ABSTRACT
This paper provides investigation into response of (forward-rotated or backrotated) piles subjected to embankment loading, in light of 2-/3-layer models. First, recent advance is briefly reviewed concerning range of parameters used to model piles subjected to excavation and embankment loading. Second, simulations of six centrifuge tests on embankment piles are exemplified. Third, 1g model tests are provided to reveal backrotation behaviour of piles, and its simulation. The study is aimed to unite the design of piles under various types of passive loading (e.g. lateral spreading, embankment, excavation, sliding slope, etc.)
INTRODUCTION
Numerical analysis indicates the challenge in gaining parameters to achieve reliable prediction of the pile response affected by excavation, and embankment loading. For instance, the on-pile pressure from centrifuge tests on piles subjected to lateral spreading is reportedly quite close to the residue strength for liquefied sand (Armstrong et al. 2014), but is sometimes inconsistent with JRA (2002) methods. The prevalent p–y curve based methods are not suitable for modelling piles subjected to lateral spreading against 3D numerical analysis, although modifying the residual strength and stiffness of the p-y curve improve the prediction.
P
wf
lm= US or UD
l = 700
Uniform loading
US=200 UD=400
Uniform loading
lm
l
P
500
wf
Triangular loading
wf
16.7o
P
l
lm=200
500
700
Figure 1: Tests on piles under an inverse triangular, uniform or arc profile of soil movement
Recently, Guo (2015, 2016) developed 2- and 3-layer theoretical models and the associated closed-form solutions. The solutions well capture nonlinear response of piles in sliding soil [e.g. lateral spreading (Dobry et al. 2003), sliding sand (Guo et al. 2017; see Fig. 1), and slope (Frank & Pouget 2008)]. In particular, the same set of input parameters well capture the 3−5 times higher bending moment induced under translational movement than rotational movement, which is not seen in other simulations. This prompts the use of the models to capture impact of embankment or excavation loading on piles.
2-/3-LAYER MODELS AND SOLUTIONS
Rigid passive piles are simulated for a stepwise, uniform soil movement ws (thus external loading ps of wsks, ks is modulus of subgrade reaction) to a sliding depth lm [see Fig. 2]. A fictitious layer is introduced to account for dragging above the underlying stable layer, and forms a 3-layer model [see Fig. 2, Guo (2016)]: a sliding layer with the modulus ks (depth 0 − lm), a transition (fictitious) layer with a linearly increasing modulus from the ks at depth lm to mks at the depth zm of maximum bending moment Mm, and a stable layer with a modulus mks in depth zm−l (l = pile embedment), respectively. Without the transition layer, the model is referred to as 2-layer model (Guo 2015). Under a rotational restraining stiffness kθ, and a lateral force H, the pile rotates rigidly about a depth zr to an angle ωr and a mudline deflection wg under the sliding movement ws, or on-pile force per unit length (FPUL) ps [= αpublm/l, and pb = α.pub], to a sliding depth of lm on the pile. As shown elsewhere, the lm is equal to (0.6−3)lexc for an excavation depth lexc, for a distance between excavation face and piles of about 0.25l. The model is underpinned by the five input parameters ks, m, pb (FPUL at pile-base level), kθ, and a factor α. of non-uniform soil movement.
Explicit expressions were developed to estimate the displacement w(z), shear force T(z) and bending moment M(z) at depth z, the maximum bending moment Mm, its depth zm of Mm and the maximum shear force Tm (sliding or stable layer). The solutions are repeated for a series of lm/l (e.g. raising by a step of 0.1) and ps = pblm/l until a final lm/l = 0.5−0.9 to gain nonlinear response (Guo 2015). It should be stressed that the solutions are contemplated for rigid piles with rotational stiffness kθ,, kθG, and kθT, otherwise other solutions (e.g. Poulos and Davis 1980; Guo 2012) should be consulted.
