Greenfield tunnelling in sands: the effects of soil density and relative depth

Tunnel construction is vital for the development of urban infrastructure systems throughout the world. An understanding of tunnelling-induced displacements is needed to evaluate the impact of tunnel construction on existing structures. Recent research has provided insight into the complex mechanisms that control tunnelling-induced ground movements in sands; however, the combined influence of relative tunnel depth and soil density has not been described. This paper presents data from a series of 15 plane-strain centrifuge tests in dry sand. The cover-to-diameter ratio, C/D, of the tunnels ranges between 1·3 and 6·3, thereby including relatively shallow and deep tunnels. The sand relative density varies between 30 and 90%, corresponding to loose and dense soils. The effects of C/D, soil density and volume loss on vertical and horizontal soil movements, shear strains and ground reaction curves are discussed. Analysis of surface and subsurface settlement trough characteristics shows that the mechanisms are non-linear and the effects of soil relative density and volume loss on deformation patterns are highly dependent on C/D. The role of soil arching in the definition of the displacement mechanisms and a discussion of the implications of the results to the assessment of damage to existing structures are also provided.


INTRODUCTION
Current needs for infrastructure development in urban areas require the construction of new tunnels. Because the excavation of new tunnels results in ground movements that affect existing surface and buried structural systems, engineers need to estimate the magnitude and distribution of greenfield displacements. In particular, prediction of the settlement trough shape is necessary because of the potential to induce differential settlements on structures; for instance, a narrow settlement trough (high curvature) with large maximum settlement has a significant potential to cause structural damage.
Ground movements due to greenfield tunnelling in undrained clay are generally well understood (Mair et al., 1993;Mair & Taylor, 1997;Grant & Taylor, 2000). However, the available data for ground movements above tunnels in coarse-grained soils from case studies (Dyer et al., 1996;Sagaseta et al., 1999;Fargnoli et al., 2013) and centrifuge tests Marshall et al., 2012) are rather limited. A greater variability of the characteristics of settlement troughs above tunnels in sands has been noted within the literature (e.g. Mair & Taylor (1997), who reported a typical range of surface settlement trough width parameter of 0·25-0·45, but values as high as 1·0 were also provided). The causes for this variability are not well known; this is one of the motivations behind the research presented here.
There have been numerous investigations of tunnelling effects on pipelines and buildings that have incorporated data related to sandy soils (Klar et al., 2007(Klar et al., , 2015Farrell et al., 2014;Giardina et al., 2015;Ritter et al., 2017). A critical component to these interaction analyses is the greenfield input; however, there remains a lack of understanding of the impact of soil density and relative tunnel depth on greenfield ground displacements for tunnels constructed in drained sandy ground.
The aim of this paper is to address these shortcomings by examining data obtained from 15 geotechnical centrifuge tests of tunnel construction in sand. The paper is structured as follows. Available empirical methods for the prediction of settlement trough shape are first summarised. An overview of the centrifuge test programme is then presented. Experimental data are used to demonstrate the main effects of cover-to-diameter ratio (C/D), soil relative density (I d ) and tunnel volume loss (V l,t ) on greenfield ground movements. In particular, ground reaction curves (i.e. the change of internal tunnel pressure with tunnel volume loss) as well as ground displacements and strains around the tunnel are presented. The influence of soil arching on the displacement mechanism and volumetric strains is also discussed. Subsequently, empirical expressions are fitted to surface and subsurface settlement through data so that the characteristics of the settlement trough profiles can be analysed. Finally, the implications of results to the assessment of damage to existing surface and buried structures are provided.

