Nonlinear soil-pile interaction induced by ground settlements: pile displacements and internal forces

In urban areas, the construction of tunnels and deep-excavations beneath and near to pile foundations can be detrimental for the superstructure and the foundation. A two-stage continuum-based nonlinear soil-pile interaction model is presented in this paper for predicting the axial and ﬂexural response of piles aﬀected by ground movements. The model accounts for the eﬀects of near-pile non-linear (hyperbolic) soil stiﬀness degradation and unloading eﬀects. The approach is used to analyse the relationship between the pile axial response (both displacements and internal forces) and greenﬁeld ground settlements for purely-frictional and ﬂoating piles in uniform ground. Both displacement and non-displacement piles are analysed by applying appropriate pre-excavation loading sequences. Results demonstrate the inﬂuence of initial safety factor, installation method, and capacity distribution (between shaft and base) on pile settlements and on critical tensile axial forces (both in terms of magnitude and depth). Dimensionless design charts are provided to estimate pile settlements and critical axial forces for the case of greenﬁeld settlements that either increase or decrease linearly with depth. These charts provide a rational and more general framework to describe excavation-induced eﬀects on piles than empirical methods. of tunnel volume loss, V l,t =1 and respectively. For comparison, (GF) for reference


INTRODUCTION
Engineers need to estimate the effects of ground movements resulting from tunnelling and deep-excavations (collectively The effect of the hyperbolic coefficient R f on the stiffness degradation is shown schematically in Figure 3c. As a result 88 of the plastic sliders, the elastic perfectly-plastic behaviour EP is given by R f = 0. For the NEP behaviour, an asymptotic 89 trend associated with a negligible tangent stiffness is obtained at large deformations when R f = 1, whereas a more gradual 90 stiffness degradation is given for R f < 1 up to the triggering of the slider limit force. 91 Finally, EP and NEP behaviour was only implemented in the vertical direction, whereas a linear elastic response (EL) 92 was considered in the horizontal direction; this assumption has been shown to be reasonable based on the analysis of 93 tunnelling problems (Basile, 2014). The proposed finite element (FEM) model was developed for groups of vertical piles of 94 length L p , diameter d p , and Young's modulus E with piles being modelled as Euler-Bernoulli beams embedded in a uniform 95 continuum. This paper limits itself to excavation-single pile interaction; elevated caps, raft foundations, and superstructure  (1) and (2) while assuming R f = 0; the elastic perfectly-plastic solution (EP) results from 102 Equations (1)-(3) imposing R f = 0 ; and the nonlinear elastoplastic solution (NEP) is given by Equations (1)-(3) for R f = 0. 103 (S + K * ) u = p + K * u cat + K * Λ * f + K * u ip ; f = (p − Su) (1) for reverse loading (2) where u is the displacement vector of the pile (consisting of three translational and three rotational degrees of freedom), 104 p is the external loading vector at the pile head, f is the vector of forces applied by the foundation nodes to the soil 105 (i.e. a vector containing the forces acting on the soil medium), S is the stiffness matrix of the pile foundation, u ip is the 4 NONLINEAR SOIL-PILE INTERACTION INDUCED BY GROUND SETTLEMENTS the pile base. R is the near-pile stiffness reduction matrix, resulting in the initial linear elastic stiffness during unloading and hyperbolic stiffness degradation for loading and reverse loading. In the NEP solutions, unless stated otherwise, the 114 values R f = 1 for the coefficient of hyperbolic near-pile soil stiffness degradation was used to be consistent with Korff et al. 115 (2016).

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The EL  6m diameter tunnel at a depth to axis level of 30m. The tunnel is assumed to be located either directly beneath the pile or 138 at tunnel-pile offset of 15m. The greenfield inputs were again computed using Loganathan & Poulos (1998).

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Results of tunnelling-induced (Tun.ind.) settlements and forces as well as post-tunnelling (Post-tun.; i.e. the combined 140 effects of initial pile loading and tunnelling-induced actions) pile axial force profiles are reported in Figures 5 and 6 for 141 SF 0 = 2; 100 and R f = 0; 0.75; 1. Figure 5 shows outcomes for the offset pile for which greenfield settlements mostly decrease 142 with depth, whereas Figure 6 displays the response of the pile directly above the tunnel where greenfield movements increase 143 with depth. Note that pre-tunnelling axial forces are negligible for SF 0 = 100, hence post-tunnelling and tunnelling-induced 144 forces match for this case. In addition, consider than tensile pile axial forces are positive in this paper.

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For both tunnel offsets, the reduction in initial pile safety factor SF 0 , associated with a greater pile head load, resulted in associated with R f .

