Tunnelling and Underground Space Empirical and semi-analytical methods for evaluating tunnelling-induced ground movements in sands

Empirical formulas and closed-form solutions provide, in many cases, good predictions of tunnelling-induced ground movements which, when combined with their computational efficiency, make them valuable for tunnel-structure interaction analyses. For sandy soils, however, there is a shortage of available methods that can adequately describe the changes in soil deformation patterns that occur as a result of tunnel volume loss, soil relative density, and geometrical parameters. In this paper, two approaches are adopted to describe centrifuge test outcomes for tunnelling in dry silica sand of varying relative density. Firstly, empirical expressions for the prediction of settlement trough shape and magnitude are presented; additionally, a set of equations is given to describe the settlement troughs using modified Gaussian curves. Next, semi-analytical expressions (modifying an elastic analytical solution for incompressible soil and ovalized tunnel) are developed to predict both vertical and horizontal movements within the ground. Results from both methods can capture the main effects that cover-to-diameter ratio, relative density, and volume loss have on surface and subsurface ground movement profiles. The proposed expressions can be used for the calibration/verification of tunnel-structure interaction analysis methods by using outputs from comparable centrifuge tests; once verified, these methods may be more widely applied to other scenarios and used within design or risk-assessment exercises.


Introduction
During tunnelling, stress relief and over-excavation result in movements within the soil that may cause deformations of subsurface structures and foundations. To estimate possible structural damage, greenfield tunnelling-induced ground movements need to be predicted and used within tunnel-structure interaction analyses, where the term greenfield indicates a tunnelling scenario characterised by the absence of surface or buried structures.
To simplify the problem, engineers generally focus on the steadystate ground movements occurring transverse to the tunnel axis at a distance of several tunnel diameters behind the tunnel face. Ground movements induced by the construction of bored tunnels in clay have been widely monitored and discussed (Grant and Taylor, 2000;Mair and Taylor, 1997), and their predictions using simple empirical methods are well-established Peck, 1969). The mechanisms governing tunnelling-induced ground displacements in sands, however, differ from those in clays, mainly because of the lack of constraint on soil volume during shear (Marshall et al., 2012). There are a few reported case studies of tunnel construction in sands and coarse-grained soils (Bilotta and Russo, 2012;Fargnoli et al., 2013;Hsiung, 2011;Sagaseta et al., 1999;Jaarsveld et al., 1999), and centrifuge testing has been used to study the excavation of tunnels in dense sands (Marshall et al., 2012;Vorster et al., 2005;Franza et al., in press). Recently, Franza et al. (in press) presented a parametric study using centrifuge test data to better understand the effects of soil relative density and tunnel relative depth on greenfield ground deformations in sands. This paper aims to integrate the available centrifuge test data related to greenfield ground movements in sands into computationally efficient methods (empirical expressions and semi-analytical solutions) to enable an accurate replication of the ground movements measured within the centrifuge tests.
Empirical and semi-analytical solutions can provide an efficient and reliable way of evaluating tunnelling related ground displacements and their characteristics. However, methods for tunnels in sands are only partially available; for example there is no satisfactory empirical framework for the prediction of horizontal movements in sands (Marshall, 2009;Farrell, 2010) and available empirical methods for settlement trough estimation do not incorporate the effect of soil relative density (Marshall et al., 2012). Semi-analytical solutions have been implemented in numerical and analytical studies of tunnel-structure interaction, especially when horizontal ground displacements significantly affect the response of the system (Basile, 2014;Dias and Bezuijen, 2017;Haji et al., 2018a;Huang and Mu, 2012;Kitiyodom et al., 2005;Franza et al., 2017;Mu et al., 2012;Zhang et al., 2011Zhang et al., , 2013, whereas empirical methods have been used for tunnel-structure interaction in the case of buildings, tunnels, and pipelines (Basmaji et al., 2017;Klar et al., 2007;Marshall, 2008, 2015;Li et al., 2015;Son, 2015Son, , 2016Yu et al., 2013). These soil-structure interaction analyses are useful within design or risk assessment procedures, where the greenfield displacement input can be decided based on the specific scenario under consideration.

