Horizontal Dynamic Stiffness and Interaction Factors of Inclined Piles
Downloaded from ascelibrary.org by Hohai University Library on 06/23/17. Copyright ASCE. For personal use only; all rights reserved.
Jue Wang, Ph.D.1; Ding Zhou, Ph.D.2; Tianjian Ji, Ph.D.3; and Shuguang Wang, Ph.D.4 Abstract: A Timoshenko beam-on-Pasternak foundation (T-P) model was developed to estimate the horizontal dynamic impedance and interaction factors for inclined piles subjected to horizontal harmonic load. The inclined pile with a fixed pile-to-cap connection was modeled as Timoshenko beams embedded in the homogeneous Pasternak foundation. The reactions of the soil on the pile tip were simulated by three springs in parallel with corresponding dashpots. The differential equations of normal and axial vibrations of the inclined pile were solved by means of the initial parameter method. The model and the theoretical derivations were validated through comparisons with results from other theoretical models for some large-diameter end-bearing piles and inclined piles. The effects of shear deformations of the soil and inclined piles, during the vibration, were highlighted by comparing the results of the T-P model with those from Euler beam-on-dynamic-Winkler foundation (E-W) model. Findings indicated that the T-P model is more accurate than the E-W model for analyzing the dynamic performance of the inclined piles with small slenderness ratios and large inclined angles. The effects of the inclined angle, slenderness ratio, and distance-diameter ratio on the dynamic impedance and interaction factors were studied by numerical examples. It is shown that the increased angle of the pile results in an increase of horizontal impedance and decreases in interaction factors. DOI: 10.1061/(ASCE)GM.1943-5622.0000966. © 2017 American Society of Civil Engineers. Author keywords: Soil-structure interaction; Inclined pile; Pile impedance; Interaction factor; Timoshenko beam; Pasternak foundation.
Introduction Inclined piles were once considered to have detrimental seismic behavior (Deng et al. 2007; Poulos 2006) and some codes even recommended avoiding them (AFPS 1990; CEN 2004) because of poor performances in a series of earthquakes that occurred in the 1990s. However, numerical (Gerolymos et al. 2008; Turan et al. 2013; Sadek and Isam 2004) and experimental studies (Okawa et al. 2002; Gotman 2013) showed the advantages of inclined piles. The ultimate reasons for failures may be attributed to incorrect design and construction techniques rather than the pile inclination itself. Because inclined piles can transmit the applied horizontal loads partly through axial compression, they provide larger horizontal stiffness and bearing capacity than conventional vertical piles. Therefore, using inclined piles in foundations for buildings, bridges, and marginal wharfs is an efficient way to resist horizontal loads such as seismic waves, vehicle impacts, and ocean waves (Gazetas and Mylonakis 1998; Padrón et al. 2010). Inclined angles commonly used in practice are between 5 and 15°, in addition to the less typical cases between 20 and 25°.
