Centrifuge Modeling Study of the Response of Piled Structures to Tunneling

The experimental package developed by Zhou et al. (2014) to model the tunneling process under greenfield and plane-strain conditions was used. This centrifuge package includes the centrifuge strong box, soil, model tunnel, and tunnel volume control system. The strong box consists of a stainless steel U-channel, a Perspex front wall, and a back aluminum wall. The inside plan dimensions of the strong box are $640\times 260\mathrm{mm}$ and the maximum height of soil within the box is 500 mm. The front wall was made of Perspex to allow the measurement of displacements using digital image analysis. A uniform dry fine silica sand known as Leighton Buzzard Fraction E was used for testing ($d_{50}=0.122\mathrm{mm}$), therefore results directly relate to tunnels in uniform sandy ground. A 90-mm-diameter model tunnel buried at a depth to axis of 225 mm was used to replicate a prototype 7.2-m-diameter tunnel with 14.4 m of cover ($C/D=2$). The tunnel comprised a rigid inner core with enlarged ends that was sealed within a flexible rubber membrane [similar to that used by Marshall and Mair (2011)]. A tunnel volume control system comprising an actuator connected to a hydraulic cylinder was used to control the volume of fluid within the tunnel (a given movement of the cylinder piston correlated to a specific change in tunnel volume). The tunnel volume loss process was conducted in 0.25% increments up to 5% and subsequently in 0.5% increments up to 10%.

Piled structures were modeled using the equivalent beam approach in which aluminum plates with varying stiffness were supported by aluminum piles ($E=70\mathrm{GPa}$). The plates had a transverse width $B=500\mathrm{mm}$ and a length $L=256\mathrm{mm}$. Four different plate thicknesses, $t$, were used: 1.6, 3, 6, and 12.3 mm. The prototype-scale axial and flexural stiffness of these plates (Table 1) encompass most real scenarios, such as the structures monitored during the Jubilee Line Extension (Farrell et al. 2014; Giardina et al. 2015).

The pile foundations, illustrated in Fig. 3(b), consist of transverse rows of seven 8-mm-diameter, flat-bottomed, aluminum alloy rods spaced at a distance of $S_l=128\mathrm{mm}$ along the structure length ($L=256\mathrm{mm}$). Piles were given a fully rough interface by bonding fraction E sand to the outer surface, resulting in a final diameter of 9 mm over the embedment depth of 135 mm. A gap of $H=38\mathrm{mm}$ existed between the plate and the soil; therefore, the model replicates a piled foundation rather than a piled-raft foundation. The piles were rigidly connected to the plate by bolting the upper (threaded) portion of the piles to the plate. Additional masses could be attached to the threaded upper portions of the piles. Pile loading tests [Fig. 3(a)] were conducted by continuously jacking equivalently prepared piles into the soil.

Two Canon PowerShot G10 14.7 MP digital cameras were used to take pictures of the soil (during the greenfield test) and the front face of the plate at each increment of $V_{l,t}$. Displacements were measured from the digital images using GeoPIV (White et al. 2003). Plate settlements were also monitored with a row of five LVDTs located along the middle of the strong box, as shown in Fig. 3.

The tests conducted for pile loading and tunneling beneath piled plates are illustrated in Figs. 3(a and b), respectively, and summarized in Table 2. Group A consists of a greenfield test; Group B are the loading tests used to characterize the pile load–settlement curve and ultimate pile capacity; Groups C and D investigated the response of piled structures to tunneling. In Group C [presented in Franza and Marshall (2016)], as a result of the varying plate thickness, the weight of the structure also varied between tests, which has been shown to have an effect on the tunnel-structure interactions (Giardina et al. 2015). Therefore, Group D tests were performed with additional masses attached to the tops of the piles (which do not affect the bending stiffness of the model structure). The additional masses were chosen to achieve a specific total weight of the superstructure, enabling a comparison of structures with the same weight but varying stiffness. The value of this specific total weight was selected to obtain a pretunneling safety factor of the foundation (i.e., 2.3) that is within the range of typical design values. Comparison of Groups C and D tests performed with the same plate thickness illustrates the effects of structure weight. Groups C and D tests are labeled according to the plate thickness and structure weight (e.g., t3.w12 refers to a 3-mm-thick plate with masses added to match the weight of the 12-mm-thick plate). Several tests were repeated three times in order to confirm repeatability of results (i.e., tests LP and tests t3 and t6 of Group C); these are indicated with a, b, or c after the test name. Finally, note that the spatial reference system and sign conventions for tunneling-induced displacements are bolded arrows in Fig. 3.

