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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech. (2009) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.775

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Cyclic shakedown of piles subjected to two-dimensional lateral loading Nina H. Levy1, ∗, †, ‡ , Itai Einav2, § and Tim Hull3, ¶

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Pile foundations are frequently subjected to cyclic lateral loads. Wave and wind loads on offshore structures will be applied in different directions and times during the design life of a structure. Therefore, the magnitude and direction of these loads in conjunction with the dead loads should be considered. This paper investigates a loading scenario where a monotonic lateral load is applied to a pile, followed by two-way cycling in a direction perpendicular to the initial loading. This configuration is indicative of the complexity of loading that may be considered and is referred to in the paper as ‘T-shaped’ loading. The energy-based numerical model employed considers two-dimensional lateral loading in an elasto-plastic soil, with coupled behaviour between the two perpendicular directions by local yield surfaces along the length of the pile. The behaviour of the soil–pile system subjected to different loading combinations has been divided into four categories of shakedown previously proposed for cyclic loading of structures and soils. A design chart has been created to illustrate the type of pile behaviour for a given two-dimensional loading scenario. Copyright q 2009 John Wiley & Sons, Ltd.

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SUMMARY

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for Offshore Foundation Systems, University of Western Australia, 35 Stirling Highway, Crawley 6009, Australia 2 School of Civil Engineering, University of Sydney, Sydney NSW 2006, Australia 3 GHD Geotechnics, Sydney NSW 2064, Australia

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INTRODUCTION

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Pile foundations may be subjected to many different types of lateral loading. These loads include wind, waves, current, ice, traffic, ship impact, mooring forces and passive loading from earth

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piles; plasticity; thermodynamics; shakedown

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KEY WORDS:

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Received 21 August 2007; Revised 30 October 2008; Accepted 15 December 2008

∗ Correspondence

to: Nina H. Levy, Centre for Offshore Foundation Systems, University of Western Australia, 35 Stirling Highway, Crawley 6009, Australia. † Email: [email protected] ‡ PhD Student. § Senior Lecturer. ¶ Principal Geotechnical Analyst.

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pressures or moving soil [1]. A vast amount of research has been undertaken for single directional lateral loading and established design procedures exist for both static and cyclic loading (e.g. [2]). In reality a pile may experience a combination of lateral loads over its design life, as well as axial and torsional loads. One example of this is when a sustained load such as a mooring force is applied in one direction and a transient cyclic load occurs such as wind, waves or a combination of these acting in another direction. If these loads are perpendicular in the lateral plane then this scenario can be referred to as ‘T-shaped’ loading. The behaviour of a pile under static lateral loads can be considered using elastic continuum or subgrade reaction methods. The continuum method represents the soil as an elastic body and solutions can be obtained using approximate algebraic solutions [3, 4] or closed-form solutions [5]. The subgrade reaction method defines the response of the soil surrounding the pile using a set of independent (Winkler) horizontal load-transfer elements [6]. A plethora of non-linear elastic pressure–displacement ( p–y) curves have been created to allow modelling of the behaviour of piles embedded in different soil types using the results of full-scale field tests or centrifuge tests (e.g. [7, 8]). Further developments involved representing the soil as an elasto-plastic medium with a yield or ultimate soil pressure defined as a function of depth [9–11]. The model defined by Einav [11] was extended to modelling a pile subjected to multi-dimensional loading using a two-dimensional yield surface by Levy et al. [12] for inclined loading and by Levy et al. [13] for two-dimensional lateral loading. Experimental work has been conducted on piles subjected to cyclic lateral loading and cyclic p–y curves have been defined for various soil types (e.g. [7, 8, 14]). Although the results from these early experimental studies are still used in design practice [2, 15], further research has been conducted into the two types of degradation of the soil–pile system postulated to occur during cyclic loading; viz ‘material’ and ‘mechanical’ degradation [16]. Material degradation occurs due to a change in soil properties, such as increases in pore pressure and changes in the soil density. Several experimental studies are available that investigate reductions in soil strength and stiffness under cyclic loading conditions (e.g. [17]). Numerical studies were undertaken into material degradation using both elastic continuum [18] and subgrade reaction methods [19, 20], with degradation factors applied to soil strength and/or stiffness. The mechanical degradation is caused by soil yield occurring along the pile (which induces plastic displacements) or by gaps developing between the pile and the soil. An experimental study completed by Rao and Rao [21] on rigid piles in soft clay observed that cyclic lateral loading can weaken piles and cause an increase in deflection, particularly at load levels beyond 50% of the static capacity. A numerical model was defined by Levy et al. [13] that considers two-dimensional lateral loading of piles. This model uses an energy-based approach, with the soil modelled as elasto-plastic with linear Winkler springs and local circular yield surfaces defined along the length of the pile. The use of a series of two-dimensional yield surfaces in this model can be compared with typical models for one-directional loading of piles that use a conceptual series of one-dimensional ‘slider’ elements [10]; the radii of the circular yield surfaces are equal to the slider thresholds. The gradual yielding along the length of the pile from top to bottom considers the previous loading on the pile, referred to by Levy et al. [13] as ‘recent load history’. Levy et al. [13] concluded that the recent load history can significantly influence the behaviour of a pile subjected to lateral loading due to the existence of (locked-in) residual displacements from previous loading stages. The impact of previous loading on the pile head displacement was found to depend on the magnitude of the applied load relative to the pile capacity, the length of the pile, the pile stiffness and also the ratio