EMPIRICAL METHODS
Tunnelling-induced ground movements are often described at a given depth by empirical formulas and ground loss parameters. Considering plane-strain conditions transverse to the tunnel direction, the magnitude of ground loss is commonly expressed with two parameters: tunnel volume loss, V l,t , and soil volume loss, V l,s . The concept of ground loss is illustrated in Fig. 1(a). Soil volume loss is defined as V l,s ¼ V s /V 0 Â 100, where V s is the volume of the settlement trough per unit length of tunnel and V 0 is the notional final area of the tunnel cross-section. In experimental, analytical and numerical studies, the ground loss is modelled through the tunnel volume loss, V l,t ¼ ΔV/V 0 Â 100; the ratio between the ground loss at the tunnel periphery, ΔV, and V 0 , expressed as a percentage. V s is generally based on surface settlements because of their relative ease of measurement. In field studies, the true value of V l,t is unknown. However, for tunnels constructed in clayey ground under undrained (constant volume) conditions, V l,t ¼ V l,s , so surface measurements can be used to evaluate volume loss at any depth. This is not the case for tunnels in drained granular soil where V l,t = V l,s ; the relationship between the two is affected by soil volumetric strains, which depend on ground conditions, the magnitude of shear strains and confining stress (Marshall et al., 2012). Finally, it should be noted that, for shallow tunnels, previous work has demonstrated that very little ground movement occurs below the tunnel invert; ground loss is distributed according to a roughly elliptical shape in clays (Rowe & Kack, 1983;Loganathan & Poulos, 1998) and is concentrated at the tunnel crown in sands (Zhou, 2014;) (see Fig. 1(b)).
The focus of this paper is on ground settlements (i.e. vertical movements). In clays, ground settlements, u z , generally conform to a standard Gaussian curve, with the maximum settlement, u z,max , and the horizontal distance of the inflection point, i, defining the curve The value of i was found to be proportional to the vertical distance between the tunnel axis depth, z t , and the depth of interest, z, through the width parameter K where the parameter K was defined in relation to the relative depth z/z t (Mair et al., 1993) K It is worth noting that Jones (2010) showed that the Mair et al. (1993) expression can overestimate the value of K for deep tunnels. Based on field data measurements, Jones proposed a logarithmic formula for the prediction of K which depends on the height above the tunnel z t À z rather than the relative depth z/z t . This results in a decrease of K at the surface as z t is increased.
In sandy soils, the use of the modified Gaussian curve rather than a standard Gaussian curve was suggested by Vorster et al. (2005) to obtain a better fit to settlements induced by shallow tunnels: where α is a fitting parameter. In particular, the additional degree of freedom represented by α allows for more effective curve fitting for narrow settlement troughs. Several studies have highlighted that, in sandy soils, the width parameter i increases with the cover-to-diameter ratio, C/D, and decreases with the magnitude of volume loss, V l,t (Sugiyama et al., 1999;Marshall et al., 2012;Zhou et al., 2014).
EXPERIMENTAL PROGRAMME To investigate the combined effect of the cover-to-diameter ratio, C/D, and the relative density, I d , a series of plane-strain tunnelling centrifuge tests was performed using a dry silica sand. The value of C/D ranged between 1·3 and 6·3, thereby including relatively shallow and deep tunnels. Tests were performed on uniform loose (I d ¼ 0·3), medium-dense (I d ¼ 0·5 and 0·7) and dense (I d ¼ 0·9) sands. 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). Most analyses in this paper are limited to the volume loss range V l,t = 0-5% (with the exception of data relating to tunnel inner pressure and ground losses); therefore collapse conditions were not necessarily reached. V l,t = 1, 2, 3 and 5% are referred to as low, medium, high and extremely high volume losses, respectively. Note that this dataset includes newly collected data from Franza (2016) and Zhou (2014) using the University of Nottingham centrifuge (groups F and Z), as well as results from Marshall et al. (2012) obtained using the Cambridge University centrifuge (group Msee Table 1). Groups F, Z and M were performed with the same soil and equivalent tunnel modelling techniques. Further details on the centrifuge equipment are provided in the next section.
It should be noted that this dataset does not account for the effects of confining stress because the tunnel depth is represented in normalised form as C/D. For instance, a tunnel with D ¼ 3 m and C ¼ 9 m has the same C/D as a tunnel with D ¼ 6 m and C ¼ 18 m. The levels of soil dilation around the deeper tunnel would be less than for the shallow tunnel, which would affect the characteristics of the settlement troughs above the tunnel. However, as the problem is limited to a reasonable domain (C/D , 6·3 and prototype z t , 20 m), the effects of stress level should be secondary.