The preparation and testing of the tests considering tunnels beneath piled structures (Groups C and D) were as follows:

The procedure for greenfield tests (Group A) was similar except that Stage 2 was omitted.

For the pile loading tests (Group B)

Greenfield ground movements are often used as an input or reference for soil-structure interaction analyses. This section presents data and a brief discussion of the data obtained from the greenfield test (GF, Group A). These data are incorporated into the modification factor analysis approach presented subsequently.

Contour plots of the greenfield displacements are displayed in Fig. 4 for a $V_{l,t}$ of 1 and 5%, which represents the investigated range of volume losses and corresponds to a typical and a very high value of volume loss encountered in practice. Displacements are normalized by $R\times V_{l,t}$; this normalization enables the identification of changes in displacement mechanisms as volume loss increases. Fig. 4 shows (1) the concentration of ground settlements at the tunnel crown, which implies a narrowing of the settlement trough shape with depth, (2) the settlement trough shape becomes narrower with tunnel volume loss, (3) the magnitude of vertical movements is greater than horizontal displacements (a different scale is adopted in the plots), and (4) horizontal displacements are mostly noticeable near the surface and negligible at greater depths.

Settlement data are commonly interpolated with empirical curves in order to quantify the effect of tunneling-related parameters on settlement trough shape. In Fig. 5, centrifuge data [particle image velocimetry (PIV)] of settlements at the surface $z/z_t=0$ and a depth $z/z_t=0.6$ (corresponding to the pile tip level in test groups C and D) were curve-fitted with Gaussian (G, dashed lines) and modified Gaussian (Vorster et al. 2005) curves (mG, solid lines) for $V_{l,t}=1$ and 5%. The figure shows that the greenfield settlements were best fitted using modified Gaussian curves (higher coefficients of determination, $R^2$), which is in agreement with Vorster et al. (2005) and Marshall et al. (2012). Analysis of the fitted curve parameters over the full range of volume loss data illustrated a narrowing of settlement trough shape with $V_{l,t}$; this result, for loose sands, agrees with that of Marshall et al. (2012) for dense sands (settlement shape does not vary considerably with $V_{l,t}$ in clays). Furthermore, the variation of settlement trough shape with relative depth $z/z_t$ is also affected by relative density due to the complex contractive and dilative behavior of the soils [this is elaborated on in Franza (2016) and Zhou et al. (2014)].

Pile load tests (Group B) were performed to determine the ultimate pile capacity of the model piles prior to tunneling in order to assess the initial safety factor of the pile foundation. The load-displacement response of nondisplacement piles was assessed from three tests with the same configuration (to check repeatability), as shown in Fig. 6. The existence of the tunnel appears to have some effect (P1, which is close to the tunnel, shows slightly lower forces than P2), however results from the two pile locations are generally consistent. The ultimate capacity of an isolated pile was assumed to be 495 N, the average load obtained at a pile head settlement of $10\%d_p$ (Fleming et al. 2009).

It is useful to describe a pile’s working load using the initial (i.e., pretunneling) safety factor. This is an important parameter that assesses the initial capacity of the system relative to a state of pile failure. For each test, the initial safety factor of the foundation, ${\mathrm{SF}}_0$, was computed as the ratio between the ultimate load capacity of the pile group and the total weight of the superstructure (accounting for the reduced $g$-level of $74g$ at the level of the plate). The ultimate load capacity of the pile group foundations in test Groups C and D was evaluated as the sum of the ultimate load capacity of each pile (495 N, based on Group B tests). This approach is valid because the pile spacing is more than $8d_p$ and the pile length to diameter ratio, $L_p/d_p$, is 15, hence block failure (likely for closely spaced slender piles) is not expected (Fleming et al. 2009).