P
H
kθG
Figure 2: Models for rigid, passive piles: model, pb and pub, kθ, and ps for nonlinear response
kθT
kθB
kθB
H
kθA
kθB
P
H
kθA
kθB
P
MODELLING PILES SUBJECTED TO EXCAVATION OR EMBANKMENT MOVEMENT
The 2-layer and 3-layer models are adopted to simulate the response of piles in ten model tests, subjected to excavation (Fig. 3, e.g. Leung, et al. 2000; Ong, et al. 2006) or embankment loading (e.g. Stewart et al. 1994; Armstrong et al. 2014). Typical parameters [of FPUL pub, modulus of subgrade reaction ks, modulus ratio of stable over sliding layers m, and rotating stiffness kθ] are deduced against measured piles response. The study reveals that (1) The ks reduces from 225su (stable piles, su= undrained shear strength) to (55−130)su (collapsed piles) due to increasing soil deformation from adjacent excavation. It is about 2.8N or 6z (MPa, N = SPT blow counts, and z = depth, m). Only (0.1−1.0)% ks (of excavation loading) is noted for consolidating and lateral spreading embankment. (2) Normalised by overburden stress (σv′), pile width (d) and coefficient of passive earth pressure (Kp), the ratio pub/(Kpσv′d) is deduced as 0.26−0.41 or 0.8−1.1 for embankment underlain by thick-clay layer or laterally spreading, and by clay-sand layers, respectively. (3) Excavation induces deformation of soil to a depth of 1.08αlexc, and on-pile FPUL ps (= αpublexc/l). The evolution of wall collapse from stable wall is associated with increasing deformation zone (with α raising from 1.2−2.3 to 2.5−2.9), friction angle of soil sliding [from (0.61−0.65)φ′ to (0.65−0.83)φ′, φ′= angle of internal friction] and ratio of rotational resistance (modulus) m (from 2.7 to 4) at constant loading depths for clay-sand layers. The stress reduces by 50% (with α = 0.63−1.1 and m = 1.4) for piles in a single sand layer but for a high stiffness (ks = 4.3 MPa). (4) Consolidating embankment involves a residual pub, smaller dimension and stiffness (with α = 0.5−0.72, ks = 15−20 kPa) but larger rotational resistance (m = 17.7) than that induced by excavation loading (of α = 1.2−2.9, ks = 0.55−2.25 MPa, and m = 2.9−4.0); and finally (5) The α (= 0.9−1.3), m (= 5.2−9.0), and ks (= 26−80 kPa) of laterally spreading sand-embankment sit in between consolidating embankment and excavation loading. Response of flexible piles is well captured using the 2-layer model (for rigid piles) and modified ks, pub and m values of (1.2−3)ks, and (−0.8)pub and(−0.8)m. The impact of pile flexibility is also assessed using new models incorporating hinges of rigid piles. The new findings are useful to design of piles, regardless of which methods. Typical examples are provided next for modelling pile response subjected to embankment loading.
l
6.5 m
Clay
ps
6.0 m
Sand
Lexc
Wall
Single pile
ld = 3 m
Mm
lm
1.5 m
Tests 5 &7
Test 6
Lexc
lm
l
10 m
Clay
ps
2.5 m
Sand
Wall
Single pile
ld= 3 m
Mm
l
6.5 m Sand
or Clay
ps
6.0 m Sand
Lexc
Wall
Single pile
ld =1− 9 m
Mm
lm
1.5 m
l
6.5 m
Sand
ps
6.0 m
Sand
Lexc
Wall
Single pile
ld =2 & 4 m
Mm
lm
1.5 m
Figure 3: Centrifuge tests on piles (behind wall) subjected to excavation in sand or clay-sand layer
PILES SUBJECTED TO LARGE EMBANKMENT MOVEMENT
Given potential dragging’ on the piles, the 3-layer model is adopted to analyse centrifuge model tests on pile groups either subjected to ‘static’ embankment loading (q), or spreading of sand embankment.
Piles nearby embankment
Stewart, et al (1994) conducted centrifuge tests on model pile groups, which consisted two rows of seven piles held in a rigid cap 2 m (all at prototype scale) above the soil surface and able to deflect freely. The piles penetrated through a soft clay layer and into an underlying dense sand stratum. Next to it, a sand embankment was constructed in six stages to a height of ~ 8.5 m (maximum). Four piles (22.5 m in length, 0.43 m in diameter) were instrumented in the pile group. Two typical Tests 9 and 11 are simulated herein.