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Most of the tunnelling-induced forces in Figure 5 are compressive because the offset pile was subjected to greenfield 152 settlements that decrease with depth. On the other hand, greenfield settlements that increase with depth can result in 153 tensile forces due to tunnelling, as shown in Figure 6. However, Figure 6 indicates that the entire pile undergoes tensile 154 post-tunnelling forces for SF 0 = 100 (see Figure 6d), whereas only the bottom part of the pile is under tension when SF 0 = 2 155 (see Figure 6c). Interestingly, the largest tensile critical force N c decreases in magnitude with the pile load P (i.e. when 156 SF 0 is reduced), whereas its location is closer to the pile tip for the highly loaded pile (SF 0 = 2) than the lightly loaded 157 pile (SF 0 = 100).

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In the following section, pile settlements and post-tunnelling tensile forces are investigated by assuming linearly 159 increasing/decreasing settlements with depth.

Studied scenarios 161
The NEP analyses in this section consider single piles of length L p = 5; 20m, diameter d p = 0.5m, and Young's modulus E 162 sufficiently large to simulate a rigid pile in a homogeneous soil with a Young's modulus E s,0 = 24MPa and a Poisson's ratio 163 ν s = 0.5. The effect of pile compressibility on the considered interaction problem is minor for most practical scenarios, and 164 the hyperbolic coefficient may be set to R f = 1 (Korff et al., 2016). Analyses were conducted using greenfield settlements 165 that linearly increase or decrease with depth z, as shown in Figure 2, which can approximate excavation-induced settlements 166 (e.g. Williamson et al. (2017b); Korff et al. (2016)).

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Both purely-frictional and floating piles (labelled FR and FL, respectively) were considered with q b,f either null or 168 proportional to τ f at the pile base. In this way, purely-frictional and floating piles are defined with respect to the mobilised 169 reaction stresses for extremely large pile settlements. Two possible τ f profiles along the pile axis were modelled: a constant 170 (e.g. FR.con) and linearly increasing (e.g. FR.inc) profile of τ f with z. In addition, to consider low shaft capacity piles due 171 to interface disturbance or shaft coating, an additional analysis was performed for a coated pile with a reduced constant 172 τ f (CO.con). A summary of the considered cases is given in Table 1, where the first three scenarios are used as the 'main 173 analyses' to demonstrate the salient features of the analysis results, and the outcomes of the remaining scenarios are included 174 as supplemental data.

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In this work, the ultimate capacity Q tot and the pile safety factor SF 0 are defined based on the very large pile settlements, 176 potentially greater than 10%d p , needed to fully mobilise both τ f and q b,f , as shown in Figure 7a. Engineering judgement 177 should be applied in specific scenarios to assess Q tot . Therefore, use of the proposed design charts presented later is predicated 178 on the existence of data which enables estimation of Q tot and SF 0 for very large pile settlements (i.e. pile settlement of at 179 least 10%d p ). Furthermore, Figures 7b and 7c sketch the pre-excavation loading sequence for non-displacement (NP; path 180 A→B) and displacement (DP; path A→D) piles.

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In the parametric study presented here, u z,gf is modelled only in terms of ∆S, which is the differential greenfield (defined in Figure 2) and D z = Q tot /K 0 is the ratio between the total pile capacity and the initial stiffness of the load-190 settlement curve (illustrated in Figure 7a). For instance, a negative value of ∆S relates to piles above the tunnel where 191 greenfield settlements increase with depth, whereas a positive ∆S relates to piles relatively far from the tunnel or adjacent 192 to deep excavations, where greenfield settlements decrease with depth. On the other hand, the parameter D z is defined 193 for the non-linear analysis (NEP) as the displacement obtained from the pile load-settlement curve P − u z (for no ground 194 movements) by the intersection between the ultimate capacity Q tot and the tangent to the initial portion of the curve with 195 a slope equal to K 0 . Alternatively, D z would be the settlement corresponding to the full mobilisation of pile capacity from 196 pile loading results in an elastic-perfectly plastic (EP) analysis.