Scope of work and applicability of results
This paper presents two efficient methods that allow the replication of greenfield tunnelling-induced ground movements measured from centrifuge tests in sand. Experimental centrifuge test data (labelled in the paper as 'EXP') were obtained using a water-filled flexible membrane model tunnel, as reported in Marshall et al. (2012) and Franza et al. (in press). Empirical (labelled as 'EMP') methods developed by previous researchers are modified to improve the calculation of settlement trough characteristics (i.e. vertical movement only) and incorporate the effects of soil relative density. Then, semi-analytical (labelled as 'SA') formulas are proposed for both horizontal and vertical ground movements. Semi-analytical results are first compared against experimental data to evaluate the accuracy of the method. Subsequently, the predictions of the proposed semi-analytical methods are compared against solutions, available in the literature, of displacements for undrained clay to demonstrate the differences that occur when tunnelling in clay and sand.
The proposed empirical and semi-analytical formulas could be useful within soil-structure interaction analyses of centrifuge tests with comparable conditions, where an input of greenfield ground movements is required (e.g. Farrell et al., 2014;Franza et al., 2018;Lee and Chiang, 2007;Ritter et al., 2017). Once verified/ calibrated using centrifuge test outcomes, the developed soil-structure interaction analysis methods could then be more widely applied to real tunnelling cases. This paper does not attempt to directly relate outcomes from the centrifuge tests to field cases; the context of the paper focuses on centrifuge studies of tunnelling problems in sand.

The concept of volume loss, its magnitude and distribution
Volume loss relates the magnitude of movements that occur as a result of tunnelling. Soil volume loss, V l s , , is defined as the ratio between the volume of the settlement trough per unit length of tunnel, V s , and the final tunnel cross-sectional area, V 0 . Tunnel volume loss, V l t , , is the ratio between the ground loss at the tunnel periphery, V , and V 0 . The relationship between V l t , and V l s , is affected by ground conditions through the influence of volumetric strains (Marshall et al., 2012;Franza et al., in press). These volume losses are sketched in Fig. 1(a). Note that V l t , is a controlled variable for experimental, analytical and numerical studies, whereas it is back-calculated in the field from ground movements.
The assumed ground loss shape at the tunnel periphery plays an important role when modelling tunnel excavation. Some analytical approaches assume a uniform convergence around the tunnel boundary, however this assumption does not describe actual conditions adequately. Centrifuge modelling has confirmed that little ground   (Mair, 1979;Potts, 1976). The actual tunnel ground loss is distributed according to a roughly elliptical shape in clays (Rowe and Kack, 1983;Loganathan and Poulos, 1998), whereas the ground loss is concentrated at the tunnel crown in sands (Marshall, 2009;Zhou, 2014) (see Fig. 1(b)). Several methods are available to estimate ground loss at the tunnel, including the empirical envelopes given by Dimmock and Mair (2007), the cavity-contraction solutions proposed by  and Mo and Yu (2017), the gap parameter from Lee et al. (1992), or the theoretical study of Vu et al. (2016).

