7/28/2019 Aseismic Design of Underground Structures

1/34

AITES

ITA

Towards animproved use

of undergroundSpace

In Consultative Status, Category II with theUnited Nations Economic and Social Council

http://www.ita-aites.org

ASSOCIATIONINTERNATIONALE DES TRAVAUX

EN SOUTERRAININTERNATIONALTUNNELLINGASSOCIATION

Aseismic Design of Underground Structures

SEISMIC EFFECTS

Title

Topic

Originally published

Abstract: This study defines the basis for the aseismic design of subsurface excavations and underground structures. It

includes a definition of the seismic environment and earthquake hazard, and a review of the analytical andempirical tools that are available to the designer concerned with the performance of underground structuressubjected to seismic loads. Particular attention is devoted to development of simplified models that appear tobe applicable in many practical cases.

Rsum: Cette tude dfinit les bases pour la conception asismique d'excavations et de structures sous-terre. On inclutune dfinition de l'environnement sismique et du risque de tremblement de terre, ainsi qu'une revue destechniques analytiques et empiriques qui sont la disposition du concepteur proccup de la performance desstructures sous terre et soumises des actions sismiques. Une attention particulire est donne audveloppement de modles simplifis qui semblent tre applicables dans la plupart du temps.

Remarks: -

in the J ournal "Tunnelling and Underground Space Technology",

Working Group: WG 9 - "Seismic Effects"

AuthorITA WG Seismic Effects

Open Session, Seminar, Workshop: -

Others: Report

Copyright 1987, Elsevier Science Limited, www.elsevier.com; All Rights reserved.

Vol. 2, Nr. 2, pp. 165 - 197, Year 1987.

Secretariat : ITA-AITES c/o EPFL - Bt. GC CH-1015 Lausanne - Switzerland

Fax : +41 21 693 41 53 - Tel. : +41 21 693 23 10 - e-mail : secretari at@ita-aites.org - www.ita-aites.org

7/28/2019 Aseismic Design of Underground Structures

2/34

FEATURE SECTION: Seismic Effects on U nde rgr oun d Structures

Aseismic Design of Underground Structures

C. M. St John and T. F. Zahrah

Ab st ra ct ~T hi s study de[ines the basis for the aseisrnic design ofsub sur fac e exc ava ti ons and un de rg ro un d st ru ctu res . It in cl ud es ade[inition of the seismic environment and earthquake hazard, and areview o] the analytical and empirical tools that are available to thedesigner concerned with the performance o[ underground structures

su bje ct ed to sei smi c loads. Pa rt ic ul ar at te nt io n is de vo te d todevelopment of sirnpli]ied models that appear to be applicable inmany practical cases.

R~ sum 6- -C et te kt ud e dk] ini t les bases p ou r la c on ce pt io n as~is rni qued'excavations et de structures sous-terre. On inclut une dklinition del'envir onnernent skisrnique et du ris que de trernblernent de terre, ainsiqu' une revue des techniques ana lytiques et ernpiriques qui so nt g~ladispo siti on du concepteur pr~occupk de la perIorrnance des structures

so us ter re et so um is es ~ des ac ti on s s~i srn ique s. Une at te nt io npa rt ic ul i~ re est do nn ~e au d~ vel opp ernen t de mod ul es sir npl i] iks qu ise rn bl ent Otre app li cab le s dan s la pl up ar t du tem ps.

Introduction

The objec tive of this r e por t is top r o v i d e a r e l a t i v e l y c o n c i s estatement of the state of the art

for the design of underground structuresi n s e is m ic e n vi r on m en t s. L i k e m a n yo t he r s t at e -o f -t h e- ar t r ep or t s, i t i si n t en d ed t o b e b r i ef a n d t o f oc us o nr e co m m en d e d p ra ct i ce . I ts i n t e nd e daudience is the practicing engineer whom a y ha ve e xte nsive e xper ie nc e in thede sign of un de r gr ound str uctur es butwho has limited awareness of the specialconsider ations necessary in a seismi callyactive environm ent.

The need to establish a consensus onse ism ic de sign pr oc edur es f or unde r-ground structures has been recognizedf or a n u m b e r o f y ea rs . I n 1980, t h eI n t er n a ti o n al T u n n e l l i n g A s so c ia t i one st ab l is he d a w o rk i n g g r o u p o n t hetopic. Since that time, the group has metregularly to discuss progress in collec-tion of case histories and pre parati on ofa ppr opr ia te doc um e nta tion a nd de signrecommendations. During this study wehave drawn heavily on the activities of that working group, and have benefitedsignif ic a ntly f r om the le ve l of inte r -national cooperation it has engendered.

To wha t e xte nt this r e por t sa tisf ies thene ed f or a se ism ic de sign m a nua l, a ndreflects the opini ons of the internati onalt u n n e l l i n g c o m m u n i t y, r e m ai n s t o b edetermined.

The remainder of the report comprisesf our se c tions, f our a ppe ndic e s, a nd abibli ogra phy. T h e ex te ns iv e us e of

This report was prepared by the authorsfor the ITA Working Group on SeismicEffects on Underground Structures, underNat io nal Science Fo unda ti on Gr an t No.CEE-8310631. Th e report was publishedoriginally by Agbabian Associates, ElSegundo, California (U.S.A.). We aregrateful

to the National Science Foundation forgranting permission to publish the report inthis publication.

Appendices reflects a desire to keep the

m a i n t ex t b r ie f, w i t h o u t l e a vi n g t h er e ade r with a n inc om ple te tr e atm e nt.Specifically, the next section, on subjectof se ism ic e nvir onm e nt, is a m plif ie d inA p p e n d i x A ; a n d t h e l a s t s e ct i on , i nwhic h sim plif ie d de sign pr oc e dur es a r er e c om m e nde d, is suppor te d by Appe n-dic es B a nd C , whic h c ove r the or e tic a lde ve lopm e nts, a nd Appe ndix D, whic hc o n t ai n s d e si g n e x am p l es . T h e t h i rdsection summarizes the current empiricalb a s e f or d e s i g n o f u n d e r g r o u n ds tr uc tu re s i n r oc k, a nd t he f ou rt hse c tion br ie fly r e vie ws the a na lytic a ltools a va ila ble to the tunne l e ngine er

c on ce rn ed w it h d es i gn i n a s ei sm ice n vi r on m en t . N ee dl es s t o s ay , t hi sreport cannot be entirely comprehensive.However, we believe it provides a basisfor unders tandi ng the issues involved ins e is m ic d e s i gn , a s w e l l a s a r a t io n a lapproach that may prove satisfactory in

many cases of practical concern.

Seismic Activity

This chapter contains a briefsu mmaryof the f unda m e ntal c once pts pe r ta iningt o t he d e fi n it i on o f t he s ei sm ic e n-

v i ro nm en t a nd t he d ev el op me nt o f se ism ic input c r i ter ia f or the de sign of unde r gr ound str uc tur e s. The subje c t ismore fully addressed in Appendix A.

S e i s m i c E n v i r o n m e n t

Seismologists typically classify earth-q u ak e s a c co r di n g t o f ou r m od es o f generation--tectonic, volcanic, collapse,or e xplosion. R e gar dle ss of the type of earthquake, an engineer concerned withd e si g n o f u n d e r g ro u n d s t ru c tu r esr e quir e s tha t the se ism ic e nvir onm e ntbe de fi ne d in a qua nti ta t ive ma nne r.Specifically,the characteristics of earth-

qua ke s a nd gr ound m otion pe r t ine nt tot he d ev el op m en t o f s ei sm ic i n pu tc r i te r ia a r e the siz e of the e a r thqua ke ,

the intensity and the frequency content

of the gr ound m otion, a nd the dur a tionof strong shaking.

Size ofearthquakeThe size of the earthquake is typically

represented for engineering purp oses interms of its magnitude. Several differentm a gnitude sca le s a re c ur r e ntly in use ,t he m os t c om m on b ei ng t he l oc alm a gn i tu d e, ML; t he s ur fa ce w av em ag ni tu de , Ms; the b od y wavem ag ni tu de , MB; a nd t he m om en tmagnitude, Mw. Definitions of each ofthese sc ale s a nd the ir a pplic a tio n a r egiven by Housner and Jennings (1982).

P h ys i ca l ly , t he m a g n it u d e h a s b ee ncorrelated with the energy released byt h e e a r t h qu a k e, a s w e ll a s t h e f au l trupture length, felt area, and maximumdisplacement. Typically the magnitudeis estimated, either in a deterministic or in a probabilistic manner, using generalor site-specificcorrelations between themagnitude and the fault rupture length.The e ngine er will use the e stim a te of m a gn i t ud e i n c o nj u nc t io n w i th e m-pir ica l at te ntuat ion re lat ion ship s tod ef in e t he i nt en si ty o f t he g r ou n dmotion experienced at a specific site ats o me d i st a nc e f ro m t he e a r th q u ak e

source.

Intensity of the ground mot ionThe intensity of the ground motion is

obtained from recorded ground motiontime histories. Several parameters, in-clud ing peak acceleration, peak velocity,pe ak di sp lac em en t, sp ec tr um int en si ty ,and root-mean-square acceleration areused; the m ost wide ly use d m e a sur e isthe peak grou nd acceleration. However,pe ak g r ou n d ac ce le ra tion is no tnecessarily a good measure of damagepo ten tial be ca us e it is of te n re pe ti ti ve

shaking wit h strong energy content thatl e ad s t o p e r m an e n t d e f or m a ti o n a n ddamage. As a result, the term "effective

Tunnelling and UndergroundSpaceTechnology,Vol. 2, No. 2, pp. 165-197, 19 87 . 0886-7798/87 $3.00 + .00Printed in Great Britain. O 1987 Pergamon Journals Ltd. 16 ~

7/28/2019 Aseismic Design of Underground Structures

3/34

pe ak ac ce le ra ti on " has been use d to ref erto an acceleration which is less than the

pe ak va lu e bu t is mo re re pr es en ta ti ve ofthe da m a ge pote ntia l ( Newm ar k a ndHal l 1982).

In view of the importance of predict-i n g t he g r ou nd m o ti on t ha t w il l b ee xper ie nc e d a t a pa r tic ula r s i te , c on-

siderable attention has been devoted tode ve loping a tte nua tion r e la tionships

ba se d on co rr el at io ns bet wee n fi el d d at a

o n g r o u nd m o t i o n a n d t he m a g n it u d eand distance of the earthquake. Ideally,such relationships should beestablishedon a site-specific basis. In the absence ofsufficient site data, use can be m ade ofregional or global relationships such asgiven by Seed and Idriss (1982). Whendoing so, care must be taken to ensurethat the correlation is based on data thatis pe r t inent both in te r ms of ge ologice nv ir on me nt a nd the e ar th qu ak emagnitude.

/-

Frequency of content of the groundmotion

The frequency content of the groundm ot i on is c om m on ly d ef in ed b y aF o ur i e r a m p l i tu d e s p ec tr um a n d / o r ar e sponse spe ctr um . B oth a re obta ine df r om c om puta tion of the r esponse of asingle-degree-of-freedom (SDOF) oscil-l at or to b as e m ot io n. T he F ou ri er a m p li t u de s pe ct ru m is a p l ot o f t heamplitude o1 the relative velocity for anunda m pe d S DOF osc il lator , a t the e ndof a strong mot ion record, as a functionof its frequency. It is less widely used ford es i gn p u rp o se s t ha n t he r es p on sespectrum, which is defined as a plot of t he m a xi m um r es po ns e of a S DO Foscillator as a function of its frequencya nd da m ping.

