M. Cubrinovski
University of Canterbury, Christchurch, New Zealand
J.J.M. Haskell
Schofield Centre, University of Cambridge, Cambridge, UK
B.A. Bradley
Geological and Nuclear Sciences (GNS), Lower Hutt, New Zealand
ABSTRACT: This paper focuses on two established (and very different) methods for analysis of piles in liquefying soils: a simplified pseudo-static analysis, and an advanced seismic effective stress analysis. The paper highlights the need for a systematic approach in the use of the pseudo-static analysis allowing for identification of key parameters and uncertainties in the analysis. Numerical simulations of shake table tests are used to illustrate some important aspects in the modelling and application of the seismic effective stress analysis.
1 INTRODUCTION
Pile foundations are often used to support engineering structures in areas where surface soils are liquefiable.
Hence the abundance of case histories from strong earthquakes on damaged piles and poor performance of pile foundations in liquefied and laterally spread-ing soils. In the 1995 Kobe earthquake, for example, a large number of bridges, buildings and storage tanks on pile foundations were severely affected by soil liq-uefaction and lateral spreading, which caused damage to the piles, loss of function or even collapse of the superstructure (JGS 1998).
Over the past 10–15 years, significant efforts have been made to improve our understanding of the behaviour of piles in liquefying soils during earth-quakes. This included benchmark experiments on full-size piles using large scale shake table tests (e.g., Tamura & Tokimatsu 2005; Cubrinovski et al. 2006;
Tokimatsu & Suzuki 2009), detailed analyses of well-documented case histories from strong earthquakes (e.g. JGS 1998; Tokimatsu & Asaka 1998) and devel-opment of new concepts and analysis procedures in an effort to explore design methodologies for piles in liquefying soils. On the analytical front, signifi-cant progress has been made across a broad range of analysis methods, from simple design-oriented approaches to the most advanced numerical proce-dures for dynamic analysis (e.g., O’Rourke et al. 1994;
Tokimatsu & Asaka 1998; Yasuda & Berrill 2000;
Finn & Thavaraj 2001; Cubrinovski & Ishihara 2004;
Cubrinovski et al. 2008).
This paper focuses on the analysis of piles in liquefying or laterally spreading soils and issues
around numerical modeling when using two represen-tative methods for analysis: a simple design-oriented approach (pseudo-static analysis), and an advanced numerical analysis (seismic effective stress analysis).
There are numerous variations in the details and devel-opment of these methods which are beyond the scope of this paper. Here, the aim is to provide an overview of important issues in the application of these meth-ods to the analysis of piles in liquefying soils and to identify areas that require further development and improvement.
2 STATEMENT OF THE PROBLEM 2.1 Response characteristics of liquefying soils Soil liquefaction involves very large changes in stiff-ness and strength of foundation soils over a very short period of time, typically during the strong ground shaking caused by an earthquake. This highly dynamic and extreme variation in stress-strain characteristics of the foundation soils is probably the first thing to acknowledge when analyzing liquefaction problems.
Strong ground shaking gives rise to a rapid build-up in excess pore water pressures and consequent reduc-tion in stiffness and strength of liquefying soils. In a period of only few seconds (or several tens of sec-onds) the stiffness of the liquefying soil may change from its initial value to nearly zero (at least temporarily in the course of shaking). The significant reduction in stiffness and strength results in a large lateral ground deformation either of cyclic nature (with peak shear strains on the order of several percent) or in the form
of lateral spreads (permanent shear strains on the order of several tens of percent). Clearly, soil liquefaction involves an extreme level of material nonlinearity and quite often a significant geometric nonlinearity due to the very large lateral displacements and associated loss of stability of the ground or supported structure.
All of the above depicts the complexity of the response with regard to its dynamic nature (time-dependent component). The process is also highly variable in space. The progressive development of liq-uefaction throughout the depth of the deposit could be very non-uniform and uniquely affected by the complex cross interaction amongst soil layers, ground response and earthquake motion characteristics.
