An assessment methodology for tunnelling beneath bridges considering soil-structure interaction

An assessmen me hodology o unnelling benea h b idges conside ing soil–s uc u e
in e ac ion
Eugene K. L. Wonga, And ea F anzab,∗, Pe e Hewi c, Giulia M. B. Viggianid
aDe elopmen Bu eau, Go e nmen o Hong Kong S.A.R., China. Fo me ly Depa men o Enginee ing, Uni e si y o Camb idge, UK
bDepa men o Ci il and A chi ec u al Enginee ing, Aa hus Uni e si y, Denma k
cGeo-en i onmen al Enginee ing, Laing O’Rou ke, UK
dDepa men o Enginee ing, Uni e si y o Camb idge, UK
Abs ac
The cons uc ion o he Tideway Tunnel in London induced g ound mo emen s ha a ec ed se e al a ch b idges along he Ri e
Thames. This s udy examines he esponses o he G os eno B idge and Pu ney B idge o unnelling-induced displacemen s, wi h
a ocus on soil-s uc u e in e ac ion (SSI) and on he ela ionship be ween ounda ion displacemen s and g een ield p edic ions.
Coupled ini e elemen (FE) analyses and wo-s age analysis me hods (TSAM) a e compa ed, suppo ed by some ield da a. The
b idges a e modelled as elas ic ames. The equi alence be ween plane-s ain and h ee-dimensional FE models is in es iga ed,
while he e ec i eness o TSAM using sp ing-based soil- ounda ion models (de i ed om hal -space heo y) o a ying complexi y
is e alua ed. F om he compa ison wi h moni o ing da a and FE esul s, TSAM is shown o be p ac ical and eliable o assessing
unnel-induced e ec s on b idges, including du ing ansien condi ions such as TBM app oach. Resul s highligh he con as ing
SSI esponse o lexible (s eel) and s i (mason y) a ch b idges: while using ull g een ield mo emen s o scaling hem down by
some educ ion ac o may easonably in o m he esponse o lexible s uc u es, he beha iou o s i e sys ems will be ma kedly
di e en as SSI signi ican ly al e s de o ma ion mechanisms. Since he b idge esponse depends on he ela i e b idge- ounda ion-
soil s i ness, i is ecommended in all cases o pe o m in e ac ion analyses om he ea ly design s ages. A wo k low using TSAM
is p oposed o guide ea ly-s age design assessmen s, in eg a ing salien s uc u al and geo echnical conside a ions.
Keywo ds: unnelling, g ound mo emen s, b idge, soil-s uc u e in e ac ion, elas ic solu ions
Highligh s1
•Tunnel-soil-b idge in e ac ion modelling: wo case s udies2
•Two-s age analysis me hod (TSAM) benchma ked agains 3
3D and plane-s ain FE models4
•App oach o educing 3D s uc u es in o plane-s ain5
model and i s limi a ions6
•B idge s i ness c i ically in luences de o ma ion and in-7
e nal o ce dis ibu ion8
•P ac ical wo k low p oposed o ea ly-s age soil-s uc u e9
in e ac ion (SSI) assessmen using TSAM10
∗Co esponding au ho
Email add ess: [email p o ec ed] (And ea F anza)
P ep in submi ed o Tunnelling and Unde g ound Space Technology Oc obe 30, 2025
1. In oduc ion11
In u ban a eas, bo ed unnelling is a common solu ion o mee 12
g owing demand o anspo a ion and u ili y in as uc u e.13
Assessing he impac o unnelling-induced g ound mo emen s14
emains a c i ical challenge o enginee s and asse owne s. Al-15
hough case s udies ha e highligh ed he impac o exca a ion-16
induced g ound mo emen s on piles and some b idges wi h17
deep ounda ions (e.g. Cheng e al.,2007;Yoo,2013;Yoo and18
Abbas,2021;Yang e al.,2025), he e is no well-es ablished19
me hodology o analysing unnel-soil-b idge in e ac ion in he20
case o shallow ounda ions. To add ess his sho coming, his21
pape builds upon he expe ience om he Thames Tideway22
p ojec . Commencing ope a ion in 2025, he p ojec in ol ed a23
25 km–long la ge-diame e unnel (8.84 m cu ing diame e ) ex-24
ca a ed unde he Ri e Thames o in e cep , s o e, and con ey25
sewage in o de o upg ade London’s sewe sys em. This unnel26
passed benea h se e al b idges, mos o which a e a ch b idges,27
including he Pu ney B idge (a he i age mason y s uc u e) and28
Black ia s B idge (a mode n s eel-conc e e s uc u e).29
The Planning Inspec o a e publicly eleased he echnical e-30
po s compiled o assess he e ec s o Thames Tideway un-31
nelling on b idges, which p o ided a use ul e lec ion o cu -32
en enginee ing p ac ices. Typically, as a i s s ep, uncoupled33
analyses a e pe o med in which g een ield unnelling mo e-34
men s, whe he de e mined om empi ical o mulae o nume -35
ical analyses, a e imposed on b idge ounda ions; he esul -36
ing s uc u al dis o ions a e hen calcula ed (e.g. Thames Wa-37
e U ili ies Limi ed,2013b,c). Whe e soil-s uc u e in e ac-38
ion (SSI) e ec s a e conside ed, plane-s ain (PS) nume ical39
analyses inco po a ing selec ed s uc u al elemen s a e ca ied40
ou by geo echnical enginee s; he compu ed ounda ion dis-41
placemen s om hese PS models a e hen passed o s uc u al42
enginee s, who apply hem as base bounda y condi ions in de-43
ailed h ee-dimensional (3D) supe s uc u e models, o analyse44
induced in e nal o ces in s uc u al membe s (e.g. Thames Wa-45
e U ili ies Limi ed,2012,2013d). This change o bounda y46
condi ions be ween wo- and h ee-dimensional models is in-47
e i ably subjec o enginee ing judgemen and app oxima ions.48
Impac assessmen s o en assume ha b idge ounda ions ol-49
low g een ield displacemen p o iles, neglec ing he in luence50
o exis ing s uc u es. This dis ega ds he e ec s o SSI and51
o e looks he abili y o he supe s uc u e s i ness in modi ying52
he ounda ion esponse. While s aigh o wa d, his app oach53
can esul in o e ly conse a i e o inaccu a e es ima es, as ex-54
plo ed in his s udy.55
The eliance on g een ield me hods in b idge assessmen s de-56
i es di ec ly om he amewo k de eloped o buildings (e.g.57
Mai e al.,1996;C oss ail,2008;Schoo e al.,2021). In58
he case o buildings, well-es ablished p ocedu es link de o -59
ma ion pa ame e s om g een ield displacemen s a ounda ion60
le el o damage ca ego ies, suppo ed by ex ensi e li e a u e61
(e.g. F anzius e al.,2006;Dimmock and Mai ,2008;Fa ell62
e al.,2014;Gia dina e al.,2015;Boldini e al.,2021). How-63
e e , unnel-b idge in e ac ion emains less well-s udied, possi-64
bly because he di e si y o s uc u al o ms and con igu a ions65
de ies gene alisa ion. F anza and DeJong (2019) concep ually66
e alua ed he e ec s o unnelling-induced mo emen s on sim-67
ple b idges on sepa a e oo ings o o hogonal a angemen s68
o pie s and decks (modelled as beam elemen s), using elas ic69
soil- ame in e ac ion analyses. Thei esul s highligh ed ha 70
he quali a i e dis ibu ion o unnelling-induced mo emen s o 71
he ounda ion is modi ied compa ed wi h g een ield shapes as72
a esul o ame ac ion, which is no eadily accoun ed o 73
by a modi ica ion ac o . A simila obse a ion was made o 74
he G os eno B idge in he s udy by Fahe y e al. (2022).75
Thei pa ame ic s udy o s eel a ch b idges highligh ed he76
challenges o gene alising isk assessmen p o ocols; b idge e-77
sponses o g ound mo emen s a e highly dependen on s uc-78
u al ac o s such as olle bea ing a el, in e nal pa ial hinges79
and expansion join s — de ails ha a e o en di icul o cap-80
u e in geo echnical models. This in insic s uc u al complex-81
i y also suppo s he sugges ion ha damage c i e ia o b idges82
subjec ed o ounda ion mo emen s a e no well es ablished.83
Fo ins ance, he Fede al Highway Adminis a ion (1985) p o-84
posed some h esholds o he ole able longi udinal angula 85
dis o ion ( a io o di e en ial se lemen o span leng h) o 86
b idges o di e en ma e ials and cons uc ion; howe e , his87
may no adequa ely accoun o exca a ion-induced ho izon al88
mo emen s. Fu he mo e, i is no s aigh o wa d o co e-89
la e s uc u al damage wi h compu ed in e nal o ces. Conse-90
quen ly, s uc u al enginee s equen ly eso o de ailed nu-91
me ical analysis o assess he s uc u al esponse o exis ing92
