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.Tunnelling and Underground Space Technology 16 2001 247293
ITAAITES Accredited Material
Seismic design and analysis of underground structures
Youssef M.A. Hashasha,, Jeffrey J. Hooka, Birger Schmidtb,John
I-Chiang Yaoa
aDepartment of Ciil and Enironmental Engineering, Uniersity of
Illinois at Urbana-Champaign, 205 N. Mathews Aenue, MC-250,Urbana,
IL 61801, USA
bParsons Brinckerhoff, San Francisco, CA, USA
Abstract
Underground facilities are an integral part of the
infrastructure of modern society and are used for a wide range
ofapplications, including subways and railways, highways, material
storage, and sewage and water transport. Underground
facilitiesbuilt in areas subject to earthquake activity must
withstand both seismic and static loading. Historically,
underground facilitieshave experienced a lower rate of damage than
surface structures. Nevertheless, some underground structures have
experiencedsignificant damage in recent large earthquakes,
including the 1995 Kobe, Japan earthquake, the 1999 Chi-Chi,
Taiwanearthquake and the 1999 Kocaeli, Turkey earthquake. This
report presents a summary of the current state of seismic analysis
anddesign for underground structures. This report describes
approaches used by engineers in quantifying the seismic effect on
anunderground structure. Deterministic and probabilistic seismic
hazard analysis approaches are reviewed. The development
ofappropriate ground motion parameters, including peak
accelerations and velocities, target response spectra, and ground
motiontime histories, is briefly described. In general, seismic
design loads for underground structures are characterized in terms
of thedeformations and strains imposed on the structure by the
surrounding ground, often due to the interaction between the two.
Incontrast, surface structures are designed for the inertial forces
caused by ground accelerations. The simplest approach is to
ignorethe interaction of the underground structure with the
surrounding ground. The free-field ground deformations due to a
seismicevent are estimated, and the underground structure is
designed to accommodate these deformations. This approach is
satisfactorywhen low levels of shaking are anticipated or the
underground facility is in a stiff medium such as rock. Other
approaches thataccount for the interaction between the structural
supports and the surrounding ground are then described. In the
pseudo-staticanalysis approach, the ground deformations are imposed
as a static load and the soil-structure interaction does not
includedynamic or wave propagation effects. In the dynamic analysis
approach, a dynamic soil structure interaction is conducted
usingnumerical analysis tools such as finite element or finite
difference methods. The report discusses special design issues,
includingthe design of tunnel segment joints and joints between
tunnels and portal structures. Examples of seismic design used
forunderground structures are included in an appendix at the end of
the report. 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Seismic design; Seismic analysis; Underground
structures; Tunnels; Subways; Earthquake design
Corresponding author. Tel.: 1-217-333-6986; fax: 1-217-265-8041.
.E-mail address: [email protected] Y.M.A. Hashash .
0886-779801$ - see front matter 2001 Elsevier Science Ltd. All
rights reserved. .PII: S 0 8 8 6 - 7 7 9 8 0 1 0 0 0 5 1 - 7
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
Technology 16 2001 247293248
Preface
This paper was developed as part of the activities of( )the
International Tunnelling Association ITA Working
Group No 2: Research. The paper provides a state-of-the-art
review of the design and analysis of tunnelssubject to earthquake
shaking with particular focus onpractice in the United States of
America. The Authorswish to acknowledge the important contribution
ofWorking Group 2 members including Mr. Yann Leblais,Animateur,
Yoshihiro Hiro Takano, Vice-Animateur,Barry New, Member, Henk J.C.
Oud and Andres Assis,Tutor and Former Tutor, respectively, as well
as theITA Executive Council for their review and approval ofthis
document.
1. Introduction
Underground structures have features that maketheir seismic
behavior distinct from most surface struc-
.tures, most notably 1 their complete enclosure in soil . .or
rock, and 2 their significant length i.e. tunnels .
The design of underground facilities to withstandseismic loading
thus, has aspects that are very differentfrom the seismic design of
surface structures.
This report focuses on relatively large undergroundfacilities
commonly used in urban areas. This includeslarge-diameter tunnels,
cut-and-cover structures and
.portal structures Fig. 1 . This report does not
discusspipelines or sewer lines, nor does it specifically
discussissues related to deep chambers such as hydropowerplants,
nuclear waste repositories, mine chambers, andprotective
structures, though many of the design meth-ods and analyses
described are applicable to the designof these deep chambers.
Large-diameter tunnels are linear undergroundstructures in which
the length is much larger than thecross-sectional dimension. These
structures can begrouped into three broad categories, each having
dis-
.tinct design features and construction methods: 1 .bored or
mined tunnels; 2 cut-and-cover tunnels; and
. .3 immersed tube tunnels Power et al., 1996 . Thesetunnels are
commonly used for metro structures, high-way tunnels, and large
water and sewage transportationducts.
Bored or mined tunnels are unique because they areconstructed
without significantly affecting the soil orrock above the
excavation. Tunnels excavated using
.tunnel-boring machines TBMs are usually circular;other tunnels
maybe rectangular or horseshoe in shape.Situations where boring or
mining may be preferable to
.cut-and-cover excavation include 1 significant excava- .tion
depths, and 2 the existence of overlying struc-
tures.
.Fig. 1. Cross sections of tunnels after Power et al., 1996
.
Cut-and-cover structures are those in which an openexcavation is
made, the structure is constructed, and fillis placed over the
finished structure. This method istypically used for tunnels with
rectangular cross-sec-
tions and only for relatively shallow tunnels 15 m of.overburden
. Examples of these structures include sub-
way stations, portal structures and highway tunnels.Immersed
tube tunnels are sometimes employed to
traverse a body of water. This method involves con-structing
sections of the structure in a dry dock, thenmoving these sections,
sinking them into position andballasting or anchoring the tubes in
place.
This report is a synthesis of the current state ofknowledge in
the area of seismic design and analysisfor underground structures.
The report updates the
.work prepared by St. John and Zahrah 1987 , whichappeared in
Tunneling Underground Space Technol. Thereport focuses on methods
of analysis of undergroundstructures subjected to seismic motion
due toearthquake activity, and provides examples of perfor-mance
and damage to underground structures duringrecent major
earthquakes. The report describes theoverall philosophy used in the
design of undergroundstructures, and introduces basic concepts of
seismichazard analysis and methods used in developing
designearthquake motion parameters.
The report describes how ground deformations areestimated and
how they are transmitted to an under-
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
Technology 16 2001 247293 249
ground structure, presenting methods used in the com-putation of
strains, forces and moment in the structure.The report provides
examples of the application ofthese methods for underground
structures in Los Ange-les, Boston, and the San Francisco Bay
Area.
This report does not cover issues related to staticdesign,
although static design provisions for under-ground structures often
provide sufficient seismic resis-tance under low levels of ground
shaking. The reportdoes not discuss structural design details and
reinforce-ment requirements in concrete or steel linings
forunderground structures. The report briefly describesissues
related to seismic design associated with groundfailure such as
liquefaction, slope stability and faultcrossings, but does not
provide a thorough treatment ofthese subjects. The reader is
encouraged to reviewother literature on these topics to ensure that
relevantdesign issues are adequately addressed.
2. Performance of underground facilities during
seismicevents
Several studies have documented earthquake da- .mage to
underground facilities. ASCE 1974 describes
the damage in the Los Angeles area as a result of the .1971 San
Fernando Earthquake. JSCE 1988 describes
the performance of several underground structures,including an
immersed tube tunnel during shaking in
. .Japan. Duke and Leeds 1959 , Stevens 1977 , Dowd- . .ing and
Rozen 1978 , Owen and Scholl 1981 , Sharma
. .and Judd 1991 , Power et al. 1998 and Kaneshiro et .al. 2000
, all present summaries of case histories of
damage to underground facilities. Owen and Scholl .1981 have
updated Dowding and Rozens work with
.127 case histories. Sharma and Judd 1991 generatedan extensive
database of seismic damage to under-ground structures using 192
case histories. Power et al. .1998 provide a further update with
217 case histories.The following general observations can be made
re-garding the seismic performance of underground struc-tures:
1. Underground structures suffer appreciably lessdamage than
surface structures.
2. Reported damage decreases with increasing over-burden depth.
Deep tunnels seem to be safer andless vulnerable to earthquake
shaking than areshallow tunnels.
3. Underground facilities constructed in soils can beexpected to
suffer more damage compared toopenings constructed in competent
rock.
4. Lined and grouted tunnels are safer than unlinedtunnels in
rock. Shaking damage can be reducedby stabilizing the ground around
the tunnel and
by improving the contact between the lining andthe surrounding
ground through grouting.
5. Tunnels are more stable under a symmetric load,which improves
ground-lining interaction. Improv-ing the tunnel lining by placing
thicker and stiffersections without stabilizing surrounding
poorground may result in excess seismic forces in thelining.
Backfilling with non-cyclically mobile mate-rial and
rock-stabilizing measures may improvethe safety and stability of
shallow tunnels.
6. Damage may be related to peak ground accelera-tion and
velocity based on the magnitude andepicentral distance of the
affected earthquake.
7. Duration of strong-motion shaking duringearthquakes is of
utmost importance because itmay cause fatigue failure and
therefore, largedeformations.
8. High frequency motions may explain the localspalling of rock
or concrete along planes of weak-ness. These frequencies, which
rapidly attenuatewith distance, may be expected mainly at
smalldistances from the causative fault.
9. Ground motion may be amplified upon incidencewith a tunnel if
wavelengths are between one andfour times the tunnel diameter.