Table 3 shows that the initial safety factor ${\mathrm{SF}}_0$ ranged between 17.4 and 2.3. In particular, the configurations that tested plates of varying stiffness and constant weight (t12 and Group D) have a ${\mathrm{SF}}_0=2.3$, indicative of a common foundation design. The tests with ${\mathrm{SF}}_0>2.3$ represent more overdesigned foundations, hence pile failure is unlikely to be induced by the stress relief caused by tunneling.

In this paper, a color scheme was adopted in figures that plot results for varying plate thickness: the darker the color, the thicker or stiffer the plate or structure. A comparison of the vertical ($u_z$) and horizontal ($u_x$) displacements of the plates with varying stiffness and constant weight (${\mathrm{SF}}_0=2.3$) is presented in Fig. 7. These displacements were measured at $V_{l,t}=1$ and 5%. For comparison, greenfield (GF) displacements at $z=0$ and $z/L_p=1$ are also shown with black lines (the variation of greenfield displacements from the depth of the pile tip to the surface was approximately linear). Structure displacements were approximately symmetric except for Tests t12 and t1.w12, which showed higher displacements on the left side and a global horizontal translation of the plate toward the right (i.e., linear trend of horizontal movements with $x$). GeoPIV measurements of greenfield settlements of low magnitude (i.e., at low volume loss or near the edges of the settlement trough) show some scatter, which is generally less than 0.05 mm. This scatter does impact some subsequent interpretations; however, its effect is not significant.

To identify the key effect of TPI, it is necessary to analyze the response of flexible superstructures, where the superstructure influence is modest (it is not possible to completely exclude its effects because a pile group with a free-head condition was not tested nor were pile head loads measured). The settlement curves of t1.w12 and t3.w12 are characterized by having both hogging and sagging regions. Furthermore, the settlements of the structure in both cases are equal to or greater than the greenfield movements at the pile tip depth; the structure settlement curves are not intermediate between the $z=0$ and $z/L_p=1$ greenfield settlement troughs. These outcomes show that the TPI mechanism is a combination of the interaction of the piles with subsurface ground movements along their lengths and the additional settlements required to remobilize capacity, causing pile settlements to be greater than greenfield values. Additionally, because piles with their tips above the tunnel settle more than the greenfield surface and piles outside this region settle less than the greenfield surface, there is an increase of the relative deflection of more-flexible piled foundations compared with shallow foundations (which would approximately deform according to surface greenfield settlement troughs).

The effects of TPSI can be obtained by examining the stiffer structures, t6.w12 and t12. The data for Test t12 show the following effects: (1) a reduction of the plate relative deflection, $\mathrm{\Delta }$, and (2) a decrease of the portion of the plate undergoing hogging deformations due to the increase of $i_{\mathrm{bldg}}$ (defined in Fig. 1) for structures centered above the tunnel. Effect 1 is due to the plate’s ability to resist the central deflection through its own stiffness and the capacity of the outer piles furthest from the tunnel. To restrain tunneling-induced downward movement of the central piles, the plate acts to reduce pile loads (i.e., it applies tensile axial reaction forces), resulting in a reduced pile settlement relative to an isolated pile scenario. This mechanism is similar to the scenario resulting from soil movements inducing negative friction along the shaft of a pile with constrained pile head movements. Due to its stiffness, the plate redistributes loads to the outer piles, which are consequently driven into the soil. If the structure is sufficiently stiff, the load redistribution may result in failure (i.e., $\mathrm{SF}=1$) of the outer piles, resulting in large vertical displacements of the entire foundation, as can be seen for Test t.12. This can be compared against Test t6.12, whose lesser stiffness is not capable of redistributing loads to the outer piles to a degree sufficient to cause pile failure. Effect 2 was also noted by Farrell et al. (2014), who performed centrifuge tests to study the deformations induced by tunneling on an equivalent plate, and Franza et al. (2017) for an equivalent piled beam. Overall, Effects 1 and 2 result in a reduction of structure distortions with plate stiffness, however load redistribution due to a stiff foundation or structure may result in significant overall settlements. Therefore, tunneling may pose a threat to the serviceability state of rigid piled structures in terms of absolute displacements (and tilting in the case of structures not centered above the tunnel).