b idges o unnel cons uc ion, which a e in o med by he ex-93
pec ed magni ude and shape o unnelling-induced ounda ion94
mo emen s p o ided by geo echnical enginee s, which may o 95
may no conside SSI. The esea ch ques ions o wha a e ade-96
qua e modelling s a egies o isk assessmen s and hei po en-97
ial pi alls a ise.98
This pape compa es di e en unnel-soil-s uc u e in e ac-99
ion app oaches, e alua ing hei s eng hs and limi a ions o 100
a ying le els o complexi y o he soil- ounda ion sys em.101
Fi s , he G os eno B idge and he Pu ney B idge case s ud-102
ies a e p esen ed along wi h an o e iew o he modelling103
app oaches used. Second, he measu ed mo emen s o he104
G os eno B idge om he Tideway Tunnel cons uc ion a e105
examined, building on Wong (2019) and Fahe y e al. (2022);106
he sho - e m, in-plane, s eady-s a e b idge de o ma ions a e107
conside ed. P edic ions om plane-s ain ini e elemen (PSFE)108
analyses using a simple soil cons i u i e model in o al s ess109
a e p esen ed; a p ocedu e o de ine an equi alen plane-s ain110
ounda ion o cap u e he lexibili y associa ed wi h he ac ual111
h ee-dimensional geome y is p oposed. Thi d, he s udy com-112
pa es ield and nume ical FE esul s wi h elas ic SSI models113
ollowing he wo-s age analysis me hod (TSAM) using di -114
e en ep esen a ions based on lumped sp ings: ully-coupled115
(hal -space heo y), locally-coupled (spa ially decoupled wi h116
local o o- ansla ional coupling), and decoupled (spa ially de-117
coupled wi hou o o- ansla ional coupling). Fou h, a TSAM118
analysis o he esponse o he Pu ney B idge o unnelling is119
p esen ed; esul s highligh he equi emen s, e sa ili y, and120
ad an ages o sp ing-based modelling app oach o a ela i ely121
s i b idge in which SSI e ec s a e expec ed o be signi ican .122
Finally, a discussion on how sp ing-based wo-s age modelling123
2
could be used o s udying he e ec s o ansien g ound mo e-124
men s in 3D by using a 3D TSAM model o he G os eno 125
B idge subjec ed o an ad ancing unnel exca a ion is p o ided;126
he esul s a e compa ed wi h ield obse a ions. Guidance is127
also gi en on how o ca y ou an in eg a ed analysis ha ac-128
coun s o bo h unnelling and soil- ounda ion in e ac ion.129
2. Case s udies130
2.1. G os eno B idge131
The G os eno B idge is loca ed in cen al London. I is a132
s eel ailway a ch b idge c ossing he Ri e Thames be ween133
Pimlico and Ba e sea, nea Vic o ia S a ion. I suppo s ailway134
acks ha ca y bo h passenge and eigh a ic. The b idge135
is suppo ed by mason y pie s embedded in London Clay. The136
Thames Tideway Tunnel passed unde he G os eno B idge a 137
a posi ion be ween i s Sou h and Cen e Pie s (Fig. 1).138
The b idge was o iginally cons uc ed in he second hal o 139
he 1800s and i unde wen se e al phases o widening, modi i-140
ca ion, and pa ial econs uc ion be ween 1865 and 1967 (Wil-141
son,1868;Fox,1868;Ke ensky and Pa idge,1967;Ke en-142
sky e al.,1967). The p esen s uc u e consis s o ou spans,143
each anging om 55 o 57 m, wi h a 50 m wide deck sup-144
po ed by en pa allel, s uc u ally sepa a e bu iden ical, s eel145
a ch ames. Each ame is made o wo a ch ibs wi h ec an-146
gula hollow c oss-sec ions. The a ches suppo he deck ia147
spand el pos s. The deck consis s o 15.9 mm hick mild s eel148
pla es, suppo ed by in e ed Uni e sal-T sec ions. The a ches149
a e pin-connec ed a each pie and connec ed monoli hically150
wi h he deck a mid-span. The deck is simply suppo ed a 151
he pie s. The pie s a e made o mason y encased in ein o ced152
conc e e and a e embedded in London Clay. Fig. 2shows he153
mechanical model o he b idge as a plane ame and indica es154
he posi ions o he a ge s used o moni o he displacemen s o 155
he b idge du ing exca a ion o he unnel.156
The wes bound ea h p essu e balance unnel bo ing machine157
(EPB TBM) wi h an ex e nal diame e o 8.84 m was launched158
om a sha on Ki ling S ee in Janua y 2019. I passed un-159
de nea h he G os eno B idge o e i e days in Ma ch 2019.160
Moni o ing da a om he Thames Tideway Tunnel p ojec , in-161
cluding se lemen , ho izon al mo emen s and displacemen s o 162
he b idge deck p o ide a e e ence o alida e model p edic-163
ions.164
The unnel was exca a ed a a dep h o app oxima ely 34 m165
below he i e bed, p ima ily h ough London Clay, bu i s in-166
e was close o o wi hin he Lambe h G oup (Thames Wa-167
e U ili ies Limi ed,2013a,d). A his loca ion, he Lam-168
be h G oup comp ises Lowe Shelly Clays and Uppe Mo -169
led Clays o he Woolwich and Reading Fo ma ions. Nea 170
G os eno B idge, he geo echnical in es iga ions by Skemp-171
on and Henkel (1957) and Ke ensky e al. (1967) p o ided he172
und ained shea s eng h p o ile wi h dep h below he London173
Clay ho izon, as shown in Fig. 3.174
A he ounda ion base le el o he G os eno B idge, ap-175
p oxima ely 4 m in o he London Clay, he und ained shea 176
s eng h is in he ange 100–200 kPa. A he small s ain le -177
els ele an o se lemen p edic ions, he und ained Young’s178
Deck
Tideway Tunnel
ELEVATION
PLAN
Sou h Pie Cen e Pie No h Pie
A ch ib
x
z
57 m
z0=34 m
y
x
B0= 13.7 m
L= 60 m
(a)
(b)
27.4 m
D i e
Ri e bed
Downs eam
Figu e 1. Layou o G os eno B idge showing (a) gene al iew and (b)
ela i e posi ions o b idge and unnel
Deck discon inuous and
simply suppo ed a pie s
( olle suppo s) Fixed deck/a ch
connec ion a mid-
span
Deck
A ch
Pie
Spand el pos s
Embedded po ion o pie modelled
as ounda ion ( oo ing)
Supe s uc u e
Founda ion
Nodes in wo-s age elas ic analysis:
Supe s uc u e nodes wi h displacemen s u
Founda ion nodes wi h displacemen s us
In e nal ic ionless hinges
Posi ion o su ey a ge s
Ri e bed
B0= 13.7 m
D= 4.3 m
Figu e 2. Plane- ame model o G os eno B idge: s uc u al membe s and
in e nal connec ions
modulus Euo London Clay is es ima ed as 500–800 Su, con-179
side ing i s co ela ion o he plas ici y index and he o e con-180
solida ion a io (Viggiani,1999). These es ima es align wi h181
he alues epo ed by Wi he s e al. (2001) who ound Eu/Su
182
in he ange 500–1500 a a s ain le el o 0.01%, and also sug-183
ges ed ha Euexceeds 75 MPa. Fo he nume ical analyses in184
his pape , we adop he Young’s modulus alue sugges ed by185
Ke ensky and Pa idge (1967), Eu=105 MPa (1000 ons/ 2),186
based on he obse a ions om a nea by building cons uc ion187
si e. Du ing he 1966–67 econs uc ion o he b idge, he same188
Young’s modulus alue was used o es ima e he immedia e se -189
lemen o i s pie s, showing e y good ag eemen wi h ield190
obse a ions.191
Be o e eaching he G os eno B idge, he Tideway TBM192
passed benea h h ee u ili y unnels, whe e he a e age olume193
loss was back-calcula ed as 0.9% (Wong,2019). In he absence194
3
0 100 200 300 400
Su (kPa)
0
5
10
15
20
25
Dep h (m)
B idge ounda ion le el
G os eno B idge
(Ke ensky e al., 1967)
Vic o ia S a ion
(Skemp on and Henkel, 1957)
Figu e 3. Und ained shea s eng h o London Clay nea G os eno B idge
o se lemen measu emen s along a ull g een ield ans e se195
c oss-sec ion, a olume loss alue VLo 1% was assumed as a196
i s app oxima ion.197
2.2. Pu ney B idge198
The Pu ney B idge is a G ade II lis ed mason y a ch b idge199
loca ed in sou hwes London, connec ing Pu ney and Fulham200
ac oss he Ri e Thames. O iginally cons uc ed be ween 1882201
and 1886, i consis s o i e segmen al s one and g ani e a ches,202
wi h spans anging om app oxima ely 34 m o 44 m and a ch203
ise o 5 m o 6 m. The s uc u e was widened in 1933, wi h a204
9.14 m wide ex ension on he downs eam side using mass con-205
c e e, while main aining he o iginal isual appea ance. The206
conc e e deck is suppo ed by longi udinal spand el walls con-207
s uc ed o b ick o he o iginal s uc u e and mass conc e e208
