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

An assessment methodology for tunnelling beneath bridges considering soil–structure interaction Eugene K. L. Wonga, Andrea Franzab,∗, Peter Hewittc, Giulia M. B. Viggianid aDevelopment Bureau, Government of Hong Kong S.A.R., China. Formerly Department of Engineering, University of Cambridge, UK bDepartment of Civil and Architectural Engineering, Aarhus University, Denmark cGeo-environmental Engineering, Laing O’Rourke, UK dDepartment of Engineering, University of Cambridge, UK Abstract The construction of the Tideway Tunnel in London induced ground movements that affected several arch bridges along the River Thames. This study examines the responses of the Grosvenor Bridge and Putney Bridge to tunnelling-induced displacements, with a focus on soil-structure interaction (SSI) and on the relationship between foundation displacements and greenfield predictions. Coupled finite element (FE) analyses and two-stage analysis methods (TSAM) are compared, supported by some field data. The bridges are modelled as elastic frames. The equivalence between plane-strain and three-dimensional FE models is investigated, while the effectiveness of TSAM using spring-based soil-foundation models (derived from half-space theory) of varying complexity is evaluated. From the comparison with monitoring data and FE results, TSAM is shown to be practical and reliable for assessing tunnel-induced effects on bridges, including during transient conditions such as TBM approach. Results highlight the contrasting SSI response of flexible (steel) and stiff(masonry) arch bridges: while using full greenfield movements or scaling them down by some reduction factor may reasonably inform the response of flexible structures, the behaviour of stiffer systems will be markedly different as SSI significantly alters deformation mechanisms. Since the bridge response depends on the relative bridge-foundationsoil stiffness, it is recommended in all cases to perform interaction analyses from the early design stages. A workflow using TSAM is proposed to guide early-stage design assessments, integrating salient structural and geotechnical considerations. Keywords: tunnelling, ground movements, bridge, soil-structure interaction, elastic solutions Highlights1 •Tunnel-soil-bridge interaction modelling: two case studies2 •Two-stage analysis method (TSAM) benchmarked against3 3D and plane-strain FE models4 •Approach for reducing 3D structures into plane-strain5 model and its limitations6 •Bridge stiffness critically influences deformation and in-7 ternal force distribution8 •Practical workflow proposed for early-stage soil-structure9 interaction (SSI) assessment using TSAM10 ∗Corresponding author Email address: [email protected] (Andrea Franza) Preprint submitted to Tunnelling and Underground Space Technology October 30, 2025 1. Introduction11 In urban areas, bored tunnelling is a common solution to meet12 growing demand for transportation and utility infrastructure.13 Assessing the impact of tunnelling-induced ground movements14 remains a critical challenge for engineers and asset owners. Al-15 though case studies have highlighted the impact of excavation-16 induced ground movements on piles and some bridges with17 deep foundations (e.g. Cheng et al.,2007;Yoo,2013;Yoo and18 Abbas,2021;Yang et al.,2025), there is no well-established19 methodology for analysing tunnel-soil-bridge interaction in the20 case of shallow foundations. To address this shortcoming, this21 paper builds upon the experience from the Thames Tideway22 project. Commencing operation in 2025, the project involved a23 25 km–long large-diameter tunnel (8.84 m cutting diameter) ex-24 cavated under the River Thames to intercept, store, and convey25 sewage in order to upgrade London’s sewer system. This tunnel26 passed beneath several bridges, most of which are arch bridges,27 including the Putney Bridge (a heritage masonry structure) and28 Blackfriars Bridge (a modern steel-concrete structure).29 The Planning Inspectorate publicly released the technical re-30 ports compiled to assess the effects of Thames Tideway tun-31 nelling on bridges, which provided a useful reflection of cur-32 rent engineering practices. Typically, as a first step, uncoupled33 analyses are performed in which greenfield tunnelling move-34 ments, whether determined from empirical formulae or numer-35 ical analyses, are imposed on bridge foundations; the result-36 ing structural distortions are then calculated (e.g. Thames Wa-37 ter Utilities Limited,2013b,c). Where soil-structure interac-38 tion (SSI) effects are considered, plane-strain (PS) numerical39 analyses incorporating selected structural elements are carried40 out by geotechnical engineers; the computed foundation dis-41 placements from these PS models are then passed to structural42 engineers, who apply them as base boundary conditions in de-43 tailed three-dimensional (3D) superstructure models, to analyse44 induced internal forces in structural members (e.g. Thames Wa-45 ter Utilities Limited,2012,2013d). This change of boundary46 conditions between twoand three-dimensional models is in-47 evitably subject to engineering judgement and approximations.48 Impact assessments often assume that bridge foundations fol-49 low greenfield displacement profiles, neglecting the influence50 of existing structures. This disregards the effects of SSI and51 overlooks the ability of the superstructure stiffness in modifying52 the foundation response. While straightforward, this approach53 can result in overly conservative or inaccurate estimates, as ex-54 plored in this study.55 The reliance on greenfield methods in bridge assessments de-56 rives directly from the framework developed for buildings (e.g.57 Mair et al.,1996;Crossrail,2008;Schoor et al.,2021). In58 the case of buildings, well-established procedures link defor-59 mation parameters from greenfield displacements at foundation60 level to damage categories, supported by extensive literature61 (e.g. Franzius et al.,2006;Dimmock and Mair,2008;Farrell62 et al.,2014;Giardina et al.,2015;Boldini et al.,2021). How-63 ever, tunnel-bridge interaction remains less well-studied, possi-64 bly because the diversity of structural forms and configurations65 defies generalisation. Franza and DeJong (2019) conceptually66 evaluated the effects of tunnelling-induced movements on sim-67 ple bridges on separate footings for orthogonal arrangements68 of piers and decks (modelled as beam elements), using elastic69 soil-frame interaction analyses. Their results highlighted that70 the qualitative distribution of tunnelling-induced movements of71 the foundation is modified compared with greenfield shapes as72 a result of frame action, which is not readily accounted for73 by a modification factor. A similar observation was made for74 the Grosvenor Bridge in the study by Faherty et al. (2022).75 Their parametric study of steel arch bridges highlighted the76 challenges of generalising risk assessment protocols; bridge re-77 sponses to ground movements are highly dependent on struc-78 tural factors such as roller bearing travel, internal partial hinges79 and expansion joints — details that are often difficult to cap-80 ture in geotechnical models. This intrinsic structural complex-81 ity also supports the suggestion that damage criteria for bridges82 subjected to foundation movements are not well established.83 For instance, the Federal Highway Administration (1985) pro-84 posed some thresholds for the tolerable longitudinal angular85 distortion (ratio of differential settlement to span length) for86 bridges of different materials and construction; however, this87 may not adequately account for