2015 Fargnoli Gragnano Boldini Amorosi

Fargnoli, V. et al. (2015). Geotechnique 65, No. 1, 23–37 [http://dx.doi.org/10.1680/geot.14.P.091]

23

3D numerical modelling of soil–structure interaction during EPBtunnelling

V. FARGNOLI�, C . G. GRAGNANO�, D. BOLDINI� and A. AMOROSI†

The paper illustrates a three-dimensional finite-element analysis aimed at back-predicting the responseof a multi-storey reinforced concrete building underpassed by a metro tunnel. The study refers to thecase of the Milan metro line 5, recently built in coarse-grained materials using an earth pressurebalance machine, for which settlement measurements along ground and building sections wereavailable. The soil behaviour is modelled using an advanced constitutive model that, when combinedwith a proper simulation of the excavation process, proves to realistically reproduce the subsidenceprofiles recorded in free-field conditions. The building is found to modify the deformative pattern atthe ground surface in relation to its stiffness and weight, reducing the differential settlements ascompared to those calculated under free-field conditions. Results of the numerical simulation carriedout, including the model of the building schematised in detail, are found to be in good agreementwith the monitoring data. They thus indirectly confirm the satisfactory performance of the adoptednumerical approach, which takes into account a unique model of the soil, the tunnel and the building– that is, the key ingredients of this interaction problem. Further analyses are also carried outmodelling the building, adopting different levels of detail. The results highlight that, for the caseunder study, the simplified approach based on the equivalent plate schematisation is inadequate tocapture the real displacement field. The overall behaviour of the system proves to be mainlyinfluenced by the buried portion of the building, including its foundation elements, which plays anessential role in the interaction mechanism.

KEYWORDS: finite-element modelling; monitoring; settlement; soil/structure interaction; tunnels

INTRODUCTIONThe construction of underground tunnels in urban areasoften requires excavation works to be carried out in closeproximity to residential buildings, cultural heritage monu-ments and underground services. The ability to predict thetunnelling-induced settlements and the associated impact onpre-existing structures represents a key aspect to estimatepotential damages and to design protective measures, whenneeded (e.g. Mair, 2008; Amorosi et al., 2012; Puzrin et al.,2012; Rampello et al., 2012).

Soil deformation and structural response are often as-sumed to be decoupled, so that the building damage istypically predicted based on free-field settlement profiles(Peck, 1969; Burland & Wroth, 1974; Burland et al., 1977;O’Reilly & New, 1982; Boscardin & Cording, 1989; Bur-land, 1995). However, such a simplified approach disregardsthe influence of the structure stiffness (Potts & Adden-brooke, 1997; Franzius et al., 2006) and the role of itsweight (Franzius et al., 2004), often leading to ratherconservative solutions in terms of estimated differentialsettlements and, consequently, of induced damage intensity.

In the last few years two-dimensional (2D) and three-dimensional (3D) numerical approaches have been developedto overcome such limitations. 2D numerical studies wereproposed by Liu et al. (2000) with reference to surfacemasonry structures, focusing on the effect of facade weight,stiffness and position with respect to the tunnel axis. Along-side the same topic, Amorosi et al. (2014) back-analysed the

interaction between the excavation of a tunnel and anancient masonry surface structure, adopting an advancedelasto-plastic constitutive model for the masonry. The depen-dency of the building response on various structural types(i.e. brick-bearing structures, open-frame and brick-infilledframe structures) and different soil conditions was discussedin detail by Son & Cording (2011).

More sophisticated 3D simulations, again with referenceto masonry surface structures, were presented in Burd et al.(2000) and Giardina et al. (2010). Soil–structure interactionstudies were also carried out, focusing on 3D framed build-ings, schematised as equivalent plates (Maleki et al., 2011)or adopting simple schemes consisting of beams, columnsand live loads acting at each floor of the reinforced concretestructure (Liu et al., 2012).

This paper proposes the results of a numerical studyaimed at investigating the soil–structure interaction duringtunnelling using a fully 3D solution scheme based on thefinite-element method. The study refers to a real casehistory, the construction of the new Milan (Italy) metro-line5, carried out in granular soils by an earth pressure balance(EPB) machine, which guaranteed surface volume losseslower than 0.4%. In the examined portion of the route (Figs1 and 3), the right tunnel of the line diagonally underpassesa nine-storey reinforced concrete structure dating back to theend of the 1950s.

In the first part of the paper the tunnelling-induced settle-ments, as observed along the segment of the route at sixtransversal ground sections and in correspondence with thereference building, are collected and interpreted. Such analy-sis allows evaluation of the ability of existing closed-formempirical solutions to back-calculate the observed free-fieldsettlement troughs along both transversal and longitudinaldirections, also providing direct information on the evolutionof the surface structure’s response during tunnelling.

Manuscript received 27 May 2014; revised manuscript accepted 23October 2014. Published online ahead of print 27 January 2015.Discussion on this paper closes on 1 June 2015, for further details seep. ii.� University of Bologna (Italy), Bologna, Italy.† Technical University of Bari, Italy.

The numerical study, described in the second part of thepaper, is performed by the code Plaxis 3D (Plaxis, 2012). Anadvanced constitutive model, called ‘Hardening soil withsmall strain stiffness’ (HSsmall, Benz, 2007), is adopted forthe soil, and a detailed numerical step-by-step procedure isused to model the main features of the tunnel excavationprocess. The surface reinforced concrete building interactingwith the metro-line is modelled in detail, taking into accountnot only its main structural components, but also its second-ary elements – that is, the external infill panels. Additionalnumerical analyses were also carried out, adopting simplifiedbuilding models to highlight the role of the different structuralcomponents in the overall behaviour of the soil–structuresystem. They include the equivalent plate schematisation anda simplified structural model only considering the buriedportion of the building together with its foundation elements.

The outcomes of the finite-element simulations are firstpresented with reference to free-field conditions and thenconsidering the pre-existence of the surface building. Theability of the proposed numerical approach to reproducerealistic results is assessed by comparing the computedsettlements with the available geotechnical and structuralmonitoring measurements. Finally, the results obtainedadopting different levels of detail in modelling the buildingare illustrated and critically discussed.

CASE HISTORY OF NEW MILAN METRO-LINE 5The new Milan metro-line 5 (Fargnoli et al., 2013) runs

from north to west of the city with a total length of 12.6 kmand 19 access stations. The monitored portion considered inthis study extends for a length of about 600 m betweenLotto and Portello stations (Fig. 1).

The twin tunnels of the line have a separation betweenthe two axes of about 15 m and a mean depth of their axesz0 ¼ 15 m. This latter reaches its maximum value of about23 m at Lotto station.

An EPB machine was selected to minimise ground move-ments in the highly populated areas affected by the construc-tion activities. The EPB shield, with a total length of about10 m and a thickness of 30 mm, is characterised by an outerdiameter of 6.69 m at the face and 6.67 m at the tail. Under

special circumstances, the maximum excavation diameter atthe face could be increased up to 6.71 m. Six pressure cellsare located on the EPB face, as shown in Fig. 2.