T h e r es po ns e s pe ct ru m, w h ic h i sc om m only plotte d in loga r ithm ic , tri-partite form, derives its popul arity fromthe f a c t tha t the S DOF osc il la tor is areasonably good analo gue for represent-i n g t he s i g ni f i ca n t r e s po ns e o f m a n ysur fa c e str ucture s. This a na logy doesn o t h o l d f or u n d e r g ro u n d s t ru c tu r es

be ca us e th ey te nd to mo ve with th eg r ou n d m as s i ns te ad o f v ib ra t in gindependently. Hence, response spectraa re g en e ra ll y l ess i m p or t an t t o t hed e s ig n er o f u n d e r g ro u n d s t ru ct u re s.Howe ve r , the y ha ve a pplic a tion in thedesign of light structures located within

a n unde r gr oun d e xc ava tion. Also, ther e sponse spe c tr a c a n be use d to de f inethe frequency content of a time-historyi n p u t f or a n u me r ic a l s i m ul a t i on o f g r o un d / st r u ct u r e r es po ns e, a n d f or a p p ro x i m at e d e fi n it i on o f t he p ea k gr ound m otion pa r a m e te r s.

Duration of strong motionT h e d u r a t i o n o f s t r o n g m o t i o n c a n

ha ve a pr of ound e ff ec t on the e xte nt of da m a ge r e sult ing f r om a n e a r thqua ke .In particul ar, it is reasonable to supposetha t the num be r of e xc ur sions into then o n l in e a r r a ng e e x pe ri e nc ed b y a n

undergro und structure and the surround-i n g m ed i a w il l c o nt r ol t he e x t en t o f per m an en t de fo rm at io n. Un fo rt un at el y,t he re is a t p re se nt n o u ni ve rs al lya cc ep te d m e th o d o f q u an t if y i ng t hedur a tion of the gr ound m otion; a nd theeffects of repeated, cyclical loading on

t he p er fo rm an ce o f u n de r gr o un dstr uctur e s a r e ve r y poor ly unde r stood.Until suc h unde r sta nding c a n be ga ine dthr ough de ta iled f ie ld inve stiga tions or

n u m e r i ca l s i m u l at i o n s , t he d e si g ne r s h o ul d e ns ur e t ha t a ny e m pi r i ca l l ybas ed de sig n cr i te ri a ar e ba se d on th eper fo r m an ce of st ru ct ur es su bj ec te d toc o m p a r a b l e l o a d i n g , i n t e r m s o f p e a k a m p li t u de , f re qu en cy c on te nt , a n ddur a tion.

Seismic Input CriteriaSeveral alternative approach es can be

used for defining seismic input criteria.

O ne a p pr o ac h i nv ol ve s t he u se o f response spectra. This approach, whichi s t he m o s t w i de l y u s e d f or s ur fa cestr uctur e s, is c ove r ed in Appe ndix A.Anothe r a ppr oa c h is to spe c if y gr oundm otion t im e historie s. I n this c a se a ne ns em b le o f m o t io n t im e h is to ri es ,rather than a single time history, should

be sp ec if ie d. Th e f am ily of m ot ion sshould ha ve the sa m e ove r a ll inte nsitya nd f r e que nc y c ontent, a nd should berepresentative of the anticipated sha kinga t t he s it e d u e t o a l l t he s i gn i fi c an tp ot en ti a l e a r th qua ke so ur ce s in th evicinity of the site. The procedure usedt o s el ec t t h e m o t i o n t i me h i st o ri e s i sdescribed by Werner (1985).

An alternative approach for specifyingseismic inp ut criteria involves the use ofseismic regionalizatio n maps of the typeu se d i n c ur re nt d es ig n co de s a ndpa r t i c ula r ly in th e se is mi c de si gng u i d el i n e s s u gg e st e d b y t he A p p l i e dTe c hnology C ounc il ( ATC 1978). Thisapproach is covered below.

Seismic regionalization mapsS ei sm ic r e gi o na l iz a ti o n m a ps a re

inte nde d to pr ovide r e pre se nta tive in-t en si ti es o f s h ak i ng f or t he r eg io n su n d er c o n si d e ra t i on , b as ed o n theirseismologic and geologic characteristics.This inte nsity f a ctor is used, toge the r with a num e r ic a l f a ctor tha t r e pre se ntslocal site effects, in order to incor poratethe inf lue nc e of the se ism ic e nvir on-m e nt in the c om puta tion of e quiva le ntforces upon which the seismic design ofthe structure is based (Berg 1982).

Although m a ny se ism ic r e giona liz a -tion maps have been developed throught he y ea rs , t he m a ps i n cl u de d i n t hede sign pr ovisions r e c om m e nde d by theApplie d Te c hnology C ounc il ( ATC - 3)are the most current (ATC 1978). Thesem a p s, w h i ch a re g e n er a l ly b a s e d o nw or k by A lg er mi ss en a nd P er ki ns( 1976), we re de ve lope d using pr oba -bil ist ic pr oc ed ur es i nc or p or a ti ng (1)identification of significant earthquakesour ce s; ( 2) a sse ssme nt of m a xi m um

c r ed i b le m a g n it u d e s a n d m a g n it u d e -recurrence laws for each source; and (3)attenu ation laws describing the intensityof sha king a s a f unc tion of m a gnitudea nd dista nc e f r om a n e pic e nte r. B asedo n t he a b ov e p r i n c ip l es , c o nt o ur s o f l o c at i o n s w i t h e q u al p r o ba b i li t i e s of

receiving specific intensities of groundshaking are produced.

T w o s e is m ic r e g io n a l iz a t io n m a pspr ov ided in AT C- 3 ar e re pr od uc ed in

F ig. 1: one c or r esponds to "ef fe ctivepea k ac ce ler at io n (E PA ), " a nd th e ot he rt o " ef fe ct i ve p e ak v el o ci t y ( EP V) ."

Nei ther of th es e pa ra me ter s ha s pr ec is ephysica l de fi ni t io ns ; ho we ve r, a co n-ceptual descript ion of their significancec an b e f o un d i n t he c o m m en t ar y of ATC - 3 ( 1978) . Although the EP A a ndEPV are rel at ed to p ea k g ro un da c c e le r a tion a nd pe a k gr ound velocity,they are not necessarily the same as or e ve n pr opor tiona l to pe a k a c c e le r a tiona n d v e lo c it y . T h e E PA e x pr es s ed i nunits ofg's (Aa)is used in ATC-3 to scalethe inte nsity of the spe c tr um sha pe too b ta i n a d es ig n s pe ct ru m. T h e EP Vexpressed as a velocity-related accelera-t ion in g's (A,,) is used (1) to adjust thespectrum shape to account for extendedd i st an ce ; a n d (2) t o r ep re se nt t hestr e ngth of sha king in the c om puta tionof equivale nt design forces.

Observed Effects ofSeismic Loading ofUnderground Structures

EJ[ects of EarthquakesT h e p re vi o us s ec ti on p ro vi d ed a

ge ne ra l intr oduc tion to the subjec t of t h e d y n a m i c e n v ir o n m en t a ss o ci a te dw i t h e a rt h q ua k es . O u r u n d e rs t a n di n go f h o w s ur fa ce s tr uc tu re s, s uc h a sbuildings, da m s, or so il sl op es , re sp on dto suc h a n e nvir onm e nt ha s de ve lope dthrough observations made both duringa n d a ft er e a rt h qu a ke s . E a rl y u n de r-s t a n d i n g o f h o w t o c o n st r u ct e ar th -qua ke - r e sista nt s tr uc tur e s wa s base dpure ly on qu a li tat ive ob se rv at io n. Mo rer ec en t ly , m e as u re m en t a n d a na l ys i s

hav e been used as the bas is for de ve lopm e nt of im pr ove d de sign pr o-cedures.

A si m ila r de ve lopm enta l pr oc ess isoc c ur r ing f or unde r gr ound struc tur es,but th e pr oc es s is fa r fr om co mp let e atpr es en t.

This section begins to follow the pathof tha t de ve lopm e nt by r e vie wing thed a t a o n p e rf o rm a n ce o f u n d e r gr o u n ds tr uc tu re s. T h i s m at er ia l h as b ee nd r a w n p r i m a r i l y f r om r ep o rt s o f t heeffects of earthquakes, but some atten-t i o n a l s o w i l l b e d e vo t ed t o r e l ev a nte xp er ie nc e o f t he p er fo rm an ce o f excavations close to large undergrounde xplosions.

Dam age Mec hanis msThe effects of earthquakes on tunnels,

m i n es , a n d o t he r l ar ge u n d e rg r o u n d

166 TUNNELLING AND UNDERGROUND SPACE TECHNOLOGY Volum e 2, Num ber 2, 1987

7/28/2019 Aseismic Design of Underground Structures

4/34

0 jO~5

0 ~ \ 2/ ~'~r~F--~M I I ~ Y O 0 5 I ~-. ~ ~ J " I ~

Note: Contours showvotue of Ao

Ao = Ef fe ct iv e peak acc etera' don , 17

(o) Effective peak acceLera'tion

f ~ d M / Z

i2 (0.304e) 0,4 - \ _.-'~ ~ h \!9.LS___z_4! 9. 2 \ / - ~ \

~.5 ( 0, 038 1) 0 .0 5t0~Q~2) o. t k ( \

Note: Contours show value of A,

( b) E f fe c t i v e p e ak v e lo c i ty

Figure 1. ,4TC-3 (1978) seismic regionalization maps.

excavations have been the subject ofseveral reports. A comprehensive reviewo f t ho se r ep or ts a n d c o mp i la t io n o f re adily a vaila ble da ta was prepa re drecently by URS/Blume and Associateso n b eh a lf o f t he N at i on al S ci en ceFounda tion a nd the Fe de ra l Highwa yAdministration Department of Trans-

po rta ti on (Ow en an d Sch oll 1981). Inthe ir revie w, e a rthqua ke da mage tounder grou nd excavations was attributedto three factors:

Fault slip. Grou nd failure. Shaking.

Damage due to fault slip occurs whent he e x ca v at i on p as se s t h ro u gha fault zone. Under such circumstancesda ma ge ge ne ral ly is re stricted to thefault zone, and may range from minorcracking of a tunnel liner to complete

c olla pse , de pe nding on the fa ult dis-pl ac em en t an d th e en gi ne er in g pr op er -t ies o f t h e m e d i u m w i t hi n w h ic h t he

e xca vation is c onstruc te d. Quite ob-viously, fault slip cannot be prevented.Hence, if an excavation crosses an activeor potentially active fault zone, speciald e s i g n /p l a n n i ng m ea su re s s h ou l d b epr ep ar ed . Ei th er th e unde rgroun dexcavation and its support system mustbe de si gn ed to ac co mm od at e th at

displacement without loss of utility, orpos t-ea rth qu ak e re pa ir pl an s an demergency safety-related plans shouldbe de ve lo pe d in adv anc e.