excavation-induced horizontal88 movements. Furthermore, it is not straightforward to corre-89 late structural damage with computed internal forces. Conse-90 quently, structural engineers frequently resort to detailed nu-91 merical analysis to assess the structural response of existing92 bridges to tunnel construction, which are informed by the ex-93 pected magnitude and shape of tunnelling-induced foundation94 movements provided by geotechnical engineers, which may or95 may not consider SSI. The research questions of what are ade-96 quate modelling strategies for risk assessments and their poten-97 tial pitfalls arise.98 This paper compares different tunnel-soil-structure interac-99 tion approaches, evaluating their strengths and limitations for100 varying levels of complexity of the soil-foundation system.101 First, the Grosvenor Bridge and the Putney Bridge case stud-102 ies are presented along with an overview of the modelling103 approaches used. Second, the measured movements of the104 Grosvenor Bridge from the Tideway Tunnel construction are105 examined, building on Wong (2019) and Faherty et al. (2022);106 the short-term, in-plane, steady-state bridge deformations are107 considered. Predictions from plane-strain finite element (PSFE)108 analyses using a simple soil constitutive model in total stress109 are presented; a procedure to define an equivalent plane-strain110 foundation to capture the flexibility associated with the actual111 three-dimensional geometry is proposed. Third, the study com-112 pares field and numerical FE results with elastic SSI models113 following the two-stage analysis method (TSAM) using dif-114 ferent representations based on lumped springs: fully-coupled115 (half-space theory), locally-coupled (spatially decoupled with116 local roto-translational coupling), and decoupled (spatially de-117 coupled without roto-translational coupling). Fourth, a TSAM118 analysis of the response of the Putney Bridge to tunnelling is119 presented; results highlight the requirements, versatility, and120 advantages of spring-based modelling approach for a relatively121 stiffbridge in which SSI effects are expected to be significant.122 Finally, a discussion on how spring-based two-stage modelling123 2 could be used for studying the effects of transient ground move-124 ments in 3D by using a 3D TSAM model of the Grosvenor125 Bridge subjected to an advancing tunnel excavation is provided;126 the results are compared with field observations. Guidance is127 also given on how to carry out an integrated analysis that ac-128 counts for both tunnelling and soil-foundation interaction.129 2. Case studies130 2.1. Grosvenor Bridge131 The Grosvenor Bridge is located in central London. It is a132 steel railway arch bridge crossing the River Thames between133 Pimlico and Battersea, near Victoria Station. It supports railway134 tracks that carry both passenger and freight traffic. The bridge135 is supported by masonry piers embedded in London Clay. The136 Thames Tideway Tunnel passed under the Grosvenor Bridge at137 a position between its South and Centre Piers (Fig. 1).138 The bridge was originally constructed in the second half of139 the 1800s and it underwent several phases of widening, modifi-140 cation, and partial reconstruction between 1865 and 1967 (Wil-141 son,1868;Fox,1868;Kerensky and Partridge,1967;Keren-142 sky et al.,1967). The present structure consists of four spans,143 each ranging from 55 to 57 m, with a 50 m wide deck sup-144 ported by ten parallel, structurally separate but identical, steel145 arch frames. Each frame is made of two arch ribs with rectan-146 gular hollow cross-sections. The arches support the deck via147 spandrel posts. The deck consists of 15.9 mm thick mild steel148 plates, supported by inverted Universal-T sections. The arches149 are pin-connected at each pier and connected monolithically150 with the deck at mid-span. The deck is simply supported at151 the piers. The piers are made of masonry encased in reinforced152 concrete and are embedded in London Clay. Fig. 2shows the153 mechanical model of the bridge as a plane frame and indicates154 the positions of the targets used to monitor the displacements of155 the bridge during excavation of the tunnel.156 The westbound earth pressure balance tunnel boring machine157 (EPB TBM) with an external diameter of 8.84 m was launched158 from a shaft on Kirtling Street in January 2019. It passed un-159 derneath the Grosvenor Bridge over five days in March 2019.160 Monitoring data from the Thames Tideway Tunnel project, in-161 cluding settlement, horizontal movements and displacements of162 the bridge deck provide a reference to validate model predic-163 tions.164 The tunnel was excavated at a depth of approximately 34 m165 below the riverbed, primarily through London Clay, but its in-166 vert was close to or within the Lambeth Group (Thames Wa-167 ter Utilities Limited,2013a,d). At this location, the Lam-168 beth Group comprises Lower Shelly Clays and Upper Mot-169 tled Clays of the Woolwich and Reading Formations. Near170 Grosvenor Bridge, the geotechnical investigations by Skemp-171 ton and Henkel (1957) and Kerensky et al. (1967) provided the172 undrained shear strength profile with depth below the London173 Clay horizon, as shown in Fig. 3.174 At the foundation base level of the Grosvenor Bridge, ap-175 proximately 4 m into the London Clay, the undrained shear176 strength is in the range 100–200 kPa. At the small strain lev-177 els relevant for settlement predictions, the undrained Young’s178 Deck Tideway Tunnel ELEVATION PLAN South Pier Centre Pier North Pier Arch rib x z 57 m z0=34 m y x B0= 13.7 m L= 60 m (a) (b) 27.4 m Drive Riverbed Downstream Figure 1. Layout of Grosvenor Bridge showing (a) general view and (b) relative positions of bridge and tunnel Deck discontinuous and simply supported at piers (roller supports) Fixed deck/arch connection at midspan Deck Arch Pier Spandrel posts Embedded portion of pier modelled as foundation (footing) Superstructure Foundation Nodes in two-stage elastic analysis: Superstructure nodes with displacements u Foundation nodes with displacements us Internal frictionless hinges Position of survey targets River bed B0= 13.7 m D= 4.3 m Figure 2. Plane-frame model of Grosvenor Bridge: structural members and internal connections modulus Euof London Clay is estimated as 500–800 Su, con-179 sidering its correlation to the plasticity index and the overcon-180 solidation ratio (Viggiani,1999). These estimates align with181 the values reported by Withers et al. (2001) who found Eu/Su 182 in the range 500–1500 at a strain level of 0.01%, and also sug-183 gested that Euexceeds 75 MPa. For the numerical analyses in184 this paper, we adopt the Young’s modulus value suggested by185 Kerensky and Partridge (1967), Eu=105 MPa (1000 tons/ft2),186 based on the observations from a nearby building construction187 site. During the 1966–67 reconstruction of the bridge, the same188 Young’s modulus value was used to estimate the immediate set-189 tlement of its piers, showing very good agreement with field190 observations.191 Before reaching the Grosvenor Bridge, the Tideway TBM192 passed beneath three utility tunnels, where the average volume193 loss was back-calculated as 0.9% (Wong,2019). In the absence194 3 0 100 200 300 400 Su (kPa) 0 5 10 15 20 25 Depth (m) Bridge foundation level Grosvenor Bridge (Kerensky et al., 1967) Victoria Station (Skempton and Henkel, 1957) Figure 3. Undrained shear strength of London Clay near Grosvenor Bridge of settlement measurements along a full greenfield transverse195 cross-section, a volume loss value VLof 1% was assumed as a196 first approximation.197 2.2. Putney Bridge198 The