10. Damage at and near tunnel portals may be sig-nificant due to
slope instability.
The following is a brief discussion of recent casehistories of
seismic performance of underground struc-tures.
2.1. Underground structures in the United States
( )2.1.1. Bay Area rapid transit BART system, SanFrancisco, CA,
USA
The BART system was one of the first undergroundfacilities to be
designed with considerations for seismic
.loading Kuesel, 1969 . On the San Francisco side, thesystem
consists of underground stations and tunnels infill and soft Bay
Mud deposits, and it is connected toOakland via the
transbay-immersed tube tunnel.
During the 1989 Loma Prieta Earthquake, the BARTfacilities
sustained no damage and, in fact, operated ona 24-h basis after the
earthquake. This is primarilybecause the system was designed under
stringentseismic design considerations. Special seismic joints
.Bickel and Tanner, 1982 were designed to accommo-date differential
movements at ventilation buildings.The system had been designed to
support earth andwater loads while maintaining watertight
connectionsand not exceeding allowable differential movements.No
damage was observed at these flexible joints, thoughit is not
exactly known how far the joints moved during
.the earthquake PB, 1991 .
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
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.Fig. 2. Section sketch of damage to Daikai subway station Iida
et al., 1996 .
2.1.2. Alameda Tubes, Oakland-Alameda, CA, USAThe Alameda Tubes
are a pair of immersed-tube
tunnels that connect Alameda Island to Oakland in theSan
Francisco Bay Area. These were some of theearliest immersed tube
tunnels built in 1927 and 1963without seismic design
considerations. During the LomaPrieta Earthquake, the ventilation
buildings experi-enced some structural cracking. Limited water
leakageinto the tunnels was also observed, as was liquefactionof
loose deposits above the tube at the Alameda portal.Peak horizontal
ground accelerations measured in the
.area ranged between 0.1 and 0.25 g EERI, 1990 . Thetunnels,
however, are prone to floatation due to poten-
tial liquefaction of the backfill Schmidt and Hashash,.1998
.
2.1.3. L.A. Metro, Los Angeles, CA, USAThe Los Angeles Metro is
being constructed in sev-
eral phases, some of which were operational during the1994
Northridge Earthquake. The concrete lining ofthe bored tunnels
remained intact after the earthquake.While there was damage to
water pipelines, highwaybridges and buildings, the earthquake
caused no da-mage to the Metro system. Peak horizontal
groundaccelerations measured near the tunnels rangedbetween 0.1 and
0.25 g, with vertical ground accelera-
.tions typically two-thirds as large EERI, 1995 .
2.2. Underground structures in Kobe, Japan
The 1995 Hyogoken-Nambu Earthquake caused amajor collapse of the
Daikai subway station in Kobe,
.Japan Nakamura et al., 1996 . The station design in1962 did not
include specific seismic provisions. Itrepresents the first modern
underground structure tofail during a seismic event. Fig. 2 shows
the collapseexperienced by the center columns of the station,
whichwas accompanied by the collapse of the ceiling slab andthe
settlement of the soil cover by more than 2.5 m.
During the earthquake, transverse walls at the ends
of the station and at areas where the station changedwidth acted
as shear walls in resisting collapse of the
.structure Iida et al., 1996 . These walls suffered sig-nificant
cracking, but the interior columns in theseregions did not suffer
as much damage under thehorizontal shaking. In regions with no
transverse walls,collapse of the center columns caused the ceiling
slabto kink and cracks 150250-mm wide appeared in thelongitudinal
direction. There was also significant sepa-ration at some
construction joints, and correspondingwater leakage through cracks.
Few cracks, if any, wereobserved in the base slab.
Center columns that were designed with very light .transverse
shear reinforcement relative to the main
.bending reinforcement suffered damage ranging fromcracking to
complete collapse. Center columns withzigzag reinforcement in
addition to the hoop steel, as inFig. 3, did not buckle as much as
those without thisreinforcement.
.According to Iida et al. 1996 , it is likely that therelative
displacement between the base and ceilinglevels due to subsoil
movement created the destructive
Fig. 3. Reinforcing steel arrangement in center columns Iida et
al.,.1996 .
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
Technology 16 2001 247293 251
horizontal force. This type of movement may haveminor effect in
a small structure, but in a large onesuch as a subway station it
can be significant. Thenon-linear behavior of the subsoil profile
may also besignificant. It is further hypothesized that the
thicknessof the overburden soil affected the extent of
damagebetween sections of the station by adding inertial forceto
the structure. Others attribute the failure to highlevels of
vertical acceleration.
.EQE 1995 made further observations about DaikaiStation:
Excessive deflection of the roof slab would
.normally be resisted by: 1 diaphragm action of the .slab,
supported by the end walls of the station; and 2
passive earth pressure of the surrounding soils,mobilized as the
tube racks. Diaphragm action was lessthan anticipated, however, due
to the length of the
station. The method of construction cut-and-cover,involving a
sheet pile wall supported excavation withnarrow clearance between
the sheet pile wall and the
.tube wall made compaction of backfill difficult toimpossible,
resulting in the tubes inability to mobilizepassive earth
pressures. In effect, the tube behavedalmost as a freestanding
structure with little or no extrasupport from passive earth
pressure. However, it is notcertain that good compaction would have
prevented thestructural failure of the column. Shear failure of
sup-porting columns caused similar damage to the Shinkan-
.sen Tunnel through Rokko Mountain NCEER, 1995 .Several key
elements may have helped in limiting the
damage to the station structure and possibly preventedcomplete
collapse. Transverse walls at the ends of thestation and at areas
where the station changed widthprovided resistance to dynamic
forces in the horizontaldirection. Center columns with relatively
heavy trans-
.verse shear reinforcement suffered less damage andhelped to
maintain the integrity of the structure. The
Fig. 4. Slope Failure at Tunnel Portal, Chi-Chi Earthquake,
CentralTaiwan.
Fig. 5. Bolu Tunnel, re-mining of Bench Pilot Tunnels,
showingtypical floor heave and buckled steel rib and shotcrete
shell Menkiti,
.2001 .
fact that the structure was underground instead ofbeing a
surface structure may have reduced the amountof related damage.
.A number of large diameter 2.02.4 m concretesewer pipes
suffered longitudinal cracking during theKobe Earthquake,
indicating racking andor compres-
.sive failures in the cross-sections Tohda, 1996 . Thesecracks
were observed in pipelines constructed by both
the jacking method and open-excavation cut-and-.cover methods.
Once cracked, the pipes behaved as
four-hinged arches and allowed significant water leak-age.
2.3. Underground structures in Taiwan
Several highway tunnels were located within the zoneheavily
affected by the September 21, 1999 Chi Chi
.earthquake M 7.3 in central Taiwan. These areLlarge horseshoe
shaped tunnels in rock. All the tunnelsinspected by the first
author were intact without anyvisible signs of damage. The main
damage occurred attunnel portals because of slope instability as
illustratedin Fig. 4. Minor cracking and spalling was observed
insome tunnel lining. One tunnel passing through theChelungpu fault
was shut down because of a 4-m fault
.movement Ueng et al., 2001 . No damage was reportedin the
Taipei subway, which is located over 100 kmfrom the ruptured fault
zone.
2.4. Bolu Tunnel, Turkey
The twin tunnels are part of a 1.5 billion dollarproject that
aims at improving transportation in themountainous terrain to the
west of Bolu between Istan-
.bul and Ankara http:geoinfo.usc.edugees . Eachtunnel was
constructed using the New Austrian Tunnel-
.ing Method NATM where continuous monitoring ofprimary liner
convergence is performed and support
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
Technology 16 2001 247293252
elements are added until a stable system is established.The
tunnel has an excavated arch section 15 m tall by16 m wide.
Construction has been unusually challeng-ing because the alignment
crosses several minor faultsparallel to the North Anatolian Fault.
The August 17,1999 Koceali earthquake was reported to have
hadminimal impact on the Bolu tunnel. The closure rate ofone
monitoring station was reported to have temporar-ily increased for
a period of approximately 1 week, thenbecame stable again.
Additionally, several hairlinecracks, which had previously been
observed in the finallining, were being continuously monitored for
additio-nal movement and showed no movement due to theearthquake.
The November 12, 1999 earthquake causedthe collapse of both tunnels
300 m from their easternportal. At the time of the earthquake, a
800-m sectionhad been excavated, and a 300-m section of
unrein-forced concrete lining had been completed. The col-lapse
took place in clay gauge material in the unfin-ished section of the
tunnel. The section was covered
.with shotcrete sprayed concrete and had bolt anchors.Fig. 5
shows a section of the collapsed tunnel after ithas been
re-excavated. Several mechanisms have beenproposed for explaining
the collapse of the tunnel.These mechanisms include strong ground
motion, dis-placement across the gauge material, and landslide.
.ORourke et al. 2001 present a detailed description ofthe tunnel
performance.
2.5. Summary of seismic performance of undergroundstructures
The Daikai subway station collapse was the firstcollapse of an
urban underground structure due toearthquake forces, rather than
ground instability. Un-derground structures in the US have
experiencedlimited damage during the Loma Prieta and
Northridgeearthquakes, but the shaking levels have been muchlower
than the maximum anticipated events. Greaterlevels of damage can be
expected during these maxi-mum events. Station collapse and
anticipated strongmotions in major US urban areas raise great
concernsregarding the performance of underground structures.It is
therefore necessary to explicitly account for seismicloading in the
design of underground structures.