o he ex ension po ion, wi h s one slabs spanning be ween209
spand els ha ing no back ill be ween hem. The a ches a e con-210
s uc ed om g ani e blocks (wi h hickness o 1.25 m) and bea 211
di ec ly on o he mass conc e e pie s and abu men s. The b idge212
does no include bea ings o mo emen join s.213
The Thames Tideway TBM passed benea h he sou he n in-214
e media e span o he Pu ney B idge, wi h a unnel axis dep h215
o 25.3 m below i e bed le el (neglec ing he p esence o 1 m216
hick allu ium laye ). This sou he n in e media e span has a217
clea span o 39.4 m and heigh ( ise) o 5.4 m. The ounding218
soil consis s o London clay up o a dep h o abou 15 m om219
he in e dep h. The ounda ions ha e a wid h o 10.3 m (in220
di ec ion ans e se o unnel axis) and leng h o 36 m (in di-221
ec ion along unnel axis), and a e embedded abou 6.1 m in o222
London Clay; he embedmen is app oxima ely 25% o he un-223
nel axis dep h. The pie heigh om he ounding le el o he224
lowe poin o he a ch is app ox. 8.2 m while pie ans e se225
wid h is 5.6 m. A unnelling olume loss o 1% was assumed,226
and he und ained Young’s modulus o he London Clay was227
aken as 105 MPa as o he G os eno B idge, an o de o 228
magni ude g ea e han he alue adop ed in Tideway P ojec ’s229
assessmen epo o Pu ney B idge (Thames Wa e U ili ies230
Limi ed,2012).231
Making e e ence o Thames Wa e U ili ies Limi ed (2012),232
we cons uc ed a plane- ame nume ical model o he b idge233
supe s uc u e. Table 1summa ises he axial and bending s i -234
ness o beam elemen s adop ed o he combined a ches and235
spand els, he deck and he pie s, o he wo b idges examined236
in his pape .237
Figu e 4. Pu ney B idge (pho o cou esy o Wandswo h Council, London)
Deck
A ch
Pie
Embedded po ion o pie modelled
as ounda ion ( oo ing)
Supe s uc u e
Founda ion
Nodes in wo-s age elas ic analysis:
Supe s uc u e nodes wi h displacemen s u
Founda ion nodes wi h displacemen s us
Ri e bed
B0= 10 m
D= 6.1 m
Figu e 5. Plane- ame model o Pu ney B idge
Table 1. C oss sec ional p ope ies o main s uc u al membe s in nume ical
models
S uc u al membe G os eno B idge Pu ney B idge
A ch I=0.298 m4I=202 m4
A=1.32 m2A=72.2 m2
Deck I=0.005 m4I=6.32 m4
A=0.761 m2A=34.5 m2
Pie I=12864 m4I=3278 m4
A=898 m2A=371 m2
Spand els I=0.004 m4(Inco po a ed
A=0.095 m2in a ch)
3. In e ac ion models238
3.1. Two-s age analysis me hod (TSAM)239
Two-s age elas ic models ha e been applied in he li e a u e240
o s udy SSI in ol ing pipelines and buildings a ec ed by un-241
nelling (e.g. Kla e al.,2005;F anza and DeJong,2019). The242
i s s age o he analysis in ol es p edic ing g een ield dis-243
placemen s a he ounda ion le el, which se e as he p ima y244
unnelling- ela ed inpu in TSAM. In he second s age, he in e -245
ac ion p oblem is sol ed by inco po a ing s uc u e- ounda ion246
4
s i ness and applying he g een ield-equi alen se o o ces247
ha would displace he soil- ounda ion by he g een ield mo e-248
men s. In his pape , linea elas ic wo-s age in e ac ion models249
a e adop ed o he b idge p oblems, wi h p elimina y esul s250
epo ed in Wong e al. (2021).251
G een ield mo emen s a he ounda ion le el can be es i-252
ma ed using semi-empi ical ela ionships, analy ical solu ions253
o nume ical me hods. In his pape , g een ield displacemen s254
in he e ical and ho izon al di ec ions a e es ima ed using he255
empi ical me hod in Wong e al. (2025). B ie ly, he ans e se256
se lemen p o ile in clay ollows a Gaussian dis ibu ion (Peck,257
1969;Mai and Taylo ,1997), while he exp ession om Mai 258
e al. (1993) is selec ed o desc ibe he linea a ia ion o he in-259
lec ion poin o se wi h dep h; ho izon al mo emen s a e cal-260
cula ed om se lemen s conside ing he a ia ion o he ocus261
poin o he o al displacemen ec o wi h ans e se o se and262
dep h. Con a y o he assump ion ha a any gi en dep h he263
displacemen ec o s poin owa ds a unique poin , Wong e al.264
(2025) demons a ed ha o case his o ies o unnel exca a-265
ions in ine-g ained soils, he ec o ocus a he g ound su ace266
is abo e he unnel cen e and i s dep h becomes shallowe wi h267
inc easing ans e se dis ance om he unnel cen e-line; be-268
low a dep h o 20% he unnel axis dep h, he ocus poin dep h269
is assumed o be a 1.54 imes he unnel dep h.270
Fo he g een ield s age 1, Eq. (1)–(2) desc ibe he s eady-271
s a e se lemen was a s anda d Gaussian cu e.272
w=wmax exp 




−x2
2i2




;wmax = π
32
VLD2
i(1)
i=0.5z −0.325z(2)
whe e wis he se lemen a a ans e se ho izon al dis ance x273
om he unnel axis, wmax is he maximum cen e-line se le-274
men , iis he ho izon al dis ance o he poin o in lexion om275
he unnel axis, VLis he olume loss, exp essed as a pe cen age276
o he nominal olume o he unnel o diame e D , and z is he277
dep h o he unnel axis.278
The ans e se ho izon al displacemen ua dis ance x om279
he unnel axis and a dep h zcan be exp essed as a unc ion280
o he se lemen wa he same loca ion, and o he no malised281
dep h o he ocus poin h=z /z , which in u n is a unc ion282
o he no malised dep h z/z and no malised o se x/i.283
u=−wx
h z −z(3)
whe e:
h=h0=3.59
4.43 +exp 





x2
1.3i2





when z=0; (4a)
h=h0−1.54 1−z
0.2z !+1.54 when 0 <z
z
<0.2; (4b)
h=1.54 when z
z
≥0.2 (4c)
I h=1, he displacemen ec o s poin owa ds he unnel284
cen e. Eq. (4) es ablishes ha , nea he g ound su ace, he285
no malised ocus dep h h<1 close o he cen e-line and h e-286
duces wi h he no malised o se x/i. In his pape , Eq. (1)–(4)287
we e applied conside ing he g ound su ace ho izon o be a 288
he ounda ion le el.289
In s age 2, he b idge esponse o unnelling is s udied by290
modelling he supe s uc u e as a plane ame es ing on oo -291
ings ied o he sp ing-based g ound model (consis ing o ei-292
he ully-coupled, locally-coupled, o decoupled sp ings as dis-293
cussed below). Fig. 6shows he in e ac ion model o he b idge294
schema ised as ei he a plane o h ee-dimensional ame. As295
he in e ac ion model is linea elas ic, he ounda ion main ains296
displacemen compa ibili y be ween he soil and he oo ings.297
The pie bases we e assumed o be igid; hus, he ounda ions298
and he displacemen s o he ounda ion soil we e desc ibed by299
single nodes a he cen es o he oo ing oo p in s, collec i ely300
e e ed o as he ounda ion mas e -nodes. Tunnelling e ec s301
we e modelled by applying an equi alen se o o ces ha dis-302
place he sp ing ounda ion model alone by he g een ield dis-303
placemen s, including g een ield o a ions when app op ia e (as304
discussed la e ).305
x,u
z,w
y,
In e ac ion be ween
ansla ion and
o a ion (FC and LC
sp ings only)
In e ac ion be ween
ounda ions (FC sp ings only)
(y, x, zdi ec ion sp ings
no shown o cla i y)
Figu e 6. In e ac i e sp ings adop ed in wo-s age elas ic solu ion analysing
plane ame in x−zplane unde s eady-s a e unnelling condi ions
The in e ac ion p oblem is sol ed wi h he ini e elemen
me hod. The equilib ium equa ion in ma ix o m desc ibing
he ame displacemen unde he e ec s o unnelling is:
(Ks +K )u=pg (5)
K =ATKsA(6)
whe e Ks and K a e he s uc u e and he ounda ion (soil-306
oo ing sys em) s i ness ma ix, espec i ely, and uis he dis-307
placemen ec o desc ibing ansla ional and o a ional deg ees308
o eedom o he s uc u e (including supe s uc u e and pie 309
base poin s); o a plane ame, he ansla ional deg ees o ee-310
dom in he xand zdi ec ions and he o a ions in he x−zplane311
a e conside ed; Ais a kinema ic cons ain ma ix ela ing he312
ansla ional and o a ional deg ees o eedom o he ounda-313
ion mas e -nodes o he ansla ional deg ees o eedom o he314
soil unde nea h he ounda ion oo p in : i.e. as he oo ing is315
assumed o be igid, compa ibili y imposed ia A equi es ha 316