Empirical methods
Greenfield tunnelling-induced settlements, u z , are often described using a standard Gaussian distribution which includes the maximum settlement, u z max , , and the horizontal distance from the tunnel centreline to the inflection point of the curve, i: where i can be related to the vertical distance between the tunnel axis depth, z t , and the depth of interest, z, through the trough width parameter K: For the soil above a new tunnel,  related the parameter K to the relative depth z z / t as follows where K s is the value of K at the surface, and i z ( / ) is the slope of i with depth when all other parameters are kept constant. For clays, 0.325. On the other hand, for sands and gravels Mair and Taylor (1997) reported a significant scatter in the available data for K s , with values ranging from 0.25 to 0.45. Based on a regression analysis of field data, Jones (2010) proposed a logarithmic formula for the prediction of K in clays that depends on the height above the tunnel z z t rather than the relative depth z z / t . The modified Gaussian curve can better represent settlements induced by shallow tunnels in sandy soils (Vorster et al., 2005): where a is an additional fitting parameter with respect to the standard Gaussian curve. Studies have illustrated that the width parameter i in sands increases with the cover to diameter ratio, C D / , (Cover, where D is tunnel diameter) and decreases with the magnitude of tunnel volume loss, V l t , (Marshall et al., 2012;Sugiyama et al., 1999;Zhou et al., 2014).
The modified Gaussian curve is more versatile than the standard Gaussian curve, however the additional fitting parameter, a, does not have a physical meaning. Therefore, Marshall et al. (2012) suggested to characterise the shape of any empirical curve with three degrees of freedom, including the modified Gaussian curve, by the offsets x and x corresponding to the points x u [ , / e ] z max , is the settlement at the inflection point of the standard Gaussian curve (i.e. if = = a x i 0.5, ). Therefore, x and x are the horizontal distances between the tunnel centreline and the point where , respectively. In their empirical approach, x and x were related to the depth of interest, z, through To predict the change in settlement trough shape that occurs in dense sands as tunnel volume loss increases for varying C D / ratios, Marshall et al. (2012) proposed the following equations: where , and = x z x z ( / ) ( / ) 0.20. Note that these expressions were obtained based on a regression of centrifuge data that assumed a linear variation of the trough shape parameters with tunnel volume loss and C D / ratio. To assess the relationship between V l s , and V l t , at the surface, Marshall et al. (2012) proposed the use of an empirical formula valid for dense sands (relative density I d = 90%) and = C D / 1.3 4.4, which can be modified into the following form by considering that the relationship should have an intercept equal to zero (i.e. where the c coefficients depend on the type of sandy soil and its relative density. For dry dense silica sand, Marshall et al. (2012) proposed A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019)  , and = 0.5. In the Appendix, the complete framework is provided to illustrate how settlement trough shape can be predicted using modified Gaussian curves. In particular, Eq. (26) relates the degrees of freedom of the modified Gaussian curve defined in Eq. (4) (a i u , , z max , ) to the parameters of the empirical method (K K V , , l s , ). This framework, which has not been detailed before, is needed to be able to use the modified Gaussian based empirical method. Note that the empirical method parameters (K K V , , l s , ) may be related to any generic empirical curve with three degrees of freedom (e.g. the yield density curve from Celestino et al., 2000).

Analytical solutions
Analytical solutions solve for ground displacements based on an assumed displacement pattern at the tunnel periphery (uniform convergence, ovalization, vertical translation, as shown in Fig. 2, including the positive sign convention). This approach relies on the fact that farfield ground movements (i.e. at some distance from the tunnel periphery) depend predominately on the ground loss distribution around the tunnel (González and Sagaseta, 2001). There are two components of tunnel deformation: uniform convergence and ovalization. The tunnel volume loss is related to the convergence parameter as whereas the ovalization mechanism is not associated with tunnel ground loss.

Elastic solution for deep tunnels
For deep tunnels, it is legitimate to neglect the influence of the free surface. The problem of evaluating the displacements due to the excavation of a circular tunnel within an infinite isotropic elastic medium was solved by Pender (1980) for the excavation of a cavity in a prestressed medium for anisotropic initial stresses.
Within an infinite space, changes in the volumetric stress produce a uniform convergence of the tunnel periphery, u , and changes in the deviatoric stress (which are not associated with tunnel volumetric changes) induce an ovalization, u max , , as shown in Fig. 2. These displacements may be normalised by the tunnel radius, R, to define the tunnel deformation components = u R / and = u R / max , . As an alternative, the relative distortion parameter = / can be used. For deep tunnels, the parameters for convergence, , and ovalization, , relate only to the soil elastic parameters and the initial stresses.

