Soil–pile interaction considering structural yielding: Numerical modeling and experimental validation

Engineering Structures 99 (2015) 319–333

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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Soil–pile interaction considering structural yielding: Numerical modeling and experimental validation Andreas E. Kampitsis a,⇑, Spyros Giannakos b, Nikos Gerolymos a, Evangelos J. Sapountzakis a a b

School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens, Greece Fugro Consultants Inc., 1000 Broadway, Suite 440, Oakland, CA 94607-4099, USA

a r t i c l e

i n f o

Article history: Received 7 October 2014 Revised 4 May 2015 Accepted 5 May 2015

Keywords: Inelastic analysis Pile-foundation system Structural yielding Distributed plasticity Fiber model Boundary element method Lateral Pushover tests 3D nonlinear FE

a b s t r a c t In the present study the accuracy of an efficient beam formulation for inelastic analysis of pile-foundation systems is validated against a series of Laboratory Pushover tests on vertical single piles embedded in dry sand under different load paths to failure in M–Q space conducted in the Laboratory of Soil Mechanics/Dynamics (LSMD) in NTUA. The obtained results are also compared to those from a fully 3D Nonlinear Finite Element (FE) simulation. The proposed beam formulation is based on the Boundary Element Method (BEM) accounting for shear deformation effect. A displacement based procedure is employed and inelastic redistribution is modeled through a distributed plasticity (fiber) model exploiting pile material constitutive laws and numerical integration over the cross-sections, while the surrounding soil is modeled through independent inelastic Winkler springs. The efficiency of the beam formulation is verified through the good agreement between the measured and the calculated results. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Pile foundation systems are extensively employed in civil engineering applications, where intense forces have to be undertaken, as for instance in bridges, high buildings, and offshore structures. Currently, these systems are designed to behave elastically for every type of loading, and hence under severe actions the damage is concentrated on the aboveground structural elements. However, recent research efforts have proved that allowing the development of a plastic hinge at the pile–soil foundation system can be beneficial [1,2]. Thus, the investigation of the material nonlinearity is important for the response of the foundation systems. According to the modeling of the mechanical behavior of the pile–soil system several approaches have been proposed. The simplest model is the Elastic-Beam-on-Winkler-Foundation [3], where the soil medium is modeled as an array of uncoupled springs while the pile as a beam element. The springs can be either linear elastic providing resistance in direct proportion to the beam deflection [3–5], or accounting for the tensionless ⇑ Corresponding author. E-mail addresses: [email protected] (A.E. Kampitsis), [email protected] (S. Giannakos), [email protected] (N. Gerolymos), [email protected] (E.J. Sapountzakis). http://dx.doi.org/10.1016/j.engstruct.2015.05.004 0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.

character of the subgrade reaction [6–10]. Moreover, several researchers have employed modified multiple-parameter approaches [11–14] in order to take into account the shear transfer between the soil layers. To account for the nonlinear nature of the soil due to high strain level (soil plastic deformations), several researches employed the concept of Elastic-Beam-on-Nonlinear-Foundation. In that formulation, the foundation load–displacement relation is assumed to follow a nonlinear law while the beam remains elastic throughout the analysis. The load–deformation relationships are described by empirical p–y curves [15–17], where the spring stiffness value is variable, allowing consideration of a non-proportional relationship between the soil resistance per unit pile length p and the lateral displacement y. In general, this method is the most commonly used in order to study the response of a loaded pile in engineering practice. Its popularity derives from the simplicity and the adequate accuracy while the main drawback of the method is that it neglects the soil continuity and the soil shearing resistance. In order to overcome these disadvantages several researchers have adopted the three-dimensional continuum model, where nonlinear behavior of the ground can be taken into consideration employing various constitutive laws. Although the continuum approach is a powerful and rigorous way of simulating the whole system

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accounting for various three-dimensional soil–pile interaction effects and the nonlinear behavior of soil, it is not widely implemented, besides research purposes, as it is intractable, mathematically complex and the computational and modeling effort required is extremely time consuming. Although the nonlinear behavior of the soil has been studied extensively, there are only few studies where the pile inelasticity has been taken into consideration through nonlinear inelastic constitutive laws for the stress–strain relations. Regarding the structural elements, distributed plasticity models are acknowledged in the literature [18–20] to capture more rigorously material nonlinearities than cross-sectional stress resultant approaches [21] or lumped plasticity idealizations [22,23]. In recent years, significant advantages have been presented in literature regarding the nonlinear inelastic analysis of beam-like structural members and various beam element models have been proposed following either the displacement-based [24] or forced-based formulations [25–27]. More specifically, Papachristidis et al. [27] proposed a force-based fiber Timoshenko beam–column element taking under consideration the interaction between axial, bending, shear and torsion. Triantafyllou and Koumousis [28] presented a displacement based hysteretic model with axial shear and bending interaction while the shear locking effects were eliminated by using the exact shape functions, while Gkimousis and Koumousis [29] presented a distributed plasticity fiber beam formulations for both displacement and force based approaches. On the contrary, only few studies have taken into account the inelastic behavior of both the beam and the foundation. Ayoub [30] presented an inelastic finite element formulation capable of capturing the nonlinear behavior of both the beam and the foundation. The element is derived from a two-field mixed formulation with independent approximation of forces and displacements and compared with the displacement based formulation. Mullapudi and Ayoub [31] expanded the research in inelastic analysis of beams resting on two-parameter foundation where the values for the parameters are derived through an iterative technique that is based on an assumption of plane strain conditions for the soil medium. Recently, Sapountzakis and Kampitsis [32] presented a BE method for the inelastic analysis of Euler–Bernoulli beam resting on two-parameter tensionless elastoplastic foundation, illustrating the important influence of both the inelastic and the tensionless character of the foundation. In the present study, the BE method for the inelastic analysis of Euler–Bernoulli beams presented by Sapountzakis and Kampitsis in [32] is extended in Timoshenko beams and validated against a series of Laboratory Pushover tests on vertical single piles embedded in dry sand under different load paths to failure in M–Q space conducted in the Laboratory of Soil Mechanics/Dynamics (LSMD) in NTUA by Gerolymos [33] and Giannakos [34]. The obtained results are also compared to those obtained from a fully 3D Nonlinear Finite Element (FE) simulation implemented in the finite element code ABAQUS [35]. The pile is characterized by a circular cross-section, while its edges are subjected to the most general boundary conditions. A displacement based procedure is employed and inelastic redistribution is approximated through a distributed plasticity (fiber) model exploiting material constitutive laws and numerical integration over the cross-sections [32,36], while the surrounding soil is modeled through independent inelastic Winkler springs. The essential features and novel aspects of the present formulation compared with previous ones are summarized as follows. i. The proposed formulation is validated against a unique series of Laboratory Pushover tests. ii. Comparison against Nonlinear 3D Finite Element simulation is performed.