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To generalise the outcomes from this work, results are presented using the dimensionless groups in Equation (4) Table 1 206 is used to refer to each scenario. surface greenfield movements (see Figure 1); thus, the ratios u pile /u gf,0 inferred from the data in Figure 8 are plotted in Figure 9 by assuming that S 0 = u gf,0 = 10D z . Subsequently, using the same layout, the normalised critical depth z c /L p and 212 the ratio between critical axial forces and pull-out pile capacity N c /Q t (positive sign assumed for tensile axial forces) are 213 shown in Figures 10 and 11, respectively. Q t is mobilised by the reverse loading at the pile shaft, as illustrated in Figure 3;

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The ratio u pile /u gf,0 shown in Figure 9 highlights that pile settlements, normalised by the constant greenfield surface

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With respect to the base capacity of the DP and NP piles with varying Q r = Q b /Q tot (Qr05.con and Qr40.con), the 241 impact of the increase in the relative base capacity is notable: increasing Q r from 5% to 40% generally shifted z i /L p towards 242 the pile base (Figure 9), while it shifted the normalised force N c /Q t towards negative (compressive) values for SF 0 < 3 243 ( Figure 11). In particular, all considered piles with Q r = 40% in Figure 11c are under compression along their entire length 244 for SF 0 < 2 (N c /Q c < 0), hence they would be at a low risk of cracking. This is not the case for the piles in Figure 11a  purely-frictional FR and floating FL piles with Q r < 10% are similar, hence the charts for Qr05.con and Qr08.inc are also 248 applicable to purely-frictional FR piles when Q b /Q tot < 10%.

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The location of the critical axial force along the pile is shown in Figure 10 using the normalised critical level depth z c /L p .

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Note that Figure 11 demonstrated that post-excavation tensile forces (N c /Q t > 0) are limited to the case of increasing 251 greenfield settlements with depth (∆S < 0); decreasing settlements with depth (∆S > 0) result in compressive excavation-252 induced axial forces for all cases (N c /Q t < 0). Consequently, a critical level depth z c is not presented for ∆S > 0. Figure 10   253 shows that z c moves towards the base of the pile for decreasing SF 0 and reaches z c /L p = 1 at SF 0 ≈ 1.5 for all cases (the 254 rate of change is notable between SF 0 = 1.5 and 3). Also, in general, the DP values of z c /L p are lower (towards the pile 255 head) than for NP for a given value of ∆S/D z . This is due to the presence of residual (post-loading and pre-excavation) 256 compressive axial forces near the pile mid-depth region for DP piles.

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Focusing again on the critical case of ∆S < 0 which can produce tensile axial forces, Figure 11 shows that, as a general 258 trend, the pile critical axial forces N c /Q t increase with the settlement rate ∆S/D z = -1 → -10, particularly for SF 0 > 2. movements, the application of a service load P of magnitude approaching the total capacity Q tot mobilises most of the 265 soil capacity and causes a degradation of the near-pile tangent soil stiffness (i.e. the reduction factor R ii is close to zero).

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Therefore, excavation-induced forces K * u cat , which relate to R ii , are negligible in magnitude for very low safety factors, 267 while N c /Q t depends mostly on the head load P .

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Results in Figures 8, 10, and 11 show that the normalised settlement magnitude ∆S/D z affects the relative response, in 269 terms of pile settlement and axial behaviour, of displacement (DP) and non-displacement (NP) piles. The difference between 270 DP and NP outcomes is particularly obvious for |∆S/D z | = 1 (except for SF 0 = 1.1, which is the lowest considered safety 271 factor), while the response of DP and NP piles is similar for |∆S/D z | = 10. The reasons for these trends are summarised 272 as follows.

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• Prior to ground movements ∆S, displacement piles DP have an unloading soil stiffness as well as residual stresses. For  Figure 3b, where tangent stiffness E is given by E s reduced by R ii ) and pile 280 settlements approach maximum greenfield settlements (z i /L p tends to 0 or 1) rather than the average settlement 281 (z i /L p ≈ 0.5).

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• For DP piles with SF 0 = 1.1 (the lowest considered safety factor), the response is close to NP behaviour because the

CONCLUSIONS
In this paper, a nonlinear two-stage continuum-based finite element model was proposed to study the problem of tunneland deep excavation-pile interaction. A hyperbolic soil model for the near-pile stiffness degradation was incorporated that 307 also considers the effects of pile unloading. The model was validated against boundary element method results based on an 308 assumed elastic perfectly-plastic soil behaviour. Pile ultimate capacity was defined based on predicted loads required for 309 extremely large pile settlements.

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Analyses were carried out to evaluate the settlements and internal forces of purely-frictional and floating piles in a uniform 311 ground for different levels of greenfield ground settlements. Greenfield settlements that increase linearly with depth were 312 used to replicate piles located above tunnels, whereas linearly decreasing greenfield settlements were used for piles more 313 distant from the tunnel or adjacent to a deep excavation. The following provides a summary of the main outcomes of the 314 research, where the term excavation is used to imply both tunnels and deep excavations.

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• Results illustrated the way in which pile safety factor can increase or decrease excavation-induced pile settlements Relatively short pile          Label dp L/dp