EXPERIMENTAL SET-UP AND SPIN-UP EFFECTS
The centrifuge tests were performed in plane-strain conditions with a model tunnel buried at varying depths  Table 1, where N is the centrifuge acceleration scale factor. The centrifuge package includes a strong box, soil, a model tunnel and a tunnel volume control system (refer to Marshall (2009), Zhou (2014 and Franza (2016) for full details). A dry silica sand known as Leighton Buzzard fraction E was used for testing. The model tunnel consisted of a cylinder with enlarged ends covered by a sealed latex sleeve. The annular gap between the sleeve and cylinder was filled with water. The tunnelling process (i.e. V l,t ) was simulated by extracting water from the annular gap using a volume control system. The centrifuge strongbox consisted of a U-channel, a poly(methyl methacrylate) (PMMA) front wall and a metallic back wall; front and back walls were designed to ensure plane-strain conditions along the tunnel direction and minimise out-of-plane deformations. The transparent PMMA 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.
During the spin-up of the model, ground movements at the tunnel periphery are induced by the stress imbalance between the model tunnel and the surrounding soil (Ritter et al., 2018). This results in the ovalisation and buoyancy of the model tunnel during the spin-up phase, as well as soil densification and a variation of the soil relative density, ΔI d (Zhou, 2014). The overall impact of these issues on greenfield test outcomes was deemed to be minimal; for instance, for test CD2·0ID50 (medium-dense sand), Zhou (2014) reported an average soil densification of ΔI d % þ1% at the surface and a maximum ΔI d % þ5% at the tunnel crown. Atkinson & Potts (1977) provided a thorough evaluation of tunnel support pressure corresponding to the state of soil failure and proposed upper and lower bound solutions for its prediction. The pre-collapse transition of tunnel pressure with volume loss (i.e. the ground reaction curve (GRC)) is also of interest. GRCs for tunnels in cohesionless soils were suggested and compared against numerical analyses by Wong & Kaiser (1991), whereas Dewoolkar et al. (2007) and Iglesia et al. (2014) conducted investigations based on centrifuge modelling of a trapdoor. There have not been extensive centrifuge test investigations of the GRC for tunnels in cohesionless soils. The data presented here give a unique illustration of the effect of tunnel C/D ratio and soil relative density on the GRC based on centrifuge test results.

GROUND REACTION CURVES
Figure 2(a) shows the variation of σ norm , the normalised (to account for the differences of soil density and tunnel diameter between tests) model tunnel pressure σ t at the tunnel axis level where ρ is the density of the soil; g is acceleration due to gravity; and D is the model tunnel diameter. In Fig. 2(b), σ t is normalised by the initial tunnel pressure, σ t,0 , to evaluate the relative reduction of the initial pressure. The reaction curve of test CD4·4ID90 was omitted because of an anomalous trend. These reaction curves are, at least partially, the consequence of soil arching (i.e. the mobilisation of shearing resistance of the soil). The general trend of the reaction curves may be interpreted using the evolution of arch mechanisms postulated by Iglesia et al. (2014) and illustrated in Fig. 3. Because the load on the tunnel is mostly due to the weight of the soil beneath the arch, the formation of the initial curved configuration allows for a drop of tunnel pressure. Then, the transition to a triangular arch and, possibly, to a rectangular mechanism for shallow tunnels results in the increase of tunnel pressure because more of the ground is without support (i.e. beneath the arch).
Figure 2(a) shows that, overall, σ norm increases with C/D and reduces with soil density for a given V l,t . The reaction curve shape is shown to be influenced by the soil relative  density in Fig. 2(b); loose soils show a more gradual decline of pressure up to high volume losses, whereas dense soils induce a steep drop of tunnel pressure within low-medium volume losses (1-2%, depending on C/D), followed by reasonably stable values under further increases of V l,t . These results are consistent with the arch mechanism displayed in Fig. 3; the greater the density, the greater the peak friction angle and, thus, the smaller the initial arch (zone 1). The trends can also be inferred from the known stress-strain response of cohesionless soils; high I d values are associated with a stiff shear response to peak strength and subsequent strain softening, whereas low I d soils have lower stiffness and exhibit strain hardening. Additionally, the minimum pressure occurs at different volume loss values depending on I d and C/D (the greater C/D and/or the looser the soil, the higher the V l,t corresponding to the minimum pressure). Finally, the data in Fig. 2(b) illustrate that collapse, which results in a load recovery stage (increase of σ t ), is initiated after 5% volume loss for shallow tunnels in dense sand (C/D ¼ 1·3; I d ¼ 0·9), whereas no load recovery was observed for C/D ! 2·4 within the investigated range of V l,t .