Damage attributed to ground failurema y be a ssociated with roc k or soi lslides, liquefaction, soil subsidence, andother phenom ena that may be triggeredby ground mo ti on . Thi s ty pe of da ma geis particularly prevalent at portals andi n s h al l ow e xc av at io ns , a n d is n o tcovered in this report. Suffice it to saythat the potential for occurrence of thistype of da ma ge should be e va lua te dthrough pa rtic ula rly c areful s ite in-

ve st iga t ion in the vic ini ty of tunne lpo rt al s an d ot he r un der gr ound sh al lo wexcavations.

D am ag e caused by s ha ki ng or vibratory moti on has been most widelyinvestigated and is the major topic of.this report. For lined tunnels, damagema y include c rac king, spa ll ing, a ndf ai lu re o f t he l in er as a d ir ec t c on -sequence of the shaking. Alternatively,v ib ra to ry m o ti o n m ay r ed uc e t hestrength of the ground, thereby placingadditional loads on the tunnel suppoff

syste m. For unlined unde rground e x-cavations in rock, such damage occursas rock fall, spalling, local opening ofrock joints, and block motion.

Na tu ra ll y, th e re sp on se of an y un de r-g r o un d e xc av at io n t o e ar t hq u ak esha king will be influe nce d by ma nyvariables. The more impor tant of theseare the shape, dimensions, and depth ofthe excavation, the properties of the soilor rock within which the excavation isc on st ru ct ed , t he p ro pe rt ie s o f a n ysupport system, and the severity of theg r o u nd s ha ki ng . S u mm a ri e s o f t he

pe rf or ma nc e of un de rgrou nd exc ava-

t io ns d u ri ng e ar th qu ak es s ho ul da c c ount-[or a ll the se varia bles. Un-fortunately, much of the data essentialf or d et ai led a na ly si s o f d am ag eexperienced during an earthquake areofte n unobta ina ble . Ac cordingly, in-v es ti gat or s o f t he p er fo rm an ce o f underground excavations have attemptedto develop direct empirical relation-s hi ps b et we en d a ma g e le vel s a ndg r ou n d m ot io n p ar am et er s. S uc hattempts are fraught with difficultiess in ce d a ma g e a ss es sm en ts m ay b ehighly subjective and the peak groundmotion experienced at a site must often

be de du ce d fr om ver y in co mp le te dat a.Therefore, it is desirable that arrays ofstrong instruments be deployed in anda ro u nd i m po r ta nt u n de r gr o un dstructures.

The empirical data baseT h e f ir st st ep i n d ev el op in g a n

empirical damage model is to define theva rious levels of da mage to be c on-sidered. Dowding and Rozen (1978)identified three levels of damage forunderground excavations in rock due toground shaking: these were no damage,minor damage, and damage. No damage

meant no new cracks or falls of rocks;rninordamage meant new crackingandminor rockfalls; and damage includedsevere cracking, major rock[alls, andclosure.

Dowding and Rozen presented resultsof c orre la tion of the e st imated pea ksurface acceleration and peak particlevelocity with reported damage. Theircorrelations are reproduced in Figs 2and 3. The nu mbers on the ordinate axisar e th e d es ig na ti on s o f th e casest ab ul at ed i n t he ir p ap er . T h e s am enumbering system also is used withinthe e xtensive ta bula tion of da ma ge

pr ep ar ed by Owe n an d Sch ol i (1981). Itshould be noted that the peak groundmotion parameters (acceleration and

Vo lume 2, Nu mber 2, 1987 TUNNELLING AND UNDERGROUNDSPACETECHNOLOGY 167

7/28/2019 Aseismic Design of Underground Structures

5/34

0. 8

07

0.G

82

05

~6c 04.o

ce

zs '~ Damage

- - 0 0= |

~ o%s s~

p __ /~ . . . . . . . . . . . . . . . . .

~' o

4-"Minor damage

g , o % - o "O.2'

"3:" t0_ ." : /| O N o d o ~

03 == 1%, ,,

o I I [ I I I I 1 I10 30 50 7 0 9 O

Ordinal.nu mber of case inappendix COwen and SchoU, {1981)

NO damage P& Near portal

o Minor damage,due to shaking S~ ShaLLow cover

Damage from shaking

Figure 2. Calculated peak surface accelera-tions and associated damage observations forearthquakes (Owen and Scholl 1981).

velocity) were not recorded at the sites oft h e e x ca v at i on s b u t w er e c a l cu l a t edusing e m pir ic a l r e la tionships suc h a sthose described in Appendix A. Strongm otion m e sur e m ents f r om instr um e ntspl ac ed in an d ar ou nd tun ne ls c oul dpr ov id e mu ch m or e re li ab le dat a in th efuture.

Review of data such as those presented

by Dowd ing an d Ro ze n su gg es ts th atn o d a m a ge s h o u l d b e e x pe ct e d i f t hepe ak su rf ac e ac ce le ra ti on s ar e le ss th ana b ou t 0 .2 g, a nd o n ly m i n or d a m ag e

56 ....

-g

~- 4 8

o Damage&&

;,~s_-~o . .. .. . .. .. .. . ~ . . . . . .'6 ,

32

) q~ el0

>~ 24 ~ oe _ _,=o

0 p =,, Minor d a m a g eo w

__ _Q _ .".

8 [ . . . . . . . e ." - = .. "~ /' t- -o - - r - - Ndomoc[

O I0 30 50 70 9 0

120

I0 0

8o ,~

G0

40

20

Or d in a l n u m b er o f c a se i n a p p e n di x COw e n a nd S c ho t t ( 1 98 1)

No damage F'zxNear portal

o Minor damage, due to shokin9 $~, ShaLLowcoverDamage from sl'Klklnq

Figure 3. Calculated peak particl e veloc iti esand associated damage observations forearthquakes (Owen and Scholl 1981).

should be e xpe r ienc e d be twe en 0.2 a nd0.4 g The c or r e sponding thr e sholds for pe ak pa rt icl e v el oc it y ar e a p pr ox i m a t e l y20 cln/s (8 in./s) and 40 c m / s ( 16 in./s) .Of these two correlations, the one basedon velocity is probably to be preferred asa d es i gn c ri t er i on b ec au se t he p ea k par ti cl e ve lo ci ty re sul t ing fr om ane a r thqua ke of a give n m a gnitude c a n bepre di ct ed to fa ll wi thi n re as ona bly

narrow limits. Moreover, experience onthe pe r f or m a nc e of m ini ng e xc a vationsa d j a ce n t t o r o c k b u r s ts h as i n d i ca t e dt h at d a m a g e i s b et t er c o rr e la t ed w i t hpe ak ve lo ci ty th an pe ak ac ce ler at io n(McGar r 1983).

I t s h ou l d b e e m ph a si z ed t ha t t hea bove r e la tionships hold f or r oc k si te so n ly , a nd m a y b e v ery d if fe re nt f or unde r gr ound str uc tur e s in soil be c a uset h e a t t e n u a t i o n o f m o t i o n w i t h d e p t hand the confinement of the structure arevery different than those for rock sites.U n f or t u na t e l y, s i m i la r r e l at i o n s h ip sh av e n o t y et b ee n d e ri v ed f or u n d er -

gr ound str uc ture s in soil .

Supporting evidenceS uppor ting e vide nc e f or se le c tion of

a n e m pir ic a l de sign c r i te r ion f or r oc ksites is provided from experience in them i n i n g i n du st ry , c iv il c on s tr uc t io ninvolving bla sting, a nd we a pons test-i n g. A s a l l ud e d t o a b ov e, t h er e a r e anum be r of c ase s in whic h unde r gr o undm ini ng e xc a vations ha ve be en da m a ge das a consequence of nearby rock bursts.The be st doc um e nte d c a se s a r e f or thede e p le vel gold m ine s of S outh Af ric a,w he re r oc k b ur st s w i th b od y w av e

m ag ni tu de s u p to 5.2 have beent ri gg er ed as a r es ul t o f e xt en si vel o ng w al l m i n i n g o f t he t a bu l ar g o l dreefs. Whether any damage accompani esa r oc k bur st de pe nds on the m a gnitudeo f t h e e ve nt a n d i ts p r o x i m i t y t o t h em i n e w o rk i n gs . E x p er i en c e i n d ic a te st h at r oc k b u rs t s w i t h e n er g y r el ea se

c or re sp on di ng to u p to a 2-2.75m a g n it u d e e a rt h qu a ke o c ca s io n al l ycause damage if they are associated witha m a j o r r u p t u r e w i t h i n a b o u t 30 m o f t h e m i n e w o rk i ng s . E ve nt s o f l a r g er m a gn i tu d e ar e a lm os t i nv ar ia bl yd a m ag i n g e no ug h t o ca us e l os s o f

pr o duc t i on an d, po ss ib ly , in jur ies orfatalities, provid ing they are sufficientlyc lo se t o m i ne w or ki n gs t o g en er at evelocities in excess of 60 cm/ s (24 in./s) .

B ec au se r o ck b u rs t s a re s i m i l a r i nc ha r ac te r to te c tonic e a r thqua ke s ( al-though the resulting duration of shakingtypically is much shorter), the records ofda m a ge to m ining e xc a va tions pr ovidedirect evidence of the likely performanceof excavations very close to a causativefault. How pert inent the experience is tothe performance of excavations remotef ro m t he s ou rc e o f a n e ar t hq u ak ede pe nds upon how im por ta nt a r ole the

dur a tion a nd dom ina nt f r e que ncy of thegr ound m otion pla y in de te r m ining thee xt en t o f d am ag e. I f t he f re qu en cy

c onte nt is r e lative ly unim por ta nt, the nt he e x pe r ie n ce g a i ne d i n t he m i n i n gindustr y is relevant. Further, data on thee ff ec ts o f g r o u n d m o t i o n i n du c ed b yhigh e xplosives a nd nuc le ar we a ponsa lso a r e of va lue . F or the pr e sent, wes ha ll d ef er a ny d is cu ss io n of t heim por ta nc e of dur a tion a nd f r e que nc yc o nt e nt a n d s i mp l y s u m m ar i ze t heempirical data base.

T h e r e qu i re m en t to m i ni m i ze t heda m a ge to unde r gr ound tunne ls due toc o n v e n ti o n a l b l a s ti n g h a s l ed t o d e-velopment of empirical design criteria.F or unline d tunne ls in r ock, La nge f or s

a n d K i hl s tr o m (1963) s ug ge st t ha tpa r ti cl e ve lo ci ti es of 30 c m / s (12 in. /s )c a use r oc k to f a l l while ve loc it ie s of 60 cm/s (24 in./s) cause the formation ofne w c r ac ks in the r ock. The se r e com -m e nda tions see m r a the r c onse r va tivewhe n c om pa r e d with the r e sults of theUnde r gr ound Explosion Te st P r ogr a m(UET), duri ng which very large chargesof high explosives were detonated with

the intent of establishing design criteriaf or c o n s tr u c ti o n o f u n d e r g r ou n d i n-sta lla t ions. Da m age , c onsist ing of in-te r m itte nt spoil ing, wa s obse r ve d f or

p a r t i c l e vel oci t ie s ab ov e 9 0 c m / s( 3 6i n ./ s ). C o nt i nu o us d a ma g e w asobse rve d f or pa r t icle ve loc it ie s a bove180 cm/s (72 in./s).

