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Soil Dynamics and Earthquake Engineering 26 (2006) 909–921

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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: gkouretz@central.ntua.gr (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, with

wavelength 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 structure

axis, 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 the

structure 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, with

wavelength 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.

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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. This

ARTICLE 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

References

[1] EQE Summary Report. The January 17, 1995 Kobe Earthquake;

1995.

[2] Sinozuka M. The Hanshin–Awaji earthquake of January 17, 1995:

performance of lifelines. NCEER-95-0015; 1995.

[3] Chen WW, Shih B-J, Chen Y-C, Hung J-H, Hwang HH. Seismic

response of natural gas and water pipelines in the Ji–Ji earthquake.

Soil Dyn Earthquake Eng 2002;22:1209–14.

[4] Wang WL, Wang TT, Su JJ, Lin CH, Seng CR, Huang TH.

Assessment of damage in mountain tunnels due to the Taiwan

Chi–Chi earthquake. Soil Dyn Earthquake Eng 2002;22:73–96.

[5] Uenishi K, Sakurai S, Uzarski SM-J, Arnold C. Chi–Chi Taiwan,

earthquake of September 21, 1999: reconnaissance report. Earth-

quake Spectra 1999;5(19):153–73.

[6] European Committee for Standardization (CEN). Eurocode 8: design

of structures for earthquake resistance—part 4: silos, tanks and

pipelines. Draft No. 2, December 2003.

[7] Wang JJ. Seismic design of tunnels. Parsons Brinckerhoff Mono-

graph 1993;7.

[8] Hashash YMA, Hook JJ, Schmidt B, Yao JC. Seismic design and

analysis of underground structures. Tunnelling Underground Space

Technol 2001;16:247–93.

[9] Hoeg K. Stresses against underground structural cylinders. J Soil

Mech Found Div, ASCE 1968;94:SM4.

[10] Hendron AJ Jr, Fernandez G. Dynamic and static considerations for

underground chambers. In: Seismic design of embankments and

caverns. New York: 1983. p. 157-97.

[11] O’Rourke MJ, Liu X. Response of buried pipelines subject to

earthquake effects. Monograph Series MCEER; 1999.

[12] Newmark NM. Problems in wave propagation in soil and rock. In:

Proceedings of the international symposium on wave propagation

and dynamic properties of earth materials. Albuquerque NM:

University of New Mexico Press; 1968. p. 7-26.

[13] Kuesel TR. Earthquake design criteria for subways. J Struct Div

ASCE 1969;ST6:1213–31.

[14] Yeh GCK. Seismic analysis of slender buried beams. Bull Seismol Soc

Am 1974;64(5):1551–62.

[15] St. John CM, Zahrah TF. A seismic design of underground

structures. Tunnelling Underground Space Technol 1987;2(2):165–97.

[16] Datta SK, O’Leary PM, Shah AH. Three-dimensional dynamic

response of buried pipelines to incident longitudinal and shear waves.

J Appl Mech ASME 1985;52:919–26.

[17] Wong KC, Datta SK, Shah AH. Three-dimensional motion of buried

pipeline I: analysis, II: numerical results. J Eng Mech ASCE 1986;

112:1319–37, 1338–45.

ARTICLE IN PRESSG.P. Kouretzis et al. / Soil Dynamics and Earthquake Engineering 26 (2006) 909–921 921

[18] Liu SW, Datta SK, Khair KR, Shah AH. Three-dimensional

dynamics of pipelines buried in backfilled trenches due to oblique

incidence of body waves. Soil Dyn Earthquake Eng 1991;10:182–91.

[19] Luco JE, de Barros FCP. Seismic response of a cylindrical shell

embedded in a layered viscoelastic half space. I: formulation; II:

validation and numerical results. Earthquake Eng Struct Dyn

1994;23:553–67, 569–80.

[20] Ogawa Y, Koike T. Structural design of buried pipelines for severe

earthquakes. Soil Dyn Earthquake Eng 2001;21:199–209.

[21] O’ Rourke MJ, El Hmadi KE. Analysis of continuous buried pipelines

for seismic wave effects. Earthquake Eng Struct Dyn 1988;110:917–29.

[22] Manolis GD, Beskos DE. Underground and lifeline structures. In:

Beskos DE, Anagnostopoulos SA, editors. Computer analysis and

design of earthquake resistant structures: a handbook. Southampton:

CMP; 1997. p. 775–837.

[23] Stamos AA, Beskos DE. Dynamic analysis of large 3D underground

structures by the BEM. Earthquake Eng Struct Dyn 1995;24:917–34.

[24] Stamos AA, Beskos DE. 3-D seismic response analysis of long

lined tunnels in halfspace. Soil Dyn Earthquake Eng 1996;15:

111–8.

[25] Manolis GD, Talaslidis DG, Tetepoulidis PI, Apostolidis G. Soil

structure interaction analyses for buried pipelines. In: Papadrakakis

M, Topping BHV, editors. Advances in simulation and interaction

techniques. Edinburgh: Civil-Comp; 1994.

[26] Manolis GD, Pitilakis K, Tetepoulidis PI, Mavridis G. A hierarchy of

numerical models for SSI analysis of buried structures. In: Cakmak

AS, Brebbia CA, editors. soil dynamics and earthquake engineering

VII. Southampton: CMP; 1995.

[27] Manolis GD, Tetepoulidis PI, Talaslidis DG, Apostolidis G. Seismic

analysis of buried pipeline in a 3D soil continuum. Eng Anal

Boundary Elem 1995;15:371–94.

[28] American Lifelines Alliance. Guidelines for the design of buried steel

pipes. New York: ASCE; 2001.

[29] Ansys Inc. ANSYS Release 8.0 Documentation; 2003.