The data show that in general, damage to tunnels isgreatly
reduced with increased overburden, and da-mage is greater in soils
than in competent rock. Da-
.mage to pipelines buckling, flotation was greater thanto rail
or highway tunnels in both Kobe and Northridge.The major reason for
this difference seems to havebeen the greater thickness of the
lining of transporta-tion tunnels. Experience has further shown
that cut-and-cover tunnels are more vulnerable to earthquakedamage
than are circular bored tunnels.
3. Engineering approach to seismic analysis and design
Earthquake effects on underground structures can .be grouped
into two categories: 1 ground shaking;
.and 2 ground failure such as liquefaction, fault dis-placement,
and slope instability. Ground shaking, whichis the primary focus of
this report, refers to the defor-mation of the ground produced by
seismic waves propa-gating through the earths crust. The major
factors
.influencing shaking damage include: 1 the shape, .dimensions
and depth of the structure; 2 the proper-
.ties of the surrounding soil or rock; 3 the properties .of the
structure; and 4 the severity of the ground
shaking Dowding and Rozen, 1978; St. John and.Zahrah, 1987 .
Seismic design of underground structures is uniquein several
ways. For most underground structures, theinertia of the
surrounding soil is large relative to theinertia of the structure.
Measurements made by Oka-
.moto et al. 1973 of the seismic response of animmersed tube
tunnel during several earthquakes showthat the response of a tunnel
is dominated by thesurrounding ground response and not the
inertialproperties of the tunnel structure itself. The focus
ofunderground seismic design, therefore, is on the free-field
deformation of the ground and its interaction withthe structure.
The emphasis on displacement is in starkcontrast to the design of
surface structures, whichfocuses on inertial effects of the
structure itself. Thisled to the development of design methods such
as theSeismic Deformation Method that explicitly considersthe
seismic deformation of the ground. For example,
.Kawashima, 1999 presents a review on the seismicbehavior and
design of underground structures in softground with an emphasis on
the development of theSeismic Deformation Method.
The behavior of a tunnel is sometimes approximatedto that of an
elastic beam subject to deformationsimposed by the surrounding
ground. Three types of
.deformations Owen and Scholl, 1981 express the re-sponse of
underground structures to seismic motions: . . .1 axial compression
and extension Fig. 6a,b ; 2
. .longitudinal bending Fig. 6c,d ; and 3 ovalingrack- .ing Fig.
6e,f . Axial deformations in tunnels are gener-
ated by the components of seismic waves that producemotions
parallel to the axis of the tunnel and causealternating compression
and tension. Bending deforma-tions are caused by the components of
seismic wavesproducing particle motions perpendicular to the
longi-tudinal axis. Design considerations for axial and bend-ing
deformations are generally in the direction along
.the tunnel axis Wang, 1993 .Ovaling or racking deformations in
a tunnel struc-
ture develop when shear waves propagate normal or
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
Technology 16 2001 247293 253
.Fig. 6. Deformation modes of tunnels due to seismic waves after
Owen and Scholl, 1981 .
nearly normal to the tunnel axis, resulting in a distor-tion of
the cross-sectional shape of the tunnel lining.Design
considerations for this type of deformation arein the transverse
direction. The general behavior of thelining may be simulated as a
buried structure subject toground deformations under a
two-dimensional plane-strain condition.
Diagonally propagating waves subject different partsof the
structure to out-of-phase displacements Fig.
.6d , resulting in a longitudinal compressionrarefac-tion wave
traveling along the structure. In general,larger displacement
amplitudes are associated withlonger wavelengths, while maximum
curvatures areproduced by shorter wavelengths with relatively
small
.displacement amplitudes Kuesel, 1969 .The assessment of
underground structure seismic
response, therefore, requires an understanding of theanticipated
ground shaking as well as an evaluation of
the response of the ground and the structure to suchshaking.
Table 1 summarizes a systematic approach forevaluating the seismic
response of underground struc-tures. This approach consists of
three major steps:
1. Definition of the seismic environment and develop-ment of the
seismic parameters for analysis.
2. Evaluation of ground response to shaking, whichincludes
ground failure and ground deformations.
3. Assessment of structure behavior due to seismic .shaking
including a development of seismic de-
.sign loading criteria, b underground structure re- .sponse to
ground deformations, and c special
seismic design issues.
Steps 1 and 2 are described in Sections 4 and 5,respectively.
Sections 68 provide the details of Steps3a, 3b and 3c.
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.Fig. 7. Deterministic seismic hazard analysis procedure after
Reiter, 1990 .
4. Definition of seismic environment
The goal of earthquake-resistant design for under-ground
structures is to develop a facility that canwithstand a given level
of seismic motion with damagenot exceeding a pre-defined acceptable
level. The de-sign level of shaking is typically defined by a
designground motion, which is characterized by the ampli-tudes and
characteristics of expected ground motions
.and their expected return frequency Kramer, 1996 . Aseismic
hazard analysis is used to define the level of
.shaking and the design earthquake s for an under-ground
facility.
A seismic hazard analysis typically characterizes thepotential
for strong ground motions by examining theextent of active faulting
in a region, the potential forfault motion, and the frequency with
which the faultsrelease stored energy. This examination may be
dif-
.ficult in some regions e.g. Eastern USA where fault-ing is not
readily detectable. There are two methods of
.analysis: a the deterministic seismic hazard analysis . .DSHA ;
and b the probabilistic seismic hazard analy-
.sis PSHA . A deterministic seismic hazard analysisdevelops one
or more earthquake motions for a site,for which the designers then
design and evaluate theunderground structure. The more recent
probabilisticseismic hazard analysis, which explicitly quantifies
theuncertainties in the analysis, develops a range of ex-pected
ground motions and their probabilities of occur-rence. These
probabilities can then be used to de-termine the level of seismic
protection in a design.
( )4.1. Deterministic seismic hazard analysis DSHA
A deterministic seismic hazard analysis involves thedevelopment
of a particular seismic scenario to sum-
marize the ground motion hazard at a site. This sce-nario
requires the postulated occurrence of a particu-lar size of
earthquake at a particular location. Reiter . 1990 outlined the
following four-step process see Fig..7 :
1. Identification and characterization of all earth-quake
sources capable of producing significantground motion at the site,
including definition ofthe geometry and earthquake potential of
each.The most obvious feature delineating a seismiczone is
typically the presence of faulting. Reiter .1990 generated a
comprehensive list of featuresthat may suggest faulting in a given
region. How-ever, the mere presence of a fault does not
neces-sarily signify a potential earthquake hazard thefault must be
active to present a risk. There hasbeen considerable disagreement
over the criteriafor declaring a fault active or inactive. Rather
thanusing the term active, the US Nuclear Regulatory
.Commission Code of Federal Regulations, 1978coined the term
capable fault to indicate a faultthat has shown activity within the
past35 000500 000 years. For non-nuclear civil infras-tructure,
shorter timeframes would be used.
2. Selection of a source-to-site distance parameter foreach
source, typically the shortest epicentralhypo-central distance or
the distance to the closest rup-tured portion of the fault. Closest
distance to rup-tured fault is more meaningful than epicentral
dis-tance especially for large earthquakes where theruptured fault
extends over distances exceeding 50km.
3. Selection of a controlling earthquake i.e. thatwhich produces
the strongest shaking level at the
.site , generally expressed in terms of a ground
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motion parameter at the site. Attenuation relation-ships are
typically used to determine these site-specific parameters from
data recorded at nearbylocations. Several studies have attempted to
corre-late earthquake magnitudes, most commonly mo-ment magnitudes,
with observed fault deformationcharacteristics, such as rupture
length and area,and have found a strong correlation. However,
theunavailability of fault displacement measurementsover the entire
rupture surface severely limits ourability to measure these
characteristics. Instead,researchers have tried to correlate the
maximumsurface displacement with magnitude to varyingresults.
Empirically based relationships, such as
.those developed by Wells and Coppersmith 1994 ,can be utilized
to estimate these correlations. An-other, more basic way to
evaluate the potential forseismic activity in a region is through
examinationof historical records. These records allow engineersto
outline and track active faults and their releaseof seismic
potential energy. The evaluation of fore-and aftershocks can also
help delineate seismic
.zones Kramer, 1996 . In addition to the examina-tion of
historical records, a study of geologic recordof past seismic
activities called paleo-seismologycan be used to evaluate the
occurrence and size of
earthquakes in the region. Geomorphic surface.landform and
trench studies may reveal the num-
ber of past seismic events, slip per event, andtiming of the
events at a specific fault. In some
14 .cases, radiocarbon C dating of carbonized roots,animal bone
fossils or soil horizons near the fea-tures of paleoseismic
evidence can be utilized toapproximate ages of the events.
4. Formal definition of the seismic hazard at the sitein terms
of the peak acceleration, velocity and
displacement, response spectrum ordinates, andground motion time
history of the maximum credi-ble earthquake. Design fault
displacements shouldalso be defined, if applicable.
A DSHA provides a straightforward framework forthe evaluation of
worst-case scenarios at a site. How-ever, it provides no
information about the likelihood orfrequency of occurrence of the
controlling earthquake.If such information is required, a
probabilistic ap-proach must be undertaken to better quantify
theseismic hazard.