Putney Bridge is a Grade II listed masonry arch bridge199 located in southwest London, connecting Putney and Fulham200 across the River Thames. Originally constructed between 1882201 and 1886, it consists of five segmental stone and granite arches,202 with spans ranging from approximately 34 m to 44 m and arch203 rise of 5 m to 6 m. The structure was widened in 1933, with a204 9.14 m wide extension on the downstream side using mass con-205 crete, while maintaining the original visual appearance. The206 concrete deck is supported by longitudinal spandrel walls con-207 structed of brick for the original structure and mass concrete208 for the extension portion, with stone slabs spanning between209 spandrels having no backfill between them. The arches are con-210 structed from granite blocks (with thickness of 1.25 m) and bear211 directly onto the mass concrete piers and abutments. The bridge212 does not include bearings or movement joints.213 The Thames Tideway TBM passed beneath the southern in-214 termediate span of the Putney Bridge, with a tunnel axis depth215 of 25.3 m below riverbed level (neglecting the presence of 1 m216 thick alluvium layer). This southern intermediate span has a217 clear span of 39.4 m and height (rise) of 5.4 m. The founding218 soil consists of London clay up to a depth of about 15 m from219 the invert depth. The foundations have a width of 10.3 m (in220 direction transverse to tunnel axis) and length of 36 m (in di-221 rection along tunnel axis), and are embedded about 6.1 m into222 London Clay; the embedment is approximately 25% of the tun-223 nel axis depth. The pier height from the founding level to the224 lower point of the arch is approx. 8.2 m while pier transverse225 width is 5.6 m. A tunnelling volume loss of 1% was assumed,226 and the undrained Young’s modulus of the London Clay was227 taken as 105 MPa as for the Grosvenor Bridge, an order of228 magnitude greater than the value adopted in Tideway Project’s229 assessment report for Putney Bridge (Thames Water Utilities230 Limited,2012).231 Making reference to Thames Water Utilities Limited (2012),232 we constructed a plane-frame numerical model for the bridge233 superstructure. Table 1summarises the axial and bending stiff-234 ness of beam elements adopted for the combined arches and235 spandrels, the deck and the piers, for the two bridges examined236 in this paper.237 Figure 4. Putney Bridge (photo courtesy of Wandsworth Council, London) Deck Arch Pier Embedded portion of pier modelled as foundation (footing) Superstructure Foundation Nodes in two-stage elastic analysis: Superstructure nodes with displacements u Foundation nodes with displacements us River bed B0= 10 m D= 6.1 m Figure 5. Plane-frame model of Putney Bridge Table 1. Cross sectional properties of main structural members in numerical models Structural member Grosvenor Bridge Putney Bridge Arch 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 Pier I=12864 m4I=3278 m4 A=898 m2A=371 m2 Spandrels I=0.004 m4(Incorporated A=0.095 m2in arch) 3. Interaction models238 3.1. Two-stage analysis method (TSAM)239 Two-stage elastic models have been applied in the literature240 to study SSI involving pipelines and buildings affected by tun-241 nelling (e.g. Klar et al.,2005;Franza and DeJong,2019). The242 first stage of the analysis involves predicting greenfield dis-243 placements at the foundation level, which serve as the primary244 tunnelling-related input in TSAM. In the second stage, the inter-245 action problem is solved by incorporating structure-foundation246 4 stiffness and applying the greenfield-equivalent set of forces247 that would displace the soil-foundation by the greenfield move-248 ments. In this paper, linear elastic two-stage interaction models249 are adopted for the bridge problems, with preliminary results250 reported in Wong et al. (2021).251 Greenfield movements at the foundation level can be esti-252 mated using semi-empirical relationships, analytical solutions253 or numerical methods. In this paper, greenfield displacements254 in the vertical and horizontal directions are estimated using the255 empirical method in Wong et al. (2025). Briefly, the transverse256 settlement profile in clay follows a Gaussian distribution (Peck,257 1969;Mair and Taylor,1997), while the expression from Mair258 et al. (1993) is selected to describe the linear variation of the in-259 flection point offset with depth; horizontal movements are cal-260 culated from settlements considering the variation of the focus261 point of the total displacement vector with transverse offset and262 depth. Contrary to the assumption that at any given depth the263 displacement vectors point towards a unique point, Wong et al.264 (2025) demonstrated that for case histories of tunnel excava-265 tions in fine-grained soils, the vector focus at the ground surface266 is above the tunnel centre and its depth becomes shallower with267 increasing transverse distance from the tunnel centre-line; be-268 low a depth of 20% the tunnel axis depth, the focus point depth269 is assumed to be at 1.54 times the tunnel depth.270 For the greenfield stage 1, Eq. (1)–(2) describe the steady-271 state settlement was a standard Gaussian curve.272 w=wmax exp      −x2 2i2     ;wmax =rπ 32 VLD2 t i(1) i=0.5zt−0.325z(2) where wis the settlement at a transverse horizontal distance x273 from the tunnel axis, wmax is the maximum centre-line settle-274 ment, iis the horizontal distance of the point of inflexion from275 the tunnel axis, VLis the volume loss, expressed as a percentage276 of the nominal volume of the tunnel of diameter Dt, and ztis the277 depth of the tunnel axis.278 The transverse horizontal displacement uat distance xfrom279 the tunnel axis and at depth zcan be expressed as a function280 of the settlement wat the same location, and of the normalised281 depth of the focus point h=zf/zt, which in turn is a function282 of the normalised depth z/ztand normalised offset x/i.283 u=−wx h zt−z(3) where: h=h0=3.59 4.43 +exp       x2 1.3i2      when z=0; (4a) h=h0−1.54 1−z 0.2zt!+1.54 when 0 <z zt <0.2; (4b) h=1.54 when z zt ≥0.2 (4c) If h=1, the displacement vectors point towards the tunnel284 centre. Eq. (4) establishes that, near the ground surface, the285 normalised focus depth h<1 close to the centre-line and hre-286 duces with the normalised offset x/i. In this paper, Eq. (1)–(4)287 were applied considering the ground surface horizon to be at288 the foundation level.289 In stage 2, the bridge response to tunnelling is studied by290 modelling the superstructure as a plane frame resting on foot-291 ings tied to the spring-based ground model (consisting of ei-292 ther fully-coupled, locally-coupled, or decoupled springs as dis-293 cussed below). Fig. 6shows the interaction model for the bridge294 schematised as either a plane or three-dimensional frame. As295 the interaction model is linear elastic, the foundation maintains296 displacement compatibility between the soil and the footings.297 The pier bases were assumed to be rigid; thus, the foundations298 and the displacements of the foundation soil were described by299 single nodes at the centres of the footing footprints, collectively300 referred to as the foundation master-nodes. Tunnelling effects301 were modelled by applying an equivalent set of forces that dis-302 place the spring foundation model alone by the greenfield dis-303 placements, including greenfield rotations when appropriate (as304 discussed later).305 x,u z,w y,v Interaction between translation and rotation (FC and LC springs only) Interaction between foundations (FC springs only) (y, rx, rzdirection springs not shown for clarity) Figure 6. Interactive springs adopted in two-stage elastic