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CYCLIC SHAKEDOWN OF PILES

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Experimental

Experimental

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between the loads during different stages of loading. An experimental study conducted by Levy et al. [22] considered the influence of loading history, both in the same direction as final loading and at 90◦ (see Figure 1). This investigation found that a clear difference has been observed between the pile reload behaviour in these different directions. These results showed excellent agreement with the results of a numerical study using the Levy et al. [13] method. The principle that a structure might fail under repeated loading by excessive plastic flow has been recognized since early in the 20th century. This principle forms the basis for shakedown theory, which states that an elastic/plastic material (whether it is a man-made structure or a soil body) will eventually either shakedown (i.e. the ultimate response will be elastic) or will fail in the sense that the response will always be plastic however many times the load is applied. The existence of a steady cycle was proven by Frederick and Armstrong [23] and a numerical method to directly evaluate the shakedown limits was first presented by Maier [24]. Martin [25] provides a summary of early work on shakedown theory and Ngo [26] presents a detailed literature review on recent research. This shakedown theory has been used for geotechnical applications including pavements [27, 28], bearing capacity for shallow foundations [29] and laterally loaded piles [16]. The shakedown limit load is generally determined using lower and upper bound solutions based on Melan’s static or Koiter’s kinematic theorems. The use of these solutions for laterally loaded piles is discussed in more detail later in this paper. In this paper the energy-based variational (EBV) model is presented and the parameters that are used to define the shakedown categories are highlighted. The four categories of shakedown are defined in terms of the dissipated energy, as defined by the energy-based model, and the pile displacement. The elastic and shakedown limits are then defined analytically in a general form that can be applied to any two-dimensional cyclic lateral loading situation. Finally, we consider a numerical example of a pile that is subjected to a static lateral load followed by two-way cycling in a perpendicular lateral direction, termed ‘T-shape’ loading. The resulting pile displacements are considered in terms of shakedown theory and the results can be divided into the four categories of

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Figure 1. Observed pile head behaviour from experimental testing and predictions made using the EBV numerical model [22]: (a) reloading in the same direction and (b) reloading at 90◦ .

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cyclic behaviour. This study shows that the Levy et al. [13] model is a useful tool for examining the behaviour of a cyclically laterally loaded pile.

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THE MODEL

W = +,

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where W is the change in virtual work, is the change in internal energy and is the change in mechanical dissipation (which is strictly non-negative) . The change in virtual work is described by integrating the product of the unbalanced force and the change in displacement along the soil–pile interface as follows: W=

L

[r x (z)u x (z)+r y (z)u y (z)] dz

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= P +

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0

L

p

p

[Sx (z, u x (z), u x (z))+Sy (z, u y (z), u y (z))] dz

fu(z) fu(z)

(3)

Yield surface

fy(z)

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where r x and r y are the unbalanced force vectors and u x and u y are the change in displacement vectors in the x and y directions, respectively, (the subscripts x and y represent two perpendicular lateral directions, as shown in Figure 2). The pile length is L and z represents the depth below the pile head. The unbalanced force vector is generally defined as zero along the length of the pile and as the imposed external forces at the pile head. For an elastic circular pile under the assumption of small strains and rotations the internal energy potential can be represented by