iii. The displacement based formulation takes into consideration inelastic redistribution along the pile axis by exploiting material constitutive laws and numerical integration over the cross-sections (distributed plasticity approach). iv. The shear deformation effect in the pile is taken into account while shear locking is avoided by employing the same order of approximation for both the rotation due to bending and the derivative of the deflection. v. The inelasticity of the soil medium is approached by employing independent linear elastic-perfectly plastic Winkler springs. vi. An incremental-iterative solution strategy is adopted to restore global equilibrium of the pile–soil system. 2. Proposed beam formulation for the inelastic analysis of piles A cylindrical pile of length l (Fig. 1a) of circular cross-section is considered, occupying the two dimensional multiply connected region X of the y–z plane bounded by the Cj (j = 1, 2, ..., K) boundary curves. In Fig. 1a, Cyz is an arbitrary bending coordinate system through the centroid of the cross-section. The normal stress–strain relationship of the pile material is assumed to be elastic–plastic– strain hardening with initial modulus of elasticity E0, shear modulus G, post-yield modulus of elasticity Et, yield stress rY0 and yield strain eY0 (Fig. 1b). The pile is partially embedded in nonlinear inelastic Winkler type soil of initial stiffness kw, yielding load PwY, and hardening modulus kwt. The pile is subjected to the combined action of arbitrarily distributed or concentrated transverse loading pz = pz(x) and bending moment my = my(x). Under the action of the aforementioned loading, the displacement field of the pile taking into account shear deformation effect  ðx; zÞ and transverse wðxÞ  is characterized by the axial u displacement components of an arbitrary point with respect to the Cyz system of axes, given as

 ðx; zÞ ¼ zhy u

 zÞ ¼ wðxÞ wðx;

ð1a; bÞ

where u(x), w(x) are the corresponding components of the centroid C and hy(x) is the angle of rotation due to bending of the cross-section with respect to its centroid. In the commonly employed case of the piles with circular or hollow circular cross section, the centroid coincides with the center of the piles’ cross section. It is worth noting that since the additional angle of rotation of the cross-section due to shear deformation is taken into account, the angle of rotation due to bending is not equal to the derivative of the deflection (i.e. hy – w0 ). Adopting the Timoshenko beam assumption, considering small strains, employing the Cauchy stress tensor and assuming an isotropic and homogeneous material, the normal stress rate is defined in terms of the corresponding strain rate as



drxx dsxz





E ¼ 0

0 G

(

deelxx

) ð2Þ

dcelxz

where d() denotes infinitesimal incremental quantities, the superscript el denotes the elastic part of the strain component and mÞ E ¼ ð1þE0mð1 . If the plane stress hypothesis is undertaken, then Þð12mÞ

E ¼ 1E0m2 holds [37] even though E is frequently considered instead of E* (E*  E0) in beam formulations [37,38]. This last consideration has been followed throughout this study, while any other reasonable expression of E* could also be used without any difficulty in many beam formulations. For the elastic response of the material, the stress–strain relationship is governed by Hooke’s law as presented in Eq. (2). When plastic flow occurs, then



dexx

dcxz

T

¼ f deelxx

T

dcelxz g þ f depl xx

dcpl xz g

T

ð3Þ

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M Q

pz

my

l Winkler Spring Mesh

C y

z i

(a) Load

(w u , P wu ) (w Y , P wY )

t

k wt kw

0 Displacement w (x)

O

0 Y0

(b)

(c)

Fig. 1. Cylindrical pile of a circular cross-section in Winkler soil (a) Normal stress–strain (b) and soil reaction–displacement (c) relationships.

where the superscript pl denotes the plastic part of the strain components. The von Mises yielding criterion and an associated flow rule for the material are considered. The yield condition is described as

f VM

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ r2xx þ 3s2xz  rY ðepl eq Þ ¼ 0

depl eq ¼ dk (dk is the proportionality factor [39]). Moreover, the plastic modulus h is defined as h ¼ drY =depl eq or drY ¼ hdk and can be estimated from a tension test as h = EtE0/(E0  Et) (Fig. 1b). According to the associated flow rule [40–42] the plastic strain rates are given as

n T @f VM dcpl xz g ¼ dk @ rxx

@f VM @ sxz

oT

ð5Þ

Using the aforementioned relation linking the yield stress rate and the proportionality factor, Eqs. (2)–(4) and exploiting the plastic loading condition (dfVM = 0), the stress rates–total strain rates relations are resolved as Delpl 8 9 zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{  >  < drxx > = 1c sym: dexx 11 ¼ dsxz > > dcxz : ; c c21 c22

c11 ¼ E0 ðhr2e þ 9Gs2xz Þ;

c21 ¼ 3E0 Grxx sxz ;

c22 ¼ Gðhr2e þ E0 r2xx Þ ð7a; b; cÞ

ð4Þ

where rY is the yield stress of the material and epl eq is the equivalent plastic strain, the rate of which is defined in [39] and is equal to

f depl xx

where [Del–pl] is the elastoplastic constitutive matrix as presented by Chen and Trahair [43] with the analytical expressions of the components expressed as

c ¼ hr2e þ E0 r2xx þ 9Gs2xz ;

re ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2xx þ 3s2xz

ð7d; eÞ

When h is set equal to zero in the above relations, the constitutive matrix presented by Baba and Kajita [44] is obtained. It is worth noting that plasticity leads to a rate form constitutive nonlinear law, thus the tangent matrix is not only a function of the material properties but also of the current stress state. The same applies to hypoelastic materials (nonlinear elasticity). The key difference between the two theories is that in plasticity the dissipated (plastic) work and the behavior upon unloading are defined by the plastic flow rule and the loading/unloading conditions (yield criterion, complementarity and consistency conditions), while in nonlinear elasticity the same constitutive matrix is used upon unloading and no residual plastic deformation occurs. 2.1. Equations of global equilibrium