1
Lecturer, College of Civil Engineering, Nanjing Tech Univ., Nanjing 211816, China; College of Mechanical & Electrical Engineering, Hohai Univ., Changzhou 213022, China. 2 Professor, College of Civil Engineering, Nanjing Tech Univ., Nanjing 211816, China (corresponding author). E-mail:
[email protected] 3 Reader, School of Mechanical, Aerospace and Civil Engineering, Univ. of Manchester, Manchester M13 9PL, U.K. 4 Professor, College of Civil Engineering, Nanjing Tech Univ., Nanjing 211816, China. Note. This manuscript was submitted on May 5, 2015; approved on March 16, 2017; published online on June 23, 2017. Discussion period open until November 23, 2017; separate discussions must be submitted for individual papers. This paper is part of the International Journal of Geomechanics, © ASCE, ISSN 1532-3641. © ASCE
Padrón et al. (2010, 2012) and Medina et al. (2014) investigated dynamic impedances of both single and group inclined piles and the kinematic interaction factors between adjacent inclined piles using a boundary element–finite element coupling model (BEM-FEM). In their studies, the pile was modeled as an elastic compressible Euler-Bernoulli beam. Their investigations contributed an understanding of the effect of inclined piles on dynamic impedances of deep foundations. Recently, Medina et al. (2015) further studied the response of the structure supported by an inclined-pile foundation. In addition to numerical methods, some simplified analytical models were also applied in the analysis of soil-pile dynamic interaction. Makris and Gazetas (1992) presented a Euler beam-on-dynamicWinkler foundation (E-W) model to account for the impedance of a single vertical pile and interaction factors between adjacent vertical piles, which can be extended to evaluate the effect of dynamic pile groups using the superposition method (Kaynia and Kausel 1982). Mylonakis and Gazetas (1998, 1999) used the E-W model to determine the dynamic interaction factors for axial and lateral vibrations of vertical piles in layered soil by considering the presence of receiver piles. Subsequently, the E-W model received widespread application because it performs with low computational complexity and reasonable agreement with rigorous solutions (Liu et al. 2014; Sica et al. 2011). Gerolymos and Gazetas (2006) developed a multiWinkler model to analyze static and dynamic stiffnesses of rigid caisson foundations with small slenderness ratios. In recent years, some researchers managed to extend the E-W model to the dynamics of inclined piles. Ghasemzadeh and Alibeikloo (2011) used the E-W model to analyze the interaction factor between two infinitely long inclined piles. As a limitation of the theoretical assumptions, the E-W model is only valid for those piles with large slenderness ratios (Wang et al. 2014a). Ghazavi et al. (2013) further considered the dynamic interaction factor of adjacent inclined piles under oblique harmonic loads. However, the E-W model neglects the shear deformations of both soil medium and pile, and it is only valid for piles with large
04017075-1
Int. J. Geomech., 2017, 17(9): 04017075
Int. J. Geomech.
Downloaded from ascelibrary.org by Hohai University Library on 06/23/17. Copyright ASCE. For personal use only; all rights reserved.
shear layer. It was confirmed by Poulos and Davis (1980) that load distribution along the inclined pile with an angle less than 30° can be reasonably assumed similar to that along the vertical pile. Therefore, the analytical expressions of springs, dashpots, and the shear layer for the vertical pile, which were based on some simplified physical models calibrated with results from numerical methods and experimental dates (Gerolymos and Gazetas 2006; Wang et al. 2014a), are introduced in the present model as follows: 8 < 1:75ðL=dÞ0:13 Es for L=d < 10 kx ffi (1a) : 1:2E for L=d 10 s Fig. 1. T-P model for inclined pile-soil dynamic interaction
1=4
cx ffi 6a0
slenderness ratios (i.e., pile with large diameter and/or short length). In fact, piles with small slenderness ratios are commonly used on firm stratum to support superstructures (Mylonakis 2001). Kampitsis et al. (2013) showed the advantages of Timoshenko theory on the kinematic and inertial interaction between soil and a column-pile with a small slenderness ratio. Recently, Wang et al. (2014a, b) presented a Timoshenko beam-on-Pasternak foundation (T-P) model to determine the dynamic interaction factors between vertical short piles with hinged-head connections in a multilayered soil medium. The parametric studies confirmed that the T-P model subjected to high frequency excitation provided improved results for the dynamic interactions of piles with small slenderness ratios. As an extension of previous works (Wang et al. 2014a, b), the T-P model was also applied to analyses of the impedance of a single inclined pile and dynamic interaction factors between adjacent inclined piles. The shear deformation and rotational inertia of inclined piles and the shear deformation of soil were simultaneously considered to overcome the limitation of the conventional E-W model (Ghasemzadeh and Alibeikloo 2011). Transformation between the local and global coordinate systems was used in the analysis because the inclined piles can transmit the horizontal loads not only through bending deformation but also through axial deformation. Therefore, horizontal and vertical interaction factors for inclined piles were coupled. The effects of slenderness ratios, pilesoil modulus ratios, inclined angles of a single pile on adjacent piles were studied in detail.