solution analysing plane frame in x−zplane under steady-state tunnelling conditions The interaction problem is solved with the finite element method. The equilibrium equation in matrix form describing the frame displacement under the effects of tunnelling is: (Kst +Kf)u=pgf (5) Kf=ATKsA(6) where Kst and Kfare the structure and the foundation (soil-306 footing system) stiffness matrix, respectively, and uis the dis-307 placement vector describing translational and rotational degrees308 of freedom of the structure (including superstructure and pier309 base points); for a plane frame, the translational degrees of free-310 dom in the xand zdirections and the rotations in the x−zplane311 are considered; Ais a kinematic constraint matrix relating the312 translational and rotational degrees of freedom of the founda-313 tion master-nodes to the translational degrees of freedom of the314 soil underneath the foundation footprint: i.e. as the footing is315 assumed to be rigid, compatibility imposed via Arequires that316 the displacements of the soil at the footing should be a linear317 function of the displacements of the pier base points (later de-318 tailed in the text); Ksis the soil stiffness matrix (defined with319 respect to all nodes at the soil-footings interface); pgf is the vec-320 tor of the equivalent greenfield tunnelling-induced forces acting321 5 on the foundation.322 As shown in Fig. 6, the superstructure is discretised into fi-323 nite elements. The superstructure stiffness matrix Kst is ob-324 tained by assembling (i) Euler-Bernoulli beams for the straight325 steel members rigid in shear, (ii) Timoshenko beams for the stiff326 concrete piers (Friedman and Kosmatka,1993) and (iii) curved327 Euler-Bernoulli beams for the arches (Litewka and Rakowski,328 1997) while (iv) the rigid footings were modelled by introduc-329 ing the kinematic constraint matrix discussed above. All inter-330 nal hinges and rollers (as illustrated in Fig. 2) were duly con-331 sidered by manipulating the finite element stiffness matrix to332 account for the release of the relevant internal forces. Full de-333 tails of the stiffness matrix are documented in Wong (2019) and334 Wong et al. (2021).335 To establish the soil stiffness matrix Ks, the soil tied to the336 footing interface was discretised into a grid of nodes (each hav-337 ing three translational degrees of freedom), whose displacement338 vector usis linked to the superstructure by the kinematic con-339 straint as us=A u. From the kinematic constraint matrix,340 it follows that Kfis a condensed stiffness matrix of Kswith341 respect to the degrees of freedom at the centroids of the pier342 bases. Therefore, the foundation system comprising soil and343 rigid footings as described by Kfconsists of roto-translational344 springs, all lumped at the bases of the piers as shown in Fig. 6345 via the kinematic constraint matrix. Interaction terms of these346 foundation springs (i.e. off-diagonal terms of Kf) depend on347 the adopted model as follows:348 •Fully-coupled (FC) springs: linear elastic half-space the-349 ory was used for the soil, resulting in a fully populated350 stiffness matrix, coupling all roto-translational behaviour351 across all footings. The soil flexibility matrix was ob-352 tained from the Boussinesq-Cerruti solution (Love,1927;353 Li and Berger,2001) and the integrated Mindlin’s solu-354 tion of a uniformly loaded rectangular area (Cheung and355 Nag,1968) (see Appendix A: Formulation of soil stiffness356 matrix for fully-coupled (FC) springs solution). The em-357 bedment of the footings was not taken into account. For a358 structure consisting of multiple individual footings, these359 were discretised into an array of nnodes distributed below360 the footprint whose three translational dofs are collected361 in us, following the procedure outlined above to generate362 a 3n×3nsoil stiffness matrix Ks. This enables the cou-363 pling between different footings across the surface of the364 soil.365 •Locally-coupled (LC) springs: the diagonal terms and lo-366 cal roto-translational terms of each footing were accounted367 for but no interaction between different piers was consid-368 ered. This was formulated in the same manner as the FC369 springs from Love (1927) and Cheung and Nag (1968),370 but by discretising a single footing instead of all the foot-371 ings across the continuum while neglecting the embed-372 ment of the footings (LC1 springs). Alternatively, the roto-373 translation stiffness of isolated, rigid, embedded/surface374 footings on the half-space or a compressible layer of fi-375 nite depth could be defined based on semi-empirical for-376 mulae (Pais and Kausel,1988;Gazetas,1991a,b) (LC2377 springs; see Appendix B: Formulation of soil stiffness ma-378 trix for decoupled (DC) or locally-coupled (LC2) springs379 solution).380 •Decoupled (DC) springs: a diagonal stiffness matrix was381 used, with springs that only react to local loads in their382 directions. The spring constants were obtained from the383 diagonal terms of the LC2 stiffness matrix.384 For practical applications, the stiffness values of locally-385 coupled and decoupled foundation springs can be readily cal-386 culated either from the literature for a given foundation layout387 (footing length, width, embedment, etc) or from 3D finite el-388 ement analyses (calculating the load-displacement relationship389 at the master-node of a footing resting on linear elastic mate-390 rial).391 The vector of tunnelling-induced forces pgf was computed392 from the vector of greenfield displacements us,gf at foundation393 nodes, established using empirical formulae or numerical re-394 sults. For FC and LC1 springs, pgf =ATKsus,gf, derived from395 the vertical and horizontal greenfield displacements at the soil-396 footing interface. For LC2 or DC springs, pgf =Kfugf; in addi-397 tion to vertical and horizontal displacements, ugf also contains398 greenfield rotations, which were computed as the first derivative399 of the greenfield settlement profile at the footing centroids.400 The TSAM analyses adopting linear elastic frames for the401 superstructure (see Fig. 2and 5) were carried out for the402 Grosvenor and Putney Bridges. Note that, if nonlinear struc-403 tural behaviour is of interest, the linear spring-based ground404 models described above could all be implemented within com-405 mercial structural computer programs for routine applications;406 in particular, LC and DC springs would be straightforward to407 calibrate and implement.408 3.2. Equivalence between 3D and PS foundations in FE analy-409 ses410 Geotechnical plane-strain analyses are frequently used for411 simplicity and efficiency. There are different methods to412 achieve the required equivalence between the plane-strain413 model and the actual three dimensional problem, while main-414 taining the relative soil-to-structure stiffness. The equivalence415 is achieved either by scaling the stiffness and the weight of the416 structure (e.g. Rampello et al.,2012) or by modifying the foun-417 dation dimensions or by a combination of the two (e.g. Callisto418 et al.,2013;Rampello et al.,2014) depending on the problem419 at hand. In fact, the deformability (i.e. the displacement un-420 der unit external load) of a foundation system (consisting of421 soil and footings) in plane-strain may be significantly differ-422 ent from that of a three-dimensional footing: an infinite strip423 foundation is more flexible than a rectangular footprint for a424 given transverse width and unit external load per longitudinal425 length (Pais and Kausel,1988;Dempsey and Li,1989;Gaze-426 tas,1991a,b). In other words, the foundation displacements and427 the rotation