Figs. 7(c and d) show the horizontal displacements of the plates, which confirm that the axial stiffness of the plate prevents significant horizontal strains of the superstructure (i.e., approximately a linear trend of displacement with offset $x$) and only marginal horizontal differential displacements arise for the flexible plate t1.w12. In general, the distribution of horizontal strains was negligible across the entire data set (Groups C and D). These outcomes agree with the findings of previous research, which indicate negligible horizontal strains for structures on continuous footings (Burland et al. 2004; Franzius et al. 2006; Farrell et al. 2014). However, these conclusions should not be generalized to structures that do not have horizontal structural elements connecting the pile heads (Goh and Mair 2014; Franza et al. 2017).

Fig. 8 shows the effect of structure weight on the settlement of Plates t1 and t6 at medium- and high-volume losses. The effect of structure weight can be evaluated by comparing tests with the same plate thickness but varying weight (e.g., t1 compared with t1.w12) because the additional weight does not contribute to structure stiffness. For Tests t1.w12 and t6.w12, the initial safety factor ${\mathrm{SF}}_0$ is 2.3, whereas for Tests t1 and t6, ${\mathrm{SF}}_0=17.4$ and 4.6, respectively. The effect of structure weight on the TPI can be evaluated for the more flexible t1 tests; the t6 tests indicate the global TPSI. The percentage increase in ${\mathrm{SF}}_0$ is different for the two plates; therefore, the comparison of the settlement increase between Plates t1 and t6 is only qualitative. In this figure, the greenfield settlement troughs at $z=0$ and $z/L_p=1$ are also plotted.

The results illustrate the importance of structure weight in TPSI problems, with greater structure weights producing higher settlements, especially in the sagging zone. Three aspects characterizing the results in Fig. 8 can be noted: (1) The zone of influence tends to be narrower for the flexible structure t1 than for the relatively rigid structure t6; this is due to the difference in bending stiffness between the two plates [Fig. 8(b), Test t1.w12 displays larger settlements than Test t1 between $x=-200$ and $+100\mathrm{mm}$, whereas the entire plate in Test t6.w12 settled more than in test t6]; (2) overall, the shape of the settlement curves is not highly affected by the structure weight; and (3) the increase in vertical displacements due to structure weight is larger at $V_{l,t}=5\%$ than at $V_{l,t}=1\%$. This response was also shown for tunnel–single pile interaction in Franza and Marshall (2017b), who showed that the settlement-volume loss curves of single piles under different vertical loads tended to diverge at high tunnel volume loss (i.e., nondisplacement piles with lower ${\mathrm{SF}}_0$ fully mobilize shaft capacity at lower $V_{l,t}$ than piles with higher ${\mathrm{SF}}_0$, whereupon they require larger settlements to mobilize base capacity). The increase in settlements with structure weight is probably associated with two phenomena that have opposing effects: (1) there is an increase in the relative deflection of the plate, $\mathrm{\Delta }$ (i.e., increase of structure deformations); and (2) there is a higher level of soil stiffness degradation, which results in an increase of the relative stiffness of the superstructure compared with the foundation-soil system. Therefore, to assess the impact of structure weight on deformations and identify the predominant phenomenon, it is necessary to analyze the variation of deformation parameters, DR and $M^{\mathrm{DR}}$, in the sagging and hogging zones (carried out in the following sections). Finally, the results of Test t1 (flexible plate of negligible weight) suggest that the use of single-pile settlements within the range of greenfield values between $z/L_p=0$ and 1, which would be predicted by a linear elastic tunnel-pile interaction analysis (Devriendt and Williamson 2011), is only valid for unloaded piles or foundations with high ${\mathrm{SF}}_0$.