Model Description Fig. 1 shows a cylindrical elastic pile embedded in a homogeneous half-space with inclined angle w 1. The head of the pile is fixed to the cap. To conveniently evaluate the dynamic impedance atop the head of the inclined pile, a horizontal harmonic excitation, Hexpiv t, in the global coordinate system, x-z, was decomposed into an axial component, Qexpiv t, and a normal component, Pexpiv t, in a local coordinate system, x0 z0 . In this case, H, Q, and P are load amplitudes, v denotes the pffiffiffiffiffiffi ffi frequency of the external load, t represents the time, and i ¼ 1. The T-P model was introduced to analyze the normal vibration of the inclined pile, which was also a further extension of previous work (Wang et al. 2014a). Timoshenko beam theory was used to simulate the transverse vibration of the pile, in which the effects of shear deformation and rotational inertia were considered. The Pasternak foundation was used to represent the elastic half-space. The normal reaction of soil against the pile-shaft was simulated by a system of uniformly distributed linear springs and dashpots, which were connected through an incompressible © ASCE
r s dVs þ 2 b s kx =v
(1b)
gx ffi 0:5kx
(1c)
in which, Es, Gs, vs, r s, b s, and Vs = Young’s modulus, shear modulus, Poisson’s ratio, mass density, damping velocity, and shear wave velocity of the soil medium, respectively; L and d = length and sectional diameter of the inclined pile, respectively; and a0 = v d/Vs = dimensionless frequency. Because axial vibration of the pile does not induce shear deformation or rotational inertia, the coefficients of springs and dashpots, which are based on the Novak’s plane-strain model (Novak and Aboul-Ella 1978), are given as follows: kz ffi 2p a0 Gs
J 1 ða0 ÞJ 0 ða0 Þ þ Y 1 ða0 ÞY 0 ða0 Þ J 20 ða0 Þ þ Y 20 ða0 Þ
4 cz ffi Gs 2 J0 ða0 Þ þ Y02 ða0 Þ v
(2a)
(2b)
where J0 and J1 = first kind of Bessel function of zero order and first order, respectively; and Y0 and Y1 are the second kind of Bessel function of zero order and first order, respectively. It is well-known that the mechanical properties at the pile head is unaffected by the boundary condition at the pile tip for the case with a large slenderness ratio, L/d. However, different from the infinite long pile, the reaction of the soil on the pile tip should be considered when it has a small slenderness ratio (L/d < 10) (Gerolymos and Gazetas 2006). The analytical expressions for normal, axial, and rotational springs in parallel with dashpots at the pile tip were suggested by Novak and Sachs (1973) as follows: " # x ð4:3 þ i2:7a0 ÞGs;tip ðd=2Þ 0 Ktip ¼ ð2:5 þ i0:43a0 ÞGs;tip ðd=2Þ3 0 (3a) z Ktip ¼ ð7:5 þ i3:4ÞGs;tip ðd=2Þ
(3b)
Formulation Normal Vibration of Inclined Pile For simplifying the analysis, the displacement of the inclined pile subjected to horizontal harmonic excitation was decomposed
04017075-2
Int. J. Geomech., 2017, 17(9): 04017075
Int. J. Geomech.
Downloaded from ascelibrary.org by Hohai University Library on 06/23/17. Copyright ASCE. For personal use only; all rights reserved.