ð6Þ

To establish global equilibrium equations, the principle of virtual work neglecting body forces is employed, that is

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Z V

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Z

ðrxx dexx þ sxz dcxz ÞdV ¼

l

Z

ðpz dw  my dw0 Þdx 

l

ðpf dwÞdx

ð8Þ

where the integral quantities represent the strain energy, the external load and foundation reaction work while d() denotes virtual quantities, V is the volume, l is the length of the pile and pf is the foundation reaction. After conducting some algebraic manipulations, the global equilibrium equations of the pile in incremental form are obtained as

GAz ðDw00 þ Dh0y Þ þ Dpf ¼ Dpz þ

9 = inside the pile pl ;

dDQ pl z dx

E0 Iy Dh00y þ GAz ðDw0 þ Dhy Þ ¼ Dmy þ

pl dD M y

dx

 l @u   du1 u dx  u  u 1 2 dx @x 0 dx 0  l Z l 2 d u2  @u  du2 and u2 ðnÞ ¼ u dx  u  u 2 2 dx @x 0 dx 0 u1 ðnÞ ¼

Z

l

GAz u1 ðnÞ ¼

 at the pile ends x ¼ 0; l

ð10a; bÞ

where D() denotes incremental quantities, while the incremental stress resultants (DQz, DMy) are given with respect to the plastic pl quantities ðDQ pl z ; DM y Þ as

E0 Iy u2 ðnÞ ¼

GAz

Z

DQ pl z

ð11aÞ

X

ð11bÞ

ð13bÞ

d u2 2

E0 Iy u2 ðnÞ ¼

l

ð14aÞ

!

 dDM pl du1 y þ u2  Dmy   DQ pl K2 dx z dx dx 0  l du2  K1 u2  E 0 I y K2 dx 0

Z

l

GAz

ð14bÞ After carrying out several integrations by parts, Eq. (14) yield

GAz u1 ðnÞ ¼

Z

l

GAz u2 K1 dx þ

Z

0

l

ðDpf  Dpz ÞK2 dx þ 0

Z

l

0

DQ pl z K1 dx



 l h il du1  K  GA K þ u K u  DQ pl 2 z 2 2 1 1 z 0 dx 0 E0 Iy u2 ðnÞ ¼

ð15aÞ

 Z l Z l du1 þ u2 K2 dx  Dmy K2 dx þ DM pl y K1 dx dx 0 0 0  l Z l h il du2 pl DQ pl K dx  D M K  E I K  K u  2 2 0 y 2 1 2 y z 0 dx 0 0 Z

l

GAz

ð15bÞ

According to the precedent analysis, the inelastic problem of Timoshenko pile embedded in nonlinear inelastic soil, reduces to establishing the displacement components Dw(x), Dhy(x) having continuous derivatives up to the second order with respect to x and satisfying the boundary value problem described by the governing differential Eq. (8) along the pile and the boundary conditions, i.e. Eq. (10), at the pile ends x = 0, l. This boundary value problem is solved employing the BEM [46], as this is developed in [32] for the solution of a single fourth order differential equation, after modifying it as follows. The motivation to use this particular technique is justified from the intention to retain the advantages of a BEM solution over a domain approach, while using simple fundamental solutions and avoiding finite differences. According to this method, let u1(x) = Dw(x), u2(x) = Dhy(x) be the sought solution of the problem. The solution of the second order 2

E0 I y

! du2 dDQ pl z GAz u1 ðnÞ ¼ GAz þ Dpf  Dpz  K2 dx dx dx 0  l du1  GAz K2  K1 u1 dx 0

2.2. Integral representation – numerical solution

2

 l du2 @u u dx  E0 Iy u  u2 dx @x dx 0

2

where the kernels are given as K1(r) = 0.5sgnr and K2(r) = 0.5|r|. Solving Eq. (9a) with respect to GAz Dw00 and substituting the result in Eq. (13a) and similarly solving Eq. (9b) with respect to E0 Iy Dh00y and substituting the result in Eq. (13b), the following integral representations are obtained

X

In Eqs. (9)–(11), A is the cross-section area, Iy is the moment of inertia with respect to z-axis and GAz is its shear rigidity according to the Timoshenko’s beam theory, where Az = jzA = (1/az)A is the shear area, jz is the shear correction factor and az the shear deformation coefficient. This coefficient is evaluated using an energy approach [45]. Moreover, ai, bi (i = 1, 2, 3) are functions specified at the pile ends. The boundary conditions, i.e. Eq. (10), are the most general ones for the problem at hand. It is apparent that all types of the conventional boundary conditions (clamped, simply supported, free or guided edge) may be derived from Eq. (10) by specifying appropriately the functions ai and bi. Finally, it is noted that by dropping the plastic quantities of the above equations, the boundary value problem of the examined topic under elastic conditions is formulated.

ð13aÞ

d u1

2

l

0

DMpl y

zfflfflfflfflfflfflfflfflfflffl Z ffl}|fflfflfflfflfflfflfflfflfflfflffl{ DMy ¼ E0 Iy Dh0y  E0 Depl xx zdX

 l du1 @u u dx  GAz u  u1 dx @x dx 0 2

l

Z

zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl ffl{ Z DQ z ¼ GAz Dcxz  G Dcpl zd X xz

2

Z 0

ð9a; bÞ

a1 DQ z þ a2 Dw ¼ Da3 b1 DMy þ b2 Dhy ¼ Db3

ð12a; bÞ

where u* = 0.5|r| is the fundamental solution [46], with r = x–n being the distance between any two points x and n, where n is a constant collocation point while x runs through the interval of the pile (0, l). Since GAz and E0Iy are independent of x, Eq. (12) can be written as

þ DQ z

along with the incremental version of the boundary conditions

2

d u1

2

differential equations d u1 =dx ¼ Dw00 , d u2 =dx ¼ Dh00y are given in integral form as [46]

In the above equations, the boundary quantities can be grouped and expressed in terms of the boundary stress resultants as

GAz u1 ðnÞ ¼

Z

l

GAz u2 K1 dx þ

0

Z

l

ðDpf  Dpz ÞK2 dx þ 0

þ ½DQ z K2 þ GAz K1 u1 l0 E0 Iy u2 ðnÞ ¼

Z 0

l

DQ pl z K1 dx ð16aÞ

 Z l Z l du1 þ u2 K2 dx  Dmy K2 dx þ DM pl y K1 dx dx 0 0 0 Z l

l  DQ pl ð16bÞ z K2 dx þ DM y K2 þ E0 Iy K1 u2 0 Z

l

GAz

0

Numerical methods requiring domain approximation of unknown quantities exhibit ‘‘locking’’ effects when Timoshenko theory is applied to cases where the Euler–Bernoulli theory could also be used (u2  u01 ) [47]. Since domain approximation is also required in the present numerical technique, locking effects are alleviated

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by employing the same order of approximation for u2 and u01 . In order to achieve explicit appearance of u01 in Eq. (16a), this integral representation is differentiated with respect to n, yielding

GAz u01 ðnÞ ¼ GAz u2 ðnÞ 

Z 0

l

The above proposed formulation is validated against a number of published Pushover tests [33,34]. Results are further compared with the predictions from 3D Nonlinear Finite Element Analyses.