Elastic solutions for shallow tunnels
A number of studies have investigated the application of the superposition of singularities method to the problem of tunnelling-induced ground displacements in a half-space. For this approach, and are considered as input parameters regardless of their relationship to soil and tunnel parameters. Their values are generally obtained by back-analysis and curve-fitting of field measurements and may not reflect realistic conditions.
The first application of the superposition of singularities method was proposed by Sagaseta (1987) to evaluate soil movements in undrained conditions. It was assumed that a shallow void of radius R v (whose area represents the tunnel volume loss, = V R v 2 ) and located within a half-space at depth z t is created and filled by the surrounding incompressible (Poisson's ratio = 0.5), linear, and isotropic soil. Verruijt and Booker (1996) used the equations of Pender (1980) within the superposition of singularities method to account for ground compressibility (through Poisson's ratio) and the tunnel ovalization deformation mechanism. In the superposition of singularities method, the input deformation parameters, and , do not induce perfect uniform contraction and ovalization of the tunnel in the half-space scenario (as they would in the full-space) due to the influence of the surface. As discussed by Pinto and Whittle (2006), this aspect is often not correctly accounted for by researchers. Exact solutions of movements induced by ground loss, ovalization and buoyancy (due to the self-weight of the removed soil) in the elastic half-space have been obtained by using the complex variable method (Fu et al., 2016;Verruijt, 1997;Strack and Verruijt, 2002;Verruijt and Strack, 2008;Zhang et al., 2018). However, the superposition of singularity method provides a sufficiently accurate approximation of the exact solution in the case of ground loss and tunnel ovalization (especially for > C D / 2) (Pinto et al., 2014). Elastic solutions can adequately match surface and subsurface greenfield deformation patterns measured in the field for clays (Ieronymaki et al., 2016(Ieronymaki et al., , 2018Pinto et al., 2014). However, the elastic approach of Verruijt and Booker (1996) over-predicts the horizontal and vertical settlements in the far field, particularly for low ovalization terms, with wider troughs and larger horizontal movements than observed in the field (Loganathan and Poulos, 1998). González and Sagaseta (2001) modified the Verruijt and Booker (1996) elastic solution for an incompressible medium ( = 0.5) to account for the effect of soil volumetric behaviour by means of a compressibility term . They introduced based on the fact that, in a nonelastic medium, the displacements attenuate with a power law that relates to the distance within the plastic zone, O r

Elastoplastic solutions for shallow tunnels
(1/ ). The compressibility parameter should be assumed equal to the mean value within the soil; is lower or greater than unity for an overall compressive or dilative behaviour of the soil, respectively. For = 1, the elastic solution of Verruijt and Booker (1996) for the incompressible medium is obtained.
The efficiency of the closed-form solution provided by González and Sagaseta (2001) was assessed against centrifuge data by Franza and Fig. 2. Tunnel displacement components after González and Sagaseta (2001).
A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019) 47-62 Marshall (2015a). Rather than calibrating the input deformation parameters predominately at the ground surface based on sparse sets of data (e.g. González and Sagaseta, 2001;Pinto et al., 2014;Ieronymaki et al., 2016), the value of = 1 was used to give the correct representation of the centrifuge model tunnel deformation mechanism (as shown in Fig. 1(b)) and a compressibility parameter = 0.9 1.3 was adopted to reflect the relationship between V l s , and V l t , in the centrifuge tests, which were done in dry dense silica sand. Fig. 3 displays the effects of the ovalization term and the compressibility parameter on the normalised surface settlement trough. According to Franza et al. (in press), the value of i in sandy soils decreases with tunnel volume loss from values of approximately z (0.5 0.7) t to z (0.35 0.5) t (depending also on the relative tunnel depth). This range is represented in Fig. 3 by the grey shaded areas and lines 1-3. Comparing the = 1 line (which is physically appropriate) with the Gaussian curves in Fig. 3a shows that this input provides an analytical surface settlement trough that is too wide, especially nearer the surface. Using a higher value of (as in Fig. 3(a)) or (as in Fig. 3(b)) can give a better match to the shape of the Gaussian curve, however this would not be realistic when considering the physical meaning of the parameters. In addition, using = 1 with average values of representative of the entire ground mass does not provide good predictions of subsurface ground movements when compared against measurements from centrifuge tests (Franza and Marshall, 2015a).

Semi-analytical solutions
Semi-analytical solutions have been developed by applying an empirical corrective term, , to displacement patterns from elastic solutions, u el , such that the semi-analytical displacement field is = u u sa el . Loganathan and Poulos (1998) proposed the evaluation of vertical and horizontal undrained movements in clays based on the elastic solution of Verruijt and Booker (1996) using an ovalization term = 0 and by substituting the modified equivalent ground loss parameter x z , to : where R is the tunnel radius, z t is the depth to the tunnel axis, x is the distance from the tunnel centreline, z is depth, is the Poisson's ratio of the soil, = V /100 is the equivalent ground loss parameter. As discussed by Pinto and Whittle (2006), the provided x z , is the outcome of a calibration process in which the corrective term was conveniently chosen to account for field observations and centrifuge model test outcomes.