The advancement is provided by 38 hydraulic jackslocated on the external perimeter of the shield body, actingon the already installed lining. The tunnel lining, set in placeinside the shield tail to support the tunnel as the machineadvances, consists of concrete precast rings characterised bya length of 1.4 m and a thickness of 30 cm. The outer andinner diameters of the lining ring are equal to 6.40 m and5.80 m, respectively. The gap behind the lining segments ispromptly filled by a two-component grouting consisting ofcement paste and grip accelerator, in order to minimise thevolume loss and the related surface settlements.

Left tunnel

Right tunnel

Lotto Station

97 m 50 m 57 m 43 m 50 m 46 m

Scale: 50 m

N

S E

W

256m

Portello Station

70 mTunnel modelled portion

GS_1 GS_2 GS_3 GS_4 GS_5GS_6

Fig. 1. Milan metro-line 5: plan view of the segment of the route between Lotto and Portello stations

D 6·69 m�

2·48 m

1·97 m

1·80 m

1·93 m

1·37 m

Fig. 2. Location of the pressure cells on the EPB face

24 FARGNOLI, GRAGNANO, BOLDINI AND AMOROSI

During the various stages of the tunnel construction,extensive geotechnical and structural monitoring was carriedout along the line by an accurate levelling survey, withrecording intervals varying between 12 and 24 h. The datapresented and discussed throughout this paper are verticaldisplacements recorded during the excavation of the firsttunnel (i.e. right tunnel) at six transversal ground sectionsand in correspondence with a nine-storey reinforced concrete

framed structure diagonally undercrossed by the tunnel (Fig.3). The 30 m high structure dates back to 1959 (Figs 4(a)and 4(b)) and is characterised by a total weight of about41 000 kN. Its plan dimensions and the position of itsmiddle-point M are reported in Fig. 3. The building inter-storey height is equal to 3.2 m, except for the ground floorand the basement floor, which have heights of 4.2 m and2.5 m, respectively.

GS_6

Ground benchmarks

Building targets

N

E

W

SScale: 10 m

B 12 m�

Left tunnel

Right tunnel

70 mTunnel modelled portion

L 30 m�

15 m

L1 L2 L3 L4 L5

T1

T3

T2

R1 R2 R3

R4 R5

Mbuilding

Fig. 3. Detail of the examined portion of the route

(a) (b)

Fig. 4. (a) General view of main left side facade and (b) detail of the garage zone on the right longitudinal side of the building

MODELLING OF SOIL–STRUCTURE INTERACTION DURING EPB TUNNELLING 25

The main structural components of the building have thefollowing dimensions: the sections of the beams are40 cm 3 45 cm at the lower floors, and 70 cm 3 20 cm and45 cm 3 20 cm within or along the perimeter of the upperfloors, respectively; the column section has dimensions of40 cm 3 40 cm; the thickness of the floor slabs is 26 cm atthe lower floors and 22 cm at the upper ones; the sections ofthe reinforced concrete interior panels are 0.2 m 3 3.26 m.

The structure is founded on five strip footings (0.65 mhigh, indicated as I, II, III, IV and V in Fig. 5), 4 m below theground surface; more specifically, the building rests on thefoundation beams I, II and III, while the garage zone (Fig.4(b)), situated at the basement floor level along the right

longitudinal side of the structure, stands on the other ones (IVand V). Three raft foundations (0.7 m high, indicated as VI,VII and VIII in Fig. 5) are located at the same level under theelevator shafts and the stairwell, both on the right longitudinalside of the building. Reinforced concrete retaining walls(40 cm thick and 3.5 m high) surround the buried portion ofthe structure along its three sides, except for the garage zone.

Several ground benchmarks (from five to nine) wereinstalled on each instrumented ground section, while build-ing targets were placed along the base of the building’slongitudinal facades and on its transversal right side (Fig. 3).

GEOTECHNICAL CONDITIONSThe city of Milan is located in the central part of the

Padana plain (northern Italy), resting on a deep glacial andalluvial Pleistocene formation. The upper part of this depositmainly consists of sand and gravel, characterised by a siltpercentage that typically increases with depth. A formationof conglomerate and sandstone underlies this upper deposit,about 40 m below the ground surface, while sand and clayare present at greater depth. The new metro-line 5 is locatedwithin the upper granular non-cohesive unit of the forma-tion, which mainly consists of gravel and sand of fluvio-glacial and alluvial origin.

Along the metro-line an extensive geotechnical investiga-tion was carried out at the design stage of the work. Inparticular, with reference to the portion considered in thispaper between Lotto and Portello stations, it included

• two core drillings (CD_1 and CD_2) to a depth of 24–30 m from the ground surface (Fig. 6), instrumented withopen pipe piezometers

• 14 and 12 SPT tests conducted along boreholes CD_1and CD_2, respectively

• three Lenfranc-type permeability tests carried out impos-ing a constant piezometric level.

Granulometric analyses were conducted on 14 disturbedsamples retrieved from the drillings. Two main granulometricgrading curves were determined: the first curve is typical ofa gravelly sand soil, S, the second one of a sandy silt, L. Atthe location under investigation the gravelly sand soilemerges as the main component of the deposit, being foundat depths between 0 and 20 m, and between 25 and 30 m,while a layer about 5 m thick of sandy silt was identifiedbetween 20 m and 25 m. The total unit volume weightsunder saturated conditions for these materials are 20 kN/m3

and 17.5 kN/m3, respectively.

1·3 m 1·3 m 1·3 m 1·3 m 1·3 m

I II III IV V

VI

1·9

m

3·1 m

VII

3·0

m

4·2 m

VIII

1·9

m

3·1 m

30·0

m

5·2 m 6·4 m 5·1 m 5·0 m

Structure aboveground level

Garage belowground level

Fig. 5. Plan view of building foundations

CD_1z: m

0

5

10

15

20

25

30

R

SGG, S(L)

SG

S, G

L(S)

S, G(L)

G, S(L)

Tunnel crown

Tunnel axis

Tunnel invert

w.t._2013

w.t._2007

g. l. m a. s. l. GS_1 GS_2 GS_3 GS_4 GS_5 GS_6

CD_2z: m

0

5

10

15

20

24

R

G(S)

L(S)

S, G(L)

G, S(L)

Horizontal scale: 10 m

Reference monitoring sections

G(S) sandy gravelG, S(L) gravel with silty sandL(S) sandy siltSG sand and gravelS, G sand with gravelS, G(L) sand with silty gravelR made ground

�

�

�

�

�

�

�

Fig. 6. Soil conditions along the examined portion of the route, depth of tunnel and position of ground monitoring sections


Results of SPT tests conducted in the gravelly sand soilwere elaborated following Skempton (1986), leading to arelative density in the range 60–80% and strength parametersequal to c9 ¼ 0 kPa and �9 ¼ 338.