ðDpf  Dpz ÞK1 dx  DQ pl z ðnÞ

 ½DQ z K1 l0

ð17Þ

Eqs. (16b) and (17) have been brought into a convenient form to establish a numerical computation of the unknown quantity. More specifically, the pile interval (0, l) is divided into L elements, on each of which u01 ; u2 are assumed to vary according to a certain law (constant, linear, parabolic, etc.). The linear element assumption is employed herein (Fig. 2) as the resulting numerical implementation scheme is both reliable and simple. It is noted that this technique requires neither differentiation of shape functions nor finite differences application. Employing the aforementioned procedure and a collocation technique, a set of 2(L + 1) algebraic equations is obtained with respect to 2(L + 4) unknowns. Two additional algebraic equations are obtained by applying the integral representation (16a) at the pile ends n = 0, l. These 2(L + 2) equations along with the four boundary conditions, i.e. Eq. (10), yield a linear system of 2(L + 4) simultaneous algebraic equations

½KfDdg ¼ fDbext g þ fDbpl g

ð18Þ

where [K] is a known generalized stiffness matrix, {Dd} is a generalized incremental unknown vector given as

8 0 u1j > > > > > u2j > > > > > > < u1i fDdg ¼ u01i > > > > > u2i > > > > DM yi > > : DQ zi

9 j ¼ 2; 3; . . . ; L > > > > j ¼ 2; 3; . . . ; L > > > > > i ¼ 1; L þ 1 > > = i ¼ 1; L þ 1 > > > i ¼ 1; L þ 1 > > > > > i ¼ 1; L þ 1 > > > ; i ¼ 1; L þ 1

3. Model validation against laboratory pushover tests and comparison with 3D nonlinear FE analysis

3.1. Laboratory Pushover tests A general closed-form expression for the failure envelope of flexible single piles under combined M–Q loading for different soil conditions was presented by Gerolymos et al. [33,48] and Giannakos [34], based on a limit equilibrium approach according to Brom’s theory for the lateral capacity of flexible piles. In this theory, the post-failure response of the pile is characterized by the formation of a plastic hinge at a certain depth that acts as a rotation pole for the above hinge rigidly deformed portion. The lateral capacity is then obtained by solving moment and force equilibriums with respect to the rotation pole, and assuming simultaneous exceedance of the ultimate lateral soil resistance and bending moment capacity [33,34,48–51]. The proposed failure envelope is normalized against the pure lateral load capacity (QY) and the pure overturning moment capacity (MY) of the pile–soil system, thus allowing the shape of the envelope to remain unaffected from any other strength parameter of the soil or the pile. For a flexible pile embedded in dry sand and taking into consideration all possible M–Q combinations at the pile head defined at the ground surface, the following expression for the failure envelope was derived:

   3=2       Q  M M   f ¼ sgnðQ Þ  þ  sgnðMÞ  1 ¼ 0 for   < 1   QY MY MY ð19Þ

while {Dbext}, {Dbpl} are known vectors representing all the terms related to the incremental externally applied loading and incremental plastic quantities, respectively. The developed incremental-iterative solution strategy in this work is similar to the one presented in [32] concerning inelastic analysis by BEM. Load control over the incremental steps is used and load stations are chosen according to load history and convergence requirements. Incremental stress resultants are decomposed into elastic and plastic parts as presented in Eq. (11) and are computed through an iterative procedure since changes in the plastic part of incremental stress resultants of two successive iterations are not negligible.

ð20aÞ

and

    M M f ¼    1 ¼ 0 for   ¼ 1 MY MY

ð20bÞ

where M, Q the moment and the lateral force acting at the pile head, respectively while MY, QY the pure overturning moment and pure lateral load capacity of the pile–soil system. In order to validate the adequacy of this proposed analytical expression, a series of Pushover tests on a vertical single pile embedded in dry sand under different load paths to failure in M–Q space, were performed. The reference point is considered at the ground surface and the pile was modeled to behave as a flexible pile. To avoid reduced-scale testing problems, the model was tested in true scale (micro-pile). A main concern was that the sand behavior is strongly dependent on the stress state and stress path. The main shortcoming of small-scale testing, which is alleviated by centrifuge model testing, is that generally involves low stresses. The significantly lower

Fig. 2. Discretization of the pile interval into linear elements, distribution of the nodal points and approximation of several quantities.

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levels of effective stress in the model result in overestimating u0 , as the latter is in fact a function of confining stress [52] rather than a constant value, and as a result the soil appears to have larger strength and dilatancy in comparison to real-scale conditions. Nevertheless, 1 g testing is a valid method, provided that the scale effects and the stress-dependent soil behavior are considered in the design of the experiments and results are interpreted appropriately [53]. The normalized nature of the proposed failure envelope allowed the use of 1 g testing for its validation, although reduced-scale testing generally involves low stresses. 3.1.1. Experimental Push-over tests The tests were conducted on a vertical pile placed in a sand mass of uniform density. The sand specimens were prepared by the air sand-raining process into a rectangular container (78 cm wide by 148 cm long by 64.5 cm deep), with the use of an electronically controlled screw-jack actuator. The desired density of the dry sand was obtained by varying three parameters: (a) the flow of sand (opening of the hopper), (b) the automatically maintained drop height, and (c) the scanning rate. Longstone sand, an industrially produced fine and uniform quartz sand with D50 = 0.15 mm and uniformity coefficient Cu = D60/D10 = 1.42, was used in the experiments. The void ratios at the loosest and densest state have been measured as emax = 0.995 and emin = 0.614, and the specific gravity as Gs = 2.64. The dry unit weight and relative density of the specimen were measured to be cd  16.2 kN/m3 and Dr = 94%, respectively. Laboratory results from tests on Longstone sand indicated mean values of peak and critical-state angles of up = 56° at very small stress levels (<10 kPa) and ucv = 32°, respectively. The material and strength characteristics of the sand have been documented in [54]. The model pile is a hollow aluminum 6063-F25 cylinder of 3 cm external diameter, 2.8 cm internal diameter, and 60 cm length. The elasticity modulus of the pile is E0 = 70 GPa and the yield stress of the aluminum is 215 MPa. The instrumentation included wire displacements transducers in order to measure the displacement and rotation at the pile–head. The pile was fixed at the base of the sandbox to ensure verticality during the sand raining process; however, its length was sufficiently large for the bending failure (plastic hinge) not to be affected by the tip boundary conditions. The load is applied to the pile at a distance e from ground surface, while the aboveground height of the pile is f. The experimental setup is portrayed in Fig. 3. For more details on the laboratory testing process the reader is referred to the studies of Gerolymos [33] and Giannakos [34].