The tunnel pressures at extremely high volume losses (V l,t . 5%) can be compared against the upper and lower bound theorem predictions of Atkinson & Potts (1977). Using these solutions with friction angles of 32°(critical state) and 45°(peak) provides ultimate upper bounds σ norm = 0·18 and 0·08, respectively, and lower bounds σ norm = 0·34 and 0·18, respectively. Atkinson & Potts (1977) showed that these predictions bounded their experimental data obtained using an air-filled model tunnel and dense sand. Many of the ultimate pressures in Fig. 2(a) fall outside these bounds; however, there are some important distinctions between the tests described here and those used by Atkinson & Potts (1977). First, the use of water within the model tunnel (which enables an accurate determination of tunnel volume loss) imposes a hydrostatic pressure distribution on the tunnel lining; hence the tunnel pressure at the crown is lower than at the axis depth (as plotted in Fig. 2(a)). This feature, as well as the constant-volume condition of the water inside the model tunnel, will have an impact on how and where failure occurs within the soil around the tunnel. In addition, for the loose soil, the strain levels required to mobilise ultimate strengths are considerable and the failure state relevant to the bound theorem predictions would not have been reached at the volume losses plotted in Fig. 2.
With respect to the trapdoor-based findings of Dewoolkar et al. (2007) and Iglesia et al. (2014), the results in Fig. 2 confirm that minimum relative loadings (σ t,min /σ t,0 ) decrease with the increase of C/D and that σ t,min slightly decreases with the increase in I d . However, it is not true that in tunnelling a higher arch efficiency, associated with the minimum loading σ t,min , is mobilised at a constant value of V l,t regardless of I d and C/D. The relationship between V l,t and σ t,min suggests that a higher V l,t is required to fully mobilise the arch (if the arch can form) for (i) relatively deep tunnels compared to shallow ones and (ii) looser soils. Additional evidence of the importance of soil arching during tunnelling in sands is provided in subsequent sections.

DISPLACEMENT AND STRAIN MECHANISMS
In this section, tunnelling-induced soil movements and strains (engineering shear strain, γ, and volumetric strain, ε v ) are presented and related to the previously mentioned arching mechanisms. The term 'shear strain' refers to engineering shear strain. Contractive volumetric strains are defined as positive in this paper (i.e. ε v . 0 is contraction; ε v , 0 is dilation) and positive horizontal displacements are oriented towards the tunnel centreline. The GeoPIV measurements of displacements were used to determine strains assuming plane-strain conditions. As statistically quantified by Marshall & Mair (2011) for test CD2·4ID90, this procedure results in calculated strain values being sensitive to small errors in the GeoPIV displacement data. They showed that for normally distributed horizontal and vertical displacement data with standard deviations in precision of 2·8 μm and 5·6 μm, respectively (based on particle image velocimetry (PIV) data), a standard deviation of 0·09% and 0·08% was obtained for shear and volumetric strains, respectively. They suggested that one standard deviation was an appropriate level of error to consider for the calculated strains.
In the plots of ground movements and strains, vertical settlements and spatial coordinates are normalised, respectively, by V l,t Â D and z t . Normalising settlements in this way enables comparison of the magnitude and distribution of ground movements between tests with different tunnel sizes and at different values of V l,t . Measurements greater or lower than the indicated contour thresholds were set equal to the closer limit value. In the regions where data were not available, the displacement and strain values were set equal to zero (e.g. around the tunnel). Based on the outcomes of Marshall & Mair (2011) and the level of scatter in the strain data, values between À0·1% and +0·1% were also set equal to zero.
The effects of cover-to-diameter ratio Figure 4 presents horizontal (left) and vertical (right) displacement data for C/D = 1·3-4·4 and I d ¼ 0·9 (dense sand) at V l,t ¼ 3%, while Fig. 5 plots the resulting shear and (ii) Vertical displacements for shallow tunnels are localised near the tunnel centreline, whereas they are more spread out for deep tunnels; this results in wider settlement troughs and a decrease of the normalised settlement magnitude for deeper tunnels. (iii) The deformation pattern becomes narrower with volume loss (i.e. more concentrated near the tunnel centreline). (iv) At low V l,t , the dense soil undergoes overall contraction because of the low magnitudes of shear strains, whereas at high V l,t , the region of soil around the tunnel crown is characterised by high shear strains which, because of the high relative density, results in dilative strains for an increment of V l,t (this feature is also highlighted later in Fig. 9 where soil volume loss is plotted against tunnel volume loss).