Standard penetration tests (SPTs) were not consideredappropriate to characterise the sandy silt layer; therefore, itsstrength parameters were assumed to be equal to c9 ¼ 5 kPaand �9 ¼ 268, based on pre-existing geotechnical character-isations carried out in the Milan area.

At the time of the geotechnical survey the hydrostaticwater level was found at an almost constant depth of about18.5 m below the ground level (Fig. 6). Six years after,during the construction stage, the corresponding level wasdetected 15 m below the ground surface, which is consistentwith the slow but continuous increase in the water tableobserved in the area in recent years.

The permeability coefficients, k, were observed to vary atdifferent depths between 5.5 3 10�3 and 1.1 3 10�2 m/s.

No geophysical investigations were specifically undertakenalong the metro route; the one nearest to the referencesegment of the line is a down-hole test performed at anearby construction site: this test resulted in the small-strainshear modulus (G0) profile shown in Fig. 7. The figure alsoillustrates the related stratigraphic profile, which proves tobe rather similar to the one of the tunnel site.

MONITORING MEASUREMENTS: ANALYSIS ANDDISCUSSION

Tunnelling-induced settlements were recorded betweenLotto and Portello stations at six transversal ground sections(i.e. GS_1, GS_2, GS_3, GS_4, GS_5 and GS_6) and incorrespondence with the surface structures interacting withthe new metro-line.

Ground monitoringTransversal settlement profiles due to the excavation of

the first tunnel are shown in Fig. 8. They refer to fullydeveloped settlements achieved when the tunnel face was ata sufficient distance from the monitoring sections.

The maximum settlement induced by the excavation neverexceeds 7 mm. Measurements are sufficiently well fitted by aGaussian distribution curve (Peck, 1969) for K values in therange 0.4–0.45, with the exception of the left-most points ofeach section, whose values are under-predicted by the empiri-cal relation (Fig. 8), probably due to the influence of nearbysurface structures located along the examined route (see Fig.1). The above interpolation allowed back-evaluation of thecorresponding volume loss, VL (%), which varies from 0.3%to 0.38%, with an average value equal to 0.33%, indicating awell-performing EPB excavation, and consistent with similarobservations reported in the literature (Leblais & Bochon,1991; Ata, 1996; Mair, 1996; Mair & Taylor, 1997).

The values of maximum settlement, K parameter andvolume loss for all the considered sections are summarisedin Table 1. The table also reports the different depths of thetunnel axis, z0, at each location and the recording date ofthe analysed settlement measurements.

The evolution of settlement above the tunnel centre-line ispresented in Fig. 9 as a function of the face distance for thetwo sections, GS_3 and GS_6, for which measurements ofvertical displacements close to the tunnel face (i.e. � 1 m)were available. The tunnel face settlement, Sv,f, at theselocations is equal to about 1 mm, indicating a satisfactoryface support during the excavation process.

Measurements were interpreted at each location by thecumulative Gaussian probability curve (Attewell & Wood-man, 1982) in order to define the longitudinal settlementtrough, assuming the volume loss and K values reported inTable 1 and considering the longitudinal inflection point, iy,equal to the transversal one, ix. As shown in Fig. 9, facesettlements are best fitted by the translated Gaussian cumu-lative curve (Mair & Taylor, 1997), obtained by equating theratio Sv,f /Sv,max to the measured one. This translated profile,however, is not able to capture the further evolution ofsettlements, predicting the achievement of steady-state con-ditions well before what is observed in situ.

Structural monitoringVertical displacements of the building were gathered during

tunnelling by monitoring targets located along the base of itsthree sides, identified by capital letters L, R and T and asequential number (see Fig. 3). The target relative distance andtheir distances from the tunnel axis are listed in Tables 2 and 3.

As expected, the vertical displacements mainly increasewith the excavation advancement, as shown in Figs 10(a)–10(c), where the evolution at each monitoring point isdisplayed at different dates. Final measurements, gatheredwhen the distance of the right tunnel face from sectionGS_6 was about 50 m, that is about 8D, range from 4.7 mmto 6.5 mm along the longitudinal left facade, from 3.5 mmto 6.6 mm along the longitudinal right facade and from4.6 mm to 5.7 mm along the transversal side.

These plots make it possible to highlight the effect of thedistance from the tunnel axis on settlement: the closer thetunnel axis (i.e. points L1, L2, L3 along the longitudinal leftside, R4, R5 along the longitudinal right side and T2, T3along the transversal one), the higher the settlement.

The progressive settlement response of the buildingevolves during tunnelling, this being particularly evidentalong its longitudinal sides: the structure, in fact, is char-acterised by a hogging-type mode of deformation when thetunnel-boring machine (TBM) face is located in correspon-

30

25

20

15

10

5

0

0 200 400 600 800

Dep

th,

: mz

Small-strain shear modulus, : MPaG0

Experimental

Computed:

Gravelly sand

Sandy silt

S_1

L

S_2

Fig. 7. Experimental and computed small-strain shear modulusprofiles with depth


dence with the middle of the building (measurements re-corded on 11 January 2013), while the deformative patternevolves in a sagging-type mode after the tunnel passage (i.e.from 12 January 2013 on).

In general, no evidence of damage was detected on this

structure at the end of the excavation works, owing to therelatively low settlements induced by EPB tunnelling.

NUMERICAL MODELGeometry and finite-element discretisation

Different numerical models were set up to simulate thetunnel excavation under free-field conditions and in thepresence of the building, the latter being modelled withdifferent levels of detail. In the interaction analyses, for thesake of simplicity, the presence of the nearby structures wasneglected.

The mesh employed in the present study is shown in Fig.11(a): it represents a soil volume 68 m wide, 30 m high and100 m long. These dimensions were selected to minimise theinfluence of the boundary conditions on the computedresults. The numerical model adopted for the free-fieldanalysis is constituted by 84 125 nodes; this number in-creases to 120 952 in the interaction analysis with thecomplete structural model. Nodes at the bottom of the mesh

8

6

4

2

0

�40 �20 0 20 40

Set

tlem

ent,

: mm

Sv

Distance from tunnel axis, : m(a)

x

8

6

4

2

0

�40 �20 0 20 40

Set

tlem

ent,

: mm

Sv

Distance from tunnel axis, : m(b)

x

8

6

4

2

0

�40 �20 0 20 40

Set

tlem

ent,

: mm

Sv

Distance from tunnel axis, : m(c)

x

8

6

4

2

0

�40 �20 0 20 40

Set

tlem

ent,

: mm

Sv

Distance from tunnel axis, : m(d)

x

8

6

4

2

0

�40 �20 0 20 40

Set

tlem

ent,

: mm

Sv

Distance from tunnel axis, : m(e)

x

8

6

4

2

0

�40 �20 0 20 40

Set

tlem

ent,

: mm

Sv

Distance from tunnel axis, : m(f)

x

Measurements:

GS_1

GS_2

GS_3

GS_4

GS_5

GS_6

Transversal troughs:

Empirical

Computed

Fig. 8. Transversal settlement troughs: measurements and best-fitting Gaussian curves (the computed profile forground section GS_6 is also shown)

Table 1. Values of maximum settlement, K parameter, volumeloss, axis depth and surveying date in the reference monitoringsections

Monitoringsections

Sv,max:mm

K VL: % z0: m Date

GS_1 5.6 0.45 0.38 20.0 16 December 2012GS_2 5.2 0.42 0.30 19.0 18 December 2012GS_3 6.0 0.45 0.37 18.0 20 December 2012GS_4 5.5 0.42 0.30 17.0 8 January 2013GS_5 6.5 0.40 0.31 16.0 10 January 2013GS_6 5.5 0.45 0.34 15.0 15 January 2013


are fixed in both vertical and horizontal directions, while thevertical boundaries are only fixed in the horizontal direc-tions. The soil domain, as well as the foundation elements,is discretised by ten-node tetrahedral elements, while two-node elastic anchor elements, three-node line beam elementsand six-node triangular plate elements are used to model thestructure and some components of the tunnel (i.e. shield andlining).

According to the in-situ stratigraphy, the soil profile isconstituted by two layers of gravelly sand (between 0 and20 m, and between 25 and 30 m) and a layer of sandy silt(between 20 and 25 m); the imposed hydrostatic water tableis 15 m below the ground surface.

The tunnel axis is located at a depth of z0 ¼ 15 m as inthe reference case of study, in correspondence with sectionGS_6 (see Fig. 6); the tunnel (D ¼ 6.7 m) underpasses thereference building, which is included in the model. Consis-tently with the real tunnel–structure relative position, theinclination of the building longitudinal sides with respect tothe tunnel axis is set as equal to 25.148 and, according tothe reference system shown in Fig. 11, the x and y coordi-nates of the structure’s middle point, M (see Fig. 3), areequal to 0 m and 35 m, respectively. The building is assumedto be directly connected to the soil at the foundation level,situated 4 m below the ground surface.

The numerical analyses were performed in terms of effec-tive stresses, assuming drained conditions for the soil owingto the relatively high permeability observed during thegeotechnical investigation.

8

6

4

2

0

�60 �40 �20 0 20 40 60 80 100 120

Set

tlem

ent,

: mm

Sv

Distance from tunnel face, : m(a)

y

8

6

4

2

0

�60 �40 �20 0 20 40 60 80 100 120

Set

tlem

ent,

: mm

Sv

Distance from tunnel face, : m(b)

y

Measurements: Longitudinal troughs:

GS_3

GS_6

Original

Translated

Computed

Fig. 9. Settlements measured above the tunnel centre-line at themonitoring sections (a) GS_3 and (b) GS_6 with original andtranslated longitudinal profiles (the computed profile for groundsection GS_6 is also shown)

Table 2. Target relative distance

Monitoring points Relative distance: m

L1–L2 5.54L2–L3 6.80L3–L4 7.09L4–L5 8.44R1–R2 7.18R2–R3 7.32R3–R4 6.97R4–R5 7.33T1–T2 4.84T2–T3 6.38

0

2

4

6

8

Set

tlem

ent,

: mm

Sv

L1 L2 L3 L4 L5

(a)

0

2

4

6

8

Set

tlem

ent,

: mm

Sv

R1 R2 R3 R4 R5

(b)

0

2

4

6

8

Set

tlem

ent,

: mm

Sv

T1 T2 T3

(c)

7 January 2013

8 January 2013

9 January 2013

10 January 2013

11 January 2013

12 January 2013

13 January 2013

14 January 2013

15 January 2013

Recording date:

Fig. 10. Structural vertical displacements recorded in correspondence with the monitoring targets: (a) L1–L5; (b) R1–R5;(c) T1–T3. Settlements recorded at targets T1–T3 before 12 January 2013 are equal to zero

Table 3. Target distance from the right tunnel axis

Monitoring points Distance from the right tunnel axis: m

L1 1.53L2 0.33L3 2.84L4 5.84L5 9.82R1 13.35R2 10.76R3 9.26R4 4.51R5 1.02T1 11.23T2 6.89T3 0.00


Soil constitutive model calibrationThe mechanical behaviour of the soil strata is described by

the hardening soil model with small-strain stiffness(HSsmall), a constitutive model capable of taking into ac-count the high soil stiffness observed at very low strain levels,its variation with strain level and the early accumulation ofplastic deformations. Details of the main aspects of itsformulation are provided in Appendix 1.

A summary of all model parameters and correspondingvalues is provided in Table 4. The total unit volume weight andthe strength parameters (c9 and �9) of the soils were deter-mined as discussed in the previous section on ‘Geotechnicalconditions’. For the sake of simplicity, the same total unitvolume weight value is assumed for the soils above and belowthe water table. The initial profile of the horizontal effectivestress was calculated using Knc

0 values defined in Table 4.The variation of the small-strain stiffness with depth is

obtained by calibrating the parameters Gref0 and m against

the down-hole experimental results, as shown in Fig. 6.The dependency of the shear stiffness on the strain level

(equation (3)) is obtained by referring to the experimentalcurves G/G0–ª proposed by Vucetic & Dobry (1991) forgranular soils (index of plasticity, IP ¼ 0) and for material oflow plasticity (index of plasticity, IP ¼ 15 %), respectively,for the layers of gravelly sand and sandy silt. The referencevalue of the Young’s modulus at small strains, E9ref

0 , is relatedto Gref

0 by the Poisson ratio for unloading/reloading, �ur. Thislatter is assumed as equal to 0.2 and 0.25 for the gravellysand and for the sandy silt, respectively. For both materialsthe coefficient of earth pressure at rest is estimated withreference to a normal consolidated state (Knc

0 ). In the absenceof laboratory experimental data, the reference unloading/reloading stiffness, E9ref

ur , is assumed to be 0.24E9ref0 for the

gravelly sand and 0.42E9ref0 for the sandy silt, corresponding

to the stiffness values observed along the decay curves at ashear strain of 0.1%. The other stiffness parameters, E9ref

50 andE9ref

oed, are assumed to be three times lower than E9refur : Finally,

Tunnel facefinal position

Tunnel axis projection

30 m

100

m

68 m

zx

y

(a)

y

z

(b)

Fig. 11. (a) Sketch of the mesh used in the interaction analysis.(b) A detail of the longitudinal section on the finite-element modelis also shown

Table 4. Soil constitutive model parameters

Parameters Name Values

Gravelly sand soil S_1 Sandy silt soil L Gravelly sand soil S_2

ª: kN/m3 Total unit volume weight 20 17.5 20

Failure parameters:

c9: kPa Effective cohesion 0 5 0�9: degrees Effective friction angle 33 26 33ł: degrees Dilatancy angle 0 0 0