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(2)

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The EBV model used for the analysis is described in detail in Levy et al. [13] for two-dimensional lateral loading. The model incorporates an elastic pile and a series of elastic soil springs in two perpendicular directions and coupled yield surfaces along the pile length to represent the soil. For rate-independent isothermal deformations the first and second laws of thermodynamics can be expressed in the form:

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fx(z)

fy(z)

fx(z)

L z

Figure 2. The concept of local yield surfaces along the pile. Copyright q



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where E P IP P = 2

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0

⎡ 2 2 2 ⎤ 2 * u y (z) * u x (z) ⎣ ⎦ dz + *z 2 *z 2

(4)

The symbol E P represents Young’s modulus of the pile and IP is the second moment of area of the pile (for a circular pile this value is the same for bending in any radial direction), both considered to be constant along the length of the pile. Sx and Sy represent the soil energy density functions, p p u x and u y are the pile displacements and u x and u y are the pile plastic displacements in the x and y directions at depth z. A soil can be defined as elasto-plastic using independent perpendicular (linear) springs in the x and y directions and a coupled yield surface at each depth (see Figure 2). The soil resistance force in the x or y direction is defined using the elastic component of the displacement (u ex or u ey ), which is represented as the difference between the total displacement and plastic displacement:

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f x (z, u x (z), u x (z)) = K (z)u ex (z) = K (z)(u x (z)−u x (z)) p

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The energy density functions can then be defined as

K (z) e 2 K (z) p u (z) = (u x (z)−u x (z))2 2 x 2 K (z) e 2 K (z) p p Sy (z, u y (z), u y (z)) = u (z) = (u y (z)−u y (z))2 2 y 2 p

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where K denotes the soil layer stiffness (also referred to as the modulus of subgrade reaction) and is a function of depth z (similar in all radial directions). The change in mechanical dissipation over the length of the pile can be defined by L = (z) dz (7)

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(8)

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where the local energy dissipation () can be defined at each depth z by: p p (z) = f u (z) u x (z)2 +u y (z)2

The ultimate resistance ( f u ) in any radial direction at each depth is represented by a Randolph– Houlsby type of mechanism (see Randolph and Houlsby [30], and Martin and Randolph [31] for modifications), as shown in Figure 2. The symbol is used to represent the change in plastic displacement. The local dissipations given by Equation (8) can be converted to a series of yield functions using a Legendre transformation [11]. The yield criterion, y, is defined at each depth by a circular envelope with a radius given by f u as follows:

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(6)

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Sx (z, u x (z), u x (z)) =

(5)

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f y (z, u y (z), u y (z)) = K (z)u ey (z) = K (z)(u y (z)−u y (z))

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p

f x (z, u x (z), u x (z))2 f y (z, u y (z), u y (z))2 y(z) = + −10 f u (z)2 f u (z)2 Copyright q


(9)


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The associated flow rule and consistency condition determined for this surface are detailed by Levy et al. [13]. This model represents the same system that can be defined using the differential equation, with the same yield condition given as Equation (9), and this can actually be derived directly from the internal energy potential defined in Equation (3). The basic governing equation that is applied to a laterally loaded pile was developed by Hetenyi [6] for a beam on elastic foundation. This equation can be expressed for two perpendicular directions as follows: 4

E P IP

* ux p + K (z)(u x −u x ) = 0 *z 4

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* uy p + K (z)(u y −u y ) = 0 *z 4

(10)

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Definition of parameters for shakedown analysis

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As discussed in the Introduction, shakedown theory has historically been applied to various structures, including foundations in elasto-plastic soil. In his thesis, Swane [16] defines the shakedown categories with regard to the behaviour of a pile subjected to single direction of lateral and moment loading and these definitions will be applied in this paper. The plastic work (W p ) that Swane considers can be defined as the plastic component of the work term given as Equation (2). The work can be broken down into elastic and plastic components as follows: L e [r x (z)u ex (z)+r y (z)u ey (z)] dz (12) W = 0

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(13)