( )4.2. Probabilistic seismic hazard analysis PSHA
A probabilistic seismic hazard analysis provides aframework in
which uncertainties in the size, location,and recurrence rate of
earthquakes can be identified,quantified, and combined in a
rational manner. Suchan analysis provides designers with a more
completedescription of the seismic hazard at a site, where
varia-tions in ground motion characteristics can be
explicitlyconsidered. For this type of analysis, future
earthquakeevents are assumed spatially and temporally indepen-
.dent. Reiter 1990 outlined the four major steps in- .volved in
PSHA see Fig. 8 :
1. Identification and characterization of earthquakesources,
including the probability distribution ofpotential rupture
locations within the source zone.These distributions are then
combined with thesource geometry to obtain the probability
distribu-tion of source-to-site distances. In many
regionsthroughout the world, including the USA, specificactive
fault zones often cannot be identified. In
.Fig. 8. Probabilistic seismic hazard analysis procedure after
Reiter, 1990 .
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
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these cases, seismic history and geological con-siderations
become critical for hazard analyses.
2. Characterization of the seismicity or temporal dis-tribution
of earthquake recurrence. Informationobtained from historical data
and paleoseismologi-cal studies can help to develop a recurrence
rela-tionship that describes the average rate at which anearthquake
of certain size will be exceeded.
3. Determination of the ground motion produced atthe site by any
size earthquake occurring at anysource zone using attenuation
relationships. Theuncertainty inherent in the predictive
relationshipis also considered.
4. Combination of these uncertainties to obtain theprobability
that a given ground motion parameterwill be exceeded during a given
time period.
The probabilistic approach incorporates the uncer-tainties in
source-to-site distance, magnitude, rate ofrecurrence and the
variation of ground motion charac-teristics into the analyses. In
areas where no activefaults can be readily identified it may be
necessary torely on a purely statistical analysis of
historicearthquakes in the region. The details of this proce-dure
are beyond the scope of this report.
4.3. Design earthquakes criteria
Once the seismic hazard at the site is characterized,the level
of design earthquake or seismicity has to bedefined. For example,
in PSHA, the designer mustselect the probability of exceedance for
the sets ofground motion parameters. Current seismic design
phi-
losophy for many critical facilities requires dual two-.level
design criteria, with a higher design level
earthquake aimed at life safety and a lower design
levelearthquake intended for economic risk exposure. Thetwo design
levels are commonly defined as maximum
.design earthquake or safety evaluation earthquakeand
operational design earthquake or function eval-
.uation earthquake , and have been employed in manyrecent
transportation tunnel projects, including the LosAngeles Metro,
Taipei Metro, Seattle Metro, and Bos-ton Central ArteryThird Harbor
Tunnels.
4.3.1. Maximum Design Earthquake .The Maximum Design Earthquake
MDE is defined
in a DSHA as the maximum level of shaking that canbe experienced
at the site. In a PSHA, the MDE isdefined as an event with a small
probability of ex-
.ceedance during the life of the facility e.g. 35% . TheMDE
design goal is that public safety shall be main-tained during and
after the design event, meaning thatthe required structural
capacity under an MDE loadingmust consider the worst-case
combination of live, dead,
and earthquake loads. Recently, some owners e.g. San
.Francisco BART have begun requiring their facilities,identified
as lifelines, to remain operational after MDElevel shaking.
4.3.2. Operating Design Earthquake .The Operating Design
Earthquake ODE is an
earthquake event that can be reasonably expected tooccur at
least once during the design life of the facilitye.g. an event with
probability of exceedence between
.40 and 50% . In an ODE analysis, the seismic designloading
depends on the structural performance re-quirements of the
structural members. Since the ODEdesign goal is that the overall
system shall continueoperating during and after an ODE and
experiencelittle or no damage, inelastic deformations must bekept
to a minimum. The response of the undergroundfacility should
therefore remain within the elastic range.
4.4. Ground motion parameters
Once an MDE or ODE is defined, sets of groundmotion parameters
are required to characterize thedesign events. The choice of these
parameters is re-lated to the type of analysis method used in
design. Ata particular point in the ground or on a structure,ground
motions can be described by three translationalcomponents and three
rotational components, thoughrotational components are typically
ignored. A groundmotion component is characterized by a time
history ofacceleration, velocity or displacement with three
sig-nificant parameters: amplitude; frequency content; andduration
of strong ground motion.
4.4.1. Acceleration, elocity, and displacement amplitudesMaximum
values of ground motion such as peak
ground acceleration, velocity and displacement arecommonly used
in defining the MDE and ODE devel-oped through seismic hazard
analysis. However, experi-ence has shown that effective, rather
than peak, groundmotion parameters tend to be better indicators
ofstructural response, as they are more representative ofthe damage
potential of a given ground motion. This isespecially true for
large earthquakes. The effectivevalue is sometimes defined as the
sustained level ofshaking, and computed as the third or fifth
highest
.value of the parameter Nuttli, 1979 . Earthquake da-mage to
underground structures has also proven to bebetter correlated with
particle velocity and displace-ment than acceleration. Attenuation
relationships aregenerally available for estimating peak ground
surfaceaccelerations, but are also available for estimating
peakvelocities and displacements. Tables 2 and 3 can beused to
relate the known peak ground acceleration toestimates of peak
ground velocity and displacement,respectively, in the absence of
site-specific data.
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Table 2Ratios of peak ground velocity to peak ground
acceleration at surface
.in rock and soil after Power et al., 1996
.Ratio of peak ground velocity cmsMoment .to peak ground
acceleration gmagnitude
.M .Source-to-site distance kmw
020 2050 50100
aRock6.5 66 76 867.5 97 109 978.5 127 140 152
aStiff soil6.5 94 102 1097.5 140 127 1558.5 180 188 193
aSoft soil6.5 140 132 1427.5 208 165 2018.5 269 244 251
a In this table, the sediment types represent the following
shearwave velocity ranges: rock 750 ms; stiff soil is 200750 ms;
andsoft soil 200 ms. The relationship between peak ground
velocityand peak ground acceleration is less certain in soft
soils.
4.4.2. Target response spectra and motion time historyThe most
common way to express the parameters of
a design ground motion is through acceleration re-sponse
spectra, which represents the response of adamped single degree of
freedom system to groundmotion. Once a target response spectrum has
been
Table 3Ratios of peak ground displacement to peak ground
acceleration at
.surface in rock and soil after Power et al., 1996
.Ratio of peak ground displacement cmMoment .to peak ground
acceleration gmagnitude
.M .Source-to-site distance kmw
020 2050 50100
aRock6.5 18 23 307.5 43 56 698.5 81 99 119
aStiff soil6.5 35 41 487.5 89 99 1128.5 165 178 191
aSoft soil6.5 71 74 767.5 178 178 1788.5 330 320 305
a In this table, the sediment types represent the following
shearwave velocity ranges: rock 750 ms; stiff soil is 200750 ms;
andsoft soil 200 ms. The relationship between peak ground
velocityand peak ground acceleration is less certain in soft
soils.
chosen, one or more ground motion time histories maybe developed
that match the design response spectra.These time histories can be
either synthetic or basedon actual recordings of earthquakes with
similar char-acteristics.
While the response spectrum is a useful tool for the .designer,
it should not be used if 1 the soil-structure
.system response is highly non-linear, or 2 the struc-ture is
sufficiently long that the motion could varysignificantly in
amplitude and phase along its length. In
.these cases, time histories St. John and Zahrah, 1987combined
with local site response analysis are typicallymore useful.
4.4.3. Spatial incoherence of ground motionFor many engineering
structures, the longest dimen-
sion of the structure is small enough that the groundmotion at
one end is virtually the same as that at theother end. However, for
long structures such as bridgesor tunnels, different ground motions
may be encoun-tered by different parts of the structure and
traveling
wave effects must be considered Hwang and Lysmer,.1981 . This
spatial incoherence may have a significant
impact on the response of the structure. There are four .major
factors that may cause spatial incoherence: 1
. .wave-passage effects; 2 extended source effects; 3ray-path
effects caused by inhomogeneities along the
.travel path; and 4 local soil site effects. The reader .should
refer to Hwang and Lysmer 1981 for details on
these factors. Recorded ground motions have shownthat spatial
coherency decreases with increasing dis-
.tance and frequency Kramer, 1996 . The generation ofground
motion time histories with appropriate spatialincoherence is a
critical task if the designer is tocompute differential strains and
force buildup along atunnel length. The designer will have to work
closelywith an engineering seismologist to identify the rele-vant
factors contributing to ground motion incoherenceat a specific site
and to generate appropriate ground
.motion time histories. Hashash et al. 1998 show howthe use of
time histories with spatial incoherence af-fects the estimation of
axial force development in atunnel and can lead to significant
longitudinal push-pulland other effects.
4.5. Wae propagation and site-specific response analysis
Research has shown that transverse shear wavestransmit the
greatest proportion of the earthquakesenergy, and amplitudes in the
vertical plane have beentypically estimated to be a half to
two-thirds as great asthose in the horizontal plane. However, in
recentearthquakes such as Northridge and Kobe, measuredvertical
accelerations were equal to and sometimeslarger than horizontal
accelerations. Vertical compo-
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
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nent of ground motion has become an important issuein seismic
designs.
Ample strong ground motion data are generally notavailable at
the depths of concern for undergroundstructures, so the development
of design ground mo-tions needs to incorporate depth-dependent
attenua-tion effects. Popular analytical procedures use
one-dimensional site response techniques, although theseanalyses
ignore the effects of all but vertically propa-gating body waves.