Centrifuge data of greenfield tunnelling
Empirical and semi-analytical methods are used in this paper to replicate results from the centrifuge tests of plane-strain tunnelling presented by Marshall et al. (2012) and Franza et al. (in press), which compile data from Marshall (2009), Farrell (2010, Zhou (2014), Franza (2016).
The centrifuge models included a strong box, soil, model tunnel, and a tunnel volume control system (Fig. 4 shows the set-up used by Marshall (2009); refer to Marshall (2009), Zhou (2014, Franza (2016) for full details). All tests used a dry silica sand known as Leighton Buzzard Fraction E. The model tunnel comprised a metallic cylinder with enlarged ends which was covered by a latex sleeve and filled with water. The plane-strain centrifuge strongboxes consisted of a Uchannel, a Perspex front wall, and a metallic back wall. The tunnel volume loss V l t , was induced by extracting water from the model tunnel using a volume control system: a constant-head standpipe, a solenoid valve, a linear actuator, a water-filled sealed cylinder and a linear variable differential transformer were used to (1) compensate, during centrifuge spin-up from 5g to N g (where g is gravity and N is the centrifuge acceleration scale factor), for volume loss within the system due to the compressibility of air trapped within the water; (2) upon reaching the target N g-level, extract a specific volume of water from the tunnel, replicating the tunnel volume loss V l t , . The transparent Perspex wall allowed for digital images to be taken at each increment of V l t , for subsequent analysis using GeoPIV (White et al., 2003) to determine subsurface ground movements.
Tests included soil with relative density (I d ) of 30% (loose), 50% (medium-dense), and 90% (dense), and cover-to-diameter ratios (C D / ) of 1.3, 2.4, 4.4, and 6.3. Table 1 summarises the geometry and soil density of the performed tests, including the adopted N g level of each test. Tests are labelled according to their C D / ratio and I d (i.e. a test with C D / = 6.3 and I d = 0.9 is labelled CD6.3ID90). Centrifuge tests were carried out using model tunnels with varying diameter. Thus, in this paper, centrifuge data are normalised with respect to the tunnel radius R. Importantly, regression analyses of empirical and semi-analytical expressions are limited to the range .

Trough width -existing methods for settlements in dense sand
The empirical approach of Marshall et al. (2012) was developed based on data from experiments in dense sand and C D / between 1.3 and 4.4. To investigate the effects of high C D / values, Franza et al. (2016) performed regression analyses of K and K for dense sand (tests CD1.3ID90, CD2.0ID90, CD2.4ID90, CD4.4ID90, and CD6.3ID90); data were curve-fitted with the surface described by Eqs. (6) and (7) proposed by Marshall et al. (2012) and scalar values of the six trough shape Fig. 3. Normalised analytical surface settlement trough: effects of (a) tunnel ovalization and (b) ground compressibility (Franza and Marshall, 2015a).
A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019)  ) were obtained. Franza et al. (2016) showed that the variation of these six shape parameters with C D / has a non-linear trend due to a transition between shallow and deep tunnels.
Results obtained using Eqs. (6), (7) and (11)  . On the other hand, Fig. 5(a) shows that the comparison is less good for test CD1.3ID90, especially at high volume losses and greater depths. This is because the empirical expressions give a qualitatively different trend of the width parameters with depth, which results in predicted values of K and K being significantly lower than the experimentally obtained values (and close to an unrealistic value of zero for K at greater depths). Similar results were obtained at low C D / values by Marshall et al. (2012), who assumed a linear variation of the trough shape parameters. In the next section, a new set of equations is
A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019) 47-62 presented that overcomes this issue at low C D / and that also accounts for the effect of soil relative density.

Trough width -new approach for settlements in sand with varying relative density
A new approach is proposed to assess settlement trough width parameters, K and K , as a function of I C D V , / , d l t , , and z z / t , based on Eq. (6). To improve performance, the non-linear trends of x x K , , s , and K s illustrated by Franza (2016) are accounted for in Eq. (12) with logarithmic functions (e.g. the partial derivatives of x and x were obtained assuming a logarithmic trend of x and x with + z z 1 / t ).
, that are dependant on the variables C D / and I d . The values of these coefficients were defined based on a regression analysis including the entire dataset of tests (groups F, M, Z) for The obtained regression analysis results are shown as markers in Fig. 6. The plots illustrate the variation of the six coefficients of Eq. (12) with C D / for varying soil density (I d is plotted using different colours and marker-shapes).
Based on the logarithmic reduction of width parameter coefficients with C D / due to a transition from shallow to deep tunnels, it may be assumed that where is used to indicate one of the obtained coefficients (i.e.  Considering that the settlement trough width decreases with soil density for relatively deep tunnels, for the sake of simplicity, the interpolation of the regression analysis results (i.e. markers in Fig. 6 (14), which are plotted as solid lines in Fig. 6. Although this method does not account for the complex pattern of results corresponding to = C D / 2.0, which is due to the variation of the arching mechanism with soil density (Franza et al., in press)