In Test 9, the soft clay layer was 18-m thick, and had an average su of 17 kPa. The pile-cap displacement wg (front/back row) and maximum moment Mm (both rows) were measured and are shown in Fig. 4(a) and (b), respectively for the increasing embankment height (or the average vertical stress q). The measured moment Mm is plotted in Fig. 4(c) against the pile-cap displacement wg for both rows. The measured bending moment profiles with depth at ‘ultimate’ state are depicted in Fig. 4(d) for front- and back-row piles. The piles (l = 22.0 m, d = 0.43 m) are simulated using following parameters: pub = 51 kN/m [= 4sudl/(0.82lm), with su = 17 kPa, d = 0.43 m, and lm = 18 m], m =17.7 (= ratio of coefficient of passive earth pressure Kp [= tan2(45+0.5φ)] over the active one Ka [= tan2(45-0.5φ)]) at ultimate state], kθ = kG = 1.12 MN⋅m/radian (with = 0.007, G = ground level), and α = 0.5 (deep sliding) (see Fig. 4(a)). Given φ ′ = 23o, γ′ = 16.5 kN/m3, and lm = 0.5l = 11 m, and q = 100 kPa, it follows an average ps of 34.2 kN/m (= 51α), and pub/(σv′d) = 0.72.
Figure 4: Predicted versus centrifuge Test 9 (Stewart et al. 1994): (a) embankment load versus maximum bending moment, (b) embankment load versus pile-head deflection, (c) pile-head deflection versus maximum bending moment, (d) bending moment profiles
Figure 5 Predicted versus centrifuge Test 9 (Stewart et al. 1994): (a) embankment load versus maximum bending moment, (b) embankment load versus pile-head deflection, (c) pile-head deflection versus maximum bending moment, (d) bending moment profiles
Figure 5 Predicted versus centrifuge Test 9 (Stewart et al. 1994): (a) embankment load versus maximum bending moment, (b) embankment load versus pile-head deflection, (c) pile-head deflection versus maximum bending moment, (d) bending moment profiles
The corresponding ps/(dq) = 0.72 (for over-consolidated clay) is slightly smaller than 0.75−0.792 deduced from measured response of piles subjected to short-term embankment loading, but is much higher than ps/(dq) = 0.277−0.35 for long-term embankment loading (Jeong et al. 1995). The ps/(σv′dKp) is equal to 0.286. The previous study (Guo 2016) indicates ks (= 15~20 kPa) and m = 12.3 (2-layer) or 17.7 (3-layer). The q–wg, q–Mm, and wg–Mm curves were thus predicted, and are shown in Fig. 4, along with the measured data. The ‘limiting’ bending moment profile was predicted for a loading depth c of 14.7 m (= 0.81lm, lm = 18 m), as plotted in Fig. 4(d). The predictions are insensitive to the modulus ks, and indicate 50% (α = 0.5) surcharge loading q being transferred onto the on-pile pressure (= ps/d) for the deep sliding. The ks may also be obtained using the scant modulus Gsec= 60su[1-0.985(c/lm)0.2] (Stewart et al. 1994), assuming a maximum shear stress mobilization ratio of c/lm. At c/lm= 0.82 with su= 17 kPa, it follows Gsec = 54.4 kPa, and ks = 18.1 kPa {= Gsec/[2(1+0.5)], Poisson’s ratio = 0.5}.
In Test 11, the clay layer was 8-m-thick with su = 11 kPa. The piles (with l = 22.0 m, d = 0.43 m) are modelled using pub = 121 kN/m (= 3.1sudlm, with su = 11 kPa, d = 0.43 m, and lm = 8 m), ks = 20 kPa, m =17.7, kθ = 26.6 MN⋅m/radian (== 0.15 for the pile-cap), and α = 0.72 (a profile of inverse triangular movement). The pub offers an average ps of 86 kN/m (= 121α), pub/(dσv′) = 1.5−2.1 and pub/(σv′dKp) = 0.68−0.93 (with φ ′ = 23o, γ′ = 16.5 kN/m3, and lm = 8−11 m). The normalised rotational stiffness was raised to 0.15, due to a higher impact of underlying sand layer on the soft clay layer. These parameters allow a good prediction of q–wg, q–Mm, and wg–Mm curves (not shown herein). As a comparison, the 2-layer predictions were made using the same parameters (but for a reduced m of 12.3), which agree well with the measured data, respectively.