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where W = W e + W p . In the current paper the mechanical dissipation will be considered rather than the plastic work, because this is one of the key parameters in our model. However, for an elastic–plastic soil with a Helmholtz free energy potential, represented purely as a function of the local linear elastic strains (or ‘local elastic displacements’ in our notation) with no free terms that depend on the local plastic strains, the plastic work and mechanical dissipation are equal (i.e. = W p ) [32]. The work and dissipation terms defined in Equation (1) are expressed in terms of the change in these parameters because the value of each of these properties at a given step depends on the loading path and therefore they are not state functions. When considering the trends in these parameters as cyclic loading progresses the total (or cumulative) values provide important information about the influence of the loading history on the pile response. The global cumulative dissipation (c ) can be defined as L c (t1 , t2 ) = c (z, t1 , t2 ) dz (14)

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where the local cumulative dissipation (c ) is defined as: c (z, t1 , t2 ) =

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p p f u (z) (u x (z))2 +(u y (z))2 dt

(15)

t1

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where t represents the time and t1 and t2 are the time limits between which we are integrating. The dissipation function defined in Equation (15) is a cumulative dissipation defined over a particular time interval and therefore the units are energy-time per unit length. This differs from Equation (8) which is an instantaneous dissipation and has units of energy per unit length. The total cumulative dissipation over the full loading history until the end of any cycle N is defined as c (0, t N ), where t N is used to represent the increment number after N full cycles. A definition for the cumulative absolute plastic displacement is given by Einav and Randolph [33]. In the current model we can define the cumulative absolute resultant plastic displacement () at each depth along the pile by: t2

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Similarly, the change in resultant pile displacement u(z) over the N th cycle at depth z is defined by u N (z) = u(z)|t=t N −u(z)|t=t N −

27

(17)

Einav and Randolph [33] use the cumulative absolute plastic displacement when developing an upper bound strain path method, as a measure of softening, or degradation, of the soil as it flows around a cylindrical object. This concept can be applied to the soil–pile system considered in the current paper, with this term representing the mechanical degradation that occurs in the soil. The accumulation of dissipation over a single cycle is used to define the category of cyclic behaviour for a given loading scenario. After a number of cycles the behaviour of the soil–pile system reaches a steady state condition. This steady state time (tSS ) can be observed as the point after which the pile displacement and dissipation either remain constant or increase linearly between subsequent cycles. These trends will be discussed in detail in the following section for each of the shakedown categories. The global dissipation accumulated over the N th cycle is defined as the integral between the N th cycle and the previous cycle, referred to as the N −th cycle, as follows:

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(16)

(19)

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Shakedown categories

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Each cyclic loading event can be categorized into one of the four response types. These are described below and are illustrated graphically using numerical examples later in this paper. The categories of shakedown for a pile foundation are defined as follows: (1) Purely elastic: When the soil resistance force remains within the elastic range for all points along the length of the pile and therefore no yielding of the soil occurs. The calculated pile behaviour will be the same for a given resultant force within this range, regardless of what percentage of the load is monotonic and cyclic. The displacements achieved after one cycle will remain the same for all future cycles. Therefore, the following conditions will be satisfied for the purely elastic case for all time t N : u N (z) = 0 (for 0zL)

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(2) Shakedown: A loading scenario such that plastic displacements initially occur in a soil and after a finite number of cycles the response stabilizes to be elastic. In this case the pile–soil system is said to shakedown. After this shakedown occurs, i.e. once steady state conditions are reached, the pile displacements along the pile will no longer increase from the maximum values reached during the previous cycle and no more dissipation will occur. At any time t N >tSS the following conditions will apply: (21a)

N = 0 =

(21b)

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TE

(for t N tSS ) (Note : N 0 (for t N 0

(22b)

N

where c is a positive constant. (4) Incremental Collapse (Ratchetting): If at any depth along the pile the displacement does not stabilize to a constant displacement, but continues to increase at a constant rate, then this will eventually result in failure of the global system. The energy dissipation of the system will also continue to increase at a constant rate. This case is defined by

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(3) Alternating Plasticity (sometimes referred to as plastic shakedown): For certain loading conditions, the deflection of the pile may stabilize after a number of cycles while dissipation still occurs at a constant rate. This defines a situation where under steady state conditions the pile behaviour follows a closed cycle, which comprises some elastic and some plastic displacements. The plastic displacements are equal in the two directions and therefore they cancel out. The alternating plasticity case is defined by

u N (z) = u N − (z) = d(z) (for any 0zL) N = N − = c>0

(23a) (23b)