M M K
In addition, to guarantee the physical sense of Eq. (4) as a settlement trough, the condition is required to satisfy > n 0, where the width parameters calculated with empirical equations are indicated as K EM , and K EM , . Further details are provided in Appendix A.
The performance of Eqs. (6), (12) and (14) against the centrifuge measurements of K and K is illustrated in Fig. 7 and 5% (the supplemental data has a figure which includes = V 2.5 Overall, the empirical predictions are shown to provide a satisfactory match with both K and K . Additionally, it can be seen that the proposed expressions are able to describe the variation of settlement trough width with soil density (which has not previously been done). Furthermore, there is a significant improvement in the prediction of subsurface values of K and K at = C D / 1.3 with the new approach (compare Figs. 5 and 7; also note that more comprehensive plots of the width parameters for are available in the supplemental data). The improved performance is likely the result of the assumption of logarithmic trends of the main variables, which takes into account the fact that [i] the process of soil stiffness degradation with strains is non-linear, and [ii] the deeper the tunnel, the more the displacement mechanism close to the surface should converge towards an elastic pattern.
In summary, Eqs. (6), (12) and (14) may be used for the prediction of the centrifuge test results of surface and subsurface settlement trough shape parameters for loose, medium-dense and dense sands.

Relationship between soil and tunnel volume loss
The relationship between V l s , and V l t , is required to fully define empirical settlement curves (e.g. the standard or modified Gaussian   Fig. 6. Outcomes of settlement trough parameter regression using new approach for varying I d. A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019) 47-62 curves) at any depth. The approach defined by Eq. (8) was extended to consider the entire data set indicated in Table 1. The soil used for all tests was the same dry silica sand, hence the values of coefficients c c , b c , and c d were assumed as in Eq. (8). Zhou (2014), in analysing test group Z, suggested that the coefficient should be a linear function of relative density. Furthermore, analyses of the full data set indicated that, for loose samples, the description of the experimental V l s , versus V l t , curves required an additional coefficient, , for These empirical expressions give quantitative guidance on subsurface soil volume losses, which has not been previously provided.
As displayed in Fig. 8, the predicted V l s , versus V l t , values (markers) give a good fit with experimental data (lines); the regression of the data set resulted in = R 0.98 2 , 0.95 and 0.94 for = z z / 0 t , 0.25 and 0.5, respectively. Note that Eq. (15) does not account for the effects of confining stress because the tunnel cover is normalised by its diameter. Therefore, judgement should be used before applying Eq. (15) to real scenarios which differ considerably to the prototype scenarios considered here. Plots of the soil volume loss against tunnel volume loss for  A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019) 47-62 = C D / 1.3, 2.0, 2.5, 4.5, 6.3 are available in the supplemental data. To illustrate the performance and limitations of the proposed empirical framework consisting of Eqs. (6), (12), (14) and (26), predictions (EMP-mG; solid lines) are compared against centrifuge data (EXP; markers) and a simpler (conventional) method using a standard Gaussian curve (EMP-sG; dashed lines) in Fig.  9 for C D z z / 1.3;2.5;4.5, / t = 0;0.5, V l t , = 2;5%, and both loose (I d = 0.3) and dense (I d = 0.9) sands. The width parameter at the surface for the standard Gaussian curves was taken as = K 0.35 (Mair and Taylor, 1997) and it was assumed that , . Results illustrate the significant influence of soil density on both the variation of settlement trough shape and the magnitude of movements. The differences between EMP-sG and EMP-mG would have important implications to the assessment of tunnel-structure interactions (e.g. in terms of maximum settlements (Mair et al., 1996), settlement trough shape (Klar et al., 2007;Giardina et al., 2015;Franza et al., 2017) or settlement trough extent (Haji et al., 2018b)).