Fig. 9 plots vertical displacements against $V_{l,t}$ for the tests in Groups C and D at Locations 1–5 (refer to Fig. 3). Image analysis (PIV) results taken at the Perspex wall and LVDT measurements made within the middle of the centrifuge container are presented. None of the plates experienced a brittle failure (i.e., a sharp increase in the rate of settlement), which was previously shown for the case of tunnel–single pile interaction in sand (Marshall and Mair 2011). This is due to two factors: (1) the use of nondisplacement piles, whose tunneling-induced settlement–volume loss curve for isolated piles with constant loads is not as steep as for displacement piles (Franza and Marshall 2017b), and (2) the load redistribution between piles, which depends on the superstructure (plate) stiffness and the magnitude of differential displacements between adjacent piles (although the thickness of Plate t1 results in a low bending stiffness, $EI$, the load redistribution may be relevant because of the large differential pile displacements). To understand the latter component, a comparison can be made to a simply supported beam loaded by a unit force at its midpoint. The maximum settlement of the beam is proportional to the cube of the span length and the inverse of $EI$. Therefore, the narrower the settlement trough (which may result from an initial loss of capacity of the central piles), the greater the load redistribution (which acts to limit the failure or movement of the central piles).

The results in Fig. 9 illustrate the influence of superstructure properties on the change of rate of settlement of the plates with $V_{l,t}$. The results in Figs. 9(d–g) for a consistent structure weight illustrate a small reduction of the rate of settlement with $V_{l,t}$, which is more marked for Test t6.w12 than for Tests t3.w12 and t1.w12 (which are approximately linear). This reduction of the rate of settlement with $V_{l,t}$ is probably due to slippage between the pile and soil and ground stiffness degradation due to tunneling and TPSI mechanisms (i.e., the structure displaces the piles because of its own stiffness with respect to the soil, inducing slippage between pile and soil and additional shearing strains at the soil-pile interface). The stiffer the structure, the greater the soil degradation and slippage. On the contrary, the rate of settlement in Test t12 is almost constant [Fig. 9(g)]; this is probably due to the combined effects of superstructure stiffness and load redistribution from the central to the outer piles (causing the outer piles to be driven into the soil as their load capacity is exceeded). Once the outer piles reach their ultimate capacity, further increments of $V_{l,t}$ result in a global settlement of the pile and plate system to satisfy the equilibrium condition of the piles. The tests performed with constant plate thickness but different ${\mathrm{SF}}_0$ are compared in Figs. 9(a and d), (b and e), (c and f); the data show a greater reduction of the rate of settlement (especially at high $V_{l,t}$) for the lower structure weights.

In general, comparison of Figs. 9(a and g) illustrates that the settlement variation of the central piles (solid black lines) induced by structure stiffness and weight is lower than for the outer piles (solid and dashed light gray lines). This has as direct consequence that the structure deformations (as DRs) are highly affected by the superstructure weight and stiffness, whereas the superstructure has a less significant impact on the maximum settlements of the structure.

The maximum structure settlement is important for the assessment of damages related to the serviceability limit state. In this and subsequent sections, the greenfield ground movements are analyzed at normalized depths of $z/L_p=0$ and $2/3$ to compare results of TPSI with terms often used in the case of shallow foundations ($z=0$) and the empirical approach used for piled structures ($z/L_p=2/3$) (Devriendt and Williamson 2011). To highlight the influence of plate stiffness and structure weight, the maximum settlements measured at each $V_{l,t}$ increment of the plates ($u_{z,\max }^{bld}$) and the greenfield settlement troughs ($u_{z,\max }^{gf}$) at $z=0$ and $z/L_p=2/3$ are displayed in Fig. 10(a), whereas the ratio ($u_{z,\max }^{bld}/u_{z,\max }^{gf}$) between the plate displacement and the surface and subsurface greenfield settlements is shown in Figs. 10(b and c), respectively. The latter figures are useful to relate the results of the global interaction with greenfield movements, which can be used as a reference term.