The effects of soil relative density Previous studies have not explicitly considered the effect of soil relative density on ground deformations caused by tunnelling. Data from centrifuge tests with varying relative density (I d ¼ 0·3, 0·5 and 0·9) at a constant C/D ¼ 2·0 are plotted for V l,t ¼ 3% in Figs 6 and 7 (readers may also refer to Figs S4 and S5 in the online supplementary data for results relating to low and high tunnel volume losses). Fig. 6 shows that, for a given value of V l,t , larger settlements should be expected in looser sands. The settlement pattern is also affected by relative density. For I d ¼ 0·9, the localised zone of large settlements is concentrated around the tunnel periphery   (indicating the delineation of an arch) and a relatively narrow settlement profile extends to the surface, whereas for I d ¼ 0·5 a larger region of ground experiences large settlements (associated with a larger arch) and the expanse of settlements at the surface increases. For the loose soil (I d ¼ 0·3), there is no clear evidence of an arch forming and the normalised settlements at the surface are wider than for the denser soil tests. A full statistical regression of settlement curves is provided in a later section.
The following observations can be made with respect to the variation of strain distribution with soil density (Fig. 7) and its relationship with ground settlements. (i) For the dense sand (CD2·0ID90), the soil arch becomes apparent at high volume loss, which reduces the tendency for tunnel volume loss to propagate to the surface. The zone of large shearing concentrated beneath the soil arch results in soil dilation (that decreases the ratio V l,s /V l,t ; discussed in detail later). (ii) For the medium dense sand (CD2·0ID50), the arching mechanism is similar to the dense sand test, although the zone of large shear strain is close to the surface at V l,t ¼ 3%, suggesting that the arch is close to failure. Its size is greater than for test CD2·0ID90 because of the lower peak soil strength associated with lower soil density, as postulated by Iglesia et al. (2014). (iii) In the loose sand data (CD2·0ID30), the soil does not mobilise a closed arch and a mechanism with near-vertical ear-shaped shear bands at the tunnel shoulders is discernible. The soil above the tunnel moves downwards as a near-rigid body (there are no shear strains close to the tunnel centreline). The inability of the loose soil to form an arch results in a settlement field for I d ¼ 0·3 that is qualitatively different than for I d ¼ 0·5 and 0·9. Wong & Kaiser (1991) identified a tunnelling-induced mode of behaviour for cohesionless soils (for at-rest stress ratios lower than unity) that is initiated by localised soil yielding at the tunnel shoulders and develops, for further stress relief withinthe tunnel, with either (i) ear-shaped shear bands intercepting the ground surface or (ii) yielding of the roof (i.e. soil above the tunnel crown). These two yielding mechanisms match well to the shear strain distributions measured in the centrifuge tests for loose and dense sands, respectively. However, Wong & Kaiser (1991) did not relate these mechanisms to the soil relative density. Figure 6 (as well as Fig. S4 in the online supplementary data) also provides insight into the distribution of horizontal movements. For the medium dense and dense sands (I d = 0·5-0·9), normalised horizontal movements can be significant both near the surface and the tunnel shoulders (similar to the mechanism illustrated in Fig. 4). The online supplementary data also demonstrate that, as volume loss increases, normalised horizontal displacements either decrease or remain relatively constant near the surface, and increase above the tunnel crown. For I d ¼ 0·3, the distribution of normalised horizontal movements is mostly induced near the surface. This difference can be related to the settlement mechanism; the onset of subsurface horizontal displacements corresponds to the concentration of settlements at the tunnel crown. Therefore, horizontal displacements near the tunnel shoulders should only be significant in medium dense/dense sands above medium-level volume losses (i.e. % 2%).

Summary of displacement mechanisms and soil arching
The soil arching phenomenon plays a major role in the definition of tunnelling-induced displacement mechanisms in sands. In particular, it can help to explain: (i) the transition from a narrow to a wide displacement field as C/D increases; (ii) the narrowing of the displacement field with V l,t ; and (iii) the complex variation of settlement profile with soil density.