Stiffness parameters:

m Power for the stress-level dependency of stiffness 0.4 0.85 0.4E9ref

50 : kPa Reference secant stiffness in standard drained triaxial test 48 000 54 250 58 944E9ref

oed: kPa Reference tangent stiffness for primary oedometer loading 48 000 54 250 58 944

E9refur : kPa Reference unloading/reloading stiffness at engineering strains 144 000 162 750 176 832

�ur Poisson ratio 0.2 0.25 0.2Gref

0 : kPa Reference shear modulus at very small strains 250 000 155 000 307 000ª0.7 Shear strain at which Gs ¼ 0.7G0 0.0001 0.0002 0.0001

Other parameters:

pref: kPa Reference stress for stiffness 100 100 100Knc

0 K0 value for normal consolidation 0.455 0.562 0.455Rf Failure ratio 0.9 0.9 0.9�tension Tensile strength 0 0 0cincrement: kPa/m Increase of cohesion with depth 0 0 0


standard values are taken for the other parameters of Table 4;for all soils the overconsolidation ratio is fictitiously assumedto be large enough to exclude yielding during compressivestress paths (i.e. excluding the activation of the cap surfaceincluded in the constitutive model).

Numerical schematisation of TBM–EPB tunnellingIn all the numerical analyses, the following sequence is

considered

• initialisation of the stress field in the soil (lithostaticconditions)

• activation of the structure in a single step (only in theinteraction analyses)

• tunnel excavation in several steps (in the first step thedisplacement field due to the weight of the building isreset to zero).

The simplified numerical procedure adopted to model thetunnel construction is illustrated in Fig. 12. The first portionof the tunnel cavity is lined by a steel shield, which extendsfor a total length of 9.8 m (i.e. seven slices) and it isconnected to the soil by way of an interface characterised bythe strength parameters of the adjacent soil. At the back ofthe shield the permanent reinforced concrete lining is in-stalled. The shield and the lining are modelled by means ofplate structural elements, characterised by isotropic linearelastic behaviour, whose properties are listed in Table 5.

The excavation is simulated by a step-by-step procedureconsisting in 43 advancements, each having the length ofone concrete lining ring (1.4 m), from y ¼ 9.8 m to 70 m.The advancement consists in removing one slice of soilinside the tunnel and imposing dry conditions. A pressure isapplied at the new tunnel face, corresponding to the esti-mated total horizontal stress acting at rest �h0(z), whichranges from 106 kPa at the tunnel crown to 185 kPa at theinvert, according to the average face pressure values re-corded during the tunnel construction.

In the intermediate zone between the shield tail and thepermanent lining, a region of 1.4 m of unlined soil issupported by a uniform pressure of 172 kPa, this lattercorresponding to the mean value of the grouting pressure asrecorded during the excavation.

In order to control the subsidence volume at the groundsurface, a fictitious contraction is applied along the shieldstarting from the second slice. Such a contraction, whichinduces greater displacements at the top of the circumfer-ence and lower ones at the bottom, is characterised by aconstant increment along each slice, aiming at reproducingin a simplified way the shield conical geometry. The applica-tion of a displacement field at the tunnel section, however,does not exclude the importance of the adopted constitutivemodel, especially for the low values of volume loss thatcharacterise the case under study (Fargnoli et al., 2014).

Numerical modelling of the buildingThe building is introduced in the numerical scheme (STR

analysis) by modelling its main structural components asfollows

• beams and columns are modelled by beam elements• plate elements with isotropic behaviour are used for the

floor slabs, reinforced concrete interior panels, elevatorshafts, stairwell and retaining walls

• foundations are represented by volume elements con-stituted by non-porous material.

A linear-elastic constitutive law is adopted for thesestructural components, whose parameters are selected consis-tently with the reinforced concrete material properties: unitvolume weight, ªc ¼ 24 kN/m3, Young’s modulus, Ec ¼25 GPa, Poisson ratio, �c ¼ 0.2.

The pressure distribution of the structure at the foundationlevel is shown in Fig. 13.

The building is characterised by the presence of infillpanels, uniformly distributed in the external frames, whichare modelled in a simplified way by means of equivalentcross-bracings (Panagiotakos & Fardis, 1996), characterisedby properties defined in Appendix 2.

Simplified structural models were also considered in thenumerical study: the building was first modelled withoutcross-bracings (STRwcb analysis), then an analysis was car-ried out limiting the modelling of the structure to its buriedportion including the foundation elements, reducing theupper portion to an equivalent load distribution (STRw

analysis). These loads were evaluated with reference to theirinfluence area and were applied at the ground floor in

Table 5. Shield and lining properties

Parameter Shield Lining

Thickness: m 0.03 0.3Unit volume weight: kN/m3 75 25Poisson ratio 0.25 0.15Young’s modulus: GPa 210 35

y

xI

II

IIIIV

V

VI

VII

VIII

40 kPa

80 kPa

120 kPa

160 kPa

200 kPa

240 kPa

Fig. 13. Pressure distribution at foundation level

Lining Grouting Shield

Face pressure

1·4 m slice

1·4 m 1·4 m 9·8 m

Fig. 12. Numerical procedure adopted to simulate TBM–EPBtunnelling


correspondence with the column’s head, the stairwell and theelevator shafts.

In a further analysis the structure was strongly simpli-fied and schematised as an equivalent plate (L ¼ 30 m andB ¼ 12 m) placed at the foundation level (STR� analysis),whose input parameters were derived adapting the ap-proach proposed by Franzius et al. (2006), as discussed inAppendix 3.

NUMERICAL RESULTS AND COMPARISON WITHMONITORING DATAResults of free-field analysis and interaction analysis with adetailed structural model

A preliminary free-field numerical analysis (FF) wasperformed to calibrate the contraction to be applied at thetunnel shield to reproduce a volume loss equal to 0.34%,corresponding to that observed at the monitored groundsection GS_6 (see Table 1) near to the building. Suchcontraction induces a maximum vertical displacement at thetunnel crown, in correspondence with the shield tail, equalto about 10 mm, which is compatible with the available gapof the adopted EPB machine.

To validate the model, the computed transversal and long-itudinal surface settlement profiles are compared to the meas-urements and the corresponding empirical curves, as shownin Figs 8(f) and 9(b). In general the comparison is satisfac-tory. The computed transversal profile (Fig. 8(f)) results infair agreement with the Gaussian distribution; however, theaccordance with the experimental measurements decreases asthe distance from the tunnel axis increases. The numericallongitudinal subsidence trough (Fig. 9(b)), which is quitesimilar in shape to the translated one, is able to capture theface and final recorded settlements; nonetheless, it predictsthe attainment of steady-state conditions at a shorter distancefrom the face when compared to what was measured.