Fig. 10(a) confirms that the plate stiffness and weight tend to decrease and increase, respectively, the maximum settlement of the superstructure within the investigated range of $V_{l,t}$. The critical maximum settlement of $10\%d_p=0.9\mathrm{mm}$ is plotted in Fig. 10(a) for guidance. The results also show that the greenfield settlements, $u_{z,\max }^{gf}$, vary almost linearly with $V_{l,t}$ and that the critical settlement of 0.9 mm was reached by all the superstructures at $V_{l,t}=1.5-2.5\%$, which is a relatively narrow interval. Furthermore, as shown by the solid lines in Figs. 10(b and c), the dimensionless parameter $u_{z,\max }^{\mathrm{bld}}/u_{z,\max }^{gf}$ ranges within the interval 0.75–1.5 (mostly between 0.85 and 1.3) for the constant structure weight cases, confirming a relatively narrow range of variability of the normalized maximum plate settlements. Therefore, if it is unlikely that pile failure will be reached (i.e., $\mathrm{SF}>1$), a preliminary assessment of maximum structure settlement could be carried out using a TPI analysis (i.e., assuming a fully flexible structure) accounting for the superstructure weight; this type of analysis would lead to a conservative but acceptable estimation of structure maximum settlements.

In the modification factor approach, the DR modification factors, $M^{\mathrm{DR}}$ (Fig. 1), are used to determine structure DR values from the greenfield settlement trough. The location of the greenfield and structure inflection points, $i$ and $i_{\mathrm{bldg}}$, may vary with tunnel volume loss, $V_{l,t}$, whereas DR is calculated based on the maximum relative deflection, $\mathrm{\Delta }$. In this study, to calculate DR and the location of $i$ and $i_{\mathrm{bldg}}$, the settlement curves of the soil and plates, obtained from test groups A, C, and D, were interpolated using modified Gaussian curves (Farrell et al. 2014; Marshall et al. 2012; Vorster et al. 2005).

The influence of the superstructure and $V_{l,t}$ on DR and $M^{\mathrm{DR}}$ is displayed in Figs. 11 and 12, which show, respectively, the effects of the structure stiffness and weight; Figs. 11(a and b) and 12(a and b) show the structure and greenfield DRs in both sagging and hogging, whereas Figs. 11(c and d) show the modification factors, $M^{\mathrm{DR},sag}$ and $M^{\mathrm{DR},hog}$.

The influence of structure stiffness can be seen by examining the structures with a constant ${\mathrm{SF}}_0$ [triangle-shaped markers in Figs. 11(a and b)]. For the flexible structure test t1.w12, the DR values are intermediate between greenfield surface and subsurface values and distributed approximately along a straight line, which indicates that the piles average the greenfield deformation pattern in terms of deformations. On the other hand, for the centrifuge tests t3.w12, t6.w12, and t12, the data follow nonlinear distributions that are characterized by decreasing values of DR with plate thickness and a gradual decrease of the increment rate with $V_{l,t}$. Although these results are qualitatively similar to the trends of the maximum settlements shown in Fig. 10, the curves in Fig. 11(a) have a marked drop in DR with plate thickness. As discussed previously, the observed nonlinear trend of superstructure deformations with volume loss could be attributed to the progressive degradation of soil stiffness and the relative pile-soil displacements induced by the superstructure.

The effect of structure weight on DR is displayed in Figs. 12(a and b); triangular markers are used for configurations with ${\mathrm{SF}}_0=2.3$, whereas square markers are used for structures with higher safety factors (lower weights). For all cases, in the sagging zone, the deflection ratios undergo a notable increase with structure weight (i.e., reduction of ${\mathrm{SF}}_0$). In the hogging zone, this increase only occurs for the flexible Plate t1; DR values essentially remain the same for the other cases. In the case of stiff structures, both square and triangular markers (heavy and light structures) follow nonlinear distributions with an asymptotic trend at high volume loss.