he displacemen s o he soil a he oo ing should be a linea 317
unc ion o he displacemen s o he pie base poin s (la e de-318
ailed in he ex ); Ksis he soil s i ness ma ix (de ined wi h319
espec o all nodes a he soil- oo ings in e ace); pg is he ec-320
o o he equi alen g een ield unnelling-induced o ces ac ing321
5

on he ounda ion.322
As shown in Fig. 6, he supe s uc u e is disc e ised in o i-323
ni e elemen s. The supe s uc u e s i ness ma ix Ks is ob-324
ained by assembling (i) Eule -Be noulli beams o he s aigh 325
s eel membe s igid in shea , (ii) Timoshenko beams o he s i 326
conc e e pie s (F iedman and Kosma ka,1993) and (iii) cu ed327
Eule -Be noulli beams o he a ches (Li ewka and Rakowski,328
1997) while (i ) he igid oo ings we e modelled by in oduc-329
ing he kinema ic cons ain ma ix discussed abo e. All in e -330
nal hinges and olle s (as illus a ed in Fig. 2) we e duly con-331
side ed by manipula ing he ini e elemen s i ness ma ix o332
accoun o he elease o he ele an in e nal o ces. Full de-333
ails o he s i ness ma ix a e documen ed in Wong (2019) and334
Wong e al. (2021).335
To es ablish he soil s i ness ma ix Ks, he soil ied o he336
oo ing in e ace was disc e ised in o a g id o nodes (each ha -337
ing h ee ansla ional deg ees o eedom), whose displacemen 338
ec o usis linked o he supe s uc u e by he kinema ic con-339
s ain as us=A u. F om he kinema ic cons ain ma ix,340
i ollows ha K is a condensed s i ness ma ix o Kswi h341
espec o he deg ees o eedom a he cen oids o he pie 342
bases. The e o e, he ounda ion sys em comp ising soil and343
igid oo ings as desc ibed by K consis s o o o- ansla ional344
sp ings, all lumped a he bases o he pie s as shown in Fig. 6345
ia he kinema ic cons ain ma ix. In e ac ion e ms o hese346
ounda ion sp ings (i.e. o -diagonal e ms o K ) depend on347
he adop ed model as ollows:348
•Fully-coupled (FC) sp ings: linea elas ic hal -space he-349
o y was used o he soil, esul ing in a ully popula ed350
s i ness ma ix, coupling all o o- ansla ional beha iou 351
ac oss all oo ings. The soil lexibili y ma ix was ob-352
ained om he Boussinesq-Ce u i solu ion (Lo e,1927;353
Li and Be ge ,2001) and he in eg a ed Mindlin’s solu-354
ion o a uni o mly loaded ec angula a ea (Cheung and355
Nag,1968) (see Appendix A: Fo mula ion o soil s i ness356
ma ix o ully-coupled (FC) sp ings solu ion). The em-357
bedmen o he oo ings was no aken in o accoun . Fo a358
s uc u e consis ing o mul iple indi idual oo ings, hese359
we e disc e ised in o an a ay o nnodes dis ibu ed below360
he oo p in whose h ee ansla ional do s a e collec ed361
in us, ollowing he p ocedu e ou lined abo e o gene a e362
a 3n×3nsoil s i ness ma ix Ks. This enables he cou-363
pling be ween di e en oo ings ac oss he su ace o he364
soil.365
•Locally-coupled (LC) sp ings: he diagonal e ms and lo-366
cal o o- ansla ional e ms o each oo ing we e accoun ed367
o bu no in e ac ion be ween di e en pie s was consid-368
e ed. This was o mula ed in he same manne as he FC369
sp ings om Lo e (1927) and Cheung and Nag (1968),370
bu by disc e ising a single oo ing ins ead o all he oo -371
ings ac oss he con inuum while neglec ing he embed-372
men o he oo ings (LC1 sp ings). Al e na i ely, he o o-373
ansla ion s i ness o isola ed, igid, embedded/su ace374
oo ings on he hal -space o a comp essible laye o i-375
ni e dep h could be de ined based on semi-empi ical o -376
mulae (Pais and Kausel,1988;Gaze as,1991a,b) (LC2377
sp ings; see Appendix B: Fo mula ion o soil s i ness ma-378
ix o decoupled (DC) o locally-coupled (LC2) sp ings379
solu ion).380
•Decoupled (DC) sp ings: a diagonal s i ness ma ix was381
used, wi h sp ings ha only eac o local loads in hei 382
di ec ions. The sp ing cons an s we e ob ained om he383
diagonal e ms o he LC2 s i ness ma ix.384
Fo p ac ical applica ions, he s i ness alues o locally-385
coupled and decoupled ounda ion sp ings can be eadily cal-386
cula ed ei he om he li e a u e o a gi en ounda ion layou 387
( oo ing leng h, wid h, embedmen , e c) o om 3D ini e el-388
emen analyses (calcula ing he load-displacemen ela ionship389
a he mas e -node o a oo ing es ing on linea elas ic ma e-390
ial).391
The ec o o unnelling-induced o ces pg was compu ed392
om he ec o o g een ield displacemen s us,g a ounda ion393
nodes, es ablished using empi ical o mulae o nume ical e-394
sul s. Fo FC and LC1 sp ings, pg =ATKsus,g , de i ed om395
he e ical and ho izon al g een ield displacemen s a he soil-396
oo ing in e ace. Fo LC2 o DC sp ings, pg =K ug ; in addi-397
ion o e ical and ho izon al displacemen s, ug also con ains398
g een ield o a ions, which we e compu ed as he i s de i a i e399
o he g een ield se lemen p o ile a he oo ing cen oids.400
The TSAM analyses adop ing linea elas ic ames o he401
supe s uc u e (see Fig. 2and 5) we e ca ied ou o he402
G os eno and Pu ney B idges. No e ha , i nonlinea s uc-403
u al beha iou is o in e es , he linea sp ing-based g ound404
models desc ibed abo e could all be implemen ed wi hin com-405
me cial s uc u al compu e p og ams o ou ine applica ions;406
in pa icula , LC and DC sp ings would be s aigh o wa d o407
calib a e and implemen .408
3.2. Equi alence be ween 3D and PS ounda ions in FE analy-409
ses410
Geo echnical plane-s ain analyses a e equen ly used o 411
simplici y and e iciency. The e a e di e en me hods o412
achie e he equi ed equi alence be ween he plane-s ain413
model and he ac ual h ee dimensional p oblem, while main-414
aining he ela i e soil- o-s uc u e s i ness. The equi alence415
is achie ed ei he by scaling he s i ness and he weigh o he416
s uc u e (e.g. Rampello e al.,2012) o by modi ying he oun-417
da ion dimensions o by a combina ion o he wo (e.g. Callis o418
e al.,2013;Rampello e al.,2014) depending on he p oblem419
a hand. In ac , he de o mabili y (i.e. he displacemen un-420
de uni ex e nal load) o a ounda ion sys em (consis ing o 421
soil and oo ings) in plane-s ain may be signi ican ly di e -422
en om ha o a h ee-dimensional oo ing: an in ini e s ip423
ounda ion is mo e lexible han a ec angula oo p in o a424
gi en ans e se wid h and uni ex e nal load pe longi udinal425
leng h (Pais and Kausel,1988;Dempsey and Li,1989;Gaze-426
as,1991a,b). In o he wo ds, he ounda ion displacemen s and427
he o a ion o an in ini e s ip oo ing a e la ge han hose o 428
a h ee-dimensional ec angula oo ing unde iden ical loads429
pe longi udinal leng h (longi udinal e e ing o ou -o -plane430
di ec ion o b idge s uc u e, pa allel o di ec ion o unnel ad-431
ancemen ).432
6
I bo h he s uc u e and he ounda ion beha iou a e de-433
sc ibed using s i ness ma ices condensed wi h espec o he434
deg ees o eedom o he ounda ion (pie base), he equi-435
lib ium equa ion o a 3D s uc u e es ing on a linea elas ic436
g ound model equi es ha :437
(Ks ,3D +K ,3D)u=K ,3D ug (7)
whe e Ks ,3D and K ,3D a e he s uc u e and ounda ion s i ness438
ma ices o he 3D s uc u e.439
A plane-s ain (PS) model may be conside ed equi alen o440
he 3D one when, o a gi en g een ield inpu ug , i yields he441
same in e ac ion ou come uas he 3D model. The equi alence442
can be achie ed ei he by scaling he s i ness o he s uc u e443
(Me hod A) o by modi ying he ounda ion layou while keep-444
ing he mechanical p ope ies o he s uc u e (Me hod B). Re-445
ga dless o he app oach adop ed, as discussed la e , a pe ec 446
equi alence be ween PS and 3D models canno be achie ed by447
ei he me hod; enginee ing judgmen is hus needed o de ine448
equi alen plane-s ain models. TSAM is well-sui ed o 3D449
analysis (e.g. plane/3D ame on 3D ounda ion), while he450
p oblem o equi alence a ises when adop ing PS ini e elemen 451
coupled nume ical analyses.452
3.2.1. Me hod A – scaling he s uc u e453