S ince the UET high e xplosive tests ,se ve r al tunne l test se c tions ha ve bee ni n c l u de d w i t h i n t he s co pe o f u n d er -gr ound nuc le a r te sts. Although m ost of the tu nnel sections have been hardened,using various types of concrete and steell ine r s, som e ha ve be e n suppor te d only

w i t h r o c kb o l ts a n d l i g ht s h ot c re t in g .Review ot the performance of all thosesections indicates that tunnels hardenedwith rockbolts may survive peak particlevelocitiesin excessof 900 cm/ s (360 in./s)

but th e th re shol d for d am a ge to unlinedt u nn e ls i s o n t he o rd er o f 1 80 cm /s(72 in./s) .

The se va lue s a r e so f a r in e xc ess of a nything tha t c ould c onve ivably r esultfrom an earthquake that one is temptedt o d i sm i ss t he p ro b le m o f s ei sm icsta bil i ty of de e p unde r gr oun d e xca va-tions a s tr ivia l . Howe ve r, ther e is oneimpor tant differencebetween the ground

m o t i o n r e s ul t i n g f r om a n e a rt h q u ak ea nd t ha t g ene rat ed b y a n ucl ear explosion. The former usually lasts for several seconds, subjecting the excava-tion to several stress cycles, while thelatter predominantly comprises a singlepul se (c om pr es si on ) la st in g so me te nsto hundreds of milliseconds. The resultsof num e r ic a l e xpe rim e nts r e por te d byDowding et al. (1983) suggest that thenum be r of s tre ss c yc le s is c r i t ica l tod e te r mi n in g h ow m uc h p er ma ne ntd e f o r ma t i o n w i ll o cc u r w i t h i n a r o ck m as s a ro u nd a t un nel w hen it issubjected to earthquake loading.

C o n c l u s i o n s

The r e sults of a t te m pts to c a ta logue

168 TUNNELLING AND UNDERGROUNDSPACE TECHNOLOGY Volume 2, Number 2, 1987

7/28/2019 Aseismic Design of Underground Structures

6/34

records of the performance of under-grou nd excavation s subjected to seismi cloading and to develop simple empiri cald es ig n c ri te ri a i nd ic at e a d a m ag et h re s ho l d o f a p pr o xi m at e ly 2 0 c m / s( 8i n. /s ). N o d am age s ho ul d beexperienced if the peak particle velocityisbeneath that threshold. This thresholdis va lid f or unde r gr ound str uc ture s inrock and may not be applicabl e for othertypes of excavations. Althoug h ther e arei m p o r t an t d if fe re nc es b e tw ee n t h eg r o u nd m o t i o n r es u lt i ng f ro m l a rg edista nt e a r thqua kes a nd r ock bur sts ,d e to n at i on o f h i gh e xp lo si ve s, o r n u cl e ar e x pl o si o ns , d at a f r om t he sesour ce s pr ovide suppor ting e vide nc ethat adoption of this threshold value asa design criteri on will be conservative. Itcan be expected that this recommendeddamage threshold will be raised as moredata becomes available.

Models of the SeismicResponse of Underground

ExcavationsOnc e de sign pr ogr esses be yond the

a p pl i c at i o n o f s i mp l e e m pi r ic a l re -lationships, such as those described inthe previous section, models become ani n t eg r al p a rt o f t he d es ig n p ro ce ss .Selection of the appropriate mo del mustbe ma de by th e de si gn er on th e ba si s ofthe type and impor tance of the structure

be in g de si gn ed an d th e qu a li ty of th ea v ai l ab l e o r o b t ai n a bl e g e o te c hn i c aldata. Early selection is to be encour aged,as the model may have data needs thatmust be satisfied during site investiga-tion.

This section provides a brief review oft he a n a ly t i c al t oo l s a v a i l ab l e t o t hedesigner concerned with the performan ceof unde r gr ound e xc ava tions subje c t toseismic loads. The analytical tools formthe ba sis of m or e or le ss c om plic a te dn u m e ri c a l m o d el s o f t h e b e h a vi o r o f geologic media and interactions betweeng eo lo gi c m ed ia a nd u n de rg r ou n dstructures. The review starts with a briefdisc ussion of a na lytica l tools use d toinvestigate relative displacements thato cc ur a lo n g f au lt s a nd o th er d is -c ontinuit ie s in r ock m asse s. S pec if icc o n s i de r a t io n i s g i v en t o m e t h o d s o f e va lua ting the pote ntia l f or displac e -m en t o n f au lt s a nd b lo ck m ot io n.S ubse quently, a t te ntion is de voted tot he s ub je ct o f w a ve p r o p ag a t i o n i nge ologic m e dia a nd a na lytica l tools f or e v a l ua t i n g s o i l / st r u c t ur e i n t er a c ti o neffects.

R e l a t i v e D i s p l a c e m e n t M o d e l s

B rie f m e ntion of the ne e d to de signu n d er g ro u n d e xc av at io ns , a n d a nys u p po r t s ys te ms , t o w i t hs t a nd f au ltdispla c e m e nt wa s m a de in the pr e vioussection. Fault displacement, whether onthe causative fault or triggered on someo t he r f au lt , i s o n e f or m o f r el at iv edisplacement. For convenience, we have

chosen to differentiate this phen omen onfrom block motion or relative motion ofr o ck m a ss i n f ra c tu r ed m e di a , w h i c hc o m pr i s es t he m o t i o n o f s o m e f i ni t ebl ock of ma ter ia l re la ti ve to it s su r-r o un d in g s. A l t ho u g h b l oc k m o t i onmay be triggered by earthquakes, it has

be en m ore wi de ly in ve sti ga ted as ap h e n om e n o n as so ci at ed wi th det on a-t io n o f h i g h e xp lo si ve s o r n u cl e ar

weapons.

Fault displacementDe signe rs of surf a ce str uc ture s a r e

c onc e r ne d with the sur f ac e m a nif e sta -tion of a causative fault. The designerso f u n d er g r ou n d s tr uc tu re s a re a ls oc onc e rne d with how tha t m a nif e sta tionm ight c ha nge with de pth. I n the a boved i s cu s s io n o f s e is m ic a c ti v it y , l i t tl ea tte ntion wa s give n to e i the r of the sede sign c onside r a tions, a l though i t wa snoted that one measure of the magnit udeo f a n e a r t h q u a k e - - t h e m o m e n t m a g n i -t u d e - i s d e fi n ed i n t er m s of t h e t o t ale l as t i c s t r a in - e n er g y r e le a se d a nd ,

t h er ef or e, i s r e l a te d t o t he f a ul t d is -pla ce m e nt an d ru ptur e ar ea . Mo res p ec i fi c al l y, t he s ei s mi c m o m e n t i sdefined as

M o = G AD , (1)

in whic h G is the she a r m odulus of therock, A the area of the rupture surface,and D the average relative displ acement( Ka na m or i a nd Ande r son 1975). Thisr e la t io n sh i p p ro vi de s o ne m ea ns o f e st im a ting the a ve ra ge f a ult displa c e -m e nt, pr ovidin g tha t the f a ult ge om e tr y

is adequately defined. A better alterna-tive is to use site-specific data.Geodetic surveying of surface move-

ments associated with large earthquakeshas provided data on how displacem entsd ec ay w i t h d i st a nc e f ro m t h e f au lt .Unf or tuna te ly, the re is m uc h less datao n t he d i s tr i b ut i o n o f r el at i ve d is -pl ac em e nt on th e fa ul t pl an e. Ho we ve r,som e insight ha s be e n ga ine d thr oughu se o f r el at iv el y s i mp l e n u m er i ca lmodels in which the fault is modeled asa disloc a tion in a se m i- inf inite e la sticmedium. For example, Pratt et al. (1979)r ep or t the resul ts of a series of

s i m u l at i o n s o f s tr ik e s l i p a n d d i p s l i pfaults with various geometries. While itis difficult to draw general conclusionsfrom the few cases they considered , theirr e sults did indic ate tha t ther e m a y bec ir c umsta nc es in whic h the displa c e -m e nt o f t h e m e d i u m a d ja c en t t o t h efault may be greater at depth than o n thesurface. However, it is general ly assumedt ha t t he r el at iv e d i sp l ac em e nt ex -pe ri en ce d unde r gr oun d is co m pa r ab leto that experienced on the surface. Thisa ssum ption c a n be c he c ked quite e a silyf or a p a rt i cu l ar f au lt g eo m et ry a n d

bo und ar y c o n d i t i o n s - u s i n g th e di s-p l a c e m e n t d i s c o n t i n u i t y m e t h o dd es cr ib ed b y C ro u ch a n d S ta rf ie ld(1983).

R e la t i ve d i s p la c em e n t s m a y b e e x-per ien ce d on fa ul ts ot he r tha n th ec a usa tive f a ult . This m a y oc c ur if thes e i sm i c al l y i n d uc e d s tr es se s a n d t he

local in-situ stress conditions are such ast o i n d u c e s h ea r f a il u re o n t h e f a ul t.A l t h o ug h q u al i ta t iv e p re d ic t io n s o f s uc h d i s pl a ce m en t u s in g n u me r ic almodels based on finite element or finited if fe re nc e m et h od s a re p o ss i bl e i npr i n c i pa l , la ck of si te dat a an d th ec om puta ti ona l e f for t r e quir e d m ili ta tea ga inst m a k ing suc h c a lc ula tions.

A s a n a lt er na ti ve , t he p r ob l em o f i nc ip i en t f au lt m ot i on c an b e i n-vestigated using the simplified approachd ev el o pe d b y J o h ns o n a nd S ch mi tz(1976). T he i r m od el is b as ed o nc al cu la ti ng t he s he ar a nd n or ma lstre sse s, on a f a ult pla ne , tha t r e sultf r om pr opa ga tion of a spher ic a l wavef r om a sour c e . C onditions of inc ipie ntslip exist if the t otal shear stress (the sumof in-situ and indu ced stress) exceeds theshear strength. The model was originallydeveloped to investigate fault movementinduc e d by a n e xplosion, whic h c a n bea de qua te ly r e pr e se nte d a s a sphe r ica lsource. Al thoug h the spherical source isn o t a g o o d i d e a l iz a t io n o f a n e ar t h-qua ke , the m ode l s t i l l should pr ovide a

ba si s for es tab lishin g an under st an din gof the m or e c r i t ic a l f a ult or ie nta tionsand locations.

Block motionF or e xc ava tions in f r ac ture d m e dia ,

a t t e nt i o n f oc us es o n c o n t a i n i ng t hef r a c tur e d m a ss or individua l bloc ks of m at er ia l d ef in ed by p re -e xi st in gfractures. However, it is convenient toinitiate the topic of analytical tools fordesign under such circumstances by firstc o n si d e ri n g t he t o p ic o f s p a l l i n g - - ap he n om e n on tha t m ay be ind uc ed byr e fl e ct i on o f a s tr es s w a ve a t a f re esurface.

Interest in the performance of under-groun d excavations in rock subjected tovery high seismic loads, such as thosei n d u ce d i n t he v i ci n it y o f a n u n de r -g ro u nd w ea po ns test, r es ul te d i ne v al u at i on o f s p al l i ng as a p os si bl eda m a ge m e c hanism . La bre c he (1983)u s ed t he r es u lt s o f w o rk b y R i n e h a r t(1 96 0) o n t he s u bj ec t o f s p a l li n g t ointe r pr et da m a ge obse r ve d in tunne lsadjacent to tests of both hi gh explosivesa n d n u cl e ar w e ap o ns . H e c o nc l u de dthat sp allin g due to tensile failure of ther oc k m a ss wa s unlike ly, e xc e pt ve ryc lose to a high e xplosive de tona tion,be ca us e th e sp al l th ic kn es s wo ul d begreater than the spacing of pre-existingf r ac tur e s. On the othe r ha nd, pse udo-s p al l in g , o r s ep ar at i on a l on g p re -e xist ing f r ac tur e s, a ppe a r e d to be a nim por ta nt da m a ge m e c ha nism.