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For the current non-degrading model the shakedown and alternating plasticity cases reach a maximum displacement, but when material degradation is considered the pile may continue to deform eventually resulting in failure. The ratio of the rates of material degradation and shakedown to steady state will determine which of these influences will govern the pile behaviour. While this interaction is beyond the scope of the current paper, material degradation can be applied using the model defined by Einav and Randolph [33] which considers degradation of the soil strength as a function of the cumulative plastic strain, as defined in Equation (16). This reduction would introduce the case where a system in the alternating plasticity or shakedown categories could be removed from a stable state to a potential failure condition as the yield surfaces along the pile contract in size. It is also important to note that the category of behaviour will be determined by the most extreme conditions along the length of the pile. Yielding occurs gradually along the length of the pile and only at complete failure will all the soil forces exceed the yielding threshold. Therefore, at some nodes along the pile the soil can still behave in an elastic manner even during incremental collapse. In other words, condition (23b) refers to the global dissipation only. It is also possible that local shakedown or alternating plasticity behaviour will occur at some points along the pile during incremental collapse.

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For each cyclic loading situation the behaviour of the system will fall into one of the abovementioned categories. Three boundaries between response categories can be defined in terms of the applied monotonic and cyclic loads; the limit of the elastic range, the so-called ‘shakedown limit’ and the incremental collapse limit. These boundaries are discussed below and a method for determining each boundary is presented.

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crossed when the resultant soil resistance force ( f (z) = f x (z)2 + f y (z)2 ) at any depth along the pile exceeds the ultimate resistance ( f u (Z)). Therefore, the system will behave in a purely elastic manner if f (Z)< f u (Z) for all 0zL. The elastic boundary can be detected by applying a monotonic load to the pile and observing the load at which this condition is no longer satisfied. The applied load that defines the elastic boundary is a single value (Pe ) and therefore the elastic limit can be represented by a circle of radius Pe in two-dimensional lateral force space. This limit can be observed as a quarter circle in Figure 3(a), which is a plot of the applied cyclic force (Pc ) versus the monotonic force (Pm ). If the soil is treated as linearly elastic perfectly plastic this can be defined as a function of depth by the equation:

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The applied load at which the soil behaviour becomes plastic, i.e. the boundary between categories (1) and (2) mentioned above, can be specified for a givenpile–soil system. This boundary will be

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Elastic boundary

u e (z) =

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(24)

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Alternating Plasticity Shakedown

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at a rate K r (i.e. K (z) = K r z) the pile displacements while still in the linearly elastic zone of behaviour at all depths along the pile can be defined by employing equilibrium: z

6F 3−4 (25) u(z) = L Kr L 2

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For a cohesive soil the yield threshold can be defined (provided that free surface effects are small) by

13 15

17

(27)

where the undrained shear strength at depth z is defined by su (z) = sur z. Now, the behaviour will cease to be purely elastic when the soil resistance force exceeds the yield threshold at any depth along the pile. The soil resistance force is defined as parabolic by Equation (26) and the yield threshold is linear according to Equation (27) and therefore there will be a maximum of two points of intersection between these curves. The depths corresponding to these points can be found by finding the depths that satisfy the equation: 6F z z

f u (z)− f (z) = 11.9Ds ur z − 2 3−4 =0 (28) L L

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where F is the lateral load applied at the pile head. The soil resistance can then be defined as the stiffness multiplied by the displacement at each depth as follows: z

6F z (26) f (z) = K (z)u(z) = 2 3−4 L L

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These are calculated as z 1 = 0 and

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If z 1 = z 2 then only one point of intersection exists, at the pile head, but if z 2 >z 1 then there will be a second intersection point and therefore between z 1 and z 2 the soil resistance force exceeds the yield threshold force. Therefore, the limiting external force (Pe ) is defined by setting z 1 = z 2 , giving the following solution:

5

11.9Ds ur L 2 L z 2 = 3− 6F 8

Pe =

7

11.9Ds ur L 2 18

(29)