into a normal component and an axial component in a local coordinate system. The horizontal impedance of the inclined pile can be obtained using the superposition method. As expected, the formulation in all steps was valid for vertical piles when w 1 was 0. By using the Hamilton principle and Timoshenko beam theory, the governing differential equations for the translational and the rotational vibrations of a loaded pile are given by ∂2 ua ðz0 ; tÞ ∂ ∂ua ðz0 ; tÞ 0 ð Þ r p Ap ¼ k G A u z ; t p p a ∂z0 ∂t2 ∂z0 ∂2 ua ðz0 ; tÞ ∂ua ðz0 ; tÞ þ c kx ua ðz0 ; tÞ gx x ∂z0 2 ∂t (4)
the system is subjected to harmonic action. Applying the differentiation chain rule to Eqs. (4) and (5), the differential equations of normal vibration and the flexural rotational vibration can be expressed as
∂2 u a ðz0 ; tÞ ∂2 u a ðz0 ; tÞ ∂ua ðz0 ; tÞ 0 r p Ip ¼ Ep I p k Gp Ap u a ðz ; tÞ ∂t2 ∂z0 2 ∂z0 (5)
where Jp = k GpAp; Sp = r pIpv 2; and Kp = r pApv 2 − kx − iv cx, as designated for brevity. From Eqs. (6) and (7) the following is found: (1) when 1/Jp !0 and Sp !0, Eqs. (6) and (7) are simplified to the equation of an Euler pile on elastic soil and (2) when gx is 0, Eqs. (6) and (7) are reduced to the Winkler model. The general solution of Eq. (6) is
where vp, r p, Ep, and Gp = Poisson’s ratio, mass density, Young’s modulus, and shear modulus of the pile, respectively; Ap, Ip, and k = 6 (1 þ vp)/(7 þ 6vp) = area of cross section, inertia moment, and shear coefficient of the pile, respectively; and ua and u a, = normal displacement and bending rotation of the pile and should experience harmonic movements ua ðz0 ; tÞ ¼ Ua ðz0 Þexpiv t and u a ðz0 ; tÞ ¼ Ha ðz0 Þexpiv t as
Kp ðJp Sp Þ d4 Ua Ep Ip Kp Jp gx þ Sp ðJp þ gx Þ d2 Ua þ Ua dz0 4 dz0 2 Ep Ip ðJp þ gx Þ Ep Ip ðJp þ gx Þ ¼0
Ha ¼
(6)
Ep Ip ðJp þ gx Þ d3 Ua Ep Ip Kp þ Jp2 dUa þ Jp ðJp Sp Þ dz0 Jp ðJp Sp Þ dz0 3
(7)
Ua ðz0 Þ ¼ A1 coshaz0 þ B1 sinhaz0 þ C1 cos b z0 þ D1 sin b z0 (8) where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vv uuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2 uu" ut Ep Ip Kp Jp gx þ Sp ðJp þ gx Þ Ep Ip Kp Jp gx þ Sp ðJp þ gx Þ Kp ðJp Sp Þ a¼t þ 2Ep Ip ðJp þ gx Þ Ep Ip ðJp þ gx Þ 2Ep Ip ðJp þ gx Þ
and
ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u #2 u u uEp Ip Kp Jp gx þ Sp ðJp þ gx Þ u Ep Ip Kp Jp gx þ Sp ðJp þ gx Þ Kp ðJp Sp Þ b ¼t þ þt 2Ep Ip ðJp þ gx Þ 2Ep Ip ðJp þ gx Þ Ep Ip ðJp þ gx Þ
A1, B1, C1, and D1 are unknown initial coefficients that can be determined on the basis of known boundary conditions. For a linear elastic Timoshenko pile, the bending moment Ma(z0 ) and shear force Qa(z0 ) of the pile are related to the displacement Ua(z0 ) and rotation Ha(z0 ) as follows: dUa ðz0 Þ 0Þ ð (9) H z Qa ðz0 Þ ¼ Jp a dz0 Ma ðz0 Þ ¼ Ep Ip
dHa ðz0 Þ dz0
and internal forces between the pile head and the pile tip can be expressed by a transfer matrix [TA] as
T
T Ua ðLÞ; Ha ðLÞ; Ma ðLÞ ¼ ½TA Ua ð0Þ; Ha ð0Þ; Qa ð0Þ; Ma ð0Þ (11) 1 where ½TA ¼ ½taðLÞ½tað0Þ . The algebraic expression for each element in ½taðz0 Þ is given in Appendix I. It is convenient to divide [TA] into four 2 2 sub matrices as
(10)
Using the initial parameter method (Wang et al. 2014b) with consideration of Eqs. (7)–(10), the relationship of both deflections
© ASCE
04017075-3
Int. J. Geomech., 2017, 17(9): 04017075
2 4
TA11
TA12
TA21
TA22
3 5
Int. J. Geomech.