I he ans e se wid h Bo he ounda ion in he PS model
is se o be he same as he ac ual wid h B0in he 3D p oblem,
i ollows ha he 3D ounda ion s i ness can be exp essed as a
unc ion o he PS s i ness ma ix:
L−1
0K ,3D =MK ,PS (8)
whe e L0is he leng h o he ounda ion in he ou -o -plane di-454
ec ion while Mis he ma ix desc ibing he ela ionship be-455
ween K ,3D and K ,PS pe uni longi udinal leng h. By manip-456
ula ing Eq. (7) and (8), i ollows ha he equi alen PS model457
is:458
(K∗
s ,PS +K ,PS)u=K ,PS ug (9)
K∗
s ,PS =M−1L−1
0Ks ,3D (10)
which is ob ained by scaling he s i ness ma ix o he s uc-
u e in he 2D model. The supe sc ip ∗deno es he equi alen
s i ness ob ained by scaling. To simpli y he scaling p oblem,
i may be assumed ha all deg ees o eedom o he 2D and 3D
ounda ion s i ness ma ices scale equally and independen ly,
in o he wo ds, Mis a scala . Unde his assump ion:
K∗
s ,PS ≈(L0·m)−1Ks ,3D (11)
No e ha m≥1 gi en ha an in ini e s ip oo ing is less s i 459
han a ec angula oo ing unde iden ical loads pe uni longi-460
udinal leng h. Eq. (11) may be p ac ically implemen ed in nu-461
me ical models by di iding he Young’s modulus o he s uc-462
u e by mwhile keeping he ac ual ounda ion wid h B0.463
3.2.2. Me hod B – scaling he ounda ion464
An al e na i e app oach is o inc ease he PS ounda ion465
wid h B om he 3D alue B0 o achie e equi alence. In his466
way, he PS ounda ion ma ix is app oxima ely he 3D ma ix467
di ided by L0while he PS s uc u e is simply ob ained as a uni 468
slice o he ull s uc u e. I ollows ha 469
(Ks ,PS +K∗
,PS)u=K∗
,PSug (12)
K∗
,PS ≈L−1
0K ,3D (13)
Ks ,PS =L−1
0Ks ,3D (14)
When adop ing di e en ans e se wid hs B o PS and 3D
models, he a io be ween 3D and PS ounda ion s i ness ma-
ices can be quan i ied by he e m
R=L0K−1
,3D K∗
,PS (15)
By de ini ion, an e ec i e equi alence achie ed by scaling he470
ounda ion (see Eq. 15) is ob ained when Ris he iden i y ma-471
ix (all diagonal elemen s Rii a e as close as possible o uni y).472
Howe e , i is impossible o sa is y Rii =1 o all elemen s be-473
cause o he di e en a ios be ween PS and 3D s i ness alues474
depending on he load di ec ions/deg ees o eedom.475
The equi alence o a plane-s ain analysis is add essed in he476
subsequen sec ion. In con as , he e is no need o conside 477
equi alence in he TSAM, gi en ha he soil s i ness ma ix is478
ob ained om 3D heo ies in he i s place.479
3.3. Plane-s ain (PSFE) and h ee-dimensional (3DFE) cou-480
pled ini e elemen models481
Two coupled nume ical analysis me hods we e used in his482
s udy o s udy he in e ac ion be ween he Tideway Tunnel483
and G os eno B idge in he sho e m: plane-s ain ini e484
elemen (PSFE) and h ee-dimensional ini e elemen (3DFE)485
models o which he p og ams PLAXIS 2D ( e sion 20) and486
PLAXIS 3D ( e sion 17) we e used, espec i ely.487
The 3DFE model ac s as a benchma k o PSFE and sp ing-488
based TSAM models; he e o e, he 3DFE analyses we e ca -489
ied ou o sho - e m, s eady-s a e unnelling only. The plane-490
s ain model PSFE is simila o he app oach p ac i ione s ypi-491
cally use o unnelling impac assessmen s o s udy ounda ion492
and supe s uc u e s eady-s a e mo emen s (e.g. Thames Wa e 493
U ili ies Limi ed,2012,2013d) and i was adop ed he e o s udy494
he equi alence be ween PSFE and 3DFE in e ac ion models by495
scaling he ounda ion leng h (Me hod A). The model geome-496
ies o bo h he PSFE and 3DFE models we e iden ical: s uc-497
u al elemen s o he b idge supe s uc u e we e modelled as498
pla e elemen s ( o PSFE) o beam elemen s ( o 3DFE), while499
spand el pos s which we e hinged a bo h ends we e modelled500
as node- o-node ancho s. On he o he hand, Pu ney B idge was501
only analysed wi h TSAM.502
The s eady-s a e condi ion ollowing he comple e passage o 503
he TBM wi hin London Clay was simula ed in o al s ess anal-504
ysis. In he analysis, he b idge was assumed o be weigh less505
as he aim o he analysis was o s udy he unnelling-induced506
7
z =34 m
1 m be ween unnel in e and bo om bounda y
H=39.4 m
B
270 m
(a) PSFE
(b) 3DFE
39.4 m
270 m
300 m
Figu e 7. Fini e elemen models o G os eno B idge and unnel exca a ion: (a) plane-s ain model PSFE and (b) h ee-dimensional model wi h plane ame 3DFE
und ained esponse o he soil and he b idge. The g ound ma e-507
ial was assumed o be linea elas ic – pe ec ly plas ic and ol-508
low he T esca ailu e c i e ion. Fig. 7shows he mesh adop ed509
o he analysis. The bo om bounda y o he g ound model510
was aken a he bo om o he London Clay ho izon, assumed511
o be 1 m below he in e o he unnel, while he ho izon-512
al ex en was se beyond he i e banks, which we e ou side513
he unnelling in luence zone, o a o al wid h o 270 m. The514
ollowing soil p ope ies we e assumed o he g ound model:515
γ=20 kN/m3,Eu=150 MPa, Su=200 kPa, K0=0.7 and516
ν=0.495, selec ed wi h e e ence o assessmen epo s unde 517
he Tideway p ojec . S anda d bounda y condi ions we e used518
( olle s on e ical bounda ies, ixed bo om bounda y). The519
model bounda y, while close o he abu men s o he b idge, did520
no a ec he analysis esul s because he deck s uc u e wi h521
olle connec ions does no ansmi longi udinal loads be ween522
spans.523
A displacemen -con olled echnique was used in he PSFE524
and 3DFE model o model unnelling. Following Cheng e al.525
(2007), a echnique was used in he PSFE and 3DFE analyses o526
model unnelling, in which he displacemen s o he nodes lo-527
ca ed a ound he unnel con e ge owa ds a single poin on he528
unnel e ical line o symme y, gene ically loca ed be ween529
he unnel axis and he in e . The poin o con e gence o dis-530
placemen s is close o he unnel cen e o deep unnels, mo -531
ing owa ds he in e o shallowe unnels. In pa icula , he532
back-analysis by Cheng e al. (2007) sugges ed a linea a ia-533
ion o he ocus poin wi h he co e - o-diame e a io; o he534
G os eno case s udy his ga e a ocus dep h a 60% he unnel535
adius below he unnel axis (z=z +0.6R).536
The b idge supe s uc u e was modelled using Mindlin pla e537
elemen s, wi h ei he ixed o hinged ends (Fig. 2). Thei bend-538
ing and axial s i ness pe uni longi udinal leng h we e ob ained539
by di iding he ac ual 3D alues o he s i ness EA and EI by540
he calib a ed leng h o he b idge ounda ion in he ou o plane541
di ec ion L=m L0, and hen he con ibu ion om he en sep-542
a a e b idge ames we e summed (PEI/Land PEA/L). The543
Young’s modulus o s eel Ewas aken as 200 GPa. The shea 544
s i ness GAswas gi en by he equi alen ec angula c oss-545
sec ion used in PLAXIS and G=2(1+ν)E(Ben ley,2020a). As546
he en ac ual b idge ames would beha e in plane-s ess con-547
di ion, i was necessa y o p e en he Poisson’s a io om s i -548
ening he plane-s ain model by se ing ν=0 (Ben ley,2020b).549
The oo ings (i.e. he po ions o he pie below g ound su ace)550
we e modelled by iangula elemen s (as was he soil) wi h he551
Young’s modulus o conc e e and mason y aken as 20 GPa.552
To make he compa ison wi h he PS ini e elemen analysis553
mo e s aigh o wa d, he ac ual ounda ion wid h was adop ed554
8
(B=B0) in Me hod A. To scale he oo ing leng h L om he ac-555
ual leng h L0, he h ee deg ees o eedom, namely, displace-556
men ualong x, displacemen walong zand o a ion abou y,557
we e indi idually conside ed. A 3D model o he ac ual ec -558
angula oo ing was i s cons uc ed using PLAXIS 3D. Uni 559
o ces (o momen ) we e applied o he oo ing in u n and he560
esul ing displacemen s we e compu ed. Nex , a PS (plane-561
s ain) model o he same oo ing wi h he ac ual wid h B0was562
cons uc ed using PLAXIS 2D. Fo each deg ee o eedom,563
he same uni o ces as in he 3D case, di ided by a ounda ion564
leng h L o ob ain he loading pe uni leng h, we e applied o565