The overall consistency of the computed profiles with theempirical solutions and with available measurements indi-cates that the adopted numerical model is amenable to beadopted for more complex interaction analyses, as illustratedin the following.

The interaction analysis, defined as STR, was carried outimposing the same excavation sequence and amount ofcontraction as defined above, but in the presence of thebuilding. It was performed considering the appropriate self-weight and stiffness of the structural elements, including inthe model the presence of the external infill panels by meansof specific cross-bracings.

The numerical final settlement troughs, as computed alongthe transversal and longitudinal directions to the tunnel axisat the foundation level (i.e. z ¼ �4 m), are compared to thecorresponding free-field predictions in Figs 14(a) and 14(b),which also show the plant position of the foundation ele-ments by dashed lines. In particular, Fig. 14 refers to thesubsidence profiles as computed at the barycentre of thebuilding (point M in Fig. 3).

As expected, the presence of the building influences thesettlement profiles, which deviate from the free-field onesalong both directions, highlighting the stiffer response ob-served in correspondence with the discrete foundation ele-ments. The two profiles overlap only outside the buildingarea. It is worth noting that the maximum vertical displace-ment and volume loss of the interaction analysis are largerthan the free-field one, due to the effect of the buildingweight. This observation is particularly evident at the stair-well (VII in Fig. 14(a)) and at the elevator shaft (VI in thesame figure), that is, in correspondence with the heaviestcomponents of the building.

Figure 15 shows the measured and calculated settlements

at the ground surface along the left and right longitudinalfacades of the building and along its transversal right side,as obtained for the tunnel face located at the middle of thestructure (point M in Fig. 3) and at the end of tunnelexcavation. In both cases, a satisfactory agreement betweenthe numerical outcomes of analysis STR and the monitoringdata is observed, confirming the capability of the proposedfinite-element simulation to provide realistic results. In con-trast, the free-field results lead to less intense settlementsand rather overestimated differential ones.

An attempt to analyse the structural response, in terms ofnormal compression forces (N) acting at the base (z ¼ 0) ofthe columns located along the longitudinal facades of thebuilding, is presented in Figs 16(a) and 16(b) together withtheir computed final settlements. The N values predictedbefore tunnelling are approximately constant, their distribu-tion being more regular along the left side of the buildingdue to the corresponding more regular column distribution.Once the excavation process has been completed, the dis-tribution of N becomes modified: in general, N decreases forthe columns that experience larger settlements, whereas itincreases for those which settle the least. This expectedpattern, which is more evident for the left facade, should beascribed to the force-transfer mechanism, which is enhancedby the presence of the cross-bracings.

Results of interaction analyses with simplified structuralmodels

Additional interaction analyses were carried out adoptingsimplified schematisation of the building, as described in theprevious subsection on ‘Numerical modelling of the building’,in order to highlight the role of the structural components onthe overall stiffness and, thus, on the computed displacementfield. In particular, the following cases were also considered.

• The stiffening role of the cross-bracings was investigated

12

8

4

0

�30 �20 �10 0 10 20 30

Set

tlem

ent,

: mm

Sv

Distance from tunnel axis, : m(a)

x

I II VII III IV V

12

8

4

0

�30 �20 �10 0 10 20 30 40 50

Set

tlem

ent,

: mm

Sv

Distance from tunnel axis, : m(b)

y

IIIVIIII

FF_ 0·32%VL �

STR_ 0·36%VL �

Fig. 14. Comparison between computed final settlement profilesof FF and STR analyses evaluated along the middle section of thebuilding in (a) the transversal and (b) longitudinal directions tothe tunnel axis (the corresponding volume loss values are alsoreported in the key)


200

400

600

800

1000

1200

1400

0

3

6

9

0 5 10 15 20 25 30

STR analysis

Nor

mal

forc

e,: k

NN

Building left side, : m(a)

l

Set

tlem

ent,

: mm

Sv

200

400

600

800

1000

1200

1400

0

3

6

9

0 5 10 15 20 25 30

STR analysis

Nor

mal

forc

e,: k

NN

Building right side, : m(b)

l

Set

tlem

ent,

: mm

Sv

200

400

600

800

1000

1200

1400

0

3

6

9

0 5 10 15 20 25 30

STR analysiswcb

Nor

mal

forc

e,: k

NN

Building left side, : m(c)

l

Set

tlem

ent,

: mm

Sv

200

400

600

800

1000

1200

1400

0

3

6

9

0 5 10 15 20 25 30

STR analysiswcb

Nor

mal

forc

e,: k

NN

Building right side, : m(d)

l

Set

tlem

ent,

: mm

Sv

Column position Sv N before tunnelling N after tunnelling

Fig. 16. STR and STRwcb analyses: normal compression force and settlement values at the base of the columnson the left (a, c) and right (b, d) longitudinal sides of the building

12

9

6

3

0

0 5 10 15 20 25 30

Set

tlem

ent,

: mm

Sv

Longitudinal left side, : m(a)

l

12

9

6

3

0

0 5 10 15 20 25 30

Set

tlem

ent,

: mm

Sv

Longitudinal right side, : m(b)

l

Measurements_middle

Measurements_end

FF (point M)

STR (point M)

FF

STR

12

9

6

3

0

0 3 6 9 12 15S

ettle

men

t,: m

mS

v

Transversal right side, : m(c)

l

Fig. 15. Comparison of monitored and computed settlements of FF and STR analyses on the longitudinal (a) left and(b) right sides and (c) on the transversal right side of the building


by the analysis denoted as STRwcb, where the buildingwas modelled without these components.

• Analysis STRw investigates the stiffening contribution ofthe buried portion of the building, characterised by thepresence of foundation elements and retaining walls, thelatter being 0.4 m thick and 3.5 m high, connectingthe foundation level to the ground floor. In this analysisthe weight of the elevated structure was accounted for bya corresponding distribution of loads, as described earlierunder ‘Numerical modelling of the building’.

• The building was finally reduced to a plate of equivalentstiffness and weight in analysis STR�, as described later,in Appendix 3.

Figure 17 summarises all the monitored and computedsettlement profiles as observed and back-predicted along thelongitudinal and transversal sides of the building.

The comparison between the STRwcb and STR resultsproves that the absence of the cross-bracings does not sig-nificantly affect the overall displacement pattern (Fig. 17),while it can play a non-negligible role on the structuralforces. As shown in Figs 16(c) and 16(d), the absence ofthese elements reduces the force redistribution process with-in the structure: at the end of excavation, in fact, the Ndistribution is characterised by a different pattern as com-pared to analysis STR (Figs 16(a) and 16(b)), with relativelyhigher values in the inner columns for the left side of thebuilding (Fig. 16(c)) and less intense actions on the externalcolumns along the right side (Fig. 16(d)).

The displacement curves obtained by the analysis STRw,carried out disregarding the above-ground portion of thebuilding, are very similar to the STR ones, thus indicating

that, in this particular case, the buried portion of thestructure provides the most relevant contribution to theoverall stiffness. In particular, the differential settlementsalong the transversal sides of the building, in correspon-dence with the foundational elements, are practically coin-cident with those computed by the complete structuralmodel.