Semi-analytical solution
This section presents a semi-analytical solution (SA) for the replication of the centrifuge test greenfield vertical and horizontal ground movements. The method is an extension of the work provided in Franza and Marshall (2015b) and Haji et al. (2018a) and is based on the elastic solution of Verruijt and Booker (1996), with the following details. (1) The case of an elastic medium for incompressible soil ( = 0.5) was adopted.
(2) The relationship = ( = 1) was used to replicate the observed experimental tunnel deformation mechanism (centrifuge data indicate negligible u x at the tunnel springline for , as shown by Franza et al. (in press)). (3) In the 'new method' introduced in this paper, two corrective terms are applied: x and z in the horizontal and vertical directions, respectively. Further details on the corrective terms are given in following sections. The semi-analytical solution for horizontal (u x ) and vertical (u z ) displacement can therefore be written as Fig. 9. Comparison between centrifuge data (EXP) and settlement troughs using modified Gaussian curves with proposed empirical formulas (EMP-mG) and conventional approach using standard Gaussian curves (EMP-sG).
A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019) is the tunnel convergence parameter. An indication of the spatial distribution of the corrective terms can be obtained by plotting the ratio between the centrifuge displacement data and the elastic analytical predictions without the corrective term applied (i.e. using Eqs. (16) and (17) c)). For dense sands, soil arching, discussed by Franza et al. (in press), induces localised vertical and horizontal movements at the tunnel crown that impact the experimental ratios in terms of total, vertical, and horizontal movements, as shown by Fig. 10. In addition, soil relative density results in significant differences between the qualitative spatial distributions of the experimental ratios (compare Figs. 10 and 11).
In the original Franza and Marshall (2015b) method, the corrective term was determined based on from a single centrifuge test data set (CD2.4ID90); thus, it was assumed = = z x . In this paper, to account for the effects of soil relative density on the ratios z and x illustrated in Figs. 10 and 11, the semi-analytical approach uses separate corrective terms in the horizontal and vertical direction ( x and z , respectively). This approach was extended to include the full data set presented in Table 1, thereby also including for the effects of cover to diameter ratio, C D / , and soil relative density, I d .

Corrective term -original method
where coefficients = + depend linearly on V l t , to describe the narrowing of the settlement trough and development of a localised zone of large displacements at the tunnel crown that occur as tunnel volume loss increases. Eq. (18) consists of two three-dimensional Gaussian functions of both spatial coordinates x and z. The first Gaussian function with coefficients c c , A 1 and c 2 allows interpolating at lower volume losses (Fig. 10(a); left-side). For the first Gaussian surface, c A is the function amplitude whereas c 1 and c 2 are the attenuation factors in the x and z directions, respectively. At higher volume losses, exhibits an additional peak in the proximity of the tunnel crown ( Fig. 10(a); rightside). To provide a good fit, an additional Gaussian function was introduced in , which has its centre at c H (0, ) 4 , amplitude c B , and attenuation factor c 3 in both spatial directions. The accuracy of Eqs. (16)-(18) in terms of vertical displacement prediction was good, however the solutions over-predicted subsurface horizontal movements because the calibration of the corrective factors was done on the basis of total displacements. This is because the vertical movements have a larger magnitude than the horizontal movements.

Corrective term -new method
The semi-analytical approach was extended to a wider set of centrifuge data, including the effects of cover to diameter ratio, C D / , and soil relative density, I d . The two corrective terms z and x given in Eq. (19) were implemented in the vertical and horizontal directions, respectively, with coefficients depending on I C D , / d , and V l t , . The need for different terms in the x and z directions was judged necessary to improve the accuracy of results in the horizontal direction compared to the 'original method'. To include for the effects of C D / and I d while allowing for settlement troughs wider than the elastic displacement field, two additional coefficients (c 5 and c 6 ) were needed. A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019)