for the formula derivation that follows in this section. Substituting the tip boundary condition 8 9 < Qa ðLÞ = : M ð LÞ ; a
¼
x Ktip
8 9 < Ua ðLÞ = : H ð LÞ ; a
The axial force in the pile can be obtained from the differential relationship Na ðz0 Þ ¼ Ep Ap ðdWa =dz0 Þ. The mechanical property between the pile head and tip can be expressed by the transfer matrix [TB] as follows: 8 9 8 9 < Wa ðLÞ = < Wa ð0Þ = ¼ ½TB (15) : N ð LÞ ; : N ð0Þ ; a
into Eq. (11), the force-displacement relationship in the normal direction at the pile head can be obtained as
Downloaded from ascelibrary.org by Hohai University Library on 06/23/17. Copyright ASCE. For personal use only; all rights reserved.
8 9 < Qa ð0Þ = : M ð0Þ ; a
¼ ½ > > > > > > > > > > > > > > > > > > > > > > > > = = = < Hðb1Þ ðLÞ > < Hð1Þ ð0Þ > < Ha ð0Þ > b ¼ ½TA þ ½TC (24) ð1 Þ > > > > > ð1 Þ ð0Þ > > > > > > > Q ð Þ a 0 Q ð Þ > > > > > > Q L b > > > > > > b > > > > > > > > ; > > : > > > > ð Þ ð Þ > ; > : 1 M 0 a : ð1Þ ð Þ ; ð Þ 0 M b Mb L 1 1 1 where ½TC ¼ ½taðLÞ½tað0Þ ½tcð0Þ½tað0Þ þ ½tcðLÞ½tað0Þ . 0 The algebraic expression for ½tcðz Þ is given in Appendix III. Substituting the boundary conditions at the tip of the receiver pile 9 8 8 9 > = < U ð1Þ ðLÞ = < Qðb1Þ ðLÞ > b x ¼ Ktip > : ð1 Þ ð Þ ; ; : M ð1 Þ ð L Þ > Hb L b
into Eq. (23) leads to
04017075-5
Int. J. Geomech., 2017, 17(9): 04017075
Int. J. Geomech.
10
T-P model Padron et al. (2010) Giannakou et al. (2006)
Ep/Es=1000
6
4
2
Ep/Es=100 0
0
5
10
15
20
inclined angle θ
25
Fig. 3. Comparison of T-P and numerical models for static stiffness of inclined pile (L/d = 15, r s/ r p = 0.7, vs = 0.4, b s = 0.05)
18
8 θ =30°
6
15 θ =20°
12
Chh/(dEs)
Khh/(dEs)
Downloaded from ascelibrary.org by Hohai University Library on 06/23/17. Copyright ASCE. For personal use only; all rights reserved.