3.1.2. Experiments examined The piles studied were subjected to displacement control lateral loading. Monotonic loading was imposed either at the pile head considered to be at the ground surface or at a specified distance from ground surface in order to produce a moment acting at the pile head. A total number of 8 experiments were performed; namely: Test 1. Lateral load at the ground surface level in order to determine the plastic yield shear force of the pile–soil system (QY = 97.15 kg; (1, 0) in M/My–Q/Qy space), Test 2. Pure moment conditions in order to validate the plastic yield moment of the pile–soil system (MY = 18.19 kg m; (0, 1)), Test 3. Lateral load applied at 32 cm above the ground surface (0.47, 0.80), Test 4. Lateral load applied at 20 cm above the ground surface (0.59, 0.63), Test 5. Retest of the pile under lateral load applied at 32 cm above the ground surface in order to check the repeatability of the experiments (0.46, 0.79), Test 6. Lateral load applied at 10 cm above the ground surface (0.75, 0.40), Test 7. Lateral load applied at 6 cm above the ground surface (0.88, 0.28), Test 8. Lateral load applied at 56 cm above the ground surface (0.28, 0.84). For the first experiment, only lateral load was applied at the pile head in order to determine the ultimate lateral load capacity QY of the pile–soil system. Similarly, for the second experiment only overturning moment was applied for the determination of the ultimate moment capacity MY. Subsequently, different combinations of moment and horizontal force at the ground surface were produced by changing the above ground height e of application of the horizontal load Q (hence M = Qe). Aiming to ensure the validity and repeatability of the testing procedure and gain confidence in the presented data, the lateral pushover test for the pile subjected to lateral load at 32 cm above the ground surface (Test 3) was repeated (Test 5). 3.2. 3D Nonlinear Finite Element model The Pushover tests described in the previous section are also modeled numerically with a fully 3D Finite Element Model taking

Disp Sensor DT3 Load

f e

Disp Sensor DT2 Disp Sensor DT1

60 cm

Longstone Sand

3 cm

(a)

(b)

Fig. 3. Pushover model setup; geometry (a) and instrumentation (b).

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into account material nonlinearities (Model 3D FE) for static analysis using the finite element code ABAQUS [35]. The pile and soil are analyzed at model scale, assuming model parameters appropriate for very small confining pressures. The view of the three dimensional Finite Element mesh used for the simulation of the pile and the sandbox is depicted in Fig. 4. Approximately 45,000 elements are used for each analysis. The soil is modeled with 8-node brick elements while the pile is modeled as elastic-perfectly plastic with 3D beam elements placed at its center and connected with appropriate kinematic constraints, namely Multi-Point Constrains (MPC) boundaries, with the nodes at the perimeter of the pile in order to model the complete geometry of the pile. Hence, the nodes of the 8-node brick elements at the perimeter of the pile at a specific elevation follow the displacement of the node of the 3D beam element at that elevation [55]. The solid elements inside the perimeter of the pile have no stiffness. In this way, each pile section behaves as a rigid disc: rotation is allowed on the condition that the disc remains always perpendicular to the beam axis, but stretching cannot occur. The element size is small enough (2 cm in the vertical direction) to capture intense plastic strain concentration at the pile–soil interfaces, but also high enough to decrease the computational time. Due to the significant loading conditions of the experiment, when the dry sand reaches the active state as the pile is laterally loaded, it collapses and flows with the pile. Thus no gap is formed and no interface elements are used in the analysis. It has been proved analytically, within the framework of limit equilibrium theory [33,34,48], that under certain conditions of soil layering the ultimate lateral resistance of a pile subjected to pure overturning moment at its head is equal to the bending moment capacity of the pile, regardless of the soil strength. Thus, the numerical behavior of the pile is validated against Test 2 which was performed in order to verify the ultimate moment capacity of the pile as derived from the alloy specifications. In Test 2, the pile was fixed to the sandbox through an aluminum plate and a lateral load was applied at 32 cm above the fixity point (Fig. 5). Due to the existence of initial gaps between the plate and the pile and to the flexibility of the equipment used for this measurement, only the value of the ultimate moment capacity was verified. For the flexural stiffness, a model of the pile alone (Shell Model) consisting of 4-node shell elements with the flexural stiffness from the specifications of the alloy used was created in ABAQUS [35], as shown in Fig. 6, which captured the shape of the yielding pile and the force–displacement curve for a load acting at 32 cm from the fixed points of the pile. The Shell Model uses the Riks method of analysis which is suitable for unstable collapse and post-buckling response [56]. As observed, the deformed shape of the pile from the Shell

Elastic-perfectly plastic 3D beam elements

8-node brick elements

Fig. 4. Mesh discretization of the 3D Finite Element Model for Pushover tests.

325

Fig. 5. Pile tested under overturning moment conditions (Test 2) in order to validate the yield moment MY of the pile.

Fig. 6. Comparison of the tested pile with the deformed configuration of the Shell Model under overturning moment conditions (Test 2).

