2-1 Introduction
Many pile-supported structures will be subjected to horizontal loads during their functional lifetime. If the loads are relatively small, a design can be made by building code provisions that list allowable loads for vertical piles as a function of pile diameter and properties of the soil. However, if the load per pile is large, the piles are frequently installed at a batter. The analyst may assume that the horizontal load on the structure is resisted by components of the axial loads on the battered piles. The implicit assumption in the procedure is that the piles do not deflect laterally which, of course, is not true. Rational methods for the analysis of single piles under lateral load, where the piles are vertical or battered, will be discussed herein, and methods are given for investigating a wide variety of parameters. The problem of the analysis of a group of piles is discussed in another publication.
As a foundation problem, the analysis of a pile under lateral loading is complicated because the soil reaction (resistance) at any point along a pile is a function of pile deflection. The pile deflection, on the other hand, is dependent on the soil resistance; therefore, solving for the response of a pile under lateral loading is one of a class of soil-structure-interaction problems.
The conditions of compatibility and equilibrium must be satisfied between the pile and soil and between the pile and the superstructure. Thus, the deformation and movement of the superstructure, ranging from a concrete mat to an offshore platform, and the manner in which the pile is attached to the superstructure, must be known or computed in order to obtain a correct solution to most problems.
2-1-1 Influence of Pile Installation and Loading on Soil Characteristics 2-1-1-1 General Review
The most critical factor in solving for the response of a pile under lateral loading is the prediction of the soil resistance at any point along a pile as a function of the pile deflection. Any serious attempt to develop predictions of soil resistance must address the stress-deformation characteristics of the soil. The properties to be considered, however, are those that exist after the pile has been installed. Furthermore, the influence of lateral loading on soil behavior must be taken into account.
The deformations of the soil from the driving of a pile into clay cause important and significant changes in soil characteristics. Different but important effects are caused by driving of piles into granular soils. Changes in soil properties are also associated with the installation of bored piles. While definitive research is yet to be done, evidence clearly shows that the soil immediately adjacent to a pile wall is most affected. Investigators (Malek, et al., 1989) have suggested that the direct-simple-shear test can be used to predict the behavior of an axially loaded pile, which suggests that the soil just next to the pile wall will control axial behavior.
However, the lateral deflection of a pile will cause strains and stresses to develop from the pile
wall to several diameters away. Therefore, the changes in soil characteristics due to pile installation are less important for laterally loaded piles than for axially loaded piles.
The influence of the loading of the pile on soil response is another matter. Four classes of lateral loading can be identified: short-term, repeated, sustained, and dynamic. The first three classes are discussed herein, but the response of piles to dynamic loading is beyond the scope of this document. The use of a pseudo-horizontal load as an approximation in making earthquake-resistant designs should be noted, however.
The influence of sustained or cyclic loading on the response of the soil will be discussed in some detail in Chapter 3; however, some discussion is appropriate here to provide a basis for evaluating the models that are presented in this chapter. If a pile is in granular soil or overconsolidated clay, sustained loading, as from earth pressure, will likely cause only a negligible amount of long-term lateral deflection. A pile in normally consolidated clay, on the other hand, will experience long-term deflection, but, at present, the magnitude of such deflection can only be approximated. A rigorous solution requires solution of the three-dimensional consolidation equation stepwise with time. At some time, the pile-head will experience an additional deflection that will cause a change in the horizontal stresses in the continuum.
Methods have been developed, as reviewed later, for getting answers to the problem of short-term loading by use of correlations between soil response and the in situ undrained strength of clay and the in-situ angle of internal friction for granular soil. Such “backbone” solutions are important because they can be used for sustained loading in some cases and because an initial condition is provided for taking the influence of repeated loading into account. Experience has shown that the loss of lateral resistance due to repeated loading is significant, especially if the piles are installed in clay below free water. The clay can be pushed away from the pile wall and the soil response can be significantly decreased. Predictions for the effect of cyclic loading are given in Chapter 3.