Figs. 11(c and d) show the effects of plate stiffness on the reduction factors, $M^{\mathrm{DR},sag}$ and $M^{\mathrm{DR},hog}$, calculated using surface (solid lines) and subsurface (dashed lines) greenfield settlements. The structure weight is constant for the data set in these figures. The data normalized with greenfield surface DRs (solid lines) show that the TPI mechanism results in $M^{\mathrm{DR}}$ values greater than unity for flexible structures, whereas structural stiffness tends to decrease the flexural deformations (as discussed previously). Except for the hogging modification factors, $M^{\mathrm{DR},hog}$, of Test t1 that are approximately constant, Figs. 11(c and d) show that the values of $M^{\mathrm{DR}}$ are characterized by a steady decrease throughout the entire range of $V_{l,t}$: the thicker the plate (higher the structural stiffness), the greater the rate of decrease. Furthermore, as displayed by the dashed lines, similar qualitative trends with plate thickness and $V_{l,t}$ were obtained for the modification factors defined using the subsurface greenfield deformations. However, because greenfield settlement troughs are wider at the surface than at depth, the magnitudes of the dashed lines are considerably lower than the solid lines. The scatter of $M^{\mathrm{DR}}$ at low volume loss is caused by the scatter in the greenfield (GF) displacements.

The dashed lines in Figs. 11(c and d) can be used to evaluate the performance of the simplified empirical TPI analyses described by Devriendt and Williamson (2011) based on subsurface greenfield settlement profiles (i.e., ${\mathrm{DR}}_{gf}$ at $z/L_p=2/3$). The modification factors $M^{\mathrm{DR},sag}$ and $M^{\mathrm{DR},hog}$ related to ${\mathrm{DR}}_{gf}$ at $z/L_p=2/3$ resulted in values of $M^{\mathrm{DR},sag}$ and $M^{\mathrm{DR},hog}<1.0$. In particular, during Test t12, $M^{\mathrm{DR},sag}<0.5$ and $M^{\mathrm{DR},hog}=0$ for ${\mathrm{DR}}_{gf}$ measured at $z/L_p=2/3$. Therefore, these data suggest that this damage assessment should be reliable and conservative for ${\mathrm{SF}}_0>2.5$ in the case of nondisplacement piles.

Finally, comparison of Figs. 10 and 11 illustrates the different range of variability of the plate normalized maximum settlements and deformations and the need to account for superstructure stiffness in the assessment of piled structure distortions. The following section presents the measured modification factors within the context of previously published design charts relating $M^{\mathrm{DR}}$ to relative structure-soil stiffness.

In the case of piled foundations, it is not possible to simplify the problem to a plane-strain condition. Therefore, to study the flexural deformations of piled structures using the modification factor framework, the relative structure-soil bending stiffness factor, ${\rho }^p$, defined in Eq. (2), was applied. To define ${\rho }^p$, it is necessary to assess the bending stiffness, $EI$, of the portion of structure corresponding to each transverse pile row, the transverse length, $B$, of the sagging and hogging zones in the greenfield condition, and the soil stiffness $E_s$. First, the plate was separated into two independent portions in the longitudinal direction to assess $EI$ corresponding to each pile row. The value of $B$ was measured from the surface greenfield settlement curves by identifying the offset of the inflection point at each volume loss value, and $E_s$ was estimated from the strains induced by greenfield displacements following the logic of Marshall et al. (2010) and Farrell (2010). The procedure consists of two parts: (1) The stiffness–shear strain relationship for the soil was defined from triaxial test data; and (2) soil shear strains induced by volume loss, $\gamma$, were assessed from the displacements measured in the greenfield condition. To account for the spatial variability of $E_s$, a representative value was assessed at a normalized depth $z/z_t=0.5$ [as suggested by Franzius et al. (2006) for shallow foundations] to account for the fact that both the soil at the piles and beneath the tip level are involved in the interaction. Additionally, an average shear strain, ${\gamma }_{\mathrm{avg}}$, from $\pm 2.5i$ along the settlement trough was used

The relationship for ${\gamma }_{\mathrm{avg}}-V_{l,t}$, measured from the greenfield test at a depth $z/z_t=0.5$, as well as the estimated secant soil stiffness $E_{\sec }$, are shown in Fig. 13; the latter curve displays an exponential reduction of soil stiffness with strain, resulting in an asymptotic trend at high volume losses. Further details on the calculation of shear stiffness are provided in Franza (2016).