R i n eh a r t (1 96 0) s h ow ed t h at t h epse ud ospa ll ve lo ci ty wi ll ap p ro a c h th ef r ee - fie ld pa r t ic le ve loc ity f or s tr esswaves that have a very sharp front. Forwaveforms and wavelengths of concern

Volum e 2, N um be r 2, 1987 TUNNELLING AND UNDERGROUND SPACETECHNOLOGY 169

7/28/2019 Aseismic Design of Underground Structures

7/34

in the de sign of unde r gr ound e xca va-tions subjected to earthquake loading,the pseudospall velocity is likely to bemuch less because the stress wave wil lhave completely engulfed the excava-tion, thereby constrainin g the movemen tof potentially unstable blocks or slabs.Hence, simple spall models have verylim ited a pplic a tion in de sign a ga instearthquake loading.

Because of the relative unimporta nce

of the dyna m ic phe nom e na , inc ludingspa ll ing or pse udospa ll ing, i t is c on-ve ntiona l to tr e at the be havior of a ne xc av at io n i n f ra ct ur ed m ed i a asps eu do st at ic ; th is is the cas e forcont inuu m modeli ng as well. However,i n t hi s c as e t h e p r i m a ry c o nc e rn i sdesign against the possibility of separa-t i on o f b l oc ks o f m a t er i al f ro m t hesurrou nding medium. Blocks of grou ndt ha t a re k i ne m at i ca ll y c ap ab l e o f movin g into the excavation are assumedto be a c ce le ra te d dif f e re ntially a t the

pe ak fr ee -f ie ld gr ound ac ce le ra ti on . Ana pp ro ac h to d ef in in g the s ha pe,dim e nsions, a nd suppor t r e quir em e ntsof such blocks is presented by Hock andBrown (1981), who primarily make useo f s i mp l e g r ap h ic a l c o ns t ru c ti o nsc o up l ed w it h l i mi t in g e q u il i b ri u mconsiderations. A more comprehensivea p p ro a ch t o d e fi n in g k i ne m at i ca l lya d mi s s ib l e b l oc ks i s p r o v id e d b y t hekeyblock theory developed by Goo dma nand Shi (1985). Some progress has beenm ad e i n u s in g t hi s m et ho d, w h ic henables all critical blocks to be identified,as a s ta rt in g p o in t fo r p re di ct i ngsuppor t r e quir e m e nts ( Goodm a n e t a l .1982).

T h e a lt er na ti v e t o a t te m p ti n g t oidentify blocks with particular geometricshapes is to rely more on precedent. Forexample, Barton (1981) has suggestedmodification of the Qsystem to accountfor seismic effects. Also, Hendr on andFernandez (1983) describe the ap plica-

t i o n o f C o r d i n g ' s (1971) m e t h od f or pr ed ict ion of th e suppor t pr es su re s fo rthe roofs of large underground excava-tions. They defined the required su pport

pr es su re (Pi) for the roof of a cavern as

p, : (1.0 + a/g) n B 3" (2)

i n w h i c h n i s a n e m p i r i c al l y d e ri v edfactor, B is the span of the cavern, 3' isthe unit weight of the material, a is theg r ou n d a cc el er at io n, a nd g is t hea cc el er at io n d ue to g ra vi ty . T hi se q ua t io n i m pl i es t ha t d et ai l s o f t hes t ru ct u re i n t he r oo f a re r el at iv el yunim por ta nt ; a r ea sona ble a ssum ption,pro vide d th at co m pr es si ve st re ss es in th er oof a re suf f ic ient to inhi bit s l ip a longthe relatively steep fractures that have a

pot en t ia l fo r d ef in ing blo ck s ki ne -m a tic a lly c a pa ble of dif f e r e ntia l m ove-ment.

T h e a l te rn at i ve t o s i m pl e d es i gn

m o de l s i s t o r es or t t o m o r e d et ai l eds i m u la t i o n u s in g o ne o f t he s ev er al

a va ila ble num e r ic a l m ode ling m e thods.The latter are relatively well developedf or a n al y si s u n d er s t at i c a n d p s eu d o-static conditions, but have been appliedo n ly r e la t iv e ly r ec en t ly t o d y n a m i c

a na ly si s o f f ra ct ur ed m ed ia . T w of unda m e nta lly dif f e r e nt a ppr oa c he s to

m ode ling of f r a c ture d m e dia ha ve be enadopted. One appr oach involves startingf r om a num e r ic a l pr oc e dur e or igina llyd ev is ed t o d e s c ri b e t he b e h av i o r o f a

c o nt i nu u m, w hi le t he o th er m od ela pp r oa ch es t he p ro b le m as o ne o f d es cr ib in g t he b eh av io r o f a d is -c ontinuum .

O n e c o n t i nu u m a p pr o ac h i nv ol ve susing special interface elements, such asd is cu ss ed b y G o o d m a n a n d S t J o h n(1977). T h i s a p p ro a c h h as t he d is -advantage that large shear displacementswill ne c essitate r e pe a te d r e z oning, or redefinition of the finite element mesh.P r ob a b ly f or t ha t r ea so n, t he l ar ged e f o r ma t i o n w av e p r o p a g a t i o n c od ess uc h a s H O N D O ( Ke y el a l. 1978),D YN A2 D ( Ha l lq u is t 1978), a ndS T E A L T H 2 D ( H of f ma n 1981) m o retypic a lly tr e a t inter f ac e s a s s l ide l ines

be tw ee n stru ct ur al l y indep en den t co m-pon en ts. A l t h o u g h th is a p p r o a c happears to have been used very success-fully to study complex impact problems,i ts a pplic a tion to pr oble m s othe r tha nv er y s i m p l e l a ye r ed g e ol o gi c m e d i aappears to have been limited.

A n a l t e r n at i v e c o n t i n u u m a p p r o a c hr el ie s o n u s i ng s pe ci al c on s ti t ut i vede sc r iptions of a f r a c tur e d m e dia tha ta c c ount f or the m e c ha nic a l pr ope r tie sof the f r a ctur es a nd the ir s pa c ing a ndor ie nta tion. The C AVS m ode l tha t wa sused by Wahi et al. (1980) to investigatethe sta bil i ty of nuc le a r wa ste isola tionc a ver ns subje c te d to sim ula te d e a r th-q uak es is an e xa mp le of such ac onsti tutive de sc r iption. S uch m ode lsr e ad i ly p e r m i t t he s i m u l at i o n o f t h ede ve lopm e nt of ne w f r a c tur e s within a

par t ic ular el em e nt or zone , but do no texplicitly represent the location of eachfracture. A ccordingly, the kin ematics of

bl oc k m ovem en t ar e ig no re d.

T o o v er co m e t he d i ff i cu l ty ind e s cr i b i ng t h e k i n em a t ic s o f b l o ck ysystems, Gund all (1971a,b) develop ed thedistinct element method. In that method,a f ra ct ur ed m e d iu m i s v ie we d a s a nassembly of interac ting particles which,in the m ost ge ner a l im pl e m e nta tions of the method, are completely free to movewith respect to each other. In its earlier im ple m e nt a tion, the blocks we re c on-sidered to be rigid and infinitely strong;the re by r e str ic t ing a ll de f or m a tions tot he f ra ct ur es a nd s ev er el y l i m i t in g

po ss ib le fa il ure mo de s. Re ce nt ge ne ra l-i za ti on s o f t he a p pr o a ch a l lo w d e-f o rm a b le b l o ck s a n d d e v e l op m e n t o f n ew f rac tu re s i n a dd it io n t o m or ec o mp r eh e ns i ve d e sc r ip t io n s o f t hem e c ha n i ca l b e h av i o r o f t he f r ac tu r es(Cundall and Hart 1983).

Although the distinct element method

is ba se d on the e qua tions of m otion of t he i n d i v i du a l p ar ti cl es , i t h a s b ee nm ost wide ly a pplie d to the solution of

pse udo sta tic pro ble m s by tre at ing t im ea s a f ic t i t ious qua nti ty use d to c ontr olthe sequence of events in a system thatm ay e xh ib it c om pl ex n on l in ea r be ha vi or . Ho we ve r, it is eq ual ly po ss ibl eto perform dynamic analyses.

S u ch a n a p p r oa c h i s d es c ri b ed h yDowding e t a l. ( 1983) who r e por t the

a p pl i ca t io n o f a c ou pl ed d is ti nc te l em e nt / fi n i te e le me nt m od el i n a ninvestig ation of the response of a cavernto ve r t ic a lly pr opa ga ting she ar waves.One of the m ost inte r e sting a spe c ts of t h ei r i n v es t i ga t i on w as t he e x te n t t ow hi ch g ro un d m ot io n r es ul te d inpr og re ss iv e sl ip on the fac es of bl oc ksa dja c e nt to the e xc ava tion. Howe ve r ,e x tr em el y h i gh a cc el er at i on s w er er e quir e d f or this to oc c ur . C ontinuingd e v e l op m e n t o f t he d i s ti n c t e l em e ntm e thod f or dyna m ic a na lyse s, c ouple dw i th s tu di es s uc h as d es cr ib ed b yD ow d in g et al ., w il l u n do u bt ed l yc ontr ibute signific a ntly to our under -s t a n d i ng o f t he b asi c: m e c h an i c s o f fractured media.

V i b ra t o ry M o t i o n

A l t h o u g h m o st o f t h e r e la t iv e d i s-pl a ce m en t eff ect s di sc us se d ab ov e re su ltfrom wave propagation from the sourcet h r ou g h g eo l og i c m ed ia , i t p ro ve sconv enien t to discuss the direct effects ofvibratory motion as a separate subject.T h i s d i sc us s io n i s d i vi d ed i n to t womain parts. The first part considers theg r o u n d m o t i o n i n t he f re e f ie ld , w i t hpa rt icu l ar at te nti on gi ve n to ho w th eg r o un d m o t io n is i n f lu en ce d b y t h eg e o lo g i c s t ru ct u re . T h e s e co n d p a rtc onside rs how unde r gr ound str uc tur e srespond to vibratory motion. The latter d i s cu s si o n i s s u b d iv i de d i n t o t hr eepa rt s. Fi rs t, re su lt s of an al ys es of li ne da nd unline d c ir c ula r tunne ls in e la sticm e di a a re s u mm a ri z ed . S ec on d , t heba se s for de vel op me nt of s imp le mo de lsf or i n ve st i ga t in g g r ou n d s tr uc tu reinteraction effects are discussed. T hird,t h e c a pa b i li t i es o f n u m er i c al m o de l st hat m ay be used to i nv es ti gat egr ound/str uc tur e inte r ac tion e ff ec ts in

greater detail are reviewed.

Free-field ground motionT h e p r o b le m o f f re e- fi el d g r o u n d

m otion, a lso known a s wa ve pr opa ga -t io n, i n a n i nf in it e h om og en eo usisotropic elastic medium was addressedas early as 1950 (Fung 1965; Desai andChristi an 1977). This section describest he f o rm u la t io n a n d s o lu t i on o f t h ethr e e - dim e nsiona l wave e qua tions a ndthe depth dependence of ground motion.