Four general types of loading are recognized above and each of these types is further discussed in the following sections. The importance of consideration and evaluation of loading when analyzing a pile subjected to lateral loading cannot be overemphasized.
Many of the load tests described later in this chapter were performed by applying a lateral load in increments, holding that load for a few minutes, and reading all the instruments that gave the response of the pile. The data that were taken allowed p-y curves to be computed; analytical expressions are developed from the experimental results and these expressions yield p-y curves that are termed “static” curves. Repeated loadings were applied as well, as will be discussed in a following section.
2-1-1-2 Static Loading
The static p-y curves can be thought of as backbone curves that can be correlated to some extent with soil properties. Thus, the curves are useful for providing some theoretical basis to the p-y method.
From the standpoint of design, the static p-y curves have application in the following cases: where loadings are short-term and not repeated (probably not encountered); and for sustained loadings, as in earth-pressure loadings, where the soil around the pile is not susceptible
As will be noted later in this chapter, the use of the p-y curves for repeated loading, a type of loading that is frequently encountered in practice, will often yield significant increases in pile deflection and bending moment. The engineer may wish to make computations with both the static curves and with the repeated (cyclic) curves so that the influence of the loading on pile response can be seen clearly.
2-1-1-3 Repeated Cyclic Loading
The full-scale field tests that were performed included repeated or cyclic loading as well as the static loading described above. An increment of load was applied, the instruments were read, and the load was repeated a number of times. In some instances, the load was forward and backward, and in other cases only forward. The instruments were read after a given number of cycles and the cycling was continued until there was no obvious increase in ground line deflection or in bending moments. Another increment was applied and the procedure was repeated. The final load that was applied brought the maximum bending moment close to the moment that would cause the steel to yield plastically.
Four specific sets of recommendations for p-y curves for cyclic loading are described in Chapter 3. For three of the sets, the recommendations that are given are for the “lower-bound”
case. That is, the data that were used to develop the p-y curves were from cases where the ground-line deflection had substantially ceased with repetitions in loading. In the other case, for stiff clay where there was no free water at the ground surface, the recommendations for p-y curves are based on the number of cycles of load application, as well as other factors.
The presence of free water at the ground surface for clay soils can be significant in regard to the loss of soil resistance due to cyclic loading (Long, 1984). After a deflection is exceeded that is based on the “elastic” response of the soil, a space develops between the pile and the soil when the load is released. Free water moves into this space and on the next load application the water is ejected bringing soil particles with it. This erosion causes a loss of soil resistance in addition to the losses due to remolding of the soil as a result of the cyclic strains. At this point the use of judgment in the design of the piles under lateral load should be emphasized. For example, if the clay is below a layer of sand, or if provision could be made to supply sand around the pile, the sand will settle around the pile, and probably restore the soil resistance that was lost due to the cyclic loading.
Pile-supported structures are subjected to cyclic loading in many instances. Some common cases are wind load against overhead signs and high-rise buildings, traffic loads on bridge structures, wave loads against offshore structures, impact loads against docks and dolphin structures, and ice loads against locks and dams. The nature of the loading must be considered carefully. Factors to be considered are frequency, magnitude, duration, and direction. The engineer will be required to use a considerable amount of judgment in the selection of the soil parameters and response curves.
2-1-1-4 Sustained Loading
If the soil resisting the lateral deflection of a pile is overconsolidated clay, the influence of sustained loading would probably be small. The maximum lateral stress from the pile against the clay would probably be less than the previous lateral stress; thus, the additional deflection due to consolidation and creep in the clay should be small or negligible.
If the soil that is effective in resisting lateral deflection of a pile is a granular material that is freely-draining, the creep would be expected to be small in most cases. However, if the pile is subjected to vibrations, there could be densification of the sand and a considerable amount of additional deflection. Thus, the judgment of the engineer in making the design should be brought into play.