Finally, the results obtained using the equivalent plateschematisation are highly unsatisfactory and on the unsafeside, since the building stiffness is found to be largely over-estimated. In fact, the displacement field at the foundationlevel is characterised by almost rigid rotations along the foursides of the structure, without indicating any sagging orhogging deformative modes.

CONCLUSIONSThis paper presents the results of a coupled geotechnical

and structural numerical study aimed at investigating theresponse of a multi-storey building affected by tunnelling-induced settlements. This topic is currently relevant as theever-increasing demand for urban space leads to under-ground developments, which represent a possible cause ofstructural damage for surface structures. The study, con-ducted using a three-dimensional finite-element code, refersto the recent construction of the metro-line 5 in Milan(Italy).

The soil behaviour is described by a non-linear elasto-plastic constitutive model (termed ‘hardening soil withsmall-strain stiffness’) calibrated with reference to in-situtests. The main aspects of the excavation process are repro-duced in the 3D numerical simulation of the EPB tunnelling.

12

9

6

3

0

0 5 10 15 20 25 30

Set

tlem

ent,

: mm

Sv

Longitudinal left side, : m(a)

l

12

9

6

3

0

0 5 10 15 20 25 30

Set

tlem

ent,

: mm

Sv

Longitudinal right side, : m(b)

l

12

9

6

3

0

0 3 6 9 12 15

Set

tlem

ent,

: mm

Sv

Transversal left side, : m(c)

l

12

9

6

3

0

0 3 6 9 12 15

Set

tlem

ent,

: mm

Sv

Transversal right side, : m(d)

l

Measurements_end STR

STRW

STRwcb

STR*

Fig. 17. Comparison of monitored and computed final settlements on the longitudinal (a) left and (b) right sides, and(d) on the transversal right side of the building. (c) The computed settlement profiles on the transversal left side arealso compared


One novel element of the proposed study is the detailadopted in modelling an existing reinforced concrete build-ing underpassed by the metro-line. The mechanical andgeometrical properties of its principal structural elements aredescribed in a realistic way, including the external infillpanels, which are schematised by means of weightless cross-bracings with equivalent stiffness.

The problem under investigation is initially discussedwith reference to the analysis of the geotechnical andstructural monitoring data, collected during the tunnellingactivities at six ground locations and along three sides ofthe reference building. It was found that the ground andthe structure experienced a maximum settlement at the endof the excavation process always lower than 7 mm, whilethe average volume loss value was equal to 0.33%, indi-cating a good performance of the adopted EPB machine.The ground vertical displacements can be well interpretedby Gaussian curves both along transversal and longitudinaldirections, in this latter case assuming a ratio between thesettlement at the tunnel face and the final settlement muchlower than the standard 0.5, due to the effect of the supportface pressure.

The paper presents and discusses a class C numericalprediction of the settlements induced by tunnelling underfree-field conditions and, subsequently, in the presence of asurface structure. The satisfactory comparison between thenumerical results and settlement measurements, includingthose along the structure, proves the reliability of theproposed finite-element model to capture the essentialmechanisms governing the problem.

As expected, the numerical results for the interactionanalysis clearly highlight the role of the structure stiffnessand weight on the settlement troughs, as compared to thefree-field ones.

It is also demonstrated that the contribution of the infillpanels appears to be negligible in terms of the overalldisplacement pattern, whereas it is shown to play a moresignificant role in the redistribution of the structural forcesacting in the vertical columns of the building during theexcavation process.

The model which considers only the buried portion of thebuilding, including the foundation elements, is found to fitthe monitoring data well: in terms of displacement pattern, itprovides almost equivalent results to those obtained by thecomplete analysis, highlighting the negligible stiffening roleof the structure above in this reference case study.

In contrast, for this particular building and foundationtypology, the equivalent plate schematisation involves a largeoverestimation of the structure stiffness, resulting in a highlyinaccurate displacement field as compared to that observedin situ.

ACKNOWLEDGEMENTSFinancial support provided by Astaldi S.p.A. in the person

of Eng. Enrico Campa is gratefully acknowledged.Special thanks go to Eng. Giuseppe Colombo of Milano

Serravalle – Milano Tangenziali S.p.A. (formerly AstaldiS.p.A.) and to Eng. Davide Fraccaroli and Eng. AlessandroCaffaro of Astaldi S.p.A. for providing the monitoring dataand for the technical support during the site activity.

APPENDIX 1. SOIL CONSTITUTIVE MODELIn the following, a summary of the main ingredients of the

hardening soil model with small-strain stiffness is provided, based onBenz (2007).

The elastic behaviour of the soil at medium strain levels (typicallyover 0.1%) is accounted for by isotropic elasticity using a stress-

dependent Young’s modulus, which is a function of the effectivestress and strength parameters according to the following expression

E9ur ¼ E9refur

c9 cos�9þ � 93 sin�9

c9 cos�9þ pref sin�9

� �m

(1)

where E9refur is the unloading/reloading Young’s modulus at the

reference pressure pref ¼ 100 kPa, c9 is the effective cohesion, �9 isthe angle of shearing resistance, � 93 is the minimum principaleffective stress and m is a constant that controls the linear or non-linear dependency of the stiffness on the above quantities. Similarexpressions are used to define the secant stiffness in standard drainedtriaxial test, E950, and the tangent stiffness for primary oedometerloading, E9oed:

The HSsmall model considers two additional stiffness parametersto take into account the soil behaviour at very small strains: thereference small-strain shear modulus, Gref

0 (i.e. the small-strain shearmodulus G0 at the reference pressure pref) and the shear strain atwhich the shear modulus is reduced to about 70% of its initial value,ª0.7.

In the case of the small-strain stiffness, the stress dependency isaccounted for by a power law which resembles the ones discussedabove for the other stiffness parameters

G0 ¼ Gref0

c9 cos�9þ � 93 sin�9

c9 cos�9þ pref sin�9

� �m

(2)

The HSsmall model implements the following modified version(Santos & Correia, 2001) of the stiffness reduction curve proposedby Hardin & Drnevich (1972)

Gs

G0

¼ 1

1þ a ª=ª0.7j j (3)

where Gs is the secant shear modulus and a is a constant equal to0.385.

The derivative of equation (3) with respect to the shear strainprovides the tangent shear modulus, Gt, which is bounded by a lowerlimit to scale back the stiffness to the value adopted at strain levels.The lower cut-off of Gt is introduced at the unloading/reloadingshear stiffness, Gur

Gt . Gur ¼Eur

2(1þ �ur)(4)

The HSsmall model is characterised by two yield surfaces whichevolve isotropically: a shear hardening yield surface, fs, which is afunction of the deviatoric plastic strain and a cap yield surface, fv,which is introduced to bound the elastic region for compressivestress paths and depends on the plastic volumetric strain. The elasticregion of the model can be further reduced by means of a tensilecut-off.