One method, discussed by Schnabel,
.et al. 1972 , applies a deconvolution procedure to asurface
input motion in order to evaluate the motion atdepth. A second
method involves applying ground mo-tions at various depths to find
the scale factors neces-sary to match the input motion. Both of
these proce-dures are repeated for a collection of soil
propertiesand ground motions to develop a ground motion spec-
.trum for the site St. John and Zahrah, 1987 . Linear,
.equivalent linear SHAKE, Schnabel et al., 1972 or
non-linear Hashash and Park, 2001; Borja et al., 1999,D-MOD,
Matasovic and Vucetic, 1995, Cyberquake,
.BRGM, 1998, Desra, Finn et al., 1977 one-dimen-sional wave
propagation methods are commonly usedto propagate waves through
soft soil deposits. Ground
motions generally decrease with depth e.g. Chang et.al., 1986 .
Performing a wave propagation analysis is
needed as the amplitude and period of vibration of theground
motion shift as the shear wave passes throughsoft soil deposits. In
the absence of more accurate .numerical methods or data, Table 4
can be used todetermine the relationship between ground motion
atdepth and that at the ground surface.
5. Evaluation of ground response to shaking
The evaluation of ground response to shaking can be . .divided
into two groups: 1 ground failure; and 2
ground shaking and deformation. This report focuseson ground
shaking and deformation, which assumesthat the ground does not
undergo large permanentdisplacements. A brief overview of issues
related toground failure are also presented.
5.1. Ground failure
Ground failure as a result of seismic shaking in-cludes
liquefaction, slope instability, and fault displace-ment. Ground
failure is particularly prevalent at tunnelportals and in shallow
tunnels. Special design consider-ations are required for cases
where ground failure isinvolved, and are discussed in Section
8.
5.1.1. LiquefactionLiquefaction is a term associated with a host
of
different, but related phenomena. It is used to describe
Table 4Ratios of ground motion at depth to motion at ground
surface after
.Power et al., 1996
Tunnel Ratio of ground motiondepth at tunnel depth to .m motion
at ground surface
6 1.0615 0.91530 0.830 0.7
the phenomena associated with increase of pore waterpressure and
reduction in effective stresses in saturatedcohesionless soils. The
rise in pore pressure can resultin generation of sand boils, loss
of shear strength,lateral spreading and slope failure. The
phenomena aremore prevalent in relatively loose sands and
artificialfill deposits.
Tunnels located below the groundwater table in liq- .uefiable
deposits can experience a increased lateral
. .pressure, b a loss of lateral passive resistance, c
.flotation or sinking in the liquefied soil, d lateral
displacements if the ground experiences lateral spread- .ing,
and e permanent settlement and compression
and tension failure after the dissipation of pore pres-sure and
consolidation of the soil.
5.1.2. Slope instabilityLandsliding as a result of ground
shaking is a com-
mon phenomena. Landsliding across a tunnel can re-sult in
concentrated shearing displacements and col-lapse of the cross
section. Landslide potential is great-est when a pre-existing
landslide mass intersects thetunnel. The hazard of landsliding is
greatest in shal-lower parts of a tunnel alignment and at tunnel
portals.
At tunnel portals, the primary failure mode tends tobe slope
failures. Particular caution must be taken if
the portal also acts as a retaining wall St. John and.Zahrah,
1987 . During the September 21, 1999 Chi Chi
earthquake in Taiwan slope instability at tunnel portalswas very
common, e.g. Fig. 4.
5.1.3. Fault displacementAn underground structure may have to be
con-
structed across a fault zone as it is not always possibleto
avoid crossing active faults. In these situations, theunderground
structure must tolerate the expected faultdisplacements, and allow
only minor damages. All faultsmust be identified to limit the
length of special designsection, and a risk-cost analysis should be
run to de-termine if the design should be pursued.
5.2. Ground shaking and deformation
In the absence of ground failure that results in large
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
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permanent deformation, the design focus shifts to thetransient
ground deformation induced by seismic wavepassage. The deformation
can be quite complex due tothe interaction of seismic waves with
surficial soft de-posits and the generation of surface waves. For
engi-neering design purposes, these complex deformationmodes are
simplified into their primary modes. Under-ground structures can be
assumed to undergo threeprimary modes of deformation during seismic
shaking: . .1 compressionextension; 2 longitudinal bending;
. .and 3 ovallingracking Fig. 6 . The simplest mode toconsider
is that of a compression wave propagatingparallel to the axis of a
subsurface excavation. Thatcase is illustrated in the figure, where
the wave isshown inducing longitudinal compression and tension.The
case of an underground structure subjected to anaxially propagating
wave is slightly more complex sincethere will be some interaction
between the structureand the ground. This interaction becomes more
impor-tant if the ground is soft and shear stress transferbetween
the ground and the structure is limited by theinterface shear
strength. For the case of a wave propa-gating normal or transverse
to the tunnel axis, thestress induces shear deformations of the
cross sectioncalled racking or ovaling. In the more general case,
thewave may induce curvature in the structure, inducingalternate
regions of compression and tension along thetunnel. The beam-like
structure of the tunnel liningwill then experience tension and
compression on oppo-site sides.
6. Seismic design loading criteria
Design loading criteria for underground structureshas to
incorporate the additional loading imposed byground shaking and
deformation. Once the groundmotion parameters for the maximum and
operationaldesign earthquakes have been determined, load
criteriaare developed for the underground structure using theload
factor design method. This section presents the
.seismic design loading criteria Wang, 1993 for MDEand ODE.
6.1. Loading criteria for maximum design earthquake,MDE
Given the performance goals of the MDE Section.4.3.1 , the
recommended seismic loading combinations
using the load factor design method are as follows:
6.1.1. For cut-and-coer tunnel structures
.UDLE1E2EQ 1
where Urequired structural strength capacity, D
effects due to dead loads of structural components,Leffects due
to live loads, E1effects due to verti-cal loads of earth and water,
E2effects due to hori-zontal loads of earth and water and EQeffects
dueto design earthquake motion.
( )6.1.2. For bored or mined circular tunnel lining
.UDLEXHEQ 2
.where U, D, L and EQ are as defined in Eq. 1 ,EXeffects of
static loads due to excavation e.g.
.ORourke, 1984 , and Heffects due to hydrostaticwater
pressure.
6.1.3. Comments on loading combinations for MDE
The structure should first be designed with ade-quate strength
capacity under static loading condi-tions.
The structure should then be checked in terms ofductility its
allowable deformation vs. maximum
.deformation imposed by earthquake as well asstrength when
earthquake effects, EQ, are con-sidered. The EQ term for
conventional surfacestructure design reflects primarily the
inertial effecton the structures. For tunnel structures,
theearthquake effect is governed not so much by aforce or stress,
but rather by the deformation im-posed by the ground.
In checking the strength capacity, the effects ofearthquake
loading should be expressed in terms ofinternal moments and forces,
which can be calcu-lated according to the lining deformations
imposedby the surrounding ground. If the strength criteria
. .expressed by Eq. 1 or Eq. 2 can be satisfiedbased on elastic
structural analysis, no furtherprovisions under the MDE are
required. Generally,the strength criteria can easily be met when
the
earthquake loading intensity is low i.e. in low.seismic risk
areas andor the ground is very stiff.
If the flexural strength of the structure lining, using .
.elastic analysis and Eq. 1 or Eq. 2 , is found to be
exceeded e.g. at certain joints of a cut-and-cover.tunnel frame
, one of the following two design
procedures should be followed:1. Provide sufficient ductility
using appropriate de-
.tailing procedure at the critical locations of thestructure to
accommodate the deformations im-posed by the ground in addition to
those caused by
. ..other loading effects see Eqs. 1 and 2 . Theintent is to
ensure that the structural strength doesnot degrade as a result of
inelastic deformationsand the damage can be controlled at an
acceptablelevel.
In general, the more ductility that is provided,the more
reduction in earthquake forces the EQ
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.term can be made in evaluating the requiredstrength, U. As a
rule of thumb, the force reduction
factor can be assumed equal to the ductility fac-.tor provided.
This reduction factor is similar by
definition to the response modification factor used .in bridge
design code AASHTO, 1991 .
Note, however, that since an inelastic sheardeformation may
result in strength degradation, itshould always be prevented by
providing sufficientshear strengths in structure members,
particularlyin the cut-and-cover rectangular frame. The use
ofductility factors for shear forces may not be ap-propriate.
2. Re-analyze the structure response by assuming theformation of
plastic hinges at the joints that arestrained into inelastic
action. Based on the plastic-hinge analysis, a redistribution of
moments andinternal forces will result.
If new plastic hinges are developed based on theresults, the
analysis is re-run by incorporating the
.new hinges i.e. an iterative procedure until allpotential
plastic hinges are properly accounted for.Proper detailing at the
hinges is then carried out toprovide adequate ductility. The
structural design in
. ..terms of required strength Eqs. 1 and 2 canthen be based on
the results from the plastic-hingeanalysis.
As discussed earlier, the overall stability of thestructure
during and after the MDE must be main-tained. Realizing that the
structures also must have
.sufficient capacity besides the earthquake effectto carry
static loads e.g. D, L, E1, E2 and H
.terms , the potential modes of instability due to
thedevelopment of plastic hinges or regions of inelas-
.tic deformation should be identified and prevented .Monsees and
Merritt, 1991 .
For cut-and-cover tunnel structures, the evaluation .of capacity
using Eq. 1 should consider the uncer-
tainties associated with the loads E1 and E2, andtheir worst
combination. For mined circular tunnels ..Eq. 2 , similar
consideration should be given tothe loads EX and H.
In many cases, the absence of live load, L, maypresent a more
critical condition than when a fulllive load is considered.