Khh/(dEs)
8
4
θ =30°
9 θ =20°
6
2 3
T-P model Padron et al. (2010) 0 0.0
0.2
0.4
0.6
0.8
T-P model Padron et al. (2010)
0 0.0
1.0
0.2
0.4
0.6
0.8
1.0
The dimensionless frequency a0
The dimensionless frequency a0
Fig. 4. Comparison of T-P and BEM-FEM models for horizontal impedance of inclined pile (Ep/Es = 1,000; L/d = 15; r s/ r p = 0.7; vs = 0.4; b s = 0.05)
8 9 8 9 8 9 ( ) < U ð1Þ ð0Þ = < Qð1Þ ð0Þ = < U ð1Þ ð0Þ = 0 b b b ½T1 þ ½T2 þ ½T3 ¼ : Hð1Þ ð0Þ ; : M ð1Þ ð0Þ ; : Hð1Þ ð0Þ ; 0 b b b (25) x x ½TA11 ½TA21 ; ½T2 ¼ Ktip ½TA12 ½TA22 ; where ½T1 ¼ Ktip x x ½T3 ¼ Ktip ½TC11 ½TC21 þ Ktip ½TC12 ½TC22 ½ Ub2 ðLÞ > > > > > > > > > > > > = < Hð2Þ ðLÞ >
9 8 ð Þ > Ub2 ð0Þ > > > > > > > > > > > ( ) > = < Hð2Þ ð0Þ > Wa ð0Þ b b ¼ ½TA þ ½TD ð2Þ ð2Þ > > > > > > > > Na ð0Þ ð Þ ð Þ L 0 Q Q > > > > b > > > > b > > > > > > > > > > ; ; : ð2Þ ð Þ > : ð2Þ ð Þ > Mb 0 Mb L
gx Jp þ gx
ð Þ
and Ub2 is the normal displacement of the receiver pile induced by the axial vibration of the source pile. The solution of Eq. (27) is
þ Fc A2 cosh h z0 þ B2 sinh h z0
1 1 1 where ½TD ¼ ½tdðLÞ½tbð0Þ ½taðLÞ½tað0Þ ½tdð0Þ½tbð0Þ . The algebraic expression for ½tdðz0 Þ is given in Appendix IV. Applying the boundary conditions at the receiver pile tip 8 9 8 9 < Qð2Þ ðLÞ = < U ð2Þ ðLÞ = b b x ¼ Ktip : M ð2 Þ ð L Þ ; : Hð2Þ ðLÞ ; b b
(28)
where Fc ¼
fz1 þ fz2 h 2 þ fz3 h 4 fz ðsÞsinð w 1 þ w 2 Þ h 4 þ d 2 h 2 λ4
0.30
to Eq. (29) leads to 0.00
ϕ =10°
-0.05
0.25
0.15 0.10 0.05 0.00 0.0
ϕ =30°
-0.10
ϕ =20°
Im[λnn]
Re[λnn]
0.20
ϕ =30°
-0.15 -0.20 -0.25
T-P model Ghasemzadeh and Alibeikloo's model (2011) 0.2
0.4
0.6
0.8
ϕ =20°
ϕ =10°
1.0
-0.30 0.0
0.2
0.4
0.6
0.8
1.0
The dimensionless frequency a0
The dimensionless frequency a0
Fig. 5. Comparison of the normal-normal interaction factors for different inclined angles (Ep/Es = 1,000; L/d = 25; r s/r p = 0.7; vs = 0.4;b s = 0.05; s/d = 3)
0.4
0.0
ϕ =30° ϕ =20°
ϕ =10°
-0.1
Im[ λan]
0.2
Re[ λan]
Downloaded from ascelibrary.org by Hohai University Library on 06/23/17. Copyright ASCE. For personal use only; all rights reserved.
ð Þ Ub2 ðz0 Þ ¼ A4 coshaz0 þ B4 sinhaz0 þ C4 cos b z0 þ D4 sin b z0
(29)
ϕ =10°
0.0
ϕ =20°
-0.2 ϕ =30°
-0.2
-0.3 T-P model Ghasemzadeh and Alibeikloo's model (2011)
-0.4 0.0
0.2
0.4
0.6
0.8
1.0
-0.4 0.0
The dimensionless frequency a0
0.2
0.4
0.6
0.8
1.0
The dimensionless frequency a0
Fig. 6. Comparison of the axial-normal interaction factors for different inclined angles (Ep/Es = 1,000; L/d = 25; r s/ r p = 0.7; vs = 0.4; b s = 0.05; s/d = 3)
© ASCE
04017075-7
Int. J. Geomech., 2017, 17(9): 04017075
Int. J. Geomech.