Because of the use of greenfield representative conditions, the structure weight effect on stresses, and thus stiffness, is neglected. The variation of $\gamma$ due to the presence of piles and the relative pile-soil displacements induced by the structure load redistribution is also neglected. Furthermore, partitioning of the building transverse length into $B_{\mathrm{sag}}$ and $B_{\mathrm{hog}}$ with respect to the surface (as for shallow foundations) neglects the presence of piles. However, calculation of the relative structure-soil stiffness in this way allows for direct comparison of modification factors, $M^{\mathrm{DR}}$, with those for shallow foundations. Also, the effects of relative stiffness on structure distortions are proportional to a logarithmic scale (Franzius et al. 2006); therefore, the accuracy of its estimation can range within an order of magnitude without introducing significant errors.

Fig. 14 displays the modification factors in sagging and hogging zones, $M^{\mathrm{DR},sag}$ and $M^{\mathrm{DR},hog}$, against the relative bending stiffness ${\rho }^p$. The values of $M^{\mathrm{DR},sag}$ and $M^{\mathrm{DR},hog}$ were defined using the measured structure deformations and surface greenfield settlements within the volume loss range of $V_{l,t}=0.5-5\%$. The envelopes proposed by Franza et al. (2017) based on the simplifying assumptions of soil linearity and perfect bonding between the pile and soil are also displayed in this figure. The Franza et al. (2017) design curves were based on soil displacement patterns applicable to undrained clays; therefore, the effects of the different soil deformation patterns obtained for sands in this study on $M^{\mathrm{DR}}$ are neglected here. This simplification is to some extent supported by Mair (2013), whose design envelopes for shallow foundations captured data related to both clays and sands.

The results in Fig. 14 show that (1) the increase in relative stiffness decreases the structure deformations (decrease of $M^{\mathrm{DR}}$), (2) the structure weight increases deformations (increases $M^{\mathrm{DR}}$), (3) in the sagging zone, the increase in volume loss results in an increase of ${\rho }_{\mathrm{sag}}^p$ and a decrease of $M^{\mathrm{DR},sag}$, and (4) in the hogging zone, an increase in volume loss causes a decrease in $M^{\mathrm{DR},hog}$ but has little impact on ${\rho }_{\mathrm{hog}}^p$. The main interaction mechanisms causing these results were described in the previous section, except for the effects of $V_{l,t}$ on ${\rho }^p$. Because the structural bending stiffness is constant for a given plate, the changes in ${\rho }^p$ must be a consequence of changes in $E_s\times B_{\mathrm{sag}}^3$ and $E_s\times B_{\mathrm{hog}}^3$ with $V_{l,t}$ [Eq. (2)]. The values of $E_s$, $B_{\mathrm{sag}}$, and $B_{\mathrm{hog}}$ were all calculated using greenfield test data; the variation of $E_s$ with $V_{l,t}$ is illustrated in Fig. 13. Values of $B_{\mathrm{sag}}$ and $B_{\mathrm{hog}}$ depend on the location of the inflection point, $i$, which decreases with $V_{l,t}$ (i.e., the settlement trough becomes narrower). For instance, ${\rho }_{\mathrm{hog}}$ is approximately constant for $V_{l,t}=0.5-5\%$ because the effects of soil stiffness degradation are countered by the increase in $B_{\mathrm{hog}}$. In general, the trends of ${\rho }_{\mathrm{sag}}^p$ and ${\rho }_{\mathrm{hog}}^p$ will be affected by the tunnel-structure eccentricity through the greenfield relationship between $i$ and $V_{l,t}$. Although the Franza et al. (2017) envelopes are based on the assumption of linear elasticity, perfect soil-pile bonding, and greenfield displacements in undrained clay, most results fall within the envelopes and, especially for the sagging zone (the primary deformation mode of the structure), the trend of $M^{\mathrm{DR}}$ from the centrifuge tests follows that of the upper and lower bounds. Despite the overall agreement, engineers should use appropriate judgment to evaluate the applicability of these design envelopes to real cases of tunnel-building interaction.