Therefore, a live load equalto zero should also be used in checking
the struc-
. .tural strength capacity using Eq. 1 and Eq. 2 .
6.2. Loading criteria for operating design earthquake, ODE
.For the ODE Section 4.3.2 , the seismic designloading
combination depends on the performance re-quirements of the
structural members. Generally
speaking, if the members are to experience little to no .damage
during the lower-level event ODE , the inelas-
tic deformations in the structure members should bekept low. The
following loading criteria, based on loadfactor design, are
recommended:
6.2.1. For cut-and-coer tunnel structures
. .U1.05D1.3L E1E2 1.3EQ 31
Where D, L, El, E2, EQ and U are as defined in Eq. .1 , 1.05 if
extreme loads are assumed for E1 and1E2 with little uncertainty.
Otherwise, use 1.3.1
( )6.2.2. For bored or mined circular tunnel lining
. .U1.05D1.3L EXH 1.3EQ 42
where D, L, EX, H, EQ and U are as defined in Eq. .2 , 1.05 if
extreme loads are assumed for EX and2H with little uncertainty.
Otherwise, use 1.3 for2EX only, as H is usually well defined.
The load factors used in these two equations havebeen the
subject of a lot of discussion. The final selec-tion depends on the
project-specific performance re-quirements. For example, a factor
of 1.3 is used for
. .dead load in the Central Artery I-93 Tunnel I-90Project
Central Artery Project Design Criteria, Bech-
.telParsons Brinckerhoff, 1992 .
6.2.3. Comments on loading combinations for ODE
The structure should first be designed with ade-quate strength
capacity under static loading condi-tions.
For cut-and-cover tunnel structures, the evaluation .of capacity
using Eq. 3 should consider the uncer-
tainties associated with the loads E1 and E2, andtheir worst
combination. For mined circular tunnels ..Eq. 4 , similar
consideration should be given tothe loads EX and H.
When the extreme loads are used for design, asmaller load factor
is recommended to avoid unnec-essary conservatism. Note that an
extreme load maybe a maximum load or a minimum load, dependingon
the most critical case of the loading combina-
.tions. Use Eq. 4 as an example. For a deep circu-lar tunnel
lining, it is very likely that the mostcritical loading condition
occurs when the maximumexcavation loading, EX, is combined with the
mini-
mum hydrostatic water pressure, H unless EX is.unsymmetrical .
For a cut-and-cover tunnel, the
most critical seismic condition may often be foundwhen the
maximum lateral earth pressure, E2, iscombined with the minimum
vertical earth load,E1. If a very conservative lateral earth
pressure
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( )Y.M.A. Hashash et al.Tunnelling and Underground Space
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coefficient is assumed in calculating the E2, thesmaller load
factor 1.05 should be used.1
. Redistribution of moments e.g. ACI 318, 1999 forcut-and-cover
concrete frames is recommended toachieve a more efficient
design.
. If the strength criteria expressed by Eq. 3 or Eq. .4 can be
satisfied based on elastic structural analy-sis, no further
provisions under the ODE are re-quired.
If the flexural strength of the structure, using elastic .
.analysis and Eq. 3 or Eq. 4 , is found to be
exceeded, the structure should be checked for its.ductility to
ensure that the resulting inelastic de-
formations, if any, are small. If necessary, the struc-ture
should be redesigned to ensure the intendedperformance goals during
the ODE.
. Zero live load condition i.e. L0 should also be . .evaluated
in Eq. 3 and Eq. 4 .
7. Underground structure response to grounddeformations
In this section, the term EQ effects due to design.earthquake
introduced in Section 6 is quantified. The
development of the EQ term requires an understand-ing of the
deformations induced by seismic waves in theground and the
interaction of the underground struc-ture with the ground.
This section describes procedures used to computedeformations
and forces corresponding to the three
deformation modes compression-extension, longitudi-.nal bending
and ovallingracking presented in Section
5.2. A brief summary of design approaches is providedin Table
6.
7.1. Free field deformation approach
The term free-field deformations describes groundstrains caused
by seismic waves in the absence ofstructures or excavations. These
deformations ignorethe interaction between the underground
structure andthe surrounding ground, but can provide a
first-orderestimate of the anticipated deformation of the
struc-ture. A designer may choose to impose these deforma-tions
directly on the structure. This approach mayoverestimate or
underestimate structure deformationsdepending on the rigidity of
the structure relative tothe ground.
7.1.1. Closed form elastic solutionsSimplified, closed-form
solutions are useful for de-
veloping initial estimates of strains and deformations ina
tunnel. These simplified methods assume the seismicwave field to be
that of plane waves with the same
amplitudes at all locations along the tunnel, differingonly in
their arrival time. Wave scattering and complexthree-dimensional
wave propagation, which can lead todifferences in wave amplitudes
along the tunnel are
neglected, although ground motion incoherence Sec-.tion 4.4.3
tends to increase the strains and stresses in
the longitudinal direction. Results of analyses based onplane
wave assumptions should be interpreted with
.care Power et al., 1996 . . .Newmark 1968 and Kuesel 1969
proposed a sim-
plified method for calculating free-field ground strainscaused
by a harmonic wave propagating at a givenangle of incidence in a
homogeneous, isotropic, elastic
.medium Fig. 9 . The most critical incidence angleyielding
maximum strain, is typically used as a safetymeasure against the
uncertainties of earthquake pre-diction. Newmarks approach provides
an order of mag-nitude estimate of wave-induced strains while
requiringa minimal input, making it useful as both an initial
design tool and a method of design verification Wang,.1993 .
.St. John and Zahrah 1987 used Newmarks ap-proach to develop
solutions for free-field axial andcurvature strains due to
compression, shear andRayleigh waves. Solutions for all three wave
types areshown in Table 5, though S-waves are typically associ-ated
with peak particle accelerations and velocities .Power et al., 1996
. The seismic waves causing thestrains are shown in Fig. 10. It is
often difficult todetermine which type of wave will dominate a
design.Strains produced by Rayleigh waves tend to governonly in
shallow structures and at sites far from the
.seismic source Wang, 1993 .Combined axial and curvature
deformations can be
obtained by treating the tunnel as an elastic beam. ab.Using
beam theory, total free-field axial strains,
are found by combining the longitudinal strains gener-ated by
axial and bending deformations Power et al.,
.1996 :
V aP Pab 2 2 cos r sincos 2C .5CP Pfor Pwaves
V aS Sab 3 sincos r cos 2C .6CS Sfor Swaves
V aR Rab 2 2 cos r sincos 2C C .R 7R .for Rayleighwaves
compressional component
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.Fig. 9. Simple harmonic wave and tunnel after Wang, 1993 .
Where:
r : radius of circular tunnel or half height of a rectan-gular
tunnel
a : peak particle acceleration associated with P-wavePa : peak
particle acceleration associated with S-waveSa : peak particle
acceleration associated with RayleighR
wave: angle of incidence of wave with respect to tunnel
axis : Poissons ratio of tunnel lining materiallV : peak
particle velocity associated with P-wavepC : apparent velocity of
P-wave propagationpV : peak particle velocity associated with
S-wavesC : apparent velocity of S-wave propagationsV : peak
particle velocity associated with RayleighR
WaveC : apparent velocity of Rayleigh wave propagationR
As the radius of the tunnel increases, the contribu-tion of
curvature deformation to axial strain increases.However,
calculations using the free-field equations ofTable 5 indicate that
the bending component of strainis, in general, relatively small
compared to axial strainsfor tunnels under seismic loading. The
cyclic nature ofthe axial strains should also be noted although
atunnel lining may crack in tension, this cracking is
usually transient due to the cyclic nature of the inci-dent
waves. The reinforcing steel in the lining will closethese cracks
at the end of the shaking, provided there
is no permanent ground deformation and the steel has.not yielded
. Even unreinforced concrete linings are
considered adequate as long as the cracks are small,uniformly
distributed, and do not adversely affect the
.performance of the lining Wang, 1993 .It should be noted that
the apparent P- and S-wave
velocities used in these equations may be closer tothose of
seismic wave propagation through deep rocksrather than the shallow
soil or rock in which a tunnel
may be located based on data from Abrahamson 1985,.1992, 1995 .
The apparent S-wave velocities fall in the
range of 24 kms while apparent P-wave velocities .fall in the
range of 48 kms Power et al., 1996 .
7.1.2. Oaling deformation of circular tunnelsOvaling
deformations develop when waves propagate
perpendicular to the tunnel axis and are therefore,designed for
in the transverse direction typically under.two-dimensional,
plane-strain conditions . Studies have
suggested that, while ovaling may be caused by wavespropagating
horizontally or obliquely, vertically propa-gating shear waves are
the predominant form ofearthquake loading that causes these types
of deforma-
.tions Wang, 1993 .