Arching mechanisms are summarised in Fig. 8. A closed arch can form in medium dense and dense sands. If an arch forms, movements caused by tunnel ground loss tend to propagate vertically within the zone below the arch, resulting in a region between the tunnel and the arch characterised by large movements/strains and narrow settlement troughs. Above the arch, ground movements propagate upwards and outwards towards the surface, resulting in wider settlement troughs. The lower the soil density, the larger the size of the soil arch and the amount of soil affected by narrowing. In the case of deep tunnels, the arch is localised near the tunnel crown and the soil deformation pattern is relatively wide. For shallow tunnels, the arching affects a proportionally large zone of soil above the tunnel, resulting in a chimney-like displacement field with narrow settlement troughs. With the increase of V l,t , the localised downwards movement of the soil between the tunnel crown and the arch as well as the evolution of the soil arch postulated by Iglesia et al. (2014) (displayed in Fig. 3) results in the narrowing of the displacement field. In loose sands, the arch does not form and two separate zones of shearing propagate from the tunnel shoulders towards the surface. If the soil arch cannot form (as for tests with I d ¼ 0·3), ground loss propagates from the tunnel towards the surface both vertically and laterally.

TUNNEL AND SOIL VOLUME LOSSES
In sands, V l,s and V l,t differ because of the contractive/dilative strains that occur. In particular, at a given depth z, due to the summation of volumetric strains of the soil mass below this depth, (i) values of V l,s greater/lower than V l,t are associated with a cumulative contractive/dilative soil response during the tunnel ground loss process, and (ii) the first derivative of the V l,s against V l,t trend being greater/lower than unity indicates an average incremental contractive/dilative volumetric strain response for the given increment of tunnel volume loss ΔV l,t . Note that the volumetric-shear strain relationship is also affected by soil   confining stress level, which is not considered in the following discussion.
The full dataset of tunnel and soil volume losses at z/z t ¼ 0 (solid lines) and 0·5 (dashed lines) is shown in Fig. 9, where C/D increases with the darkness of lines. The ratio V l,s /V l,t (i) increases with C/D due to a lower shear strain level associated with high C/D and (ii) decreases with V l,t because of higher shear strains (and dilation) at high volume losses (as suggested by Marshall et al. (2012)). In addition, these new data illustrate the trend and enable quantification of the decrease of V l,s /V l,t as I d increases (due to the greater dilation in denser sands). Fig. 9 demonstrates that the relative density has a significant influence on the variation of V l,s /V l,t . Some context to these results may be obtained by considering the work of Ritter et al. (2017), who provided centrifuge test data on the effects of overlying structures (increased vertical stresses and varying shear strains) on these outcomes. For the low-rise buildings considered, the variation in V l,s /V l,t was lower than the influence of I d measured in this study. Figure 9 also illustrates the effect of relative depth, z/z t . V l,s is greater at z/z t ¼ 0 than at the subsurface level z/z t ¼ 0·5 for all tests (except CD1·3ID90 at V l,t . 5). This indicates that the overall soil behaviour within the range z/z t = 0-0·5 is contractive, regardless of the initial soil density; this can be explained by the low levels of shear strains in this region. Most of the dilative soil response is concentrated nearer the tunnel,that is, at z/z t . 0·5 (as illustrated in Figs 5 and 7).
It should be noted that V l,s , V l,t at high volume losses for the shallowest tunnels (C/D = 1·3-2·5) in loose sands, as displayed in Fig. 9(a). This is somewhat contradictory to the expected response, since shearing of loose soils should result in contractive volumetric strains. It must be that, for these shallow tunnel cases, the lower levels of confining stress within the ground above the tunnel enabled dilation to occur at higher volume losses.

ANALYSIS OF SETTLEMENT TROUGHS
In this section, the variation of settlement trough shape and magnitude with the main physical variables of the problem are assessed. The GeoPIV data were interpolated in Matlab with modified Gaussian curves using a least-squares regression technique.

Maximum settlements
Predictions of maximum settlement are often used within preliminary assessments of the risk of tunnelling to existing structures or infrastructure. The normalised maximum settlements, u z,max /D, measured at z/z t ¼ 0 and 0·5 are plotted against V l,t in Fig. 10 for I d = 0·3, 0·5 and 0·9. The adopted normalisation allows comparison of results obtained with tunnels of different size. The results show that the value of u z,max /D generally increases with V l,t , decreases with C/D and reduces with I d both at the surface and subsurface levels. However, the variation of u z,max /D with I d and C/D are not monotonic; they are complicated by the combined effects of soil arching (associated with the narrowing of settlement troughs close to the tunnel crown) and the dilative/ contractive behaviour of the soil. For instance, the differences in the rate of variation with V l,t between loose and dense samples are probably due to the higher efficiency of the dense soil to create and maintain an arch, whereas u z,max /D being lower in CD1·3ID30 than in test CD2·0ID30 may be due to the absence of the soil arch for shallow tunnels in loose sand and the overall state of contraction of the soil.

