, a global coordinate system rotatedby 301 relatively to the structure axis was used, and thefollowing dynamic displacement boundary conditions wereapplied on each shell node:
uy0;i ¼ Amax sin b sin2pL
z0i � Ct� �� �
, (49)
uz0;i ¼ Amax cos b sin2pL
z0i � Ct� �� �
, (50)
uy0;i ¼ yx0;i ¼ yy0;i ¼ yz0;i ¼ 0, (51)
with Amax ¼ 1.0 and the random angle b considered to be751.
The boundary conditions described above imply that, asin the analytical solution, the shell fully conforms to theground motion and, consequently, that kinematic andinertia soil-structure interaction effects are ignored. Thebuildup of ground motion is gradual (Fig. 8), using atransition time interval equal to 8 wave periods, so asnumerical pseudo-oscillations from the sudden applicationof a large amplitude displacement are avoided. The abovedisplacements are not synchronous at all nodes; supposingthat the wave front reaches z ¼ 0 at t ¼ 0, it will arrive atnodes with zi 6¼0 with a time lag of ti ¼ zi/(C/cosj).Applied displacements are zero before that time instant.
Fig. 9 compares analytical with numerical strainamplitude predictions. Absolute rather than algebraicvalues of strains are presented in order to simplify thecomparison. It may be observed that the analyticalpredictions match not only the peak values but also thedistribution of strain amplitudes over the entire cross-section of the shell.
0 90 180 270 360
polar angle θ (degrees)
0
0.2
0.4
0.6
0
0.2
0.4
0.6
ε α γ
Fig. 9. Comparison of analytical and numerical results alo
5.2. Soft soil over bedrock
A 3-D shell, identical to the one used in the uniformground analysis presented above (Fig. 8), is now consideredto rest within a soft soil layer with CS ¼ 100m/s, overlyingthe seismic bedrock that features a shear wave propagationvelocity of CR ¼ 365m/s. The seismic waves are assumedto impinge at the soil–bedrock interface with an angleaR ¼ 201 so that, according to Eq. (37), the angle of wavepropagation in the soft soil layer will be aS ¼ 751. Todescribe the propagation of the refracted shear wave in thesoft soil layer, a rotated coordinate system has to bedefined, as before: The axis z0 of the new coordinate systemis rotated relatively to the undeformed axis of the structurez (Fig. 8) by an Eulerian angle yyz ¼ �aS ¼ �751 and theaxis x0 relatively to the original axis x by yxy ¼ j*
¼ 301,where j* is the random angle formed by the axis of thestructure and the vertical plane defined by the propagationpath. In this rotated coordinate system, the displacementfield resulting from the harmonic wave propagation can beanalytically described as
ux0;i ¼ Amax sin b� sin
2pL
z0i � CSt� �� �
, (52)
uy0 ;i ¼ Amax cos b� sin
2pL
z0i � CSt� �� �
, (53)
uz0;i ¼ yx0;i ¼ yy0;i ¼ yz0;i ¼ 0, (54)
where the amplitude of the shear wave in the soft soil layeris Amax ¼ 1.0, the wavelength is L ¼ 10m and the randomangle of the particle motion relatively to the propagationplane z0y0 of the refracted wave is taken as b*
¼ 601.As in the previous case of uniform ground, the above
displacements are not synchronous at all nodes. Supposingthat the wave front reaches z0 ¼ 0 at t ¼ 0, it will arrive atnodes with z0ia0 with a time lag of ti ¼ z0i=CS, whileapplied displacements are zero before that time instant.
0
0.2
0.4
0.6
ε h
0 45 90 135 180 225 270 315 360
polar angle θ (degrees)
0
0.2
0.4
0.6
ε vM
analyticalnumerical
ng a random cross-section for the uniform ground case.
ARTICLE IN PRESS
0 90 180 270 360
polar angle θ* (degrees)
0
0.2
0.4
0.6
0
0.2
0.4
0.6
0 45 90 135 180 225 270 315 360
polar angle θ* (degrees)
0
0.2
0.4
0.6
0
0.2
0.4
0.6analyticalnumerical
ε hε v
M
ε α γ
Fig. 10. Comparison of analytical and numerical results along a random cross-section for the soft soil case.
Table 1
Maximum normalized seismic strains for underground structures in
uniform ground
Strain component Design value St. John and Zahrah [15]
ea/(Vmax/C) 0.50 0.50
g/(Vmax/C) 1.00 1.00
eh/(Vmax/C) 0.50 0.50
evM/(Vmax/C) 0.87/(1+nl) [1.00/(1+nl)]a
e1/(Vmax/C) 0.71 [1.00]a
e3/(Vmax/C) �0.71 [�1.00]a
aComputed from superposition of the respective ea, eh and g.