he PS model (Fig. 8). The calib a ed ounda ion leng hs L ha 566
ga e iden ical displacemen s in he PS and 3D models o each567
deg ee o eedom we e ob ained (Table 2). The combina ions568
o Band Lwhich ga e iden ical displacemen s be ween PS and569
3D solu ions a e illus a ed in Fig. 9, in which he shaded a -570
eas e e o a de ia ion wi hin 10%. A ounda ion leng h o 571
L=1.32 L0(L0=60 m) was adop ed. Wi h his geome y,572
he PSFE analysis should gi e he same ho izon al oo ing dis-573
placemen as a ull 3D model while he e ical and o a ional574
mo emen s we e wi hin 6% o he co esponding 3D esul s.575
3D analysis 2D plane-s ain analysis
B0
L0
Fx
Fz
MyFx/ L
Fz/ L
My/ L
B
Displacemen s
u3d, w3d, y,3d
Displacemen s
u2d, w2d, y,2d
Thickness o comp essible
elas ic laye H
Figu e 8. App oach o es ablish equi alence be ween PS and 3D ounda ions
0.8 0.9 1 1.1 1.2 1.3 1.4
B/B0
0.6
0.8
1
1.2
1.4
1.6
L/L0
y,2d = y,3d
u2d = u3d
w2d = w3d
E o < 10%
Adop ed ( L = 1.32 L0, B = B0)
Figu e 9. Combina ions o Land Bgi ing equi alen plane-s ain and 3D
ini e elemen solu ions
4. Resul s o he G os eno B idge case s udy576
4.1. G een ield mo emen s577
Be o e add essing he in e ac ion p oblem, g een ield mo e-578
men p edic ions o he G os eno B idge a e discussed.579
Table 2. Calib a ed leng h Lo oo ing o scaling supe s uc u e s i ness in
PSFE analysis
Deg ee o eedom Calib a ed oo ing leng h L†
Ho izon al displacemen u80.38 m
Se lemen w71.07 m
Ro a ion y72.86 m
† Ac ual oo ing leng h L0=60 m
Fig. 10 compa es g een ield g ound mo emen s along he ho i-580
zon a he base o he ounda ion (app oxima ely 4 m dep h),581
ob ained om he PSFE and 3DFE models in g een ield condi-582
ions (wi hou he b idge), wi h hose ob ained om he empi -583
ical me hod o Wong e al. (2025) desc ibed ea lie . All p edic-584
ions assume a unnelling olume loss o 1%.585
FE p edic ions o he g een ield se lemen and he g adien 586
o he se lemen ough a e in excellen ag eemen wi h he em-587
pi ical alues a he Sou h and Cen e Pie s o he G os eno 588
B idge — he wo pie s loca ed wi hin he unnel’s in luence589
zone. This ag eemen is impo an , as hese mo emen s a e he590
main d i e s o soil-s uc u e in e ac ion in his case (as dis-591
cussed la e ).592
-50 0 50
x (m)
-5
0
5
10
15
20
Se lemen w (mm)
Sou h Pie Cen e Pie
(a)
-50 0 50
x (m)
-5
0
5
Ho izon al mo emen u (mm)
Sou h Pie Cen e Pie
PSFE
3DFE
Empi ical
(b)
Figu e 10. G een ield mo emen s a ounda ion le el o unnelling olume
loss o 1%: (a) se lemen and (b) ans e se ho izon al mo emen
Howe e , his igu e also shows ha he ex en o he ho izon-593
al displacemen p o ile om he FE analyses is wide han ha 594
o he empi ical cu e, which a ec ed he magni ude o ho i-595
zon al mo emen s a he Sou h and Cen e Pie s, despi e simila 596
peak alues. The g een ield ho izon al mo emen s om he FE597
analyses a e la ge han om he empi ical p edic ions by abou 598
35% and 70% a he loca ions o he Sou h and Cen e Pie s, e-599
spec i ely. The FE ho izon al displacemen p o ile is wide and600
educes mo e g adually away om he unnel axis compa ed o601
he empi ical p edic ion. These di e ences could be ele an 602
9
o 3D and ansien condi ions, o e s a powe ul and e icien 973
amewo k o e alua ing he e ol ing esponse o b idges du -974
ing TBM ad ancemen . Such analyses a e aluable o bo h975
design and cons uc ion moni o ing, as hey in o m he in e -976
p e a ion o eal- ime da a and help iden i y in e media e s ages977
o inc eased ulne abili y ha may be los in a s eady-s a e anal-978
ysis.979
7. Guideline o u u e isk assessmen s o b idges on shal-980
low ounda ions981
TSAM has he ad an age o enabling quick sensi i i y s ud-982
ies o he g een ield displacemen ield, s uc u al con igu a-983
ions (and nonlinea i ies) and he esul ing SSI ha can mod-984
i y he ounda ion mo emen dis ibu ion. In addi ion o his985
e sa ili y and he possibili y o pe o m ei he s eady-s a e o 986
ansien analysis o plane- ame o h ee-dimensional b idges,987
he compa ison wi h coupled ini e elemen analysis has demon-988
s a ed a simila p edic i e capabili y o TSAM wi h i s in in-989
sic 3D na u e which bypasses he app oxima ions equi ed by990
equi alen plane-s ain models. The e o e, TSAM is always991
ecommended in place o uncoupled g een ield analysis (im-992
posing g een ield displacemen s di ec ly o ounda ions).993
A he p elimina y phases o an assessmen , due o he lack994
o a gene alised me hod o es ima e ela i e b idge- ounda ion-995
soil s i ness, in e ac ion models a e almos always needed o 996
b idge assessmen in he p esence o unnelling, as opposed997
o building assessmen s o which ecognised empi ical ap-998
p oaches a e a ailable. On he one hand, o ela i ely s i 999
b idges, in e ac ion models able o accoun o SSI e ec s on1000
ounda ion mo emen s a e necessa y. E en o ela i ely lex-1001
ible b idges (e.g. G os eno B idge) which con o m closely o1002
g een ield mo emen s, i is di icul o assess whe he SSI is1003
negligible a p io i. In his con ex , TSAM is a sui able ool o1004
assess he ele ance o SSI, and decide how o p oceed u he 1005
in he design p ocess.1006
Rega ding he complexi y o he lumped-sp ing models o 1007
he ounda ion, e en he simples in e ac ion me hod wi h Win-1008
kle decoupled (DC) sp ings can p oduce ealis ic esul s o 1009
s uc u al mo emen s and in e nal o ces. Such decoupled1010
sp ings can be eadily implemen ed in mos comme cial s uc-1011
u al compu e p og ams o a TSAM. Unce ain ies ega d-1012
ing Winkle sp ing cons an s can be add essed by a sensi i -1013
i y s udy, conside ing a ailable analy ical and empi ical exp es-1014
sions o accoun o embedmen , shape, as well as he bed ock1015
dep h- o- oo ing wid h (e.g. Gaze as,1991a;Pais and Kausel,1016
1988). The e o e, TSAM based on Winkle g ound models is1017
sugges ed, as a i s s ep, o assess he le el o ela i e soil-1018
s uc u e s i ness and he expec ed de o ma ion mechanism o 1019
he b idge, which is ele an o he design o moni o ing wo k1020
and he expec ed se iceabili y/ul ima e s a es o be analysed in1021
de ail.1022
The obse a ion ha he s eel G os eno b idge was ela-1023
i ely lexible wi h espec o he unde lying s i soil should1024
no be ex apola ed o gene alised o all s eel a ch b idges. All1025
he in e ac ion analyses in his s udy inco po a ed olle con-1026
nec ions (be ween deck and pie s) as well as ic ionless hinges1027
(be ween a ches and pie s) ha educed he supe s uc u e eac-1028
ion o ces ac ing on he pie s. Wong (2019) demons a ed ha 1029
he b idge esponse o unnelling would become signi ican ly1030
s i e i he in e nal hinges we e emo ed and, hus, SSI is sen-1031
si i e o he deg ee o ixi y o in e nal connec ions in s eel a ch1032
ames. S uc u al de ails and in e nal connec ions can be ead-1033
ily included in TSAM.1034
The inal ecommenda ion is o e alua e i he in e ac ion1035
p oblem can be conside ed in an elas ic egime; o his, we1036
ha e sugges ed: (i) es ima ing he ex en o he plas ic zone1037
and he linea i y o he g een ield soil mo emen s wi h VL(us-1038
ing a g een ield ini e-elemen analysis o unnelling), along1039
wi h (ii) an assessmen o unnelling-induced mobilised shea 1040
s eng h a he ounda ion (using TSAM).1041
Fig. 18 summa ises he geo echnical and s uc u al design1042
p ocesses, including TSAM applicabili y checks, in a low1043
cha .1044
S i beha iou
Re ine s uc u al model
wi h ealis ic a ches,
beams, deck and
spand els (possibly in
3D)
TSAM (e.g. decoupled
sp ings) wi h main s uc u al
componen s (e.g. pie s and
a ches only)
Plane ame model o b idge
supe s uc u e
Take g een ield ounda ion
mo emen s and assess
s uc u al esponse
conse a i ely
DC analysis ma ches
uncoupled GF closely?