Model is remarkably similar to the deformed shape of the tested pile. Thus, the flexural stiffness from the specifications (Shell Model) was adopted as the flexural stiffness of the pile in the 3D FE Model (Fig. 7). It can be observed that the Shell Model with Riks method of analysis captures satisfactorily the experimental response. In the same figure, the elasto-plastic behavior of the 3D FE Model is also depicted. 3.2.1. Soil constitutive model Having verified the pile structural behavior, the soil–pile system behavior of the 3D FE Model is calibrated against Test 1 (lateral load at the ground surface). The description of the stress–strain relationship for soil behavior is based on an extended version of the constitutive model CT for sand [55]. The original model which incorporates a nonlinear kinematic hardening law and associated plastic flow rule, is a reformulation of the single surface plasticity model by Armstrong and Frederick [57], available in the material library of ABAQUS [35], appropriately modified to account for

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20

Moment (kgm)

16

12

8

Experimental Results Shell FE Model 3D FE Model

4

0 0

0.02

0.04

0.06

0.08

0.1

Lateral Displacement (m) Fig. 7. Experimental and numerical moment–displacement curves at pile head for Test 2 (overturning moment conditions) in order to validate the structural response of the pile.

g

Fig. 9. Numerical moment–curvature relationships.

Fig. 8. Variation of the friction angle u with the octahedral plastic shear strain coct and the mean effective stress p.

Fig. 10. Experimental and calibrated (proposed beam and 3D FE models) force– displacement curves at pile head for Test 1 (lateral loading acting at the ground surface).

Table 1 Pile properties for the proposed beam model. Model properties

Symbol

Length Area/moment of inertia Young’s Modulus Shear modulus Yield stress Number of elements

l X/Iy E0 G

Cross-sectional discretization

Values

0.6 m 9.11  105 m2/9.6  109 m4 7.0  104 MPa 26.9  103 MPa rY0 215 MPa 150 (120 embedded length – 30 free length) 100 quadrilateral cells and 3  3 Gauss integration scheme for each cell

strain softening. According to the model, the uniaxial yield stress rY is given by:

C

rY ¼ þ r0 c

ð21Þ

where r0 is the value of stress at zero plastic strain (usually 0.1– 0.3rY), while C (which is equal to the soil Young’s Modulus Es) and c are hardening parameters that define the maximum transition of the yield surface and the rate of transition, respectively. In the Mohr–Coulomb failure criterion rY is equal to:

Table 2 Ultimate load capacity of the pile–soil system. Plastic yield components

Experimental measurements

Proposed beam model

3D FE model

QY (kg) MY (kg m)

97.15 18.19

96.80 18.22

102 18.16

rY ¼

pffiffiffiffiffiffiffi 3J 2

ð22Þ

where J2 is the second deviatoric stress invariant which is a function of the mobilized friction angle u. Thus, the yield surface of the constitutive model CT is determined to fit the Mohr–Coulomb failure response in a triaxial loading test for both compression and extension conditions assuming linear interpolation for the intermediate stress states. For this reason, the parameter k is introduced which is a function of Lode angle and takes values from zero to unit. The case of k = 0 corresponds to pure triaxial extension conditions while k = 1 corresponds to pure triaxial compression conditions. In the extended version of the model, isotropic strain softening is also implemented as a parameter in the model. For the problem

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Load (kg)

e = 32 cm

60 cm

Longstone Sand

Fig. 11. Experimental and computed force–displacement curves at pile head for Tests 3 and 5 (lateral force at 32 cm above the ground level).

Load (kg)

e = 20 cm

60 cm

Longstone Sand

Fig. 12. Experimental and computed force–displacement curves at pile head for Test 4 (lateral force at 20 cm above the ground level).

examined herein, strain softening is introduced by reducing the mobilized friction angle u (or equivalently, by increasing parameter c) with the increase of the octahedral plastic shear strain and the mean effective stress (effective pressure), as depicted in Fig. 8. Initially, from 0% to 5% strain, the friction angle of the soil is equal to the peak friction angle upeak which is a function of the mean effective stress p according to:

upeak ¼ 56:1  2:8 ln p for Dr  90%

ð23Þ

From 15% strain and onwards, the friction angle of the soil is equal to the friction angle at critical state ucs = 32° and from 5% to 15% strain linear interpolation is performed for the values of the friction angle. The expression for upeak was based on results from normally consolidated drained triaxial tests. The upeak = 56° corresponds to a relative density of about 90%. Despite the fact that the relative density in the pushover experiments was measured to be slightly larger than 90% (Dr  94%), in the absence of relevant triaxial loading test data, Eq. (23) for upeak–p relation at Dr = 90% was adopted in the analyses.

The remaining model parameters are calibrated through the experimental data for small confining pressures. The empirical correlation of the Young’s Modulus distribution with depth for the Longstone Sand of Loli et al. [58] is utilized for the calibration of model parameter C, leading to:

C ¼ Es ¼ Er pm ¼ 2000p0:4

ð24Þ

where Er the reference Young’s Modulus of the soil, p the mean effective stress and m a parameter that defines the rate of increase of Es with depth. The constitutive model is encoded in ABAQUS through the user subroutine USDFLD (User Defined Field), which relates the model parameters to the mean effective stress, octahedral plastic shear strain and the Lode angle, at every loading step. It should be stated that analysis of the experiments via a Mohr– Coulomb yield criterion in conjunction with a non-associated plastic flow rule was also attempted, but the solution exhibited numerical instabilities at lateral displacements greater than 3 cm failing to converge without severely sacrificing accuracy.

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Load (kg)

e = 10 cm

60 cm

Longstone Sand

Fig. 13. Experimental and computed force–displacement curves at pile head for Test 6 (lateral force at 10 cm above the ground level).

Load (kg)

e = 6 cm

60 cm

Longstone Sand

Fig. 14. Experimental and computed force–displacement curves at pile head for Test 7 (lateral force at 6 cm above the ground level).

3.3. Inelastic-Beam-on-Winkler-Foundation model The Inelastic-Beam-on-Nonlinear-Winkler-Foundation formulation proposed herein is utilized for the simulation of the experiments with a vertical discretization of Dx = 0.5 cm. The structural stress–stain response of the pile material is assumed to be bilinear elastic-perfectly plastic, following the aluminum alloy 6063-F25 properties (Table 1), while a von Mises yield criterion is adopted [39–42]. Fig. 9 presents the moment–curvature relationship as obtained from the proposed fiber beam model. The equivalent curve used in concentrated plasticity models is also depicted for comparison purposes. It is observed that although the stress–strain constitutive law is a perfectly plastic one, the moment–curvature relationship gradually leads to the full plastification of the piles cross section following a curvilinear law, contrary to the lumped plasticity models.