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The ‘shakedown limit’ is defined as the applied load above which shakedown will not occur, i.e. the response does not become elastic. Two methods for determining this limit for piles are discussed in the literature [16]; the optimization method and the extrapolation method. These methods both require an analysis of many different loading combinations in order to assess which one satisfies a particular criterion that defines shakedown behaviour. An optimization method uses Melan’s Theorem, one of the fundamental theorems of shakedown, with linear programming. This method involves determining whether a time-independent, selfequilibrating distribution of residual stress can be obtained such that the sum of the residual and elastic stresses does not exceed the ultimate stress at any point along the pile. If this condition is satisfied then shakedown has occurred. This approach has not been used in the analysis presented in this paper. The extrapolation method considers the behaviour of the plastic work within the soil as it changes with loading. As defined earlier in this paper this value is determined to be either zero or constant after a finite number of cycles (under steady state conditions) and from this information the category of behaviour can be defined. The extrapolation method is used in this paper with dissipation considered instead of plastic work, as discussed in the previous section. In our problem the shakedown limit is found to be closely related to the elastic boundary and in order to shakedown the cyclic displacements along the length of the pile must reduce to below u e . For the ‘T-shape’ loading scenario, if the cyclic load exceeds the allowable elastic load then there will continue to be plastic displacements and therefore dissipation. Therefore, an upper bound solution can be defined for the shakedown boundary as Pc = Pe , as shown in Figure 3(b). Based on the results of the detailed parametric study for T-shape loading presented later in this paper, this boundary appears to apply up to approximately Pm /Pult >0.75, after which time shakedown occurs at lower cyclic loads. This means that the magnitude of the applied monotonic load does not determine whether shakedown occurs until it reaches close to the pile capacity (in our case Pm /Pult >0.75). However, it will influence the evolution of the soil resistance forces and as a consequence the magnitude of the pile displacements. A series of contours of normalized pile head displacements (u(0)/D) are included in Figure 3(b) at spacings of 0.1, and a discussion on these contours is included in the numerical analysis presented later in this paper. While this upper bound solution is a useful guide to the shakedown boundary, it is recommended that for high monotonic loads a full analysis be undertaken to determine the pile behaviour. A similar principle can be applied to any loading case where a monotonic load is applied followed by cyclic (1 or 2 way) in any horizontal direction (i.e. not necessarily perpendicular).

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9

F

Shakedown limit

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F

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The extrapolation method described above for determining the shakedown limit can also be applied to the incremental collapse limit by considering the change in pile displacement instead of the change in plastic work. This limit depends on the proportion of the load that is monotonic and cyclic, which in this paper is represented as the angle () at which the maximum resultant load acts. As the pile behaviour is influenced by the states of the soil forces along the length of the pile, it is difficult to analytically define the incremental collapse limit. However, a full analysis of a pile completed using the EBV method will give a clear indication of whether incremental collapse occurs. As well as the parameters defined in Equation (23) the behaviour of the soil forces as they follow the local yield surfaces along the pile also provides an indication of whether alternating plasticity or incremental collapse is happening. This is illustrated in the numerical example presented in the following section. The incremental collapse boundary determined from the numerical analysis is included as a dotted line in Figure 3(a) based on analytical solutions, and Figure 3(b) using the detailed numerical analysis.

O

7

Incremental collapse boundary

O

5

PR

3

While the monotonic loading history provides an initial displacement, which may include a plastic component, once cycling commences the factor that determines whether shakedown will occur is whether the displacements reduce below u e along the pile in the direction of cyclic loading, as an upper bound first approximation.