of an infinite strip footing are larger than those of428 a three-dimensional rectangular footing under identical loads429 per longitudinal length (longitudinal referring to out-of-plane430 direction of bridge structure, parallel to direction of tunnel ad-431 vancement).432 6 If both the structure and the foundation behaviour are de-433 scribed using stiffness matrices condensed with respect to the434 degrees of freedom of the foundation (pier base), the equi-435 librium equation of a 3D structure resting on a linear elastic436 ground model requires that:437 (Kst,3D +Kf,3D)u=Kf,3D ugf (7) where Kst,3D and Kf,3D are the structure and foundation stiffness438 matrices of the 3D structure.439 A plane-strain (PS) model may be considered equivalent to440 the 3D one when, for a given greenfield input ugf, it yields the441 same interaction outcome uas the 3D model. The equivalence442 can be achieved either by scaling the stiffness of the structure443 (Method A) or by modifying the foundation layout while keep-444 ing the mechanical properties of the structure (Method B). Re-445 gardless of the approach adopted, as discussed later, a perfect446 equivalence between PS and 3D models cannot be achieved by447 either method; engineering judgment is thus needed to define448 equivalent plane-strain models. TSAM is well-suited to 3D449 analysis (e.g. plane/3D frame on 3D foundation), while the450 problem of equivalence arises when adopting PS finite element451 coupled numerical analyses.452 3.2.1. Method A – scaling the structure453 If the transverse width Bof the foundation in the PS model is set to be the same as the actual width B0in the 3D problem, it follows that the 3D foundation stiffness can be expressed as a function of the PS stiffness matrix: L−1 0Kf,3D =MKf,PS (8) where L0is the length of the foundation in the out-of-plane di-454 rection while Mis the matrix describing the relationship be-455 tween Kf,3D and Kf,PS per unit longitudinal length. By manip-456 ulating Eq. (7) and (8), it follows that the equivalent PS model457 is:458 (K∗ st,PS +Kf,PS)u=Kf,PS ugf (9) K∗ st,PS =M−1L−1 0Kst,3D (10) which is obtained by scaling the stiffness matrix of the structure in the 2D model. The superscript ∗denotes the equivalent stiffness obtained by scaling. To simplify the scaling problem, it may be assumed that all degrees of freedom of the 2D and 3D foundation stiffness matrices scale equally and independently, in other words, Mis a scalar. Under this assumption: K∗ st,PS ≈(L0·m)−1Kst,3D (11) Note that m≥1 given that an infinite strip footing is less stiff459 than a rectangular footing under identical loads per unit longi-460 tudinal length. Eq. (11) may be practically implemented in nu-461 merical models by dividing the Young’s modulus of the struc-462 ture by mwhile keeping the actual foundation width B0.463 3.2.2. Method B – scaling the foundation464 An alternative approach is to increase the PS foundation465 width Bfrom the 3D value B0to achieve equivalence. In this466 way, the PS foundation matrix is approximately the 3D matrix467 divided by L0while the PS structure is simply obtained as a unit468 slice of the full structure. It follows that469 (Kst,PS +K∗ f,PS)u=K∗ f,PSugf (12) K∗ f,PS ≈L−1 0Kf,3D (13) Kst,PS =L−1 0Kst,3D (14) When adopting different transverse widths Bfor PS and 3D models, the ratio between 3D and PS foundation stiffness matrices can be quantified by the term R=L0K−1 f,3D K∗ f,PS (15) By definition, an effective equivalence achieved by scaling the470 foundation (see Eq. 15) is obtained when Ris the identity ma-471 trix (all diagonal elements Rii are as close as possible to unity).472 However, it is impossible to satisfy Rii =1 for all elements be-473 cause of the different ratios between PS and 3D stiffness values474 depending on the load directions/degrees of freedom.475 The equivalence of a plane-strain analysis is addressed in the476 subsequent section. In contrast, there is no need to consider477 equivalence in the TSAM, given that the soil stiffness matrix is478 obtained from 3D theories in the first place.479 3.3. Plane-strain (PSFE) and three-dimensional (3DFE) cou-480 pled finite element models481 Two coupled numerical analysis methods were used in this482 study to study the interaction between the Tideway Tunnel483 and Grosvenor Bridge in the short term: plane-strain finite484 element (PSFE) and three-dimensional finite element (3DFE)485 models for which the programs PLAXIS 2D (version 20) and486 PLAXIS 3D (version 17) were used, respectively.487 The 3DFE model acts as a benchmark for PSFE and spring-488 based TSAM models; therefore, the 3DFE analyses were car-489 ried out for short-term, steady-state tunnelling only. The plane-490 strain model PSFE is similar to the approach practitioners typi-491 cally use for tunnelling impact assessments to study foundation492 and superstructure steady-state movements (e.g. Thames Water493 Utilities Limited,2012,2013d) and it was adopted here to study494 the equivalence between PSFE and 3DFE interaction models by495 scaling the foundation length (Method A). The model geome-496 tries of both the PSFE and 3DFE models were identical: struc-497 tural elements of the bridge superstructure were modelled as498 plate elements (for PSFE) or beam elements (for 3DFE), while499 spandrel posts which were hinged at both ends were modelled500 as node-to-node anchors. On the other hand, Putney Bridge was501 only analysed with TSAM.502 The steady-state condition following the complete passage of503 the TBM within London Clay was simulated in total stress anal-504 ysis. In the analysis, the bridge was assumed to be weightless505 as the aim of the analysis was to study the tunnelling-induced506 7 zt=34 m 1 m between tunnel invert and bottom boundary H=39.4 m B 270 m (a) PSFE (b) 3DFE 39.4 m 270 m 300 m Figure 7. Finite element models of Grosvenor Bridge and tunnel excavation: (a) plane-strain model PSFE and (b) three-dimensional model with plane frame 3DFE undrained response of the soil and the bridge. The ground mate-507 rial was assumed to be linear elastic – perfectly plastic and fol-508 low the Tresca failure criterion. Fig. 7shows the mesh adopted509 for the analysis. The bottom boundary of the ground model510 was taken at the bottom of the London Clay horizon, assumed511 to be 1 m below the invert of the tunnel, while the horizon-512 tal extent was set beyond the river banks, which were outside513 the tunnelling influence zone, for a total width of 270 m. The514 following soil properties were assumed for the ground model:515 γ=20 kN/m3,Eu=150 MPa, Su=200 kPa, K0=0.7 and516 ν=0.495, selected with reference to assessment reports under517 the Tideway project. Standard boundary conditions were used518 (rollers on vertical boundaries, fixed bottom boundary). The519 model boundary, while close to the abutments of the bridge, did520 not affect the analysis results because the deck structure with521 roller connections does not transmit longitudinal loads between522 spans.523 A displacement-controlled technique was used in the PSFE524 and 3DFE model to model tunnelling. Following Cheng et al.525 (2007), a technique was used in the PSFE and 3DFE analyses to526 model tunnelling, in which the displacements of the nodes lo-527 cated around the tunnel converge towards a single point on the528 tunnel vertical line of symmetry, generically located between529 the tunnel axis and the invert. The point of convergence of dis-530 placements is close to the tunnel centre for deep tunnels, mov-531 ing towards the invert for shallower tunnels. In particular, the532 back-analysis by Cheng et al. (2007) suggested a linear varia-533 tion of the focus point with the cover-to-diameter ratio; for