Comparison of semi-analytical solution predictions with centrifuge data
An indication of the accuracy of the semi-analytical formulas given by Eqs. (16), (17) and (19) (solid lines, labelled 'SA') is presented in Fig. 12 by comparing outputs with centrifuge data (markers, labelled 'EXP') from six tests with varying relative density and cover-to-diameter ratio ( = = I C D 30, 90%; / 1.3, 2.5, 4.5 . The displacement values were normalised by V R ( ) l t , to allow comparison of data at varying volume losses and for tunnels of different sizes. In Fig. 12, the left y-axis displays the depth of the displacement profiles, while the scale of the normalised displacements is shown on the right y-axis, which differs for vertical and horizontal ground movements.
The 'new method' expressions presented here are able to replicate, with reasonable accuracy, the surface and sub-surface horizontal and vertical movements of the centrifuge data in sand as they are affected by tunnel volume loss, relative density and relative depth. The limitation of the 'original method' of Franza and Marshall (2015b), in terms of horizontal displacement prediction, was significantly improved by including the horizontal corrective term.
In addition, to illustrate the differences in tunnelling-induced displacement fields in clays and sands, the movements obtained from the semi-analytical expressions of Loganathan and Poulos (1998) for clays in undrained conditions (dashed lines, labelled 'S.A. Clay') are plotted against the 'new method' expressions (solid lines, labelled 'S.A. Sand') in Fig. 13. The soil deformation pattern in both vertical and horizontal directions for sands is significantly affected by the magnitude of tunnel volume loss, whereas it is constant with V l t , for clays. The differences in the settlement troughs for sands and clays are significant at depths closer to the tunnel axis, particularly at high volume losses, whereas the differences are smaller near the surface. The magnitude of horizontal displacements increases with depth in clays, whereas it decreases in sands, leading to wider subsurface horizontal movements in clays than in sands. These differences in horizontal ground movement patterns are due largely to the different tunnel ovalization mechanisms, where sandy soils have negligible inwards movement at the tunnel springline and clays have an oval-shaped tunnel contraction. These results should not be generalised to real excavations performed with tunnel boring machines that use tail-skin grouting, for which there could be outwards horizontal displacements at the tunnel periphery for large grout pressures (Bilotta and Russo, 2012;Dias and Kastner, 2013).

Conclusions
This paper deals with the analysis of centrifuge test data of greenfield tunnelling in sands and the development of efficient empirical and semi-analytical equations that can be used to replicate the centrifuge test displacements. The empirical method, based on the framework from Marshall et al. (2012), can be used to determine vertical movements; the new equations give improved performance at low cover-todiameter ratios and extend the previous work by also considering the effect of soil relative density. Empirical expressions were also provided that give the relationship between surface and subsurface soil volume losses for varying soil relative density. Finally, a framework for the prediction of settlement trough shape based on the modified Gaussian Fig. 11. Total ( ), vertical ( z ), and horizontal ( x ) ratios between centrifuge and elastic displacements for CD2.5ID30.
A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019) 47-62     A. Franza and A.M. Marshall Tunnelling and Underground Space Technology 88 (2019) 47-62 curve was provided (Appendix A), which provides a straightforward approach that can be readily applied by researchers and engineers. When subsurface and horizontal ground movements are used for tunnel-structure interaction analyses, it is necessary to assume a consistent soil deformation pattern. A computationally efficient semi-analytical solution involving the modification of an existing elastic analytical solution for incompressible soil was presented for the replication of vertical and horizontal ground movements obtained from centrifuge testing. This solution is able to account for the complex effects of tunnel volume loss, cover-to-diameter ratio, and soil relative density observed in sands, something that has not previously been achieved.
With appropriate judgement, the proposed empirical and semianalytical equations can be used to define the input for tunnel-soil-structure interaction analyses. These inputs are most directly applicable to soil-structure interaction tests done with comparable centrifuge test methods. Once validated with the centrifuge test outcomes, the soilstructure interaction analyses could be more widely applied to other scenarios and used within design and risk-evaluation exercises. and 4%: comparison between experimental and semi-analytical profiles.

A.1. Equations of the modified Gaussian curve
The modified Gaussian curve is defined in Eq. (4). It has three degrees of freedom (a and i defining the shape and u z max , defining the magnitude), where the conditions > > a n 0, 0 and i 0 guarantee the physical sense of the equation as a settlement trough. In the empirical approaches, the modified Gaussian curve shape was defined through x and x . From the definition of x and x , we may obtain Combining Eqs. (5), (20) and (21) The lower limit of K K / was obtained from and 4%: comparison between profiles in clays against those in loose (a,c,e,f) and dense (b,d,f,h) sands.