8 9 8 9 < U ð2Þ ð0Þ = < Qð2Þ ð0Þ = b b 0 Wa ð0Þ ½T1 ¼ þ ½T2 þ ½T4 0 Na ð0Þ : Hð2Þ ð0Þ ; : Mð2Þ ð0Þ ; b b
Comparison with Numerical Models Verification of T-P Model for Single Inclined Pile
(30) where
"
x Ktip
TD1;1
TD1;2
TD2;1
TD2;2
#
"
TD3;1
TD3;2
TD4;1
TD4;2
#
Substituting Eq. (16) and the boundary conditions at the head ð Þ ð Þ of the receiver pile, Qb2 ð0Þ ¼ 0 and Hb2 ð0Þ ¼ 0), into Eq. (30), the axial-normal interaction factor l an can be obtained as: ð Þ Ub2 ð0Þ ðT42;1 þ T42;2 > ð ð a þ ! a a z U sinh > > > > > > > > > > > = < ðUa þ !a3 Þcoshðaz0 Þ > fa2 g ¼ > > > U b þ ! b 3 sinð b z0 Þ > > > > > > > > >
> ; : U b ! b 3 cosð b z0 Þ >
(34)
Appendix II. Algebraic Expression for [tb(z0 )] The algebraic expression for [tb(z0 )] is given as follows: 2 ½tbðz0 Þ ¼ 4
9T 8 > Wa þ Jp !a3 sinhðaz0 Þ > > > > > > > > >
> > > 3 0 = < Wa þ Jp !a coshðaz Þ > fa3 g ¼ > > > W b þ Jp ! b 3 sinð b z0 Þ > > > > > > > > > > ; : W b J ! b 3 cosð b z0 Þ >
(35)
3
coshð h z0 Þ
sinhð h z0 Þ
h Ep Ap sinhð h z0 Þ
h Ep Ap coshð h z0 Þ
5
(37)
Appendix III. Algebraic Expression for [tc(z0 )] The algebraic expression for [tc(z0 )] is given as follows:
p
½tcðz0 Þ ¼ fc1 ; c2 ; c3 ; c4 gT
9T 8 > ðUa2 þ !a4 ÞEp Ip coshðaz0 Þ > > > > > > > > > > > 2 4 0 > = < ðUa þ !a ÞEp Ip sinhðaz Þ > fa4 g ¼ > > > U b 2 ! b 4 Ep Ip cosð b z0 Þ > > > > > > > > > > ; : U b 2 ! b 4 E I sinð b z0 Þ >
where
(38)
0 0 0 0 fc1 g ¼ z Fa sinhðaz Þz Fa coshðaz Þ
(36)
z0 Fb sinð b z0 Þz0 Fb cosð b z0 Þg
p p
8 9T > Fa ðUa þ !a3 Þz0 coshðaz0 Þ þ ðU þ 3!a2 Þsinhðaz0 Þ > > > > > > > > > > > > 2 0 3 0 0 = < Fa ðU þ 3!a Þcoshðaz Þ þ ðUa þ !a Þz sinhðaz Þ > fc2 g ¼
> > > Fb U b ! b 3 z0 cosð b z0 Þ þ U 3! b 2 sinð b z0 Þ > > > > > > > > >
> ; : F U 3! b 2 cosð b z0 Þ U b ! b 3 z0 sinð b z0 Þ >
(39)
(40)
b
8
9T
2 0 > ðaz0 Þ þ W þ 3Jp !a2 sinhðaz0 Þ > F z a W þ J ! a cosh > > a p > > > > > >
> > > = < Fa W þ 3Jp !a2 coshðaz0 Þ þ a W þ Jp !a2 z0 sinhðaz0 Þ > fc3 g ¼ > F b W J ! b 2 z0 cosð b z0 Þ þ W 3J ! b 2 sinð b z0 Þ > > > > > b p p > > > > > >
> ; : F W 3J ! b 2 cosð b z0 Þ b W J ! b 2 z0 sinð b z0 Þ > b
p
p
8 9T > Fa Ep Ip 2aðU þ 2a2 !Þcoshðaz0 Þ þ a2 ðU þ a2 !Þz0 sinhðaz0 Þ > > > > > > > > > > > > 2 2 0 0 2 0 = < Fa Ep Ip a ðU þ a !Þz coshðaz Þ þ 2aðU þ 2a !Þsinhðaz Þ > fc4 g ¼
> > > Fb Ep Ip 2 b U 2 b 2 ! cosð b z0 Þ b 2 U b 2 ! z0 sinð b z0 Þ > > > > > > > > >
> > ; : 2 2 2 0 0 0 Fb Ep Ip b U b ! z cosð b z Þ þ 2 b U 2 b ! sinð b z Þ
© ASCE
(41)
04017075-13
Int. J. Geomech., 2017, 17(9): 04017075
(42)
Int. J. Geomech.