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Table 5 .Strain and curvature due to body and surface waves
after St. John and Zahrah, 1987
Wave type Longitudinal strain Normal strain Shear strain
Curvature
V V V a1P P P P2 2 2P-wae cos sin sincos sincos l n 2C C C CP P
P PV V V a1P P P P
for 0 for 90 for 45 0.385 for 3516lm lm m 2C C 2C CP P P max
P
V V V aS S S S2 3S-wae sincos sincos cos K cos l n 2C C C CS S S
SV V V aS S S S
for 45 for 45 for 0 K for 0lm n m m m 22C 2C C CS S S S
V V V aRayleigh wae R P R P R P R P2 2 2 cos sin sincos K sincos
l n 2Compressional C C C CR R R R
component V V V aR P R P P R P
for 0 for 90 for 45 K 0.385 for 3516lm n m m m 2C C 2C CR R R
R
V V aShear R S R P R S 2 sin cos K cos n 2component C C CR R
R
V V aR S R S R S for 90 for 0 K for 0n m m m 2C C CR R R
2 .C C 21 p sThe Poissons ratio and dynamic modulus of a soil
deposit can be computed from measured P- and S-wave propagation
velocities in an elastic medium: or Cm P22 .C C 1p s
. . .2 1 1 12m m m2 2 C ; E C ; and G C , respectively.S m P m
S( . .1 1m m
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Table 6 .Seismic racking design approaches after Wang, 1993
Approaches Advantages Disadvantages Applicability
Dynamic earth pressure 1. Used with reasonable 1. Lack of
rigorous For tunnels with minimalmethods results in the past
theoretical basis soil cover thickness
2. Require minimal 2. Resulting in excessiveparameters and
racking deformationscomputation error for tunnels with3. Serve as
additional significant burialsafety measures 3. Use limited to
certainagainst seismic types of groundloading properties
Free-field racking 1. Conservative for 1. Non-conservative for
For tunnel structures withdeformation method tunnel structure
stiffer tunnel structure more equal stiffness to ground
than ground flexible than ground2. Comparatively easy to 2.
Overly conservative forformulate tunnel structures3. Used with
reasonable significantly stiffer thanresults in the past ground
3. Less precision withhighly variable groundconditions
Soilstructure interaction 1. Best representation of 1. Requires
complex and All conditionsfinite-element analysis soilstructure
system time consuming
2. Best accuracy in computer analysisdetermining structure 2.
Uncertainty of designresponse seismic input3. Capable of solving
parameters may beproblems with several times thecomplicated tunnel
uncertainty of thegeometry and ground analysisconditions
Simplified frame analysis 1. Good approximation of 1. Less
precision with All conditions except formodel soilstructure
interaction highly variable ground compacted subsurface
2. Comparatively easy to ground profilesformulate3. Reasonable
accuracyin determiningstructure response
Ground shear distortions can be defined in two ways,as shown in
Fig. 11. In the non-perforated ground, themaximum diametric strain
is a function of maximumfree-field shear strain only:
d max . . 8d 2
The diametric strain in a perforated ground is fur-ther related
to the Poissons ratio of the medium:
d . .2 1 . 9max md
Both of these equations assume the absence of thelining,
therefore ignoring tunnelground interaction. Inthe free-field, the
perforated ground would yield amuch greater distortion than the
non-perforated,
sometimes by a factor of two or three. This provides areasonable
distortion criterion for a lining with littlestiffness relative to
the surrounding soil, while the
Fig. 10. Seismic waves causing longitudinal axial and bending
strains .Power et al., 1996 .
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Fig. 11. Free-field shear distortion of perforated and
non-perforated .ground, circular shape after Wang, 1993 .
non-perforated deformation equation will be appropri-ate when
the lining stiffness is equal to that of themedium. A lining with
large relative stiffness shouldexperience distortions even less
than those given by Eq. . .8 Wang, 1993 .
7.1.3. Racking deformations of rectangular tunnelsWhen subjected
to shear distortions during an
earthquake, a rectangular box structure will undergo .transverse
racking deformations Fig. 12 . The racking
deformations can be computed from shear strains inthe soil such
as those given in Table 5.
7.1.4. Numerical analysisNumerical analysis may be necessary to
estimate the
free-field shear distortions, particularly if the
sitestratigraphy is variable. Many computer programs areavailable
for such analyses such as 1-D wave propaga-tion programs listed in
Section 4.5, as well as FLUSH . .Lysmer et al., 1975 , and LINOS
Bardet, 1991 . Mostprograms model the site geology as a
horizontally lay-ered system and derive a solution using
one-dimen-
.sional wave propagation theory Schnabel et al., 1972 . .Navarro
1992 presents numerical computations for
ground deformations and pressures as a result of body .shear and
compression wave as well as surface .Rayleigh and Love waves. The
resulting free-fieldshear distortion can then be expressed as a
shear straindistribution or shear deformation profile with
depth.
7.1.5. Applicability of free field deformation approachThe
free-field racking deformation method has been
used on many significant projects, including the San .Francisco
BART stations and tunnels Kuesel, 1969
and the Los Angeles Metro Monsees and Merritt,.1991 . Kuesel
found that, in most cases, if a structure
can absorb free-field soil distortions elastically, no spe-cial
seismic provisions are necessary. Monsees and
.Merritt 1991 further specified that joints strained intoplastic
hinges can be allowed under the Maximum
.Design Earthquake MDE , provided no plastic hingecombinations
are formed that could lead to a collapsemechanism, as shown in Fig.
13.
The free-field deformation method is a simple andeffective
design tool when seismically-induced ground
distortions are small i.e. low shaking intensity, verystiff
ground, or the structure is flexible compared to the
.surrounding medium . However, in many cases, espe-cially in
soft soils, the method gives overly conservativedesigns because
free-field ground distortions in softsoils are generally large. For
example, rectangular boxstructures in soft soils are typically
designed with stiffconfigurations to resist static loads and are
therefore,
.Fig. 12. Typical free-field racking deformation imposed on a
buried rectangular frame after Wang, 1993 .
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Fig. 13. Structure stability for buried rectangular frames after
Wang,.1993 .
less tolerant to racking distortions Hwang and Lysmer,.1981;
TARTS, 1989 . Soilstructure interaction effects
have to be included for the design of such structures .Wang,
1993 . A comparison of the free field deforma-tion approach with
other methods for seismic rackingdesign is given in Table 6.
7.2. Soil structure interaction approach
The presence of an underground structure modifiesthe free field
ground deformations. The following para-graphs describe procedures
that model soil structureinteraction.
7.2.1. Closed form elastic solutions for circular tunnels,axial
force and moment
In this class of solutions the beam-on-elastic founda- .tion
approach is used to model quasi-static soil-struc-
ture interaction effects. The solutions ignore dynamic .inertial
interaction effects. Under seismic loading, thecross-section of a
tunnel will experience axial bendingand shear strains due to free
field axial, curvature, andshear deformations. The maximum
structural strains
.are after St. John and Zahrah, 1987 :
The maximum axial strain, caused by a 45 incidentshear wave,
Fig. 9, is:
2A / fLLa . 10max 2 4E AE A l c2l c2 /K La
Where
L wavelength of an ideal sinusoidal shear wave ..see Eq. 15
Fig. 14. Induced forces and moments caused by seismic waves
Power. .et al., 1996 , a Induced forces and moments caused by
waves
.propagating along tunnel axis, b Induced circumferential forces
andmoments caused by waves propagating perpendicular to tunnel
axis.
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K longitudinal spring coefficient of mediumain force per unit
deformation per unit length
..of tunnel, see Eq. 14A free-field displacement response
amplitude of
.an ideal sinusoidal shear wave see Eqs. 17 ..and 18
A cross-sectional area of tunnel liningcE elastic modulus of the
tunnel liningl
.f ultimate friction force per unit length betweentunnel and
surrounding soil
The forces and moments in the tunnel lining causedby seismic
waves propagating along the tunnel axis areillustrated in Fig. 14a.
The maximum frictional forcesthat can be developed between the
lining and thesurrounding soils limit the axial strain in the
lining.
.This maximum frictional force, Q , can be esti-max fmated as
the ultimate frictional force per unit length
.times one-quarter the wave length, as shown in Eq. 10 .Sakurai
and Takahashi, 1969 .
The maximum bending strain, caused by a 0 incidentshear wave,
is:
22A /Lb . r 11max 4E I 2l c1 /K Lt
Where
I moment of inertia of the tunnel sectioncK transverse spring
coefficient of the mediumt
in force per unit deformation per unit length of ..tunnel see
Eq. 14
r radius of circular tunnel or half height of arectangular
tunnel
Since both the liner and the medium are assumed tobe linear
elastic, these strains may be superimposed.
Since earthquake loading is cyclic, both extremes posi-.tive and
negative must be evaluated. The maximum
shear force acting on a tunnel cross-section can bewritten as a
function of this maximum bending strain:
32E I Al c / 2L
V Mmax max4 /LE I 2l c1 /K LtE I b2 l c max . 12 / /L r
A conservative estimate of the total axial strain andstress is
obtained by combining the strains from the
axial and bending forces modified from Power et al.,.1996 :
ab a b . . 13max max
Again, these equations are necessary only for struc-tures built
in soft ground, as structures in rock or stiffsoils can be designed
using free-field deformations. Itshould be further noted that
increasing the structuralstiffness and the strength capacity of the
tunnel maynot result in reduced forces the structure mayactually
attract more force. Instead, a more flexibleconfiguration with
adequate ductile reinforcement or
.flexible joints may be more efficient Wang, 1993 .7.2.1.1.
Spring coefficients. Other expressions of maxi-
mum sectional forces exist in the literature SFBART,.1960;
Kuribayashi et al., 1974; JSCE, 1975 , with the
major differences involving the maximization of forcesand
displacements with respect to wavelength. JSCE .1975 suggests
substituting the values of wavelengththat will maximize the
stresses back into each respec-tive equation to yield maximum
sectional forces. St.