D

1


29 31 33 35 37 39 41

EC

R

O R

27

C

25

N

23

A series of test cases were considered in order to check the validity of the category descriptions and boundaries given above. The pile used for the analysis was a solid steel circular pile with Young’s modulus (E P ) of 210 GPa and a diameter (D) and length (L) of 1 and 25 m, respectively. The choice of pile stiffness and geometry is not intended to model a particular situation in the field, but was chosen so as to ensure a ‘rigid’ pile response for ease of checking with closed-form solutions. The soil was modelled as normally consolidated clay, with an undrained shear strength (su ) of zero at the ground surface increasing at a rate of 1 kPa/m with depth. Young’s Modulus of the soil (E S ) was zero at the surface increasing at a rate of 100 kPa/m. The soil layer stiffness (or modulus of subgrade reaction) is defined using the equation presented by Ashford and Juirnarongrit [34]. The pile capacity was calculated as 966 kN and all applied forces will be considered as a percentage of this value. The load required to induce a plastic response (Pe ) was calculated as 480 kN, corresponding to approximately 50% of the pile capacity. A series of loading combinations were considered for a T-shape analysis, which was defined by a monotonic load (Pm ) applied in the x direction followed by a two-way cyclic load (Pc ) in the y direction, as shown in Figure 4. Each load case is expressed in terms of the maximum resultant load (P) as a proportion of the pile capacity and the angle () at which this load acts (see Figure 4). In general, P/Pult values of 0.5, 0.6, 0.7 0.8, 0.9 and 0.98 and values of 10, 30, 45, 60, 80 and 90◦ were selected, and near the boundaries between shakedown categories a higher concentration of tests was included. Each test included as many cycles of loading as needed to ensure that steady state conditions were achieved. The results for each loading case were inspected and based on the displacement and dissipation trends the behaviour was categorized into the four categories of cyclic behaviour, as shown in

U

21

NUMERICAL ANALYSIS

TE

19

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Fy0 Pult

P = Pm + Pc 2

Pc

2

Pm

Pult

Fx0

F

-Pc

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Figure 4. Definition of loading combination for ‘T-shape’ loading.

9 11 13 15 17 19 21

0 0 5.45E-07 0.030

0 0 0.015 0.203

PR

45 45 60 60

R

EC

Figure 3(b), where each category is represented by a different symbol. The numerical results confirm the locations of the elastic and shakedown boundaries discussed previously, and the upper bound for the shakedown boundary appears to be close to an exact solution until a monotonic load of about 75% of the pile capacity is reached. The incremental collapse boundary can only be defined using the numerical data. In order to examine the numerical results more closely an example of a typical test from each of the four categories has been selected. It should be noted that the magnitudes of the applied loads differ for these tests and it is the general behaviour of the soil response and pile deformation that will be compared. The tests that were selected are summarized in Table I. For the four test cases the maximum normalized pile head displacement (u(0)/D) and cumulative global dissipation (c (0, t N )/Pult D) after N cycles are plotted against the cycle number N in Figure 5. The slopes of the lines on these plots represent the change in displacement and dissipation over each full cycle, giving the normalized parameters u N (0)/D and N /Pult D that can be compared with Equations (20)–(23) in order to determine the category of behaviour. After 100 cycles the category of behaviour was clearly observable for all the tests. The yielding behaviour of the soil along the length of the pile influences the pile head loaddisplacement response. The soil resistance forces at the pile head are plotted in Figure 6 together with the yield surface to illustrate the soil behaviour for the four different categories. For the soil properties selected the soil starts yielding at a location close to the pile head and therefore these plots are representative of the most extreme case of yielding observed along the length of the pile. As implied by the definition, the soil resistance forces remain elastic throughout loading

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0.45 0.6 0.7 0.9

C

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N /Pult D (100 cycles)

N

3

u(0)/D (100 cycles)

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(◦ )

TE

Purely elastic (E) Shakedown (S) Alternating plasticity (AP) Incremental collapse (IC)

P/Pult

D

Category

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Table I. Typical loading combinations for each category of cyclic behaviour.

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0.03

0.4

0.024

0.3

0.02 ∆ uN(0)= 0

0.012 0.1

ΦN = 0 0.012

0.2 0.008 0.1

0.004

0.006 ΦN = 0 20

40 60 80 cycle number, N

0 (b)

3.5

0.75

10

3

u(0)/D

1.5

ΦN = constant

D

2

Φc (0,tN)/PultD

0.45

TE

u(0)/D

2.5

50

40

PR

8

∆ uN(0)= 0

0 100

ΦN = constant

0.6

0.3

50 cycle number, N

O

0 (a)

0

0 100

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0

F

0.05

6

4

30

∆ uN(0)= constant

20

1

0.15

2

10

EC

0.5

0 0

50 cycle number, N

0 (d)

50

0 100

cycle number, N

R

(c)

0

0 100

5 7 9 11

C

N

3

for the purely elastic case, which means that there will be no change in the pile displacement in the x direction during cyclic loading. For tests that fall into the shakedown category the soil resistance forces reach yield either during monotonic loading, if Pm >Pe , or during the first cycle of loading (in the example case shown in Figure 6(b) Pm