4.2 Piles embedded in lateral spreading embankment
Three centrifuge tests (Armstrong et al. 2014) were conducted to examine the response of embankment-pile subjected to lateral spreading. Each test comprised of two identical approach embankments (of dry, dense Monterey sand, Dr = 100%), which are separated by a 12-m wide channel. The embankments were 8 m high at the crest and 11 m high at the model container wall (or an average of 10 m in current modelling), with a crest width of 12 m and side slopes of 2:1 (horizontal to vertical) in all directions. Below the embankment was a compacted, non-plastic silt layer 1.3 m thick, a 5-m-thick loose sand layer (Dr =30%), a second silt layer 0.7 m thick, and a 17-m-thick, dense sand layer (Dr = 75%). One embankment from each centrifuge test (referred to as Kobe-1×6, Kobe-2×4, and Sine-2×4) included a pile group that extended into the dense sand layer. The Kobe-1×6 test was conducted on a single row of six closed-ended, aluminium piles with an outer diameter of 0.72 m and a flexural stiffness (EI) of 174 MN⋅m2. The Kobe-2×4 and Sine-2×4 group tests both had two rows of four closed-ended, aluminium piles, with do (outside diameter) = 1.22 m, a center-to-center spacing of 3do, and EI = 1,876 MN⋅m2. An aluminium-epoxy pile cap connected the piles together in each test. Input motions were applied to the model base in the longitudinal direction. Model Kobe-1×6 and Kobe-2×4 tests were subjected to a modified version of the ground motion recorded at a depth of 83 m at Port Island in the 1995 Kobe Earthquake, with a peak base acceleration of 0.8g and 0.7g, respectively. Model Sine-2×4 was first subjected to a shaking event contained a total of 20 sine wave cycles with packets at 0.2g, 0.3g, and 0.5g, respectively.
Kobe 1×6: m = 5.2, ks =26 kPa, pub= 471.1 kN/m, kθ/(ksl3)= 0.0045, α =0.9
Kobe 2×4: m = 9, ks = 79.5 kPa, pub= 730.2 kN/m, kθ/(ksl3)= 0.019, α =1.3 (pb= 949.3 kN/m)
SINE 2×4: m = 9, ks =71 kPa, pub= 730.2 kN/m, kθ/(ksl3)= 0.0016, α =1.0
Loose sand
Embankment
Dense sand
~18
5
10
5
l
lm
ps
ks
mks
(1−m)ks
zm
Mo= ωrkG
Qm1
Mm
Qm3
Figure 5: Predicted versus numerical solutions (Armstrong et al. 2014) of (a) maximum shear force versus embankment displacement, and (b) maximum bending moment versus pile-head deflection
An equivalent static analysis (ESA) was conducted on the centrifuge tests to gain in sequence (1) the embankment displacement and displacement profile for a range of pile/bridge restraining forces, (2) the pile/bridge restraining forces for a range of imposed ground displacements, and (3) the point of compatibility in the forces and displacements between the two steps. The analysis overestimated the embankments deformations (without piles) (Armstrong et al. 2014), and bending moment of the piles (Fig. 5) against the centrifuge tests. It is thus important but difficult to select input parameters (e.g. undrained shear strength).
In simulating the piles of Kobe 2×4 test, the 3-layer solutions adopt: pub = 730.2 kN/m, m = 9 (φ′ = 30o), ks = 70.5 kPa [= 51d1.5m(lm/l)21.5, d = 1.22 m], and kθ = 1.3 MN⋅m/rad (see Fig. 5). The surcharge q was calculated for the embankment height he of 8 to 11-m (above the loose sand). The lateral spreading FPUL ps was estimated as 162.26 kN/m (= 0.78σv′d,, with σv′= γ′he,, γ′ = 17 kN/m3, and he = 10 m) in light of ps/(σv′dKp) = 0.26 (with φ ′ = 30o). The normalised on-pile pressure ps/(dq) is estimated as 0.78, as with short-term embankment loading (Jeong et al. 1995). The pub is obtained using l = 22.5m, lm = 5 m, and α =1.0 with pub/(σv′d) = 0.6. The current predictions agree well the numerical solution of restraining force−embankment displacement relationship, and pile-head displacement − maximum bending moment curve, respectively in Fig.5; and (2) the profiles of bending moment in Fig.6.