Settlement trough shape
The influence of C/D and I d on the settlement trough shape is illustrated in Fig. 11 where settlements, normalised by the maximum settlement at the tunnel centreline, are fitted with modified Gaussian curves. Results indicate that: (i) the higher the C/D, the wider the surface and subsurface settlement trough; (ii) the influence of soil density on the settlement trough shape is modest for shallow tunnels (C/D ¼ 2·0), whereas greater effects are induced for relatively deep tunnels (C/D % 4·5); and (iii) higher volume losses result in narrower settlement troughs.
A quantitative assessment of the trough shape can be obtained using the values of i (hence K ¼ i/(z t À z)) from the fitted modified Gaussian curves. First, K s (i.e. the value of K at the surface) is plotted against V l,t in Fig. 12. The results demonstrate: (i) decreasing values of K s with I d and V l,t ; (ii) a gradual reduction of the rate of change of K s with V l,t (resulting in a non-linear trend); and (iii) the greater the value of C/D, the greater the impact of I d , with soil density having a negligible effect for shallow tunnels (C/D ¼ 1·3). However, owing to the complex combination of arching mechanisms and volumetric strains (previously discussed), the trend in K s with I d for C/D ¼ 2·0 is not monotonic and the results do not fully agree with point (i).
Subsurface values of K are shown in Fig. 13 for V l,t = 1, 3 and 5%, I d ¼ 0·3 and 0·9 (loose and dense sands). The plotted data confirm that, for any depth z/z t : (i) the greater the value of V l,t , the lower the width parameter K; and (ii) the effect of density on settlement trough width is significant for high cover-to-diameter ratios. In addition, Fig. 13 displays that K at a given tunnel volume loss is approximately constant with depth for the shallower tunnels, whereas the width parameter increases non-linearly with depth for the relatively deep tunnels.

IMPLICATIONS OF RESULTS -DAMAGE ASSESSMENT OF SURFACE AND BURIED STRUCTURES
Greenfield displacements are often used as an input for soil-structure interaction analyses or for preliminary risk assessments. In addition, the data presented here will be of interest to the various researchers that use dry silica sand within centrifuge tests for the study of tunnel-structure interaction problems (e.g. Vorster et al., 2005;Farrell et al., 2014;Giardina et al., 2015;Ng et al., 2015). In this section, the measured settlement troughs in the context of the damage assessment of surface and buried structures are discussed. The width parameter at the surface (K s ) plotted in Fig. 12 varies between about 0·25 and 1·25, which contrasts considerably against a reference range of K s = 0·25-0·45 (Mair & Taylor, 1997). This variation of K s has the potential to significantly alter the outcomes of soil-structure interaction analyses. For instance, if a surface structure spanning the entire sagging region and a fixed tunnel depth are assumed, the relative soil-structure stiffness given by Giardina et al. (2015) (which is inversely proportional to the cube of the length in sagging/hogging) would vary by a factor of (2 Â 0·25) 3 -(2 Â 1·25) 3 = 0·13-15·6 in the sagging regionthat is, two orders of magnitude. This range can result in the structure ranging from fully flexible to almost rigid.
The implications of these results may also be evaluated by considering the greenfield curvature, χ, of a settlement trough above the centreline of the tunnel. For pipelines buried above a tunnel, the maximum bending moment induced by the tunnel for greenfield conditions can be calculated as M max,gf ¼ EIχ by assuming the pipeline behaves as a beam and that its deformed profile follows greenfield displacements (i.e. neglecting soil-structure interaction), where EI is bending stiffness. The curvature of the fitted modified Gaussian curves was calculated as χ mg,sand ¼ 2αu z,max /ni 2 (Klar et al., 2015). Figure 14 plots the measured values of curvature in a dimensionless form (to allow comparison between tests) using Dχ mg,sand , where D is tunnel diameter. Data shown in Fig. 14 illustrate that changes in C/D and V l,t can affect χ by several orders of magnitude. Following this, it is interesting to compare these data (for sand) to curvatures based on tunnels in clay. To do this, the maximum curvature of the ground above a tunnel in clay, χ clay , at z/z t ¼ 0 and 0·5 was evaluated using V l,t ¼ V l,s with equations (1)-(3). The curvature from the sand data was then normalised against the clay data as χ mg,sand /χ clay , as shown in Fig. 15. Figure 15 indicates that the increase in greenfield maximum curvature in sands with respect to clays is significant for shallow tunnels with C/D = 1·3-2·0, whereas the effects of soil density are limited. This latter observation is due to the counteracting effects of the narrowing of the settlement trough and reduction in V l,s /V l,t with I d .