T h e m o ti o n o f a c on t in uu m b od ymust obey the equation

Oa,j0oel = ~ +Xi i= 1 , 2, 3 (3)

170 TUNNELLING AND UNDERGROUND SPACETECHNOLOGY Volume 2, Number 2, 1987

7/28/2019 Aseismic Design of Underground Structures

8/34

where p = mass density of thecontinuum,

ai = particle acceleration,oi7 = stress field, andXi = body force per un it

volume.

In the theory of elasticity, the aboveequat ion i s known as the Eulerian

equation of motion of a continuu m. Ifwe limit ourselves to the linear theory o rinfinitesimal displacement theory, wecan write the following relationships

between st rain , e0; par tic le disp lace -ment, u,; particle velocity, vi; andparticle accelerat ion , ai:

1eli = -~ (ui, i + ui, i) (4)

Oui Ovi 02uivi = ~ , ~i =0-7 =Or - - T " (5 )

In addition to the above equati ons, thetheory of linear elasticity is based onHooke 's law. For a homogeneousisotropic material, this is

0 6 = h ekk i i + 2G eii (6)

whe re ~ and G a re cal led L ame ' sconstants. The stress field oii can be

eliminatedby substitutingEqua tion (6)

into Equation (3) and using Equation(4) to obtain the well-known Navier'sequation

02UiG ui, f i+ ( X+ G ) ui, i i+ X i= p "ff~"" (7)

The above equation can be cast indifferent forms and its general sol utio nfor the case of a steady state harmonicm ot i on c an be e as ily c al cu la te d(Achenbach 1975). In the next sectionsome types of waves that satisfy theabove equat ion of mot ion are con-sidered.Pl ane el as ti c wa ve s

Several types of waves can propagatein an elastic medium. Their existencecan be demon stra ted from the basic fieldequation (Equation (7)), which, in the

absence of body force, is

02UiO ~ = G u i . #+ ( h+ G ) ui. ii . (8)

In the following discussion, displace-ment components u~, u2, us will bereferred to by u, v, and w; t hey represent,respectively, the motion parallel to thed i rect ion o f wave propagat ion , themotion in the horizontal plane normalto the direction of wave propagation,and the motion in the vertical planen or ma l to t he d ir ec ti on of wavepr op ag at ion .

One type of particle motion can bedefined by

u = As in ~ ( x + c t ) ,

v = w = 0

Substi tution of Equations (4-9) into thefield equation, leads to the relationship

P C~ = ), + 2G

o r

--7 2GCp = P

where Cp has been subst ituted for c andrepresen ts the wave veloc ity. Thepa tt er n of mo ti on expressed by Eq ua tio n(9) remains unchanged when ( x + ct)remains constant and L is the wave-length. The particle velocity is in thedirection of propag ation , namel y the x-direction. Hence, this motion is said to

represen t a compressional wave or P-wave.

A second type o f mot ion can be

defined by

u = 0

271v = A sin ~ - ( x + c t) (12)

w= 0

which represents a train of pla ne wavesof wavelength L propaga ting in the x-d ir ec ti on w it h a ve loc it y c. T h e

substitution of Equation (12) into thefield equation yields a value for the

wave velocity, Cs, given by

Cs = ~ (13)

T h e p ar ti cl e v eloc ity is i n t he y-direction and is perpendicular to thedirection of propag ation, namely the x-d irect ion . Such a mot ion is said torepresent transverse or shear waves

(S-waves).A t hi rd type of m ot io n, w hi ch

represents transverse waves can also bedefined by

u = 0

v = 0 (1 4 )

271"w = A s i n ~ - ( x + Cs t )

This wave is similar to the previouswave except that the particle motion isin the z-direction. In order to differen-tiate between the two motions, one isreferred to as transverseho rizo ntal (SH),and the other as transverse vertical (SV),depending on whether the wave is

pr opaga t ing in a hor iz on ta l or a vert ical(9) plane, respectively.

For all of the above waves, the mo ti onrepresented by Equations (9), (12) and(14) are called pla ne waves, since at anyinstant of time the wave crests lie inpa ra ll el pl anes . Th es e waves may exis ton ly in an unbounded e last ic con-tinu um. In a finite body, a plane wave

(10) w il l be re fl ec ted wh en i t h it s t he

bo un da ry . If the re is an ot he r elas ticmed ium be yond the boundary, refractedwaves occur in the second medium. Thepr ob le m of re fl ec ti on an d re frac ti on is

(11) addressed below.Of course, arbitrarily incident plane

waves can propagate within a medium.

F or t he se w av es , th e g ov e rn i ngequat ions of mot ion can be foundelsewhere (Achenba ch 1975).

Surface wavesI n a dd it io n to the waves t ha t

pr op ag at e within an elastic me di um ,i.e. body waves, it is possible to haveanother type of waves--that is, thosethat propagate over the surface of themedi um and penetrate to only a minorextent into the inter ior of the body.These are called surface waves. For thesetypes of waves, it is characteristic thatthe amplitude of displacement in themedium decreases exponentially withincreasing distance from the bounda ry.

One type o f sur face wave is theRayl eigh wave, which occurs on the freesurface of a homogeneous, isotropic,s e m i - in f i ni t e m e d i u m. I n a t wo -dime nsi onal elastic half-space with y >_0 and a stress-free surface at y = 0, themot io n can be defined by the real part ofthe fol lowing expressions

u : ,4 e-br exp [ik (x - ct)]

v : B e-byexp [ik (x - ct)] (15)

w = 0

where i is the imaginary number x/ -l,and A and B are complex constants. Thecoefficient b is considered to be a realan d p os it iv e con st an t so tha t t heamplit ude of the wave decreases ex-po ne nt ia ll y wi th in cr ea si ng y, an d tendsto zero as y approaches infinity. Theconstants in the above expressions arechosen such tha t the d isp lacementequat ions sat isfy the equat ions of motion and the boundary conditionsonthe free surface.

Th e proof of the existence of Rayle ighwaves can be fou nd in books on classicaltheory of elasticity (Fung 1965), and isnot repeated here. However, an illustra-tion of the elliptical retrograde-typemotion and a discusion of the relativepr op ag at io n velocit ies of compress io nal,shear and Rayleigh waves are inclu dedwithin Appendix A. The i l lustrationshows that [or the Rayleigh waves thepa rt ic le mot ion is in the pl an e of wave

V ol um e 2, N um be r 2, 1987 TUNNELLING AND UNDERGROUNDSPACETECHNOLOGY 171

7/28/2019 Aseismic Design of Underground Structures

9/34

pr op aga tion. Surface waves wi thmotion perpendicular to the directionof propagation can occur if the shearwave velocity in the upper layer is lessthan that in the lower stratum. Thesewaves are known as Love waves. Again,the equations of motion governing

these types of waves can be derivedanal ytic ally (Achenbac h 1975).

Re fl ec ti on an d relracti on o] p la newaves

To illustra te the problem of reflectionand refraction of plane P and S waves,consider a homogeneous isotropicelastic medium occupying a half spaceand with a free surface. Plane P waves

hitting the free boundary are reflectedinto the medium as plane P waves andpl an e S waves. Si mi la rly in ci de nt SVwaves are reflected as both P and SVwaves.

If the medi um consists of two or morelayers, then incident P waves propagat-ing in one layer are reflected into P andSV waves and refracted into the adjacen tlayer as P and SV waves. The same holdsfor incident SV waves. The SH wavesbehave diffe rently. A tr ain of SH waveswil l n ot g en er at e P waves a t t heinterface; it is reflected and refracted asSH waves.

Am pl i f i ca t i on o] SH wavesBody and surface waves are created by

disturbances caused by an earthquake.The amplitude and frequency contentof the earthquake mot ion depend on thesource and the transmission path as wellas site characteristics. Along the trans-miss ion path, body waves are influen ced

by the geometry a nd ma te ri al prop er ti esof the medium. They are reflected andrefracted between layers of differentmaterial properties--a phenomenonthat resul ts in a local decrease o r increase of the wave amplitude andaffects the frequency content of theresulting motions.

For the p rac t ic ing eng ineer, thepr ob le m is to de te rm in e the char-acteristics of the ground motion at a site(surface and/or underground motion)

on the basis of the motion recorded atother sites. In view of the complexity of

the waves propagation problem, i t isnot possible at present to solve thegeneral problem, which includes bodywaves (P- and S-waves) and surfacewaves. Therefore, consideration has

been restricted here to t he case of vertical

pro pagation of ho ri zo nt al ly po larizedshear waves in a horizontally layeredme di um -- a case for which an analyticalso lut io n can be easily derived using one-dimensional wave theory. While thisapproximation has i ts l imitations inrepresenting the actual problem, i t isbased in pa rt on the ob se rv at io n thatbo dy waves re ac hi ng the si te from thesource of the disturbance arrive, ingeneral, with nearly vertical incidenceto the g round sur face , and no t in a

straight line from the source to the site(Tsai and Housner 1970).

A con t inuum so lu t ion to the one-dimensional wave equation can be usedto analyze the free-field response of a

horizontally layered site subjected tovertically incident shear waves. T heanalysis is carried out in the frequencydomain by utilizing the Fourier Trans-form of tire input motion to representthe mot ion as the superposi tion o f harmonic s ignals o f d if ferent fre-

quencies. The frequency-dependentt ransfer funct ion of the system isobtained by comp uti ng the response ofthe sys tem to u n it h a rm o nic i n p ut

motion. The t ime-dependent systemresponse to the actual input motion isthen obtained as the inverse Fourier Transform of the product of the systemtransfer funct ions and the var ious

harmonic s ignals tha t compr ise theinput motion. The above procedure iscarried out when the motion is defineda t th e b ase o f t he soi l l ayers. A

deco nvol utio n procedure can be used tocompute the subsurface motion oncethe surface mo tio n is defined.

The theoret ica l derivat ion of theequations for the above procedure areinvolved and beyond the scope of thisreport. The y can be found in Desai andChristian (1977). The result of thisexercise is to define the amplificationfacto] or the ratio of the amplitude ofm o tio n a t t he free sur face to t heam pl i tu d e o f m o tio n at r ock/ so ilinter face . A typ ical shape for theamplification factor of a uniform soillayer above rock is shown in Fig. 4. For

other cases computer programs such asFL USH ( Ly sm er et al. 1 97 5) an dSHAKE (Schnabel et al. 1972), which

are based on the above procedure, ca n beused. These codes are discussed below.

Seismic analysis ofundergroundstructures

A wide range of analytical tools hasbeen used to in vest ig at e the be ha vi or of

underg round excavations subjected toseismic loading. Because they can beanalysed in closed form, particular atte ntion has been devoted to analysisoflined and unli ned circular tunnels. Theemphasis o f tha t work has been oninves tiga ting the results of pla ne waves

propagating pe rp en di cul ar to thelongitudinal axis of the tunnel. For the

case of waves propagat ing along theaxis, use has been made of simplifiedmodels in which the tunnel l iner isideal ized as a beam on an e last icfoundation.

More recently, attention has turnedtowards the use of a number of differentn u mer ica l p ro cedu re s t ha t enableground/structur{ interaction problems

to be studied in either the time domainor frequency domain. The followingdiscussion comprises a brief review ofthese three areas of investigation.