G.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921 917
Again the buildup of ground motion is gradual (Fig. 8), soas to avoid numerical pseudo-oscillations from the suddenapplication of a large amplitude displacement.
Fig. 10 compares analytical with numerical strainamplitude predictions. Absolute rather than algebraicvalues of strains are presented again, in order to simplifythe comparison. It may be observed that the analyticalrelations predict with reasonable accuracy the peak values,as well as the distribution of strain amplitudes over theentire cross section of the shell. Observed differences arerather small, and can be attributed to the low CS/CR ratio(i.e. CS/CR ¼ 0.275) and the low aR angle considered in theanalysis, which are close to the limits of application of theanalytical solutions for soft soil over bedrock. In a similaranalysis, not shown here for reasons of briefness, where theshear wave velocity ratio was decreased to CS/CR ¼ 0.09the differences between the analytical and the numericalpredictions practically diminished.
6. Design strains
6.1. Uniform ground conditions
Seismic strain amplitudes derived previously wereexpressed as functions of the following angles: the angleof incidence j, the angle of motion vector b, and the polarangle y. In order to conclude to a set of design seismicstrains, which should be superimposed to strains resultingfrom various static loads, the analytical expressions mustbe properly maximized with respect to the above unknownangles, which form the random problem variables, as theycannot be a priori known. A strict mathematical max-imization procedure is rather cumbersome, as it has toaccount simultaneously for all independent random pro-blem variables, and consequently maximization of strainswas accomplished numerically. In more detail, the strainamplitudes were normalized against eo ¼
VmaxC
, their values
were computed numerically for a total of equally spaced90� 360� 90 combinations of input j, y and b values, andthe peak was identified as the design strain. Angles j and bvaried in the 0Cp/2 range while angle y varied in the 0C2prange.The same methodology is applied for the computation of
the principal (major and minor)
e1;3 ¼ea þ eh
2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiea � eh
2
� 2þ
g2
� 2r, (55)
which represents the maximum normal strain applied to apoint, and the von Mises strain amplitudes
evM ¼1
1þ nl
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2a þ e2h � eaeh þ
3
4g2
r, (56)
which is mostly employed in failure criteria for steelpipelines. However, note that these seismic strain measuresare merely indicative of the relative magnitude of seismicstrains, and have been computed for the sake ofcomparison with the respective normal and shear straincomponents. In actual design practice their computationshould also take into account strains from static loadcombinations (e.g. self-weight, soil overburden load,internal pressure).
ARTICLE IN PRESS
Table 2
Maximum normalized seismic strains for underground structures in soft
soil
Strain component Design value
ea/(Vmax/CS) 0.5 �CS/CR
g/(Vmax/CS) 0.43 �CS/CR+0.98
eh/(Vmax/CS) 0.36 �CS/CR+0.5
evM/(Vmax/CS) [0.38 �CS/CR+0.85]/(1+nl)e1/(Vmax/CS) 0.5 �CS/CR+0.5
e3/(Vmax/CS) �0.5 �CS/CR�0.5
G.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921918
In summary, the resulting design seismic strain ampli-tudes are given in Table 1. To compare with current designpractice, Table 1 also lists the analytical relations of St.John and Zahrah [15] which form the basis of designguidelines for buried pipelines [6,28] and tunnels [7] today.It is reminded that these relations apply strictly tomaximum axial (ea), hoop (eh) and shear (g) strains. Thus,the principal (e1, e3) and von Mises (evM) strains appearing(in brackets) in the same column were approximatelyevaluated by superposition of the corresponding maximumnormal and shear components, despite the fact that theyare not concurrent on the cross section, while ignoringstrains due to static loads for the sake of comparison.
Note that, since the direction of wave propagation israndom, the peak particle velocity Vmax that is used in thecomputation of design strains in Table 1 corresponds to themaximum amplitude of the 3-D resultant of motion on theground surface, which is equal to
Vmax ¼ max
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2
L þ V2T þ V2
V
q, (57)
where VL, VT and VV are the velocity time historycomponents of the seismic motion in any two horizontaland the vertical direction. However, actual seismic record-ings show that the above 3-D peak particle velocity is, moreor less, equal to the peak particle velocity of the strongercomponent of the recorded motion, i.e.