2.2 Soil-pile interaction in liquefying soils
In the course of rapid build-up of excess pore pres-sures and development of liquefaction (temporal and spatial), piles are generally subjected to two significant lateral loads arising from the ground movement (kine-matic load) and vibration of the superstructure (inertial load). Both these loads are oscillatory in nature with magnitudes and spatial distribution dependent on the ground motion characteristics, soil density, presence of non-liquefied crust at the ground surface, predom-inant periods of the ground and superstructure, and the relative stiffness of the foundation soil and the pile, among others. In view of the significant varia-tion of these loads and rapidly changing stiffness and strength characteristics of the foundation soils in the process of development of liquefaction, it is useful to distinguish between several different phases in the soil-pile interaction (e.g., Tokimatsu & Asaka 1998;
Cubrinovski & Ishihara 2004). Such strategy has been adopted in many design codes where, for example, the cyclic phase and lateral spreading phase of the response are considered by two separate analysis pro-cedures. As illustrated schematically in Figure 1, in the two separate analyses different combinations of kinematic and inertial loads, and also different char-acteristics (stiffness and strength) of the foundation soils are used in order to depict representative scenar-ios for the analysis of the pile during a specific phase of the evolving seismic response.
Putting aside for a moment the complex issues around the combined kinematic and inertial effects on the pile response, one may identify three gen-eral scenarios for the pile response in terms of the displacement of the pile relative to that of the free field ground. A schematic illustration of the three types of responses is shown in Figure 2 for the so-called stiff-pile-behaviour, flexible-pile-behaviour and reverse-pile-behaviour respectively. As implied by the size of the cyclic and permanent ground deforma-tion described in Secdeforma-tion 2.1, cyclic lateral ground displacements in liquefied soils and lateral spreading displacements in particular could be very large. Hence, cases where either flexible-pile-behaviour or reverse-pile-behaviour occurs, i.e. lateral pile displacement (UP) is either similar to the ground displacement (UG)
Figure 1. Schematic illustration of loads on pile (and char-acteristics of foundation soils) during strong ground shaking (cyclic phase) and post-liquefaction lateral spreading.
Figure 2. Schematic illustration of reverse-pile-behaviour, flexible-pile-behaviour and stiff-pile-behaviour based on the relative displacements between the pile and free field soil.
or greater, imply serious damage to the pile and unac-ceptable performance in the case of strong ground motions. In other words, the stiff-pile-behaviour where the pile foundation has the capacity to resist the ground movement, and hence control the deformation and damage to the pile, is the preferred type of response, from a performance viewpoint.
2.3 Analysis and design issues
Undoubtedly, the analysis and design of piles in lique-fiable soils are burdened by the extreme complexity of the phenomena considered and unknowns in the anal-ysis. A rigorous analysis of the problem would need to address the following issues (not an exhaustive list though):
– Temporal and spatial characteristics of liquefaction – Effects of excess pore water pressures and liq-uefaction on ground response and stress-strain characteristics of foundation soils
– Soil-pile interaction in liquefying soils including characteristic loads (kinematic and inertial loads) and deformation mechanism
– The need to estimate inelastic deformation and damage to piles
– Assessment of the response and performance of the soil-pile-foundation-structure system including all critical components and the system as a whole
– Uncertainties in the ground motion (earthquake load) and system characteristics
There is no universal analysis method that rigor-ously addresses all the above issues, but rather different methods focus on different aspects of the problem and provide a specific contribution in the assess-ment. Some issues in the application of two principal methods for analysis of piles in liquefying soils are discussed in the following sections.
3 SIMPLIFIED PSEUDO-STATIC ANALYSIS 3.1 Characteristics and objectives
The pseudo-static analysis is a conventional method for analysis of seismic problems that has its version for analysis of piles in liquefying soil. As a practical design-oriented approach based on conventional engi-neering concepts, this method is commonly adopted in the seismic design codes. Even though there are significant differences between different pseudo-static analysis approaches, in concept they are all similar, and hence share common features and problems in their application. In this context, the method used herein should be taken as a representative of the pseudo-static analysis that illustrates key features of this analysis approach.
The adopted pseudo-static method was designed to satisfy the following requirements in the analysis:
(a) To capture the relevant deformation mechanism for piles in liquefying soils (combined kinematic and inertial loads, and degradation of soil stiffness and strength due to liquefaction).
(b) To permit estimation of inelastic deformation and damage to piles (conventionally, foundations are designed to remain in the elastic range of deforma-tions, but this may be very restrictive and expen-sive strategy for pile foundations in liquefying soils).
(c) To address the uncertainties (and unknowns) asso-ciated with liquefaction and soil-pile interaction in liquefying soils.