No Yes
Calcula e b idge
displacemen s and
in e nal o ces
Re-pe o m TSAM wi h
coupled o decoupled
sp ings
TSAM no applicable
Yes No
Check mobilised
s esses a ounda ions
in TSAM esul s
End
PSFE wi h calib a ed
oo ing leng h 3dFE
Calcula e b idge
displacemen s and
in e nal o ces
TSAM
Flexible beha iou
S uc u al
Geo echnical
G een ield check using PSFE analyses
(i) plas ic zone a ound unnel
(ii) linea i y o GF mo emen s a
ounda ion oo p in
Figu e 18. P ocess o unnelling impac assessmen o b idges on shallow
ounda ions
16

8. Conclusions1045
This pape p esen s a me hodology o analysing he in e ac-1046
ion be ween a bo ed unnel and an exis ing b idge on shallow1047
ounda ions: he wo-s age analysis me hod (TSAM), which de-1048
couples he geo echnical and s uc u al aspec s o he p oblem1049
bu e ains hei in e ac ion h ough sp ing-based ounda ions.1050
Fi s , his wo k builds on insigh s om he Thames Tideway1051
Tunnel p ojec and examines he G os eno B idge and Pu ney1052
B idge as case s udies. The p edic ions o wo-s age modelling1053
o a ying le els o complexi y ( ully coupled, locally cou-1054
pled, and decoupled sp ings) we e benchma ked agains ield1055
da a and mo e igo ous coupled plane-s ain (PSFE) and h ee-1056
dimensional ini e elemen (3DFE) models. Resul s illus a e1057
how s uc u al s i ness, soil esponse, and in e ac ion mod-1058
elling assump ions in luence he p edic ion o b idge displace-1059
men s du ing unnelling. The main conclusions a e gi en be-1060
low.1061
•The TSAM amewo k o e s a p ac ical balance be ween1062
simplici y and accu acy, making i sui able o p elimina y1063
assessmen s o unnelling impac s on exis ing b idges. A1064
compa ison o TSAM and FE models demons a es ha 1065
sp ing-based wo-s age me hod can p o ide esul s ha a e1066
consis en wi h hose o mo e ad anced coupled FE anal-1067
ysis, cap u ing essen ial aspec s o he s uc u al and SSI1068
beha iou s, wi h minimal equi emen s o calib a ion o 1069
he sp ing cons an s, gi en eadily a ailable exp essions1070
om he hal -space heo y. Also, TSAM can be ex ended1071
o cap u e 3D e ec s du ing unnel ad ancemen , a oiding1072
he compu a ional demands o 3D FE analysis o he ap-1073
p oxima ions equi ed by equi alen plane-s ain FE mod-1074
els. TSAM esul s closely ma ched moni o ing da a in1075
bo h s eady-s a e and ansien condi ions in he G os eno 1076
case s udy.1077
•Resul s ha e con i med he need o mo e beyond g een-1078
ield assump ions and accoun o in e ac ion e ec s, es-1079
pecially when dealing wi h s i b idge s uc u es. Fo 1080
ela i ely lexible b idges, such as he G os eno B idge,1081
he in e ac ion esponse closely ollows g een ield pa -1082
e ns due o he limi ed s uc u al es ain and in e nal1083
a icula ion. In such cases, uncoupled g een ield analy-1084
ses may yield simila de o med shapes o coupled mod-1085
els, al hough TSAM s ill p o ides a be e es ima e o in-1086
e nal o ces and mobilised capaci ies. Fo ela i ely s i 1087
b idges, such as he Pu ney B idge, g een ield assump ions1088
signi ican ly o e -p edic in e nal ac ions and mis ep esen 1089
he de o ma ion mechanism. TSAM esul s ha e showed1090
ha SSI can change bo h he magni ude and di ec ion o 1091
pie mo emen s. This has implica ions o se iceabili y,1092
c ack con ol, and condi ion moni o ing s a egy.1093
•E en simpli ied TSAMs using decoupled (Winkle )1094
sp ings, such as hose o mula ed by Gaze as (1991a), lead1095
o ealis ic p edic ions o s uc u al displacemen s and in-1096
e nal o ces, making hem sui able o p elimina y assess-1097
men s. These models a e easy o implemen in comme cial1098
so wa e and p oduce adequa e i s es ima es o he e ec s1099
o SSI.1100
•The applicabili y o linea TSAM mus be checked by1101
e alua ing plas ici y and mobilised shea s eng h benea h1102
ounda ions. Fo he G os eno B idge case, es ima es1103
based on TSAM and PSFE indica ed ha he elas ic ange1104
emains alid o unnel olume losses up o 2–3%, which1105
is mo e han adequa e o ypical u ban unnelling p ojec s1106
in s i clays. P ac ical checks combining a g een ield FE1107
analysis and he compa ison o shea s eng h wi h es i-1108
ma es om TSAM we e ecommended.1109
•The equi alence be ween plane-s ain and h ee-1110
dimensional coupled ini e elemen models o s eady-s a e1111
assessmen was in es iga ed, and a calib a ion p ocedu e1112
o de ining he equi alen ounda ion wid h in plane-1113
s ain analysis was p oposed. As pe ec equi alence1114
is no achie able, we sugges ed ha he ounda ion1115
wid h should be selec ed based on he dominan mode1116
o s uc u al esponse, using enginee ing judgemen .1117
Fo he G os eno B idge case s udy, he ounda ion1118
wid h was calib a ed o ma ch ho izon al s i ness, a he1119
cos o o e -es ima ing he o a ional mo emen s. This1120
likely explains he highe in e nal o ces and mo emen s1121
p edic ed by he plane-s ain FE model compa ed o he1122
h ee-dimensional FE analysis.1123
TSAM o e s a eliable and e icien amewo k o e alua -1124
ing unnel-soil-b idge in e ac ions. I enables sensi i i y s ud-1125
ies on g een ield displacemen ields, ounda ion s i ness, and1126
supe s uc u e con igu a ions, and i suppo s bo h s eady-s a e1127
and ansien assessmen s. I is sugges ed as a pa hway om1128
g een ield displacemen es ima ion o in eg a ed s uc u al anal-1129
ysis, p o iding a use ul ool o geo echnical and s uc u al de-1130
signe s o make a ional isk assessmen s and plan moni o ing1131
wo ks.1132
Fu u e wo k in his ield could ex end TSAM amewo ks o1133
inco po a e nonlinea b idge beha iou and p obabilis ic inpu 1134
pa ame e s o isk-in o med assessmen o co e he unce ain-1135
ies in he supe s uc u e and geo echnical domains.1136
Acknowledgemen s1137
The i s au ho ’s pos g adua e s udies o ming he basis o his1138
pape was pa ially suppo ed by he Geo echnical Enginee ing1139
O ice, Ci il Enginee ing and De elopmen Depa men o he1140
Hong Kong S.A.R. Go e nmen and he Hong Kong Ins i u ion1141
o Enginee s A hu & Louise May Memo ial Schola ship.1142
The au ho s would like o acknowledge Fe o ial Ag oman1143
and Laing O’Rou ke JV o p o iding hem wi h access o he1144
moni o ing da a.1145
CRediT au ho s a emen 1146
E.K.L. Wong: Concep ualiza ion, Me hodology, So wa e,1147
Valida ion, Fo mal analysis, In es iga ion, Da a Cu a ion, W i -1148
ing – O iginal D a , W i ing – Re iew & Edi ing, Visualiza-1149
ion. A. F anza: Concep ualiza ion, Me hodology, So wa e,1150
17
Valida ion, W i ing – O iginal D a , W i ing – Re iew & Edi -1151