.John and Zahrah 1987 suggest a maximization method .similar to
the JSCE 1975 approach, except that the
spring coefficients K and K are considered functionsa tof the
incident wavelength:
.16G 1 dm m .K K 14t a L .34m
where G , shear modulus and Poissons ratio ofm mthe medium,
ddiameter of circular tunnel or height
.of rectangular structure . .These spring constants represent 1
the ratio of
.pressure between the tunnel and the medium, and 2the reduced
displacement of the medium when thetunnel is present. The springs
differ from those of aconventional beam analysis on an elastic
foundation.Not only must the coefficients be representative of
thedynamic modulus of the ground, but the derivation ofthese
constants must consider the fact that the seismicloading is
alternately positive and negative due to the
.assumed sinusoidal wave Wang, 1993 . When usingthese equations
to calculate the forces and momentsfor tunnels located at shallow
depths, the soil springresistance values are limited by the depth
of cover andlateral passive soil resistance.
7.2.1.2. Idealized sinusoidal free field wae parametersfor use
in soilstructure interaction analysis. Matsubara
.et al. 1995 provide a discussion of input wavelengthsfor
underground structure design. The incident wave-length of a ground
motion may be estimated as:
.LT C 15s
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.Fig. 15. Lining response coefficient vs. flexibility ratio,
full-slip interface, and circular tunnel Wang, 1993 .
where T is the predominant natural period of a shearwave in the
soil deposit, the natural period of the siteitself, or the period
at which maximum displacements
.occur Dobry et al., 1976; Power et al., 1996 . .Idriss and Seed
1968 recommend that:
4h .T , h is the thickness of the soil deposit 16CS
if ground motion can be attributed primarily to shearwaves and
the medium is assumed to consist of a
uniform soft soil layer overlying a stiff layer St. John.and
Zahrah, 1987 .
The ground displacement response amplitude, A,represents the
spatial variations of ground motionsalong a horizontal alignment
and should be derived bysite-specific subsurface conditions.
Generally, the dis-
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placement amplitude increases with increasing wave- .length
SFBART, 1960 . Assuming a sinusoidal wave
with a displacement amplitude A and a wavelength L,A can be
calculated from the following equations:
For free-field axial strains:
V2A s . sincos. 17L CS
For free-field bending strains:
2 a4 A s 3 . cos . 182 CL S
7.2.2. Oaling deformations of circular tunnelsIn early studies
of racking deformations, Peck et al.
.1972 , based on earlier work by Burns and Richard . .1964 and
Hoeg 1968 , proposed closed-form solu-tions in terms of thrusts,
bending moments, and dis-placements under external loading
conditions. The re-sponse of a tunnel lining is a function of the
compress-ibility and flexibility ratios of the structure, and
the
.in-situ overburden pressure h and at-rest coeffi-t .cient of
earth pressure K of the soil. To adapt to0
seismic loadings caused by shear waves, the free-fieldshear
stress replaces the in-situ overburden pressureand the at-rest
coefficient of earth pressure is assigned
.a value of 1 to simulate the field simple shearcondition. The
shear stress can be further expressed asa function of shear
strain.
The stiffness of a tunnel relative to the surroundingground is
quantified by the compressibility and flexibil-
.ity ratios C and F , which are measures of the exten-sional
stiffness and the flexural stiffness resistance to
.ovaling , respectively, of the medium relative to the .lining
Merritt et al., 1985 :
2 .E 1 rm l .C 19 . .E t 1 12l m m
2 . 3E 1 Rm l .F 20 .6E I 1l m
where E modulus of elasticity of the medium, Im .moment of
inertia of the tunnel lining per unit width
for circular lining R, and tradius and thickness ofthe tunnel
lining.
Assuming full-slip conditions, without normal sepa-ration and
therefore, no tangential shear force, thediametric strain, the
maximum thrust, and bending
.moment can be expressed as Wang, 1993 :
d 1 . K F 211 maxd 3
E1 m .T K r 22max 1 max6 .1m
E1 m 2 .M K r 23max 1 max6 .1m
where
.12 1m .K . 241 2 F56m
These forces and moments are illustrated in Fig. 14b.The
relationship between the full-slip lining response
.coefficient K and flexibility ratio is shown in Fig.
15.1According to various studies, slip at the interface is
only possible for tunnels in soft soils or cases of
severeseismic loading intensity. For most tunnels, the inter-face
condition is between full-slip and no-slip, so bothcases should be
investigated for critical lining forcesand deformations. However,
full-slip assumptions un-der simple shear may cause significant
underestimationof the maximum thrust, so it has been
recommendedthat the no-slip assumption of complete soil
continuity
be made in assessing the lining thrust response Hoeg,.1968;
Schwartz and Einstein, 1980 :
Em .T K rK r 25max 2 max 2 max .2 1m
where
. . F 12 12 Cm m1 2 . 12 2m2 .K 1 . 262 . . F 32 12 Cm m
5 2C 8 6 68m m m2
As Fig. 16 shows, seismically-induced thrusts in-crease with
decreasing compressibility and flexibilityratios when the Poissons
ratio of the surroundingground is less than 0.5. As Poissons ratio
approaches
.0.5 i.e. saturated undrained clay , the thrust responseis
independent of compressibility because the soil is
.considered incompressible Wang, 1993 .The normalized lining
deflection provides an indica-
tion of the importance of the flexibility ratio in lining
.response, and is defined as Wang, 1993 :
d 2lining . K F . 271d 3freefield
According to this equation and Fig. 17, a tunnellining will
deform less than the free field when the
flexibility ratio is less than one i.e. stiff lining in soft
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.soil . As the flexibility ratio increases, the lining de-flects
more than the free field and may reach an upperlimit equal to the
perforated ground deformations. Thiscondition continues as the
flexibility ratio becomes
.infinitely large i.e. perfectly flexible lining . .Penzien and
Wu 1998 developed similar closed-form
elastic solutions for thrust, shear, and moment in thetunnel
lining due to racking deformations. Penzien .2000 provided an
analytical procedure for evaluatingracking deformations of
rectangular and circular tun-nels that supplemented the previous
publication.
In order to estimate the distortion of the structure,
alining-soil racking ratio is defined as:
structure .R . 28 freefield
In the case of circular tunnel, R is the ratio of
liningdiametric deflection and free-field diametric
deflection.Assuming full slip condition, solutions for thrust,
mo-ment, and shear in circular tunnel linings caused
bysoil-structure interaction during a seismic event are
.expressed as Penzien, 2000 :
n n .d R d 29lining freefield
12 E Idn l lining . .T cos2 30 /3 2 4 .d 1l6E Idn l lining . .M
cos2 31 /2 2 4 .d 1l24E Idn l lining . .V sin2 32 /3 2 4 .d 1l
The lining-soil racking ratio under normal loadingonly is
defined as:
.4 1mn .R 33n . 1
.12 E I 56l mn . . 343 2 .d G 1m l
The sign convention for the above force componentsin circular
lining is shown in Fig. 18. In the case of noslip condition, the
formulations are presented as:
.d Rd 35lining freefield
24E Id l lining . .T cos2 36 /3 2 4 .d 1l
.Fig. 16. Lining thrust response coefficient vs. compressibility
ratio, .no-slip interface, and circular tunnel Wang, 1993 .
6E Id l lining . .M cos2 37 /2 2 4 .d 1l24E Id l lining . .V
sin2 38 /3 2 4 .d 1l
where
.4 1m .R 39 .1
.24E I 34l m . . 403 2 .d G 1m l
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.Fig. 17. Normalized lining deflection vs. flexibility ratio,
full slip interface, and circular lining Wang, 1993 .
.The solutions of Penzien 2000 result in values ofthrust and
moment that are very close to those of
.Wang 1993 for full-slip condition. However, value ofthrust
obtained from Wang is much higher compared
to the value given by Penzien in the case of no slip see.example
3 in Appendix B . This observation was also .noted by Power et al.
1996 . The reason for the
difference is still under investigation.
7.2.3. Racking deformations of rectangular tunnelsShallow
transportation tunnels are usually box shaped
cut-and-cover method structures. These tunnels haveseismic
characteristics very different from circular tun-nels. A box frame
does not transmit static loads asefficiently as a circular lining,
so the walls and slabs ofthe cut-and-cover frame need to be
thicker, and there-fore stiffer. The design of cut-and-cover
structures re-quires careful consideration of soil-structure
interac-tion effects because of this increased structural
stiff-ness and the potential for larger ground deformationsdue to
shallow burial. Seismic ground deformationstend to be greater at
shallow depths for two reasons:
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.1 the decreased stiffness of the surrounding soils due .to
lower overburden pressures; and 2 the site ampli-
fication effect. The soil backfill may also consist ofcompacted
material with different properties from thein-situ soil, resulting
in a different seismic response .Wang, 1993 .
The structural rigidity of box structures significantlyreduces
computed strains, often making it overly con-servative to design
these structures based on free-field
.strains Hwang and Lysmer, 1981 . While closed-formsolutions for
tunnel-ground interaction problems areavailable for circular
tunnels, they are not available forrectangular tunnels because of
the highly variable geo-metric characteristics associated with
these structures.For ease of design, simple and practical
procedureshave been developed to account for dynamic
soil-struc-
.ture interaction effects Wang, 1993 .A number of factors
contribute to the soil-structure
interaction effect, including the relative stiffnessbetween soil
and structure, structure geometry, inputearthquake motions, and
tunnel embedment depth.The most important factor is the stiffness
in simpleshear of the soil relative to the structure that
replaces
.it, the flexibility ratio Wang, 1993 .Consider a rectangular
soil element in a soil column
under simple shear condition, as shown in Fig. 19.When subjected
to simple shear stress the shear strain,or angular distortion, of
the soil element is given by .Wang, 1993 :
. . 41s H Gm
After rearranging this equation, the shear or