Circular and noncircular tunnelsA considerable body of literature is

d ev oted to t he d ev elop ment andapplication of analytical solutions tothe problem of plane waves propagati ngin an e last ic medium; normal to atunn el axis. Intera ction of the wave andthe tunnel causes a distortion of thecross-sectional shape and stress con-centrations over and above those result-

ing f rom the in-situ stresses existingpr io r to ex cavation. I nt er ac ti on can a lsot ake the f or m o f en tr apm ent and

8Uo

g

Q.

E

~O0I i

, , / 3= oi i

i^l

I t '~ = o o s

"III

I

!

/ X

oo

t o0I

I

i

I

I

I

f ij~ ,l

i !

Z 71 \ i i ~ p I I ~ \\ / / \ ~ x \ \ / / ~, \ /

I I [ L I i i

0 ~ 4 6 8 IO t2 14 i6 18

F r e q u e n c y (Hz)

Figure 4. Ampl if icat io n cu rv e/ or un il orm layer wi th rigi d rock (rnodi[ied [rom Desai andChristian 1977).

172 TUNNELLING AND UNDERGROUND SPACETECHNOLOGY Volume 2, Number 2, 1987

7/28/2019 Aseismic Design of Underground Structures

10/34

circulation of the seismic waves aroundt he t un ne l. H ow ev er , t hi s is o n l y

po ss ib le wh en wa ve le ng th s are les s thanthe tunnel's radius (Glass 1976) and the

circulating waves appear to be heavily

da m pe d be ca use they r a dia te e ne r gyinto the solid (Cundall 1971a,b).

Using c lose d-f orm solutions, Mowand Pao (1971) investigated the inter-action of steady state P-, SV-, and SH-waves with cylindrical cavities. For P-waves p r op a ga t in g n or ma l t o t helongitudina l a xis, the y de m onstr a te dthat the peak dy namic stress concentra-tions were approx. 10-15% higher than

those resulting from static stress equalto the peak free-field stress; and thatthese str ess c onc entr a tions oc c ur f or wavelengths that are approx. 25 timesthe c a vity dia m e te r . The str ess c on-centrations resulti ng from SV- and SH-wa ve s a lso we r e a f e w pe r c e nt highe r t h a n t he s t at i c e qu i va le n t. T h e i m -

po rt an ce of th es e re su lt s is no t so m u chthat the dynamic effects are small, but

that static or pseudostatic analyses area d eq u at e f or w av el e ng t hs t y p i ca l l ya ss o ci a te d w i t h e a rt h q u ak e - in d u ce dgr ound m otion.

R e sults pr e se nte d by Mow a nd P a oindic a te d tha t ther e will be ve ry l i t t leconcen trati on of stress if the wavel engt his short in compa rison to the diameter of the cavity. Such short wavelengths areunlikely to be importan t [or earthq uakeloading, except very near to the source,but ca n be i m p or ta nt for ex ca va tio nssubjected to loading from conventionalor nuc le a r e xplosions. F or ve r y shor twavelengths, the wall of the excavationacts like a plane-free surface at whichthe stress wave is reflected as a wave ofopposite s ign. He nc e , inc om ing c om -pr es si on wa ve in du ce , upon re fl ec ti on ,tensile stresses and create stress con-c en tr at io ns t ha t i nt er ac t w it h t her ef le ct io n. T h e p re se nc e o f t e ns i lestresses raises the possibility of sp alli ng( a phe nom e non tha t ha s bee n c ove re dabove).

T h e r eal p ro b le m o f s p a il i ng a tu n d er g r ou n d e x ca v at i on s is m o rec o mp l ex t ha n w as c o ns i de re d b yR i ne h ar t , s i nc e t he i n c o m i n g s tr es screates stress concentra tions that interact

w i th t he r ef le ct io n. T h e p r o b le m o f i n t er a ct i o n c a n b e i n ve s ti g at e d q u i t esim ply in c lose d f or m . Typic a l r e sultsf ro m a n u m b e r o f r e ce nt c a l cu l a t io n su s i n g a c o m p u t e r c od e d e v el o p ed b yGa r ne t e t a l . ( 1966) a r e r e pr oduc e d inF igs 5 a nd 6, in whic h the r e la tionship

be tw ee n ti me , stres s, an d di sta nc e fr omthe tunnel wall is illustrated for the caseof a tr ia ngula r pla ne P - wa ve e ngulf ingthe ope ning. T he tota l dur a tion of thew a ve f or m i s e q u al t o t h e t r av el t i meacross eleven tunnel diameters, with thestr e ss r ising l ine a r ly to a pe a k in onetunnel diameter. At time zero, the waveha s just r e a c he d the wa ll of the tunne l;i ts f r ont c a n be se e n c le a r ly in F ig. 5.The front is indeed reflected; however, if

~ = 0P e ok s t r e s s

= I .O

t = I

~ = 2

C i r c u t o r t u n n e l

Fi gu re 5. /1 tr ia ng ul ar wa ve wi th wav e]ro nt an d tot al le ng th eq ua l to one tu nn el di am et er an deleven tunnel diameters, respectively.

the wavelength is greater than about tent u n n e l d i am e te r s, t h e i n d u ce d r a d ia lstr ess r e m a ins c om pr e ssive . AlthoughF ig. 6a indic a tes tha t the induc e d ho opstre ss is te nsile , this is to be e xpec te dsince the P-wave induces a biaxial stressstate in which the peak confining stressis related to the peak stress by the factorv / ( l - v ) .

The case of lined circular tunnels canalso be analysed in closed form. Resultsc o m p a r ab l e t o t h o se f or t h e u n l i n e dtunne l a r e r e produc e d in F ig. 6b. Wha tis noticeable in these figures is that thereis a min or increase in the radial stress inthe r oc k a nd a m a r ke d c onc e ntr a tion of hoo p stress in the liner. Thi s is observed

be ca us e th e l in er pr op er ties we re ch os ens o a s t o m ak e t he l i ne r a p p e ar s ti ff r e la tive to the roc k m e dium . Whe the r aliner will significantly interact with them e d i u m d e p en d s u p o n t he c o m pr e ss i -

bil i ty ra t io an d th e fl ex ib ili ty ra t io( H e n d r o n a n d F e rn a nd e z 1983). O f

these, the f le xibil i ty r at io is the m or ei m p o r t a n t b e ca u se i t i s r e l at e d t o t h eability of the liner to resist distortion.

The flexibility ratio, F, is defined by

2E (1 - v~)RsF-

E~ (1 +v) ts

i n w hi ch E a nd v .are t he Y ou ng ' sm o d u l us a n d P o is s on ' s r at i o o f t h em ed iu m a nd Et, vt, R, a nd t are,r es p ec ti v el y , t h e Y o u n g' s m o d u l us ,Poisson's ratio, radius, and thickness of the liner.

Several investigators have discussedthe r e la tionship be twe en the f le xibil i tyr a t i o a n d t he e x t e n t t o w h i c h a l i n e r m o d if i e s a t u n ne l r e sp o ns e t o e i t he r

s ta t ic or dyna m ic loa ds ( f or e xa m ple ,P ec k et a l. 1972; a n d E i ns t ei n a n dS c hwa r tz 1979). The y c onc lude d tha tthe l ine r c a n be c onside r e d pe rf e ctlyflexible if the flexibil ity rati o exceeds 20.I n tha t c a se the l ine r c onf or m s to thed is to rt io ns i mp os ed o n it b y them e di u m. I f, o n t he o t he r h an d, t hef le xibil i ty r a t io is low, the n the l ine r

will resist the distortion of the m edium.W h et h e r t h er e i s a c o n ce n t ra t i on o f stress in the liner depends mai nly on therelative elastic modul us of the liner andm e dium .

For the case illustrated in Fig. 6b theelastic modulus of the liner is twice thatof the medium. However, the liner has av er y h i g h f l ex i b il i t y r a t i o ( a pp r ox .1000). Accordingly, the distortion of them e dium is substa ntia l ly unr e str a ine d.In general, it would be conservative toch eck t ha t t he l in er is c ap ab le of withstanding the unrestrained distortionof the medium.

S ev er al c l os e d- f or m s o l ut i o ns a rea va i la bl e f or e s ti m at i ng g r o un d /structure interacti on for circular tunnels.The solutions m or e c om m only use d for static design of tunnel liners have beenr ev i ew ed b y D u dd e ck a n d E r d ma n n(1982). They are based on the assump-t i on t h at t he l i n er b e h a ve s a s a t h inshell. In fact, the more general solutiono f a c on ce nt ri c e la st ic r i n g o f a nythickness can be derived quite simply;the necessary equations for the dynamiccase are given by Garnet et al. (1966).U se o f t he s t at i c s o l u t io n s h o ul d b epe rf ec tl y ac ce pt ab le fo r ev al uat ing ther es po ns e t o w av e le n gt h s t yp i ca ll ya ssoc ia te d with e a r thqua ke s, pa r tic u-larly if the static overstress is increased

Volum e 2, Nu mbe r 2, 1987 TUNNELLING AND UNDERGROUND SPACETECHNOLOGY 173

7/28/2019 Aseismic Design of Underground Structures

11/34

o

-6

ry

g

o ' ~.\~.~ s\o~ o %.~e

(o) Un li ne d t un ne l

x

o 3.\me s ~ c~ o ~ ~me

( b) L in ed t un ne l, E L, ,, , /E m,d ~um : 2 . 0

I JL in , r = 0 . 1 8 , l )r n. di u m = 0 . 2 5

Fig ure 6. R ad ial and circ um] eren tial ho op stres s his tor ies in the wa ll of a n un li ne d tu nn el (a) and a l ine d tu nn el (b ). T he stress p rof ile s a re/ or theline ,'tB in Fig. 5.

10- 15% a bove the pe a k dyna m ic f re e-field stress.

A note of caution in regard to the useof any of the lined tunnel solut ions is inor de r . As O'R oa r k e t a l . ( 1984) pointout, the re a r e dif f er e nc e s be twee n theca se o f e xt er na l l o a di n g o f a l i ne dtunne l a nd e m pla c e m e nt of a l ine r in a

pre viou sly st re ss ed m e diu m . P r ovid ingt he s ur ro un di ng m ed i um r em ai nse lastic, the l ine r s tr esses im m e dia te lya f te r insta ll a t ion c a n be c onse rva tive lyestimated by assumin g that the processeso f e x ca v at i on a nd l i n er i n st a l la t io noc c ur sim ulta ne ously. I n pr a c tice , theliner is frequently installed after at least50% of the e la stic displa c e m e nt of the

medium has already taken place and theline r loa ds a r e c or r e spondingly lower .

To e va lua te the e f fe ct of e a r thqua ke

l oa di ng , t he s o lu t io n f or e xt er na ll o ad i n g s h o ul d b e u se d. S in ce b o th

m e d i um a n d l i ne r a r e a ss u me d t o b elinearly elastic, the post-excavation an dearthquake-induced stresses, or thrustsa n d b e n d i n g m o m e nt s , c an b e s u pe r -i m p o s ed t o e s t i m at e t he t o ta l l o ad s .R e me m b er , h o we ve r , t h at t h e e a rt h -q u a ke l o a di n g i s cy cl ic a n d t h at t hedesigner is concerned with the states of l i n e r a n d m e d i u m a t b o t h e x t r e m e s o f the cycle.