These data can be incorporated into simplified soilstructure interaction analyses which consider the effect of structure stiffness. For example, the greenfield information χ can be used with the reduction factor based approach proposed by Klar et al. (2005) to estimate maximum bending moment within a pipeline buried above a tunnel. For this method, the tunnelling-induced pipeline bending moment can be evaluated using M max ¼ c M max,gf ¼ c EI χ, where c is a reduction factor that can be estimated using the rigidity factor R ¼ EI/i 3 r 0 E s , where r 0 is the radius of the pipe and E s is the Young's modulus of the soil.

CONCLUSIONS
Centrifuge modelling of greenfield tunnelling in a uniform cohesionless soil was performed to study displacement and strain mechanisms around tunnels. The effects of key geometrical and tunnelling parameters in loose, medium dense and dense sands were considered, providing new insights into the combined influence of tunnel relative depth and soil density. The following list summarises the key findings presented in the paper.
(a) The variation of ground reaction curves with relative tunnel depth and soil density was in general agreement with trapdoor centrifuge tests reported by Dewoolkar et al. (2007) and Iglesia et al. (2014). However, in contrast to the trapdoor results, it was shown that the greater the cover-to-diameter ratio (C/D) and/or the looser the soil, the higher the tunnel volume loss corresponding to the minimum tunnel pressure. (b) It was illustrated that vertical and horizontal displacement patterns are a consequence of the soil's ability to develop/maintain an arching mechanism. Qualitative soil arching mechanisms were proposed that related to C/D and soil density. (c) The new data for loose and medium dense sand showed a similar pattern to previously published data for dense sand (Marshall et al., 2012) in that settlement trough width increases with C/D, whereas it decreases with an increase in tunnel volume loss. Soil density was shown to have a noteworthy effect on settlement trough shape, especially for relatively deep tunnels (C/D . 4). It was also demonstrated that the variation of settlement trough shape with C/D and soil density is non-linear; this was related to soil arching and a transition from relatively shallow to deep tunnels.
(d ) For the first time, a set of experimental data for a fine silica sand was provided that allows soil volume loss (V l,s ) to be related to tunnel volume loss (V l,t ) as a function of both soil relative density and normalised depth (z/z t ). It was shown that, for a given magnitude of V l,t , the looser the soil, the greater the value of V l,s . The variation of V l,s was shown to be significant and should be accounted for in design. (e) Charts were provided that summarise normalised maximum settlements, central curvatures and width parameters from the experimental dataset. The implications of these data to the assessment of damage to surface and buried structures were provided. Judgement should be used before applying these outcomes to real cases which differ considerably from the conditions considered in the tests presented here.

NOTATION
C cover: distance from surface to tunnel crown c reduction factor D tunnel diameter E s Young's modulus of the soil EI pipe bending stiffness g acceleration due to gravity I d relative density i distance between the centreline and the inflection point K trough width parameter K s trough width parameter for surface settlements M max maximum tunnelling-induced pipeline bending moment N centrifuge acceleration n shape parameter used in modified Gaussian curve R rigidity factor r 0 radius of pipe V 0 area of tunnel cross-section (per m length) V l,s volume loss of soil V l,t volume loss of tunnel V s area of settlement trough (per m length) u x horizontal movement u z vertical movement u z,max maximum settlement x horizontal offset distance from tunnel centreline z depth, measured from ground surface z t depth of tunnel axis α shape parameter used in modified Gaussian curve γ shear strain ΔV ground loss at tunnel periphery ε v volumetric strain ρ density of soil σ norm normalised tunnel pressure at the axis level σ t tunnel pressure at the axis level σ t,0 initial tunnel pressure value σ t,min minimum tunnel pressure value χ curvature