ing, Supe ision. P. Hewi : In es iga ion, Resou ces, Da a1152
Cu a ion. G.M.B. Viggiani: Concep ualiza ion, Me hodology,1153
Resou ces, W i ing – Re iew & Edi ing, Supe ision, P ojec 1154
adminis a ion.1155
Appendix A: Fo mula ion o soil s i ness ma ix o ully-1156
coupled (FC) sp ings solu ion1157
This appendix desc ibes he p ocedu e ollowed in assem-1158
bling he soil s i ness ma ix Ksoil. Elas ic solu ions such as1159
Lo e (1927) and Cheung and Nag (1968) gi e he displace-1160
men s in a homogeneous elas ic hal -space unde he applica-1161
ion o o ces on he su ace. The solu ions a e o he o m1162
Fq =u(16)
whe e Fis he soil lexibili y ma ix, qis he ec o o applied1163
e ical and ho izon al o ces and uis he ec o o ho izon al1164
and e ical displacemen s on he su ace o he hal -space.1165
Lo e (1927) gi es he displacemen s a a poin (x,y) on he1166
su ace o he hal -space unde a concen a ed ho izon al o ce1167
H0as1168
ux=H0
2πG(1−ν
+νx2
3) (17a)
uy=νH0
2πG(xy
3) (17b)
uz=(1 −2ν)H0
4πG(x
2) (17c)
and he displacemen s unde a concen a ed e ical o ce V0as
ux=−(1 −2ν)V0
4πG
x
2(18a)
uy=−(1 −2ν)V0
4πG
y
2(18b)
uz=(1 −ν)V0
2πG (18c)
whe e is he plan dis ance be ween he poin o o ce ap-1169
plica ion and he poin a which displacemen s a e e alua ed1170
(Fig. 19a). The exp essions in Eq. (17a) o (18c) a e used o1171
ob ain he o -diagonal e ms o he soil lexibili y ma ix Fbu 1172
a e unde ined a he poin o o ce applica ion ( =0).1173
The diagonal e ms o Fmay be de e mined by in eg a ing1174
he o ces o e a ec angula a ea and e alua ing he o al dis-1175
placemen s acco dingly. Cheung and Nag (1968) gi e he dis-1176
placemen s benea h a ec angula a ea o dimensions a×bunde 1177
applied ho izon al and e ical o ces H0and V0as1178
ux=2H0(1 −ν2)
aπE[Csinh−11
C+sinh−1C]+2H0ν(1 +ν)
aπE(sinh−1C)
(19a)
uz=2V0(1 −ν2)
aπE[Csinh−11
C+sinh−1C] (19b)
whe e C=a/band Eis he Young’s modulus o he elas ic1179
hal -space (Fig. 19b). This gi es he diagonal e ms and com-1180
ple es F.1181
The soil s i ness ma ix Ksis he e o e gi en by he in e se1182
o he soil lexibili y ma ix:1183
Ks=F−1(20)
Fo a s uc u e consis ing o mul iple indi idual oo ings1184
which a e disc e ised in o an a ay o nnodes, ollowing he1185
p ocedu e ou lined abo e gene a es a 3n×3nsoil s i ness ma-1186
ix Ks. This enables he coupling be ween di e en oo ings1187
ac oss he su ace o he soil.1188
x
y
z
V
H
(x,y)
(a)
x
y
z
V
H
ab
x= y= 0
Rec angula a ea o
load applica ion
(b)
Figu e 19. Displacemen s on su ace o homogeneous elas ic hal -space: (a)
displacemen s a loca ions o he han poin o o ce applica ion and (b)
displacemen s a poin o o ce applica ion
18
Appendix B: Fo mula ion o soil s i ness ma ix o decou-1189
pled (DC) o locally-coupled (LC2) sp ings solu ion1190
This appendix desc ibes he o mula ion o s a ic ounda-1191
ion sp ing cons an s ollowing Gaze as (1991a) and Gaze as1192
(1991b). Fo a igid oo ing on he su ace o an in ini e elas ic1193
hal -space, he ele an sp ing cons an s o he h ee ele an 1194
deg ees o eedom o a 2D p oblem is gi en by Eq. (i)–(iii) in1195
Table 5. The sp ings a e unc ions o he plan geome y o he1196
oo ing and soil p ope ies.1197
Table 5. S a ic sp ing s i ness o igid ounda ion on su ace o elas ic
con inuum Gaze as (1991a) and Gaze as (1991b)
Mode S i ness (Load pe uni Dep h o zone
displacemen ) o in luence
H→ ∞
Ve ical (z)Kz=[GL/(1–ν)](0.73 (i) 2.5–7.5 B
+1.5χ0.75) wi h χ=A/(2L)2
Ho izon al (x)Kx=[GL/(2–ν)] (ii) 1–3 B
(2 +2.5χ0.85)
Ro a ional K y =[G/(1–ν)]I0.75
y(L/B)0.25 (iii) B
(abou y) (2.4+0.5B/L)
H,∞
Ve ical (z)Kz=GL
1−ν[0.73 +1.54( B
L)0.75] (i )
(1 +B/H
1+2B/L)
See Fig. 20 o no a ions.
L
PLAN SECTION
x
y
x
z
G,
n
B
B
A ea A
L> B
H
^^^
Bed ock
Figu e 20. Founda ion on su ace o elas ic con inuum
The p esence o a igid bounda y a ec s he sp ing s i ness1198
in di e en ways o di e en modes o mo emen . The e ec 1199
o he igid bounda y may be assessed by conside ing he zone1200
o in luence. Gaze as (1991b) gi es he app oxima e ange o 1201
zone o in luence o di e en modes o mo emen s. Fo ex-1202
ample, o a e ical applied load, he zone o in luence anges1203
om 2.5B o a ec angula oo ing o 7.5B o a s ip oo ing.1204
I is no able ha he e ec o a igid bounda y has he g ea es 1205
e ec on he e ical s i ness. I inc eases he e ical s i ness1206
by a ac o o (1 +B/H
1+2B/L). Fo ho izon al and o a ional mo e-1207
men s, he e ec o he igid bo om bounda y may easonably1208
be igno ed i he dep h o he igid bounda y is g ea e han he1209
zone o in luence. The ounda ion s i ness ma ix is he e o e1210
w i en as:1211
K =
Kx0 0
0Kz0
0 0 K y

(21)
The sp ing s i nesses o a oo ing embedded in an in ini e1212
elas ic hal -space a e gi en by Eq. ( )–( ii) in Table 6. Em-1213
bedmen subs an ially inc eases he sp ing cons an s compa ed1214
wi h he case wi hou embedmen , in pa icula he o a ional1215
s i ness.1216
Gaze as (1991a) also gi es an addi ional e m o he cou-1217
pling be ween ansla ion and o a ion o an embedded oo ing.1218
Hence he ounda ion s i ness ma ix o an embedded oo ing1219
is w i en as:1220
K =
Kx,emb 01
3Kx,emb d
0Kz,emb 0
1
3Kx,emb d0K y,emb

(22)
Table 6. S a ic sp ing s i ness o igid ounda ion embedded in elas ic
con inuum a e Gaze as (1991a)
Mode S i ness (Load pe uni displacemen )
Ve ical (z)Kz,emb =Kz[1 +(2/21)(D/B)(1 +1.3χ)] ( )
[1 +0.2(Aw/A)2/3]
Ho izon al (x)Kx,emb =Kx[1 +0.21(D/B)0.5] ( i)
{1+0.69[(h/B)(Aw/L2)]0.4}
Ro a ional K y,emb =K y{1+2.52(d/B) ( ii)
(abou y) [1 +(2d/B)(d/D)−0.2(B/L)0.5]}
See Fig. 21 o no a ions.
L
PLAN SECTION
x
y
x
z
G,
n
B
B
A ea A
L> B
Dd
Side wall con ac
a ea Aw
h= D–d/2
Figu e 21. Founda ion embedded in elas ic con inuum
19
Re e ences1221
Ben ley, 2020a. Ma e ial Models - PLAXIS 2D CONNECT Edi ion V20.1222
PLAXIS manuals , 218–221.1223
Ben ley, 2020b. Re e ence Manual - PLAXIS 2D CONNECT Edi ion V20.1224
PLAXIS manuals , 250–251.1225
Boldini, D., Losacco, N., F anza, A., DeJong, M.J., Xu, J., Ma shall, A.M.,1226
2021. Tunneling-Induced De o ma ion o Ba e F ame S uc u es on Sand:1227
Nume ical S udy o Building De o ma ions. Jou nal o Geo echnical and1228
Geoen i onmen al Enginee ing 147, 04021116. doi:10.1061/(ASCE)GT.1229
1943-5606.0002627.1230
Callis o, L., Rampello, S., Viggiani, G.M.B., 2013. Soil–s uc u e in e ac ion1231
o he seismic design o he Messina S ai B idge. Soil Dynamics and1232
Ea hquake Enginee ing 52, 103–115. doi:10.1016/j.soildyn.2013.1233
05.005.1234
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