If the soil resisting lateral deflection of a pile is soft, saturated clay, the stress applied by the pile to the soil could cause a considerable amount of additional deflection due to consolidation (if positive pore water pressures were generated) and creep. An initial solution could be made, the properties of the clay could be employed, and an estimate could be made of the additional deflection. The p-y curves could be modified to reflect the additional deflection and a second solution obtained with the computer. In this manner, convergence could be achieved. The writers know of no rational way to solve the three-dimensional, time-dependent problem of the additional deflection that would occur so, again, the judgment and integrity of the engineer will play an important role in obtaining an acceptable solution.
2-1-1-5 Dynamic Loading
Two types of problems involving dynamic loading are frequently encountered in design:
machine foundations and earthquakes. The deflection from the vibratory loading from machine foundations is usually quite small and the problem would be solved using the dynamic properties of the soil. Equations yielding the response of the structure under dynamic loading would be employed and the p-y method described herein would not be employed.
With regard to earthquakes, a rational solution should proceed from the definition of the free-field motion of the near-surface soil due to the earthquake. Thus, the p-y method described herein could not be used directly. In some cases, an approximate solution to the earthquake problem has been made by applying a horizontal load to the superstructure that is assumed to reflect the effect of the earthquake. In such a case, the p-y method can be used but such solutions would plainly be quite approximate.
2-1-2 Models for Use in Analyses of Single Piles
A number of models have been proposed for the pile and soil system. The following are brief descriptions for a few of them.
2-1-2-1 Elastic Pile and Soil
The model shown in Figure 2-1(a) depicts a pile in an elastic soil. A model of this sort has been widely used in analysis. Terzaghi (1955) gave values of subgrade modulus that can be used to solve for deflection and bending moment, but he went on to qualify his recommendations. The standard equation for a beam was employed in a manner that had been suggested earlier by such writers as Hetenyi (1946). Terzaghi stated that the tabulated values of subgrade modulus could not be used for cases where the computed soil resistance was more than one-half of the bearing capacity of the soil. However, a recommendation was not included for the computation of the bearing capacity under lateral load. Nor were any comparisons given between the results of computations and experiments.
The values of subgrade moduli published by Terzaghi have proved to be useful and provide evidence that Terzaghi had excellent insight into the problem. However, in a private
the paper and only did so in response to numerous requests. The method illustrated by Figure 2-1(a) serves well in obtaining the response of a pile under small loads, in illustrating the various interrelationships in the response, and in giving an overall insight into the nature of the problem.
The method cannot be employed without modification in solving for the loading at which a plastic hinge will develop in the pile.
(a) (b) Mt
Pt
Mt Pt
Mt Pt
Mt Pt
(c) (d)
Figure 2-1 Finite Element Model of Pile Under Lateral Loading, (a) 3-Dimensional Mesh, and (b) Mesh Cross-section of 3-D Mesh,
(c) Brom’s Model, (d) MFAD Model
2-1-2-2 Elastic Pile and Finite Elements for Soil
The case shown in Figure 2-1(b) is the same as the previous case except that the soil has been modeled by finite elements. No attempt is made in the sketch to indicate an appropriate size of the map, boundary constraints, special interface elements, most favorable shape and size of elements, or other details. The finite elements may be axially symmetric with non-symmetric loading or full three-dimensional models. The elements may be selected as linear or nonlinear.
In view of the computational power that is now available, the model shown in Figure 2-1(b) appears to be practical to solve the pile problem. The elements can be three-dimensional and nonlinear. However, the selection of an appropriate constitutive model for the soil involves not only the parameters that define the model but methods of dealing with tensile stresses, modeling layered soils, separation between pile and soil during repeated loading, and the changes in soil characteristics that are associated with the various types of loading.
Yegian and Wright (1973) and Thompson (1977) used a plane-stress model and obtained soil-response curves that agree well with results at or near the ground surface from full-scale experiments. The writers are aware of research that is underway with three-dimensional, nonlinear, finite and boundary elements, and are of the opinion that in time such a model will lead to results that can be used in practice. More discussion on the use of the finite-element method is presented in a later chapter where p-y curves are