The associate flow rule is adopted for the cap yield surface fv,whereas a non-associate rule is employed for fs, adopting aformulation inspired by the well-known stress-dilatancy theory.

APPENDIX 2. DEFINITION OF THE CROSS-BRACINGS’PROPERTIES

The width, bw, of the cross-bracings is defined with reference tothe expression proposed by Mainstone (1971)

bw ¼ 0.175(ºhhw)�0.4dw (5)

where dw is the diagonal length of the panel, hw is the panel heightand the parameter ºh is equal to

ºh ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEwtw sin(2Ł)

4EcI chw

4

s(6)

with Ew and Ec being the Young’s moduli of the infill panel and ofthe reinforced concrete structural elements surrounding the panel,respectively; Ł is the angle formed by the diagonal of the infill panelwith respect to the horizontal axis; tw is the panel thickness (equal to40 cm); and Ic is the moment of inertia of the columns adjacent tothe infill panel. A reduced value of Ew ¼ 3 GPa was entered inequation (6) instead of the effective Young’s modulus of the infillpanels (about equal to 6 GPa) in order to take into account thediffuse presence of voids (doors or windows) on the building


facades, which contribute to reducing the overall stiffness of thestructures (Melis & Ortiz, 2003).

The cross-bracings are modelled as weightless one-dimensionalnode-to-node anchor elements reacting solely to axial stresses andcharacterised by an axial stiffness equal to Kw ¼ Ew 3 bw 3 tw. Anelasto-plastic constitutive law is selected for such elements tointroduce a limit value of the tensile strength equal to zero.Furthermore, the maximum value of the compression strength isevaluated according to the following expression

F lim ¼ 1.30�crAw (7)

where �cr is the shear cracking stress of the panel, assumed equal to0.2 MPa, and Aw is its transversal area, evaluated as the product ofthe panel length, lw, per its thickness, tw.

APPENDIX 3. DEFINITION OF THE EQUIVALENTPLATE PROPERTIES

The axial (EcA)building and bending (EcJ)building stiffnesses of thebuilding were calculated by considering that the structure consistedonly of floor slabs and was oriented with the longitudinal sidesparallel to the tunnel axis (such an hypothesis does not significantlyaffect the second moment of area of the slab):

(EcA)building ¼Xn

1

(EcA)slab (8)

(EcJ )building ¼Xn

1

(EcJ )slab ¼ Ec

Xn

1

(J slab þ AslabH2m) (9)

where n is the reference level of the building; Aslab and Jslab are thecross-sectional area and the second moment of area of the slab ateach level, respectively; and Hm is the vertical distance between theslab’s and the structure’s neutral axes (the latter assumed to belocated in correspondence with the structure’s centroid).

The building foundation system was neglected in this simplifiedapproach (Franzius et al., 2006). The computed axial and bendingstiffness for each slab are reported in Table 6, together with thethickness and Hm values.

The input parameters of the plate element used in the finite-element analysis were then evaluated as

tfe ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12(EcJ )building

(EcA)building

s(10)

Efe ¼(EcA)building

tfe

(11)

with tfe and Efe being the equivalent thickness and the Young’smodulus, respectively.

The unit volume weight of the plate element, which is equal to2.92 kN/m3, was calculated as the ratio between the total buildingweight (excluding the weight of the retaining walls modelled inSTR�analysis, equal to about 2500 kN) and the plate volume(B 3 L 3 tfe).

NOTATIONAslab cross-sectional area of slab

Aw transversal area of infill panelsB structure width

bw width of cross-bracingsc9 soil effective cohesion

cincrement increase of soil cohesion with depthD tunnel diameter

dw diagonal length of infill panelsEc Young’s modulus of reinforced concreteEfe Young’s modulus of equivalent plate

E9refoed reference tangent stiffness of soil for primary oedometer

loadingEur unloading/reloading Young’s modulus of soil

E9refur reference unloading/reloading stiffness of soilEw Young’s modulus of infill panels

E9ref0 reference Young’s modulus of soil at small strains

E9ref50 reference secant stiffness of soil in standard drained

triaxial testFlim maximum compression strength of cross-bracings

fs shear hardening yield surfacefv cap yield surface

Gs secant shear modulus of soilGt tangent shear modulus of soil

Gur unloading/reloading shear modulus of soilG0 shear modulus of soil at small strains

Gref0 reference shear modulus of soil at small strainsH structure height

Hm vertical distance between the slab’s and structure’s neutralaxis

hw height of infill panelsIc moment of inertia of the columnsIP index of plasticityix transversal inflection pointiy longitudinal inflection point

Jslab second moment of area of slabK trough width parameter

Kw axial stiffness of cross-bracingsK0 coefficient of earth pressure at rest

Knc0 coefficient of earth pressure at rest in a normal

consolidated statek permeability coefficientL structure length

M structure middle-pointm power of stress-level dependency of stiffnessN normal compression force in building columnsn reference level of building

pref reference pressureRf failure ratioSv settlement

Sv,max maximum settlementSv,f settlement at tunnel facetfe thickness of equivalent platetw thickness of infill panels

VL volume lossz0 depth of tunnel axis

�max maximum angular distortionª shear strain, unit volume weight for soilªc unit weight of volume for reinforced concrete

ª0.7 shear strain at which Gs ¼ 0.7G0

˜max maximum relative deflection(˜/L)max maximum deflection ratio

�smax maximum differential settlementŁ angle formed by diagonal of infill panel with respect to

horizontal axisŁmax maximum slopeºh parameter depending on geometrical and material properties

of infill panels and reinforced concrete structural elements�c Poisson coefficient of reinforced concrete�ur unloading/reloading Poisson coefficient

�tension tensile strength� 93 minimum principal effective stress�cr shear cracking stress of the infill panels�9 effective friction angleł dilatancy angleø rotation

Table 6. Stiffness properties of the slabs at each level

Level, n Slabthickness: m

Hm: m EcAslab: kN EcJslab: kNm2

Basement floor 0.26 15.70 1.95 3 108 1.10 3 106

Ground floor 0.26 13.20 1.95 3 108 1.10 3 106

First floor 0.22 9.00 1.65 3 108 6.66 3 105

Second floor 0.22 5.80 1.65 3 108 6.66 3 105

Third floor 0.22 2.60 1.65 3 108 6.66 3 105

Fourth floor 0.22 0.60 1.65 3 108 6.66 3 105

Fifth floor 0.22 3.80 1.65 3 108 6.66 3 105

Sixth floor 0.22 7.00 1.65 3 108 6.66 3 105

Seventh floor 0.22 10.20 1.65 3 108 6.66 3 105

Eighth floor 0.22 13.40 1.65 3 108 6.66 3 105

Ninth floor 0.22 16.60 1.65 3 108 6.66 3 105


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