the534 Grosvenor case study this gave a focus depth at 60% the tunnel535 radius below the tunnel axis (z=zt+0.6R).536 The bridge superstructure was modelled using Mindlin plate537 elements, with either fixed or hinged ends (Fig. 2). Their bend-538 ing and axial stiffness per unit longitudinal length were obtained539 by dividing the actual 3D values of the stiffness EA and EI by540 the calibrated length of the bridge foundation in the out of plane541 direction L=m L0, and then the contribution from the ten sep-542 arate bridge frames were summed (PEI/Land PEA/L). The543 Young’s modulus of steel Ewas taken as 200 GPa. The shear544 stiffness GAswas given by the equivalent rectangular cross-545 section used in PLAXIS and G=2(1+ν)E(Bentley,2020a). As546 the ten actual bridge frames would behave in plane-stress con-547 dition, it was necessary to prevent the Poisson’s ratio from stiff-548 ening the plane-strain model by setting ν=0 (Bentley,2020b).549 The footings (i.e. the portions of the pier below ground surface)550 were modelled by triangular elements (as was the soil) with the551 Young’s modulus of concrete and masonry taken as 20 GPa.552 To make the comparison with the PS finite element analysis553 more straightforward, the actual foundation width was adopted554 8 (B=B0) in Method A. To scale the footing length Lfrom the ac-555 tual length L0, the three degrees of freedom, namely, displace-556 ment ualong x, displacement walong zand rotation about y,557 were individually considered. A 3D model of the actual rect-558 angular footing was first constructed using PLAXIS 3D. Unit559 forces (or moment) were applied to the footing in turn and the560 resulting displacements were computed. Next, a PS (plane-561 strain) model of the same footing with the actual width B0was562 constructed using PLAXIS 2D. For each degree of freedom,563 the same unit forces as in the 3D case, divided by a foundation564 length Lto obtain the loading per unit length, were applied to565 the PS model (Fig. 8). The calibrated foundation lengths Lthat566 gave identical displacements in the PS and 3D models for each567 degree of freedom were obtained (Table 2). The combinations568 of Band Lwhich gave identical displacements between PS and569 3D solutions are illustrated in Fig. 9, in which the shaded ar-570 eas refer to a deviation within 10%. A foundation length of571 L=1.32 L0(L0=60 m) was adopted. With this geometry,572 the PSFE analysis should give the same horizontal footing dis-573 placement as a full 3D model while the vertical and rotational574 movements were within 6% of the corresponding 3D results.575 3D analysis 2D plane-strain analysis B0 L0 Fx Fz MyFx/ L Fz/ L My/ L B Displacements u3d, w3d, ry,3d Displacements u2d, w2d, ry,2d Thickness of compressible elastic layer H Figure 8. Approach to establish equivalence between PS and 3D foundations 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 ry,2d = ry,3d u2d = u3d w2d = w3d Error < 10% Adopted ( L = 1.32 L0, B = B0) Figure 9. Combinations of Land Bgiving equivalent plane-strain and 3D finite element solutions 4. Results for the Grosvenor Bridge case study576 4.1. Greenfield movements577 Before addressing the interaction problem, greenfield move-578 ment predictions for the Grosvenor Bridge are discussed.579 Table 2. Calibrated length Lof footing for scaling superstructure stiffness in PSFE analysis Degree of freedom Calibrated footing length L† Horizontal displacement u80.38 m Settlement w71.07 m Rotation ry72.86 m † Actual footing length L0=60 m Fig. 10 compares greenfield ground movements along the hori-580 zon at the base of the foundation (approximately 4 m depth),581 obtained from the PSFE and 3DFE models in greenfield condi-582 tions (without the bridge), with those obtained from the empir-583 ical method of Wong et al. (2025) described earlier. All predic-584 tions assume a tunnelling volume loss of 1%.585 FE predictions of the greenfield settlement and the gradient586 of the settlement trough are in excellent agreement with the em-587 pirical values at the South and Centre Piers of the Grosvenor588 Bridge — the two piers located within the tunnel’s influence589 zone. This agreement is important, as these movements are the590 main drivers of soil-structure interaction in this case (as dis-591 cussed later).592 -50 0 50 x (m) -5 0 5 10 15 20 Settlement w (mm) South Pier Centre Pier (a) -50 0 50 x (m) -5 0 5 Horizontal movement u (mm) South Pier Centre Pier PSFE 3DFE Empirical (b) Figure 10. Greenfield movements at foundation level for tunnelling volume loss of 1%: (a) settlement and (b) transverse horizontal movement However, this figure also shows that the extent of the horizon-593 tal displacement profile from the FE analyses is wider than that594 for the empirical curve, which affected the magnitude of hori-595 zontal movements at the South and Centre Piers, despite similar596 peak values. The greenfield horizontal movements from the FE597 analyses are larger than from the empirical predictions by about598 35% and 70% at the locations of the South and Centre Piers, re-599 spectively. The FE horizontal displacement profile is wider and600 reduces more gradually away from the tunnel axis compared to601 the empirical prediction. These differences could be relevant602 9 to 3D and transient conditions, offers a powerful and efficient973 framework for evaluating the evolving response of bridges dur-974 ing TBM advancement. Such analyses are valuable for both975 design and construction monitoring, as they inform the inter-976 pretation of real-time data and help identify intermediate stages977 of increased vulnerability that may be lost in a steady-state anal-978 ysis.979 7. Guideline for future risk assessments of bridges on shal-980 low foundations981 TSAM has the advantage of enabling quick sensitivity stud-982 ies of the greenfield displacement field, structural configura-983 tions (and nonlinearities) and the resulting SSI that can mod-984 ify the foundation movement distribution. In addition to this985 versatility and the possibility to perform either steady-state or986 transient analysis of plane-frame or three-dimensional bridges,987 the comparison with coupled finite element analysis has demon-988 strated a similar predictive capability of TSAM with its intrin-989 sic 3D nature which bypasses the approximations required by990 equivalent plane-strain models. Therefore, TSAM is always991 recommended in place of uncoupled greenfield analysis (im-992 posing greenfield displacements directly to foundations).993 At the preliminary phases of an assessment, due to the lack994 of a generalised method to estimate relative bridge-foundation-995 soil stiffness, interaction models are almost always needed for996 bridge assessment in the presence of tunnelling, as opposed997 to building assessments for which recognised empirical ap-998 proaches are available. On the one hand, for relatively stiff999 bridges, interaction models able to account for SSI effects on1000 foundation movements are necessary. Even for relatively flex-1001 ible bridges (e.g. Grosvenor Bridge) which conform closely to1002 greenfield movements, it is difficult to assess whether SSI is1003 negligible a priori. In this context, TSAM is a suitable tool to1004 assess the relevance of SSI, and decide how to proceed further1005 in the design process.1006 Regarding the complexity of the lumped-spring models for1007 the foundation, even the simplest interaction method with Win-1008 kler decoupled (DC) springs can produce realistic results of1009 structural movements and internal forces. Such decoupled1010 springs can be readily implemented in most commercial struc-1011 tural computer programs for a TSAM. Uncertainties