The method could be applied to a pile group includ-ing a superstructure, and hence it could address some aspects in the performance of the soil-pile-foundation-structure system. Here, a single-pile model is used for the sake of simplicity and clarity of argument.
3.2 Soil-pile model
A typical beam-spring model representing the soil-pile system in the pseudo-static analysis is shown in Figure 3 (Cubrinovski & Ishihara 2004; Cubrinovski et al. 2009a). The model can easily incorporate a strat-ified soil profile (multi-layer deposit) with liquefied layers of different thicknesses sandwiched between a crust of non-liquefiable soil at the ground surface and an underlying non-liquefiable base layer. As illus-trated in the figure, the proposed model incorporates
Figure 3. Beam-spring model for pseudo-static analysis of piles in liquefying soils (Cubrinovski and Ishihara, 2004;
Cubrinovski et al. 2009a).
simple non-linear load-deformation relationships for the soil and the pile. The soil is represented by bilin-ear springs, the stiffness and strength of which can be degraded to account for effects of nonlinear behaviour and liquefaction. The pile is modelled using a series of beam elements with a tri-linear moment-curvature relationship. Parameters of the model are summarized in Figure 3b for a three-layer configuration in which a liquefied layer is sandwiched between a surface layer and a base layer of non-liquefiable soils.
In the model, two equivalent static loads are applied to the pile: a lateral force at the pile head (F), represent-ing the inertial load on the pile due to vibration of the superstructure, and a horizontal ground displacement (UG) applied at the free end of the soil springs (for the liquefied layer and overlying crust), representing the kinematic load on the pile due to lateral ground movement (cyclic or spreading) in the free field. As indicated in Figure 3, it has been assumed that the non-liquefied crust at the ground surface is carried along with the underlying liquefied soil and that it under-goes the same ground displacement as the top of the liquefied layer, UG.
3.3 Influence of soil stiffness and strength
The application of the method to the analysis of piles in liquefying soils is burdened by the aforementioned uncertainties associated with soil liquefaction, soil-pile interaction in liquefying soils and the need to reduce a very complex dynamic problem to a sim-ple equivalent static analogy. Questions posed to the user are ‘what stiffness and strength to adopt for the liquefied soil’, ‘how to combine dynamic kinematic and inertial loads in a static analysis’ and ‘what is the sensitivity of the pile response to a certain model parameter’, among others. A thorough discussion on these issues may be found in (Cubrinovski et al. 2009a, 2009b) whereas herein the focus is placed on only one albeit important aspect of the problem.
The load-deformation relationships for the soil springs shown in Figure 3b are characterized essen-tially by two groups of parameters: one related to the stiffness, and the other to the strength (ultimate pressure) of the soil. Before going into a detailed exam-ination of the variation of these parameters, it is useful to consider the sensitivity of the pile response to the stiffness and strength of the soil. For this purpose, Haskell (2009) conducted a comprehensive sensitiv-ity study in which a wide range of soil-pile systems, loading conditions and load-deformation characteris-tics for the soil and the pile were considered. One of the important outcomes from this study is summarized in Figure 4 where a conceptual illustration of the influ-ence of stiffness and strength of the liquefied soil on the pile response is comparatively shown. The size of the horizontal bars on the left-side of the figure indicates the degree of sensitivity of the pile response to the soil stiffness or strength respectively; the solid symbol on the load-deformation relationships on the right-hand side indicates the particular deformation (load) level of the spring for which the respective sensitivities apply.
For example, the plots at the top of the figure indicate a large influence of soil stiffness on the pile response and no influence of soil strength when the spring defor-mation is below the yield level. Conversely, the plots at the bottom of the figure indicate that soil strength strongly influences the pile response when yielding in the soil occurs. While this transition from soil stiffness to soil strength controlled pile response with increas-ing deformation is intuitive, the abovementioned study (Cubrinovski et al. 2009b) provides quantification of these effects.
It was argued in Section 2.2 that the stiff-pile-behaviour is the preferred type of pile response, because this type of behaviour (where the pile resists the ground movement) provides a controlling mech-anism for the pile deformation (damage). Figure 2 clearly shows that the stiff-pile-behaviour results in a large relative displacement between the free field soil and the pile which implies yielding in the soil spring. With reference to the schematic plots shown in Figure 4, the stiff-pile-behaviour is represented by the bottom plots where effects of soil stiffness are negligible while the ultimate soil pressure (strength) strongly influences the pile response. Hence, the focus in the modelling should be placed on the ultimate soil pressure.