As with Kobe 2×4, the groups Kobe 1×6 and Sine 2×4 were simulated and shown in Fig.5. A design level of m = 5.2 (about 60% ultimate value of 9) was adopted for the Kobe 1×6 to compensate the impact of flexibility associated with the small diameter. A similar ratio ps/(σv′dKp) of 0.28 (based on /(σv′ = 170 kPa and φ ′ = 30o) [or pub/(dσv′) = 0.66] was adopted, which offers ps = 105.3 kN/m (Kobe 1×6) and 176.3 kN/m (Sine 2×4), respectively. The current predictions agree well the numerical solutions. The pile-head displacement (at c = lm = 5 m) is predicted as 1.05 m (Kobe 1×6), 0.75 m (Kobe 2×4) and 0.56 m (Sine 2×4), which agree well with measured values of 1.03 m, 0.75 m and 0.57 m, respectively. The agreement is observed (Fig. 6) for the profiles of bending moment as well. The current solutions are much simple and efficient to apply than other approaches, and have a good accuracy. Note that the stipulated α value can only be verified with evolution of the pile response with the soil movement, which is not available
Figure 6: Predicted versus measured (Armstrong et al. 2014) bending moment profiles at a pile-head displacement of (a) 1.03 m (Kobe 1×6), (b) 0.75 m (Kobe 2×4) and (c) 0.57 m (Sine 2×4)
Piles Subjected to backrotation
The existing tests, analytical and numerical simulations have been confined to forward rotation (e.g. Bransby & Springman 1997; Juirnarongrit & Ashford 2003; Armstrong et al. 2014). However, it was back-rotation that incurred failure of most of the piles during nature disasters (Knappett & Madabhushi 2009; Fraser 2013; Haskell et al. 2013, Guo 2020). The tests were conducted using the apparatus shown previously (Guo et al 2017) on 2 piles in line, without an axial loading P (= 0) or with P of 294 N/pile, respectively, at the sliding depth (SD) lm of 0.57l (i.e. 0.4 m). The model piles were subjected to a uniform translation of the sand (at the loading location, Fig .1) to a total movement wf of 140 mm. The piles were made of aluminum tube, 1.2 m in length, 32 mm in diameter (d32), and 1.5 mm in wall thickness (t). The bending stiffness EpIp is 1.28×106 kNmm2. The model sand has a dry density γ′ of 16.27 kN/m3 and an internal (residual) frictional angle φ of 38°. The d32 piles were socketed into an aluminum cap. The bending capacity Moy of the pile-cap connections is 5−10% of the pile body (My). The tests provide profiles of bending moment, shear force, soil reaction, and deflection, and maximum bending moment Mm, maximum shear force Qm, pile-rotation angle r, and pile-head deflection wg, and ks of 20−35 kPa.
The model piles display features of sway, sliding and back-rotation, as with in-situ piles. The pile deflection wg, rotation angle ωr and bending moment need to be curbed to prevent formation of hinges. They are estimated as (Guo 2016) (i) ks = 25 kPa (= 2.1Gs, and Gs = 12 kPa). (ii) m = 17.7 (= Kp/Ka, ultimate state). (iii) pub = 10 kN/m (= sgγ′Kp2dl), using γ′ = 16.5 kN/m3, φ = 38o, d = 32 mm, l = 0.7 m, and sg = 1.53 (average). The pub and ks values for rotating piles are reduced to 5.0−5.6 kN/m and 14−16 kPa, respectively to simulate translating piles. The parameters are ‘identical’ to those of forward rotating piles but a negative kθ (with = -0.04). They are presented together as m/ks(kPa)/pb(kN/m)/ of 17.7/25/5.6/-0.04, and 17.7/16/5/-0.04, respectively for no-loading and with vertical loading, respectively.
Figure 7: Yielding and sliding piles − Predicted versus measured development of 2-pile in-line groups: (a) Mmi − wf, (b) Qmi − wf, (c) wg−Mmi, and (d) ω r−Mmi
The test piles are subjected to soil movement only and no shear force at head-level (with H = 0). The My is estimated grossly using yielding stress of 350 MPa (the aluminum) as 0.4 kNm [= 350×103×1.28/(70×106)/ (0.5d) m, d = 0.032 m, Young’s modulus of 70 GPa]. The connection Moy (= My/ωr) is estimated as 20−60 Nm using /ωr = 0.05−0.2. The Moy/My ratio is much lower than 0.25−0.6 gained from steel pile-to pile-cap connection due to stress concentration at the pile and cap connection.
Conclusions
This paper provides investigation into response of piles subjected to embankment loading, and backrotated piles in light of 2-/3-layer models. Recent advance is briefly reviewed concerning range of parameters used to model piles subjected to excavation and embankment loading. Simulations of six centrifuge tests on embankment piles are elaborated. 1g model tests are provided to reveal backrotation behaviour of piles, and its capture using a negative stiffness in the 2-/3- models. The study is aimed to unite the design piles under passive loading (e.g. lateral spreading, embankment, excavation, sliding slope, etc.).
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