It has to be noted that since the proposed formulation is a general purpose displacement based fiber model, a vast amount of cross sectional materials commonly used in engineering practice, (i.e. constructive steel, cement, reinforced concrete, aluminum, etc.) can be treated, considering the appropriate constitutive relations of the materials. The current formulation can be easily enriched with any of the well known J1, J2-plasticity yield criteria (Mises–Huber, Tresca, Mohr–Coulomb, Drucker–Prager, etc.) in order to accurately capture the inelastic response of the soil–foundation system. The constitutive relation of the lateral soil reaction against the deflecting pile is similar to the one for structural pile behavior (bilinear elastic-perfectly plastic). The small-amplitude stiffness kw for the Winkler soil springs are approximated by [59], as:

kw ¼ 1:2Es

ð25Þ

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Load (kg)

e = 56 cm

60 cm

Longstone Sand

Fig. 15. Experimental and computed force–displacement curves at pile head for Test 8 (lateral force at 56 cm above the ground level).

1.2

Pwu was based on Brom’s coarse-grained soil, as:

Pwu ¼ kw wy ¼ 3K p czd

M / MY

0.8

0.4

Analytical Expression Experimental Results Repeated Test Proposed Beam Model 3D FE Model 0

0

0.4

0.8

1.2

Q / QY Fig. 16. Comparison of the failure envelope from the analytical expression Gerolymos [27] and Giannakos [28] for piles embedded in dry sand with the results from (a) the LSMD experiments, (b) the proposed beam model and (c) the 3D FE analysis.

Table 3 Depth (cm) of plastic hinge from the ground surface. Experiment number

Measured depth experiment

Calculated depth – proposed beam model

Calculated depth – 3D FE model

Test 1 Tests 3 and 5 Test 4 Test 6 Test 7 Test 8

24 15

22.5 12

22 10

18 21 22 11

14 19 20.5 8.5

14 18 20 8

in which Es the initial (at very small strains) Elasticity modulus of the soil which is assumed to vary parabolically with depth according to Eq. (24). The computation of the ultimate lateral soil reaction

[49]

analytical

expression

for

ð26Þ

in which Kp [ = tan2 ð45 þ u=2Þ] is the passive earth pressure coefficient, c the unit weight of soil, z the depth from ground surface, d the pile diameter and wy is the value of pile displacement at initiation of yielding in the soil. The hardening modulus kwt has been taken equal to zero, thus, it is assumed that the yield, PwY, and ultimate, Pwu, values of lateral soil reaction coincide. As stated on Eq. (26), for cohesionless soils, Broms assumed that the ultimate lateral resistance is a factor n (in Broms’ case n = 3, a conservative approach) times the Rankine passive pressure. Many researchers have proposed different values for the factor n. For example, Fleming et al. [60] proposed a value equal to the passive earth pressure coefficient n = Kp. Pender [61] proposed a value of n = 5 times the Rankine passive pressure because it compares well results from finite element analysis. Additionally, Cubrinovski et al. [62] based on full-scale pile tests measured a value of n = 4.5. Higher values of n have also been proposed. These results apply for full scale piles. Back analysis of Test 1 with the use of Eq. (26) for the ultimate lateral soil reaction dictated a friction angle of u = 64° for the Longstone sand. Undoubtedly, this value for the friction angle is unrealistically large and incompatible with Eq. (23) and seems to be a fitting parameter rather than a parameter with physical meaning. Recent research efforts [34,63] however, indicated that the ultimate lateral resistance is severely underestimated by Brom’s expression. Indeed, according to some recent findings by the third and second authors [33,34], the multiplier of 3 in the above equation should be replaced by 5 or 6 (with respect to the peak friction angle) to yield consistent results with measured response. Under this condition, the equivalent friction angle of u = 64° for Winkler analysis reduces to u = 53–56° which is by far more realistic and compatible with Eq. (23). 4. Results and comparison In this section, the obtained results from all three methods and their comparison are presented. In Fig. 10, the force–displacement curves at the pile head derived from the proposed model for Test 1

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is presented as compared with the experimental results and those from the calibrated 3D FE model. It is observed that both the stiffness and the maximum lateral capacity of the pile of the calibrated 3D model match the experimental values. On the contrary, the beam model matches well the maximum lateral capacity of the pile, but has a stiffer response than the experiment and the 3D numerical analysis. Moreover, in Table 2 the ultimate lateral QY (Test 1) and moment MY (Test 2) capacities of the pile–soil system obtained from both numerical models are compared with those measured from the laboratory tests. The calculated results indicate that the numerical models are calibrated to capture accurately the response of the system in case of Tests 1 and 2. Having calibrated both numerical models, the validation of the proposed beam formulation is performed through the examinations of the response of each individual test (Tests 3–8). In Figs 11–15 the calculated lateral force acting at various heights on the pile with the corresponding displacement at the ground surface obtained from the proposed beam model are presented and compared with the measured values from the experiment and the calculated results from the 3D FE analysis. Both the stiffness and the maximum force values from the two numerical models compare well with the measured results from the experiments. More specifically, it is observed that the beam model exhibits a stiffer behavior compared to the 3D FE model, as expected since it does not capture the strain softening behavior of the soil, but it captures accurately the ultimate capacity of the system. In the same figures (Figs. 10–15), the tangent stiffness at low load levels is also presented as obtained from the experimental data and the numerical analyses. Results from the proposed inelastic beam on foundation and the fully 3D FE models almost coincide and compared well with the tangent stiffness at low loads obtained from the experimental tests. In general, from the conducted investigation, it is deduced that the proposed beam model can be employed providing minimum calculation effort while retaining

good precision in the obtained results for the soil–pile inelastic systems instead of executing complicated 3D analyses. Similar trend is observed in Fig. 11, where the lateral force–horizontal displacement curves at the pile head from Tests 3 and 5 are compared with the calculated response from both numerical methods. The measured experimental results show quite satisfactory agreement between the original and repeated test, indicating that the experimental conditions are repeated with good accuracy in every experiment. Furthermore, Fig. 16 depicts (a) the failure envelope of the analytical expression for pile embedded in dry sand for the case when the lateral force and the bending moment at the pile head act toward the same direction as compared with the points of overturning moment and lateral force (M–Q) at failure, normalized with the values of pure moment MY and pure lateral loading QY capacities respectively as derived from (b) the laboratory experiments, (c) the proposed beam model and (d) the 3D FE model. Despite small-scale effects, it is observed that both the measured points from the experiments and the calculated points from the two numerical analyses almost coincide with the proposed failure envelope. The green triangular symbol on the figure corresponds to the repeated Test 5, performed in order to check the repeatability of the experiments for pile under lateral load applied at 32 cm above the ground surface (Test 3). It should be stated that the failure criterion of the closed-form solution (Eq. (26)) is different from those used in the numerical model. However, it has been shown that closed form expressions based on Brom’s limit equilibrium theory for the failure envelope of a laterally loaded flexible pile, in their normalized versions with respect to the unilateral capacities MY and QY [34,48], are approximately valid for any considered numerical model (3D Finite Element Model or beam on Winkler foundation model). The main difference between different models lies in the determination of the absolute values of the unilateral ultimate capacities. Indeed,