3.4 Ultimate soil pressure on the pile
In the adopted model, the ultimate soil pressure from the crust of non-liquefied soil at the ground surface per unit width of the pile is approximated as
where pp(z) is the Rankine passive pressure while αC
is a scaling factor accounting for the difference in the lateral pressure between a single pile and an equiva-lent wall. Results from experimental studies indicate
Figure 4. Conceptual illustration of sensitivity of the pile response (size of horizontal bars) to stiffness and strength of liquefied soil as a function of spring deformation (rela-tive displacement UG− UP); Haskell 2009, Cubrinovski et al.
2009b.
Figure 5. Uncertainties in the ultimate pressure from a crust of non-liquefied soil on the pile.
that αC typically takes value in the range between 3 and 5; a value of 4.5 has been back-calculated from benchmark lateral spreading experiments on full-size piles (Cubrinovski et al. 2006). In many guidelines, however, a value of αC= 3 has been adopted based on the study of Broms (1964) which is based on active-pile-loading (in which the soil provides resisting force to the pile deformation), from a range of measured val-ues between αC= 3 and 6. Hence, the uncertainty in the ultimate pressure form the crust layer on the pile illustrated in Figure 5.
The ultimate pressure from the liquefied soil is similarly estimated as
where Srand αLare the residual strength and the scal-ing factor respectively for the liquefied soil. Note that αLis different from the corresponding parameter αC
for the crust, because the interaction and mobilization of soil pressure on the pile is different for liquefied and non-liquefied soils. The value of αL is highly uncer-tain and could be approximated anywhere between 1 and 6 depending on the underlying assumptions in
Figure 6. Empirical correlation for the residual strength of liquefied soils (after Seed and Harder, 1991).
Figure 7. Uncertainties in the ultimate pressure from the liquefied soil on the pile.
the calculations. The residual strength could be esti-mated using empirical correlation, such as that based on the SPT blow count proposed by Seed and Harder (1991), shown in Figure 6. A large scatter exists in this correlation indicating significant uncertainty in the value of Srfor a given (N1)60cs value. For exam-ple, for (N1)60cs= 10, the value of Srcan be anywhere between 5 kPa (lower bound value) and 25 kPa (upper bound value). By combining the uncertainties in αL
and Sr, the possible variation in the ultimate pressure from the liquefied soil could vary by a factor of 30 for this case, as illustrated schematically in Figure 7.
The relative contribution or influence of the ulti-mate soil pressure from the crust and liquefied soil could be very different depending on the thickness of the crust. This is illustrated in Figure 8 where the sensitivity of the pile response (φ/φref) on the ultimate pressure parameters of the liquefied soil (Sr
and αL) is plotted for two different scenarios, one without crust (HC= 0 m) and the other with a 1.5 m thick crust (HC= 1.5 m). The effects of liquefied soil parameters on the pile response substantially decrease with an increasing thickness of the crust, as the crust load represents a greater proportion of the total load and hence becomes the dominant load component.
Clearly, uncertainties in model parameters could be quite important and should not be evaluated in an iso-lated manner but rather by considering the ‘system response’.
Clearly, there are significant uncertainties in the pseudo-static analysis associated with the ultimate
Figure 8. Sensitivity of the pile response (φ/φref) on the ultimate pressure parameters of the liquefied soil (Srand αL) as a function of the crust thickness (HC).
pressure from the crust and liquefied soil on the pile.
In order to address these uncertainties, a system-atic approach is needed including the development of a design strategy, selection of relevant deforma-tion mechanism and identificadeforma-tion of key parameters influencing the pile response. Then, by focussing on the critical uncertainties in the analysis, a range of relevant pile responses could be estimated. Needless to say, additional uncertainties, such as those related to the inertial load, need to be considered, however, the same systematic approach as above is generally applicable (Cubrinovski et al. 2009b). This approach provides means for more consistent use of the pseudo-static analysis and proper treatment of the uncertainties in the analysis, which is a key issue in the assessment of piles in liquefying soils.
4 SEISMIC EFFECTIVE STRESS ANALYSIS
4 SEISMIC EFFECTIVE STRESS ANALYSIS