regard-1012 ing Winkler spring constants can be addressed by a sensitiv-1013 ity study, considering available analytical and empirical expres-1014 sions to account for embedment, shape, as well as the bedrock1015 depth-to-footing width (e.g. Gazetas,1991a;Pais and Kausel,1016 1988). Therefore, TSAM based on Winkler ground models is1017 suggested, as a first step, to assess the level of relative soil-1018 structure stiffness and the expected deformation mechanism of1019 the bridge, which is relevant to the design of monitoring work1020 and the expected serviceability/ultimate states to be analysed in1021 detail.1022 The observation that the steel Grosvenor bridge was rela-1023 tively flexible with respect to the underlying stiffsoil should1024 not be extrapolated or generalised to all steel arch bridges. All1025 the interaction analyses in this study incorporated roller con-1026 nections (between deck and piers) as well as frictionless hinges1027 (between arches and piers) that reduced the superstructure reac-1028 tion forces acting on the piers. Wong (2019) demonstrated that1029 the bridge response to tunnelling would become significantly1030 stiffer if the internal hinges were removed and, thus, SSI is sen-1031 sitive to the degree of fixity of internal connections in steel arch1032 frames. Structural details and internal connections can be read-1033 ily included in TSAM.1034 The final recommendation is to evaluate if the interaction1035 problem can be considered in an elastic regime; for this, we1036 have suggested: (i) estimating the extent of the plastic zone1037 and the linearity of the greenfield soil movements with VL(us-1038 ing a greenfield finite-element analysis of tunnelling), along1039 with (ii) an assessment of tunnelling-induced mobilised shear1040 strength at the foundation (using TSAM).1041 Fig. 18 summarises the geotechnical and structural design1042 processes, including TSAM applicability checks, in a flow1043 chart.1044 Stiff behaviour Refine structural model with realistic arches, beams, deck and spandrels (possibly in 3D) TSAM (e.g. decoupled springs) with main structural components (e.g. piers and arches only) Plane frame model of bridge superstructure Take greenfield foundation movements and assess structural response conservatively DC analysis matches uncoupled GF closely? No Yes Calculate bridge displacements and internal forces Re-perform TSAM with coupled or decoupled springs TSAM not applicable Yes No Check mobilised stresses at foundations in TSAM results End PSFE with calibrated footing length 3dFE Calculate bridge displacements and internal forces TSAM Flexible behaviour Structural Geotechnical Greenfield check using PSFE analyses (i) plastic zone around tunnel (ii) linearity of GF movements at foundation footprint Figure 18. Process of tunnelling impact assessment for bridges on shallow foundations 16 8. Conclusions1045 This paper presents a methodology for analysing the interac-1046 tion between a bored tunnel and an existing bridge on shallow1047 foundations: the two-stage analysis method (TSAM), which de-1048 couples the geotechnical and structural aspects of the problem1049 but retains their interaction through spring-based foundations.1050 First, this work builds on insights from the Thames Tideway1051 Tunnel project and examines the Grosvenor Bridge and Putney1052 Bridge as case studies. The predictions of two-stage modelling1053 for varying levels of complexity (fully coupled, locally cou-1054 pled, and decoupled springs) were benchmarked against field1055 data and more rigorous coupled plane-strain (PSFE) and three-1056 dimensional finite element (3DFE) models. Results illustrate1057 how structural stiffness, soil response, and interaction mod-1058 elling assumptions influence the prediction of bridge displace-1059 ments during tunnelling. The main conclusions are given be-1060 low.1061 •The TSAM framework offers a practical balance between1062 simplicity and accuracy, making it suitable for preliminary1063 assessments of tunnelling impacts on existing bridges. A1064 comparison of TSAM and FE models demonstrates that1065 spring-based two-stage method can provide results that are1066 consistent with those of more advanced coupled FE anal-1067 ysis, capturing essential aspects of the structural and SSI1068 behaviours, with minimal requirements for calibration of1069 the spring constants, given readily available expressions1070 from the half-space theory. Also, TSAM can be extended1071 to capture 3D effects during tunnel advancement, avoiding1072 the computational demands of 3D FE analysis or the ap-1073 proximations required by equivalent plane-strain FE mod-1074 els. TSAM results closely matched monitoring data in1075 both steady-state and transient conditions in the Grosvenor1076 case study.1077 •Results have confirmed the need to move beyond green-1078 field assumptions and account for interaction effects, es-1079 pecially when dealing with stiffbridge structures. For1080 relatively flexible bridges, such as the Grosvenor Bridge,1081 the interaction response closely follows greenfield pat-1082 terns due to the limited structural restraint and internal1083 articulation. In such cases, uncoupled greenfield analy-1084 ses may yield similar deformed shapes to coupled mod-1085 els, although TSAM still provides a better estimate of in-1086 ternal forces and mobilised capacities. For relatively stiff1087 bridges, such as the Putney Bridge, greenfield assumptions1088 significantly over-predict internal actions and misrepresent1089 the deformation mechanism. TSAM results have showed1090 that SSI can change both the magnitude and direction of1091 pier movements. This has implications for serviceability,1092 crack control, and condition monitoring strategy.1093 •Even simplified TSAMs using decoupled (Winkler)1094 springs, such as those formulated by Gazetas (1991a), lead1095 to realistic predictions of structural displacements and in-1096 ternal forces, making them suitable for preliminary assess-1097 ments. These models are easy to implement in commercial1098 software and produce adequate first estimates of the effects1099 of SSI.1100 •The applicability of linear TSAM must be checked by1101 evaluating plasticity and mobilised shear strength beneath1102 foundations. For the Grosvenor Bridge case, estimates1103 based on TSAM and PSFE indicated that the elastic range1104 remains valid for tunnel volume losses up to 2–3%, which1105 is more than adequate for typical urban tunnelling projects1106 in stiffclays. Practical checks combining a greenfield FE1107 analysis and the comparison of shear strength with esti-1108 mates from TSAM were recommended.1109 •The equivalence between plane-strain and three-1110 dimensional coupled finite element models for steady-state1111 assessment was investigated, and a calibration procedure1112 for defining the equivalent foundation width in plane-1113 strain analysis was proposed. As perfect equivalence1114 is not achievable, we suggested that the foundation1115 width should be selected based on the dominant mode1116 of structural response, using engineering judgement.1117 For the Grosvenor Bridge case study, the foundation1118 width was calibrated to match horizontal stiffness, at the1119 cost of over-estimating the rotational movements. This1120 likely explains the higher internal forces and movements1121 predicted by the plane-strain FE model compared to the1122 three-dimensional FE analysis.1123 TSAM offers a reliable and efficient framework for evaluat-1124 ing tunnel-soil-bridge interactions. It enables sensitivity stud-1125 ies on greenfield displacement fields, foundation stiffness, and1126 superstructure configurations, and it supports