(a)

(b)

(c)

(d)

(e)

(f)

Lateral Force 40kg (Inelastic)

Lateral Force 54.5kg (Just before collapse)

Lateral Force 15kg (Elastic)

Fig. 17. Cross-sectional distribution of von Mises stresses (kPa) at the hinge point of Test 4, accounting for (a–c) or ignoring (d–f) shear deformation effect.

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k = 0.0

k = 0.1

k = 0.3

k = 0.5

k = 0.7

k = 0.8

k = 1.0

Fig. 19. Cross-section with the contours of the active and passive stress states in terms of the state parameter k at the maximum measured lateral loading of the single pile in Test 1: k = 1 corresponds to pure triaxial compression loading condition (passive state) and k = 0 to pure triaxial extension loading condition (active state), while k  0.5 sets the boundaries between the active and the passive state.

Fig. 18. Contours of the plastic strain magnitude at failure (plotted on the undeformed FE mesh) for: (a) Test 1 (pure lateral load), (b) Test 4 (lateral load at 20 cm above the ground surface), and (c) Test 8 (lateral force at 56 cm above the ground surface).

according to Brom’s theory and for a given axial load, MY is assumed to be independent of the shear force developed in the pile and should be known a priori. In the beam-on-Winkler-foundation model proposed herein, MY is the bending moment resultant of the normal stresses acting on the pile section at failure conditions considering their interaction with the associated shear stresses through a von Mises yield criterion. In most cases, including those for flexible piles, the interaction between normal and shear stresses is negligible, therefore, the bending moment capacity is for all practical purposes unaffected by the associated shear force and the pile fails in bending, not in shear. In Table 3, the measured depths of the formation of the plastic hinge from the experiments are presented in comparison to the calculated ones from the proposed and the solid models. Plastic hinge was measured as the distance from the soil surface to the point of maximum strain on the pile. It is observed that both numerical methods predict well the decrease of the depth of the plastic hinge with the increase of the bending moment acting at the pile head. Hence, the maximum depth of plastic hinge is measured at 24 cm (Test 1) from the ground surface while the minimum one is located at the pile head (0 cm) when only bending moment acts at the pile head. Since this is a mesh-dependent problem, and the mesh in the simplified beam model is denser (0.5 cm in the vertical direction) than the 3D FE model (2 cm), the simplified beam model presents a slightly better accuracy for the depth of the plastic hinge. As it was observed by Giannakos [34] and

Gerolymos et al. [48], when the bending moment acts on the opposite direction from the lateral load then the depth of the plastic hinge increases further up to the point where the bending moment dominates the response of the soil–pile system (plastic hinge at 0 cm). Additionally, in Fig. 17 the cross-sectional distributions of von Mises stresses rVM accounting for (a–c) or ignoring (d–f) shear deformation effect, at the depth of the plastic hinge are presented for three load levels as calculated from the proposed beam model with respect to Test 4. The first load level (a, d) corresponds to an elastic behavior, while the remaining ones refer to inelastic response. By comparing the results, it is concluded that accounting the shear deformation effect leads to slight increase of the developed stresses and consequently to more rapid spread of plasticity leading to the formation of the plastic hinge (collapse) for lower values of the applied load. What is of great interest is the plastic strain distribution into the soil medium with respect to the height of load application. In Fig. 18 the contours of the plastic strain magnitude of the 3D model at failure, plotted on the undeformed mesh is presented, for the experimental setup of: (a) Test 1 (lateral load at the ground surface), (b) Test 4 (lateral load at 20 cm above the ground surface), and (c) Test 8 (lateral force at 56 cm above the ground surface). The plastic strains are plotted on the same scale. It is observed that larger plastic strains develop around the pile for the case of the pure lateral loading. Similarly, in Fig. 19 the contours of the active and passive stress states are depicted, in terms of the state parameter k at the maximum measured lateral loading of the single pile in Test 1. The case of k = 1 corresponds to pure triaxial compression loading condition (passive state), k = 0 corresponds to pure triaxial extension loading condition (active state), while k  0.5 corresponds to the response of the soil in direct shear test. 5. Conclusion In the present study the accuracy of an efficient beam formulation for inelastic analysis of pile-foundation systems is validated against a series of Laboratory Pushover tests on vertical single piles embedded in dry sand under different load paths to failure in M–Q space conducted in the Laboratory of Soil Mechanics/Dynamics in NTUA. The obtained results are also compared to those from a fully 3D Nonlinear Finite Element (FE) simulation. The main conclusions are summarized as follows: 1. Overall, the efficiency of the developed beam formulation is verified as it captures accurately enough the response of a single pile embedded in dry sand.

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2. The ultimate capacities are predicted precisely, while the initial stiffness is over estimated compared to both the experimental and 3D solid results which take into account the strain softening behavior of the soil. 3. The results from the proposed Boundary Element formulation compares well with those from the fully 3D FE model with material nonlinearities. 4. The proposed model obtains accurate results with minimum computational effort, providing a simple and effective tool in comparison to fully three dimensional analyses. 5. The proposed analytical expression of the failure envelope for flexible piles embedded in dry sand by Gerolymos [33] and Giannakos [34] is validated also numerically. 6. A new parameter is introduced in the CT constitutive model which accounts for strain softening by reducing the mobilized friction angle u with the increase of the octahedral plastic shear strain and the mean effective stress p. 7. The decrease of the depth of plastic hinge with the increase of the bending moment acting on the same direction with the lateral load is captured accurately.

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