Appendix IV. Algebraic Expression for [td(z0 )] The algebraic expression for [td(z0 )] is given as follows: 3 Fc coshð h z0 Þ Fc sinhð h z0 Þ 7 6
7 6 7 6 Fc U h þ ! h 3 sinhð h z0 Þ Fc U h þ ! h 3 coshð h z0 Þ 7 6 0 ½tdðz Þ ¼ 6 7
3 0 6 Fc W h þ Jp ! h 3 sinhð h z0 Þ Fc W h þ Jp ! h coshð h z Þ 7 7 6 5 4
Fc U h 2 þ ! h 4 Ep Ip coshð h z0 Þ Fc U h 2 þ ! h 4 Ep Ip sinhð h z0 Þ Downloaded from ascelibrary.org by Hohai University Library on 06/23/17. Copyright ASCE. For personal use only; all rights reserved.
2
b s ¼ radiation damping of the soil; ð jÞ Ha, Hb ¼ rotation angle of source pile and rotation of receiver pile induced by vibration of the source pile in the j direction; l nn, l an ¼ dynamic interaction factor in the normalnormal and axial-normal direction, respectively; vp, vs ¼ Poisson’s ratio of the pile and soil, respectively; r p, r s ¼ mass density of the pile and soil, respectively; f ¼ angle between the centerline direction of two piles; w 1, w 2 ¼ inclined angle of the source pile and receiver pile, respectively; and v ¼ frequency of the harmonic excitation.
Acknowledgments The financial supports from the National Natural Science Foundation of China (Grant 51678302) and Fundamental Research Funds for the Central Universities (Grant 2016B15014) are gratefully acknowledged. This work is also supported in part by the scholarship from the China Scholarship Council (CSC) under Grant 201408320149.
Notation The following symbols are used in this paper: Ap ¼ cross-section area of the pile; a0 ¼ dimensionless frequency of the excitation; Ch ¼ horizontal damping ratio at the pile head; cx, cz ¼ normal and axial damping along the pile shaft, respectively; d ¼ sectional diameter of the inclined pile; Ep, Es ¼ Young’s modulus of the pile and soil along the pile shaft, respectively; Gs, Gs,tip ¼ Shear modulus of the soil along the pile shaft and under the pile tip, respectively; gx ¼ shear stiffness of the soil along the pile shaft; Ip ¼ inertia moment of the pile; K z h ¼ horizontal dynamic stiffness at the pile head; K ¼ axial base impedance of the pile; tip x Ktip ¼ normal base impedance of the pile; kx, kz ¼ normal and axial stiffness along the pile shaft, respectively; L ¼ length of the inclined pile-shaft; ð jÞ Ma, Mb ¼ bending moment of a source pile and rotation of a receiver pile induced by the vibration of the source pile in the j direction, respectively; Na ¼ axial force of a source pile; ð jÞ Qa, Qb ¼ shear force of a source pile, and rotation of a receiver pile induced by the vibration of the source pile in the j direction, respectively; ð jÞ