both steady-state1127 and transient assessments. It is suggested as a pathway from1128 greenfield displacement estimation to integrated structural anal-1129 ysis, providing a useful tool for geotechnical and structural de-1130 signers to make rational risk assessments and plan monitoring1131 works.1132 Future work in this field could extend TSAM frameworks to1133 incorporate nonlinear bridge behaviour and probabilistic input1134 parameters for risk-informed assessment to cover the uncertain-1135 ties in the superstructure and geotechnical domains.1136 Acknowledgements1137 The first author’s postgraduate studies forming the basis of this1138 paper was partially supported by the Geotechnical Engineering1139 Office, Civil Engineering and Development Department of the1140 Hong Kong S.A.R. Government and the Hong Kong Institution1141 of Engineers Arthur & Louise May Memorial Scholarship.1142 The authors would like to acknowledge Ferrovial Agroman1143 and Laing O’Rourke JV for providing them with access to the1144 monitoring data.1145 CRediT author statement1146 E.K.L. Wong: Conceptualization, Methodology, Software,1147 Validation, Formal analysis, Investigation, Data Curation, Writ-1148 ing – Original Draft, Writing – Review & Editing, Visualiza-1149 tion. A. Franza: Conceptualization, Methodology, Software,1150 17 Validation, Writing – Original Draft, Writing – Review & Edit-1151 ing, Supervision. P. Hewitt: Investigation, Resources, Data1152 Curation. G.M.B. Viggiani: Conceptualization, Methodology,1153 Resources, Writing – Review & Editing, Supervision, Project1154 administration.1155 Appendix A: Formulation of soil stiffness matrix for fully-1156 coupled (FC) springs solution1157 This appendix describes the procedure followed in assem-1158 bling the soil stiffness matrix Ksoil. Elastic solutions such as1159 Love (1927) and Cheung and Nag (1968) give the displace-1160 ments in a homogeneous elastic half-space under the applica-1161 tion of forces on the surface. The solutions are of the form1162 Fq =u(16) where Fis the soil flexibility matrix, qis the vector of applied1163 vertical and horizontal forces and uis the vector of horizontal1164 and vertical displacements on the surface of the half-space.1165 Love (1927) gives the displacements at a point (x,y) on the1166 surface of the half-space under a concentrated horizontal force1167 H0as1168 ux=H0 2πG(1−ν r+νx2 r3) (17a) uy=νH0 2πG(xy r3) (17b) uz=(1 −2ν)H0 4πG(x r2) (17c) and the displacements under a concentrated vertical force V0as ux=−(1 −2ν)V0 4πG x r2(18a) uy=−(1 −2ν)V0 4πG y r2(18b) uz=(1 −ν)V0 2πGr (18c) where ris the plan distance between the point of force ap-1169 plication and the point at which displacements are evaluated1170 (Fig. 19a). The expressions in Eq. (17a) to (18c) are used to1171 obtain the off-diagonal terms of the soil flexibility matrix Fbut1172 are undefined at the point of force application (r=0).1173 The diagonal terms of Fmay be determined by integrating1174 the forces over a rectangular area and evaluating the total dis-1175 placements accordingly. Cheung and Nag (1968) give the dis-1176 placements beneath a rectangular area of dimensions a×bunder1177 applied horizontal and vertical forces 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) where C=a/band Eis the Young’s modulus of the elastic1179 half-space (Fig. 19b). This gives the diagonal terms and com-1180 pletes F.1181 The soil stiffness matrix Ksis therefore given by the inverse1182 of the soil flexibility matrix:1183 Ks=F−1(20) For a structure consisting of multiple individual footings1184 which are discretised into an array of nnodes, following the1185 procedure outlined above generates a 3n×3nsoil stiffness ma-1186 trix Ks. This enables the coupling between different footings1187 across the surface of the soil.1188 x y z V H (x,y) r (a) x y z V H ab x= y= 0 Rectangular area of load application (b) Figure 19. Displacements on surface of homogeneous elastic half-space: (a) displacements at locations other than point of force application and (b) displacements at point of force application 18 Appendix B: Formulation of soil stiffness matrix for decou-1189 pled (DC) or locally-coupled (LC2) springs solution1190 This appendix describes the formulation of static founda-1191 tion spring constants following Gazetas (1991a) and Gazetas1192 (1991b). For a rigid footing on the surface of an infinite elastic1193 half-space, the relevant spring constants for the three relevant1194 degrees of freedom for a 2D problem is given by Eq. (i)–(iii) in1195 Table 5. The springs are functions of the plan geometry of the1196 footing and soil properties.1197 Table 5. Static spring stiffness for rigid foundation on surface of elastic continuum Gazetas (1991a) and Gazetas (1991b) Mode Stiffness (Load per unit Depth of zone displacement) of influence H→ ∞ Vertical (z)Kz=[GL/(1–ν)](0.73 (i) 2.5–7.5 B +1.5χ0.75) with χ=A/(2L)2 Horizontal (x)Kx=[GL/(2–ν)] (ii) 1–3 B (2 +2.5χ0.85) Rotational Kry =[G/(1–ν)]I0.75 y(L/B)0.25 (iii) B (about y) (2.4+0.5B/L) H,∞ Vertical (z)Kz=GL 1−ν[0.73 +1.54( B L)0.75] (iv) (1 +B/H 1+2B/L) See Fig. 20 for notations. L PLAN SECTION x y x z G, n B B Area A L> B H ^^^ Bedrock Figure 20. Foundation on surface of elastic continuum The presence of a rigid boundary affects the spring stiffness1198 in different ways for different modes of movement. The effect1199 of the rigid boundary may be assessed by considering the zone1200 of influence. Gazetas (1991b) gives the approximate range of1201 zone of influence for different modes of movements. For ex-1202 ample, for a vertical applied load, the zone of influence ranges1203 from 2.5Bfor a rectangular footing to 7.5Bfor a strip footing.1204 It is notable that the effect of a rigid boundary has the greatest1205 effect on the vertical stiffness. It increases the vertical stiffness1206 by a factor of (1 +B/H 1+2B/L). For horizontal and rotational move-1207 ments, the effect of the rigid bottom boundary may reasonably1208 be ignored if the depth of the rigid boundary is greater than the1209 zone of influence. The foundation stiffness matrix is therefore1210 written as:1211 Kf= Kx0 0 0Kz0 0 0 Kry  (21) The spring stiffnesses for a footing embedded in an infinite1212 elastic half-space are given by Eq. (v)–(vii) in Table 6. Em-1213 bedment substantially increases the spring constants compared1214 with the case without embedment, in particular the rotational1215 stiffness.1216 Gazetas (1991a) also gives an additional term for the cou-1217 pling between translation and rotation for an embedded footing.1218 Hence the foundation stiffness matrix for an embedded footing1219 is written as:1220 Kf= Kx,emb 01 3Kx,emb d 0Kz,emb 0 1 3Kx,emb d0Kry,emb  (22) Table 6. Static spring stiffness for rigid foundation embedded in elastic continuum after Gazetas (1991a) Mode Stiffness (Load per unit displacement) Vertical (z)Kz,emb =Kz[1 +(2/21)(D/B)(1 +1.3χ)] (v) [1 +0.2(Aw/A)2/3] Horizontal (x)Kx,emb =Kx[1 +0.21(D/B)0.5] (vi) {1+0.69[(h/B)(Aw/L2)]0.4} Rotational Kry,emb =Kry{1+2.52(d/B) (vii) (about y) [1 +(2d/B)(d/D)−0.2(B/L)0.5]} See Fig. 21 for notations. L PLAN SECTION x y x z G, n B B Area A L> B Dd Side wall contact area Aw h= D–d/2 Figure 21. Foundation embedded in elastic continuum 19 References1221 Bentley, 2020a. Material Models - PLAXIS 2D CONNECT Edition V20.1222 PLAXIS manuals , 218–221.1223 Bentley, 2020b. Reference Manual - PLAXIS 2D CONNECT Edition V20.1224 PLAXIS manuals , 250–251.1225 Boldini, D., Losacco, N., Franza, A., DeJong, M.J., Xu, J., Marshall, A.M.,1226 2021. Tunneling-Induced Deformation of Bare Frame Structures on Sand:1227 Numerical Study of Building Deformations. 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