Because of the availabilityof relativelysimple closed-form analytical solutionsf or l ine d a nd unline d c ir c ula r tunne ls,t he c o nd i ti o ns r es u lt i ng f ro m p l an ew av e p r op a ga t in g n or ma l o r n ea r-

nor m a l to the tunne l a xis ar e r e lative lywe ll unde r stood. Muc h less a t te ntionh as b ee n d e vo t ed t o i n v e s t i g a ti n g t h e

beh av ior of ex ca va ti on s, suppor ted orunsupported, of different shapes. How-

e ve r, the ge ne r a l c onc lusions r e ac hedfo r t he c ir cu la r t un ne ls s ho ul d bea pplic a ble .

M os t i m po r ta n tl y , we e xp ec t t her e sp o n se t o e a r t hq u a k e l o a d in g t o b enear enoug h pseudostatic and we expectgr ound/st r uc tur e inte r ac tion e ff ec ts tobe re lat iv el y u n i m po r t a n t pr ovi ding th eg r o u n d s u p p o r t s ys te m i s r e l at i ve l yflexible. In practice, the ground supportis generally flexibleand the conservativea p p ro a c h o f a s s u mi n g t ha t t he l i ne r e xpe rie nc es the unr e stra ine d de f or m a -tion of the m e dium c a n be a dopte d. I f this a ppr oa c h r e sults in the c onc lusion

that special provisions need to be madet o p ro v id e a d eq u at e s af et y, t he n i twould be a ppr opr ia te to c onduc t m or e

174 TUNNELLING AND UNDERGROUND SPACE TECHNOLOGY Volum e 2, Num ber 2, 1987

7/28/2019 Aseismic Design of Underground Structures

12/34

thorough ground/structure interactioncalculations using one of the numerical

model ing tools discussed below.

Simple ground structure interactionmodels

I f the f le xibil i ty r a tio of a l ine r, a sdefined above, is low, then the liner isstiff compared to the medium and willresist the distortions imposed on it byt he m ed i um . O f c ou rs e, i t w i l l b ec o ns er v at i ve t o d es ig n t he l i n e r t owithstand the unrestrained distortionsof the medium. However, this approac hm a y be unduly c onse rvative f or s t if f liners, and the liner may become verydif f ic ult to de sign. I n suc h c a se s, the

g r o u nd / s t r uc t u r e i nt er ac ti on i s i m -por ta nt an d sh oul d be co ns id er ed in th edesign.

Lit t le a t te ntion ha s be e n de vote d tod e ri v in g a na l yt i ca l s ol u ti on s f or gr ound/str uc tur e inter a ction pr oble m sfor the case of waves propa gating alon gthe axis of the structure. This is due, in

pa rt , to th e fac t th at sev era l as sum ptionsor approximations are needed to derivea solution for a simple ground/st ructurem ode l. The se a ssum ptions r e str ic t thea pplic a tion of the r e sults to a l im ite dclassof problems. This ground/structurei n te ra ct i on p r ob l em h as f ir st b ee naddressed in the design of the Tran s-Ba yTube of the S a n F r anc isc o B ay Ar e aRapid Transport (Parsons Brinckerhoff 1960) system and, later, by the Jap anSociety of Civil Engi neers (1975, 1977).

The analytical procedure for estimat-ing strains and stresses experienced by astructure that resists ground motion are

ba se d on : (a) th e th eo ry of wa vepr o p a g a t i on in an in fi ni te , hom o-geneous, isotropic, elastic medium; and(b) the theory of an elastic beam on ane la stic f ounda tion. The be a m the or y isnecessary to account for the effects of interaction between the ground and thestructure. The details of this procedureand the assumptions made to arrive at a"closed-form solution" are discussed inde ta il in A ppe ndix C . Its a pplic a tion ind e si g n i s s u m ma r i ze d b el o w, u n d er " R e c om m e n d e d P ro ce d ur es f or P re -l i mi n ar y D es ig n o f U n de r gr o u ndStructures."

A m a in a ss u mp t io n i n t he a bo vepr oc ed ure is th at th e st ru ct ur e iss u pp o rt e d b y a n e la st ic f o u nd a t io ncharacterized by a foundation modulus.The latter is defined as a spring constantpe r un it le ngt h of st ru ct ur e. Un -f o rt u na t el y , t h er e i s n o u n i ve r s al l yagreed upon approac h for the derivationof the foundatio n modulus; and differentpr oc ed ur es m ay yi el d wi de ly di ff er en ta ns we rs . O n e a p p r oa c h , p r es e nt e d i nA p pe n di x C, is b as ed o n t he t wo -d i m e n s i on a l , p l a n e s t r a in s o l u t i on t othe Ke lvin 's proble m . The a ppr oa c h, ineffect, neglects the w idth of the structureand, therefore, its transverse stiffness.

A m or e ge ne r a l a ppr oa c h would be touse a num e r ic a l solution to de r ive the

f o u n d a t i o n m o d u lu s . N u m e ri c a l s o l u-t i on s r e qu i re t he u se o f a c o mp u t er p r o g r a m , su ch as a la rg e ge ne ra l-p u r p os e fi ni te e lem e nt co de ; th ey ar edescribed below. Regardless of how the

f o un d at i on m o du l us is o bt ai ne d, ar a n ge o f v a lu e s, r a th e r t h a n a s i ng l ev al ue , s h o ul d b e u se d i n p a ra m et r ica na ly s es t o e st i ma te b o un d s o n t hestrains and stresses experienced by thes t r uc t u re a n d g r o u n d m e d i u m d u e t odyna m ic loa ding.

We believe that simple models for theg r o u nd / s t ru c t u re i n te ra ct i on , w he nu se d i n c o nj u n ct i o n w i th r el at iv el ys i m p l e s t r uc t ur a l d e si g n m o d el s f or

l i ne rs , a re g en er al l y a de q ua t e fo r p r e l i m i n a r y de si gn of u nd e r g r ou nde xc a va tions with inte r na l s tr uc tur e s or supports that resist ground deformation.Of course, there will be many instancesi n w h ic h t he s tr uc tu re i s e it he r t ooc o m p le x o r t oo i m p o rt a n t t o r el y onsuc h si m ple pr oc edur e s a lone. I n the sec as es , o n e o f t he n u m e ri c a l m e th o d sdiscussed below should be used.

Numerical modeli ng o] groundstructure interaction

I n r e c e nt ye a r s, num e r ic a l m ode lingt e c hn i q u es h a ve s ee n a t r em e n do u sgr owth a nd ha ve be e n f ound to be ve ryuseful as tools for analysis. As opposedt o c l o se d -f o rm a n al y t i ca l s o l ut i on s ,which exist for a relatively small class of pr oblem s, num er ica l me tho ds ca n beused for analysis and design of complexstr uc tur e s. A la r ge num be r of public a -t io ns h av e c ov er ed t he d if fe re ntn u m er i ca l m et h od s u se d t o a na ly sewave propagation and ground/structurei n te ra ct i on p ro b le m s ( Des ai a ndChrist ian 1977). Herein, an overview ofthe dif f e r e nt num e r ic a l m e thods a va il-a ble is pr e sente d, f ollowe d by a ver y

br ie f s um m a ry of so me po p u l a r co m-pute r pr ogra m s us ed for th e dynam icanalysis of underground structures.

T h e n u m e ri c a l m e t ho d s o f a n al y si sfall u nd er o ne of t he f ol lo wi ngcategories: (a) finite difference method;(b) finite element method; (c) bound aryi n te g ra l e q ua t io n m et h od ; a nd ( d)m e t h o d o f c ha ra ct er i st i cs . T h e u se -fulness, validity and app lica tion of eachof these methods greatly depends on thetype of problem under consideration.

The f inite difference method was them a i n m e th o d o f a n al y si s b ef or e t hedevelopment of finite element methods.The method involves a discretization ofthe gove r ning e qua tions of m otion f or t he s o il / st r uc t ur e s ys te m. T h e d is -c r et i za t io n i s b a s ed o n r e p l ac i n g t h econtinuous derivatives in the governingequations by the ratio of changes in thev ar ia bl es o ve r a s ma ll , b u t f in it e,inc r em e nt. The dif f er e ntia l e qua tionsa r e, thus, t r a nsf orm e d into diff e re ncee q u at i o n s. T h e m e t h od o f s o l ut i o n o f these e qua tions f or tr a nsient a na lysisc an b e b as ed o n e it he r a n i m pl i ci t

s ch em e o r a n e x pl i ci t s ch em e. T h e

implicit scheme requires the solution ofa s et o f s i m ul t a ne o us e q ua t io n s a n d

la r ge stor a ge m a y be ne ede d. Explic itsc hem e s a r e r e la tive ly str a ightf or wa r d

and m ay require less effort than implici tschemes. For certain types of p/oblems,i t is possible to obta in unc onditio na lly

sta ble im plic it sc he m e s. The c hoic e of the best solutio n scheme depends on thepar t i cu l ar a ppli ca tion.

The f inite dif f er e nc e m e thod c an bedifficult to apply when n onhomoge neitya nd nonline a r it ie s e xist ; howe ve r , thisdifficulty can be overcome using the so-called integrated finite difference tech-n i qu e s. A n o t he r s i t u a t i o n c o m m o n i nw av e p r o p a g a t i on p r o bl e m s i nv o lv esinf inite m e dia . Ac c ordingly, ther e is an ee d t o c re at e a p p r o p r i a t e b o u n d ar yc on di t io ns t ha t w il l s im ul at e t heph ys ica l b eh av ior of th e ac tu al pr ob le m .The most popular approach is the use ofviscous dashpots to eliminate boundaryreflections.

I n t he f in it e element method, th ec o nt i n uu m is d is cr et iz ed i nt o a ne quiva le nt system of sm a lle r c ontinua ,which are called finite elements. Eache le m en t i s a s si g ne d c on st i tu ti v e o r material properties and its equations ofs tate a r e f or m ula te d. S ubse que ntly thee le me nt s a re a ss em bl ed to o bt ai nequatio ns for the total structure. As inthe case of the finite difference method,t he s o l u t i o n s c he m e c a n b e b as ed o nei th er a n i mp li ci t o r an e xp li ci tf o r m ul a t i on . I n e i th e r c as e, a f i ni t ed i ff er en ce a p p r o x i m a t i o n i s u s ed t orepresent the time dimension. T he mainadvantage of the finite element methodis that arbitrary boundaries and materialinhom oge ne ity c a n be a c c om m oda te deasily. As in the finite difference method,e ne r gy- a bsor bing bounda r ie s a r e use dt o a p p r o x i m a t e t h e w a ve p r o p a g at i o nin a n inf inite m e dium .

Th e boundary integral equationmethod involves numerical solution ofa set of integral equations that connectthe boundary, or surface, tractions to the

bou nda r y di sp lac em en ts . It is ba se d ons o lu t io n o f i nt eg ra l, r at he r t ha ndif fe r e ntia l , e qua tions. I t r e quir e s thediscretization of only the surface of the

bo dy an d th e su rf ac e of th e ex ca va ti oninto a nu mber of segments or elements.The numerical solution is first obtaineda t t he b o u n da r y s eg me nt s ; t h en t hesolution a t dif f e r e nt points within themedium is obtained from the solution att he b o un d ar y . I n t hi s m et ho d, t hei n f in i t e m e d i u m c an b e h a n d l ed v er ye a sily be c ause the inte gr a l e qua tionapplies for a load applied on an infiniteor se m i- inf inite m e dium . This m e thodi s m o st p o p u la r f or t he a na ly si s o f