Vmax � maxðVL;max; VV;max; VT;maxÞ. (58)
The first thing to observe in Table 1 is that the maximumprincipal strain (e1) and the von Mises strain (evM)predicted by the new relations for seismic wave actionexclusively are about 42% to 74% higher than the axialand the hoop strain components which form the basis fordesign today. This observation has little practical signifi-cance for buried pipelines with peripheral joints, as long asthese joints have reduced strength relative to the pipelinematerial and consequently they will fail under the action ofaxial strain ea. However, they suggest that in all other cases(e.g. continuous tunnels or steel pipelines with spiralwelding) seismic design should be based on e1 and evM,rather than on ea and eh.
Comparison with the St. John and Zahrah relationsshows further that the widely used approximate procedure
0 0.05 0.1 0.15 0.2 0.25
CS/CR
0
0.2
0.4
0.6
0.8
1
1.2
(a)
εh
εα
γ
VmaxCS
�
Fig. 11. Variation of strain amplitudes with the CS/CR ratio, for underground
von Mises strains.
to predict seismic strains in the structure from free fieldground strains is accurate as far as the normal (axial ea andhoop eh) and the shear strains are concerned. Nevertheless,if this procedure is used to compute the major principaland the von Misses strains (e1 and evM), the proposeddesign values are overestimated by 41% and 15%respectively.
6.2. Soft soil over bedrock
In this case, the random problem variables become four,as the angle of incidence at the soil–bedrock interface(aR ¼ 0Cp/2) should also be accounted for. Using avariation of the computer code written for the simulta-neous maximization of strains for the uniform ground case,the maximum design strains are computed as functions ofthe CS/CR ratio, and accordingly drawn in Fig. 11.All data can be fitted with reasonable accuracy by simple
linear relations, which are listed in Table 2. Furthermore,Table 3 lists the design strains computed herein for atypical case with CS=CR ¼
15, and compares them to
analytical strain predictions based on St. John and Zahrah[15]. As the latter refer strictly to uniform groundconditions with shear wave velocity equal to C, twoalternative predictions are considered: the first for C ¼ CR
and the second for C ¼ CS.Focusing first upon the proposed design seismic strains,
observe that the maximum principal strain (e1) is con-siderably higher than the axial strain (ea) and approxi-mately the same as the hoop strain (eh). The differencefrom the axial strain is proportional to the CR/CS ratio,and in the present application it amounts for 500%. On the
0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
CS/CR
ε1
εvM
(b)
structures in soft soil: (a) axial, shear and hoop strains, (b) principal and
ARTICLE IN PRESS
Table 3
Comparison of design seismic strains and strains proposed by St. John and Zahrah [15] for CS=CR ¼ 1=5
Strain component Design value St. John and Zahrah [15]
for C ¼ CR for C ¼ CS
ea/(Vmax/CS) 0.10 0.10 0.50
g/(Vmax/CS) 1.06 0.20 1.00
eh/(Vmax/CS) 0.57 0.10 0.50
g/(Vmax/CS) 1.06 0.20 1.00
evM/(Vmax/CS) 0.92/(1+nl) [0.20/(1+nl)]a [1/(1+nl)]
a
e1/(Vmax/CS) 0.60 [0.20]a [1.00]a
e3/(Vmax/CS) �0.60 [�0.20]a [�1.00]a
aComputed from superposition of the respective ea, eh and g.
G.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921 919
other hand, the von Mises strain (eVM) is higher than bothea and eh. In this case also, the differences increase with theCR/CS ratio, and take the values of 820% and 60%,respectively for the soil and bedrock properties consideredin the present application. The second observation is thatnearly all design seismic strain components are under-predicted when the St. John and Zahrah relations are usedin conjunction with the shear wave velocity of the bedrock,while they are overpredicted when the shear wave velocityof the soft soil is used instead. The differences rangebetween 0% and �470% in the first case and between400% and �14% in the second.
Finally, a closer examination of Tables 1–3 reveals thatsolutions derived for uniform ground may be approxi-mately used for structures in soft soil over bedrock as well,under one condition: axial strains (ea) are computed basedon the shear wave velocity of the bedrock, while hoop (eh)and shear strains (g) are computed from the soft soil shearwave velocity. This finding applies to the relations foruniform ground proposed herein, but also for the differentapproximate relations developed earlier.
In summary, it appears that the presence of soft soil inthe vicinity of the structure has a significant effect on allseismic strain components that cannot be predicted whenusing the shear wave propagation velocity of the bedrock.The aforementioned effects apply both to continuous andsegmented structures, with peripheral joints, and increaseas the shear wave velocity contrast between the bedrockand the soil cover becomes larger.
7. Conclusions
A 3-D thin shell strain analysis has been presented,regarding long cylindrical underground structures (buriedpipelines and tunnels) subjected to seismic shear (S) waveexcitation. Similarly to the majority of analytical solutionsused today (e.g. [7,12,13,15]), inertia and kinematic soil-structure interaction effects are neglected, while seismicwaves are assumed to be harmonic, propagating with aplane front. Two distinct cases were considered, namely‘‘uniform ground’’ and ‘‘soft soil over bedrock’’. Note that,currently available solutions apply strictly to uniform
ground conditions, and draw upon free field strains tocompute peak normal and shear strains only in thestructure section.In general terms, two are the major gains from the
previous investigation. The first is to derive a set ofanalytical solutions for the distribution over the entirestructure section of the normal (axial and hoop) and theshear strain components, that can be used for thecalculation of their principal and von Mises counterparts.The second gain is to take into account the effect of softsoil conditions on seismic strains in a systematic way, andclarify the use of an apparent bedrock velocity of wavepropagation that is recommended by currently applieddesign guidelines [6,28]. The derived solutions allow aconsistent seismic analysis of continuous, as well as,segmented cylindrical underground structures, under var-ious ground conditions.In more practical terms, it has been shown that, for
structures in ‘‘uniform ground’’:
(a)
The maximum principal (e1) and the von Mises (evM)strains computed with 3-D shell theory for seismicwave action exclusively are 42–74% higher than therespective axial and hoop strains, which form the basisfor design today.(b)
The approximate procedure used to predict structurestrains from free field ground strains is justified onlyfor normal (axial and hoop) and shear strains.However, superposition of these peak strains mayprove overly conservative, as they do not develop atthe same position on the cross section. For instance,the seismic major principal strain computed in this wayis about 41% higher than the value computed with the3-D shell theory.For structures in ‘‘soft soil over bedrock’’, it is furtherconcluded that:
(c)
The maximum seismic principal (e1) and von Mises(evM) strains computed with the 3-D shell theory areagain higher than the respective axial and hoop straincomponents, only that now the difference is larger. ThisARTICLE IN PRESSG.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921920
is especially true for the axial strain where the abovedifference may reach the same order of magnitude asthe shear wave velocity ratio CR/CS.
(d)
Axial design strains can be indeed predicted fromsolutions developed for uniform ground, using as inputthe shear wave velocity of the bedrock. On thecontrary, this procedure severely underpredicts thedesign seismic hoop and shear strains. At firstapproximation, these strains can be computed usingthe shear wave velocity of the soft soil instead.Conclusions (a) and (c) above imply that the seismicdesign of continuous underground structures (e.g. tunnelsor concrete pipelines) or steel pipelines with spiral weldingshould be based on total (static plus seismic) e1 and evM,rather than on normal strain components (ea and eh). Thisrequirement, which admittedly complicates computations,does not apply in the presence of peripheral joints or welds,with reduced strength relative to the pipeline material,which will rather fail under the action of ea.
The methodology outlined herein is currently extendedto P and Rayleigh waves. In parallel, a systematic analysisof the damages to tunnels and pipelines from Kobe, Chi-Chi and Duzce earthquakes is underway, seeking aquantitative verification of the theoretical findings throughwell documented case studies.
Acknowledgments
This research is partially supported by ‘‘EPEAEK II-Pythagoras’’ grant, co-funded by the European SocialFund and the Hellenic Ministry of Education. Professor G.Gazetas of NTUA offered valuable comments as memberof the Doctoral Thesis Committee of the first author. Thesecontributions are gratefully acknowledged.
Appendix. Nomenclature
Amax: peak particle displacement of the seismic motionC: wave propagation velocity in uniform groundCR: wave propagation velocity in bedrockCS: wave propagation velocity in soft soilD: diameter of the underground structureEl: Young’s modulus of the structure materialEm: Young’s modulus of the soil mediumF: flexibility ratioL: wavelength in uniform groundLR: wavelength in bedrockLS: wavelength in soft soilt: timets: thickness of the cross-sectionT: harmonic wave periodu: displacementamax: peak particle acceleration of the seismic motionaR: angle of incidence in the soft soil-bedrock interfaceaS: angle of propagation in the soft soil layer
b: angle formed by the peak particle velocity vectorand the propagation plane
g: shear straingl: specific weight of the structure materialea: axial straineh: hoop strainevM: von Mises straine1,3: principal strainsy: polar angle in the cylindrical coordinate system of
the cross-sectionyxy: Eulerian angle of rotation of coordinate system
axisyx0,i: rotationnm: Poisson’s ratio of the soil mediumnl: Poisson’s ratio of the structure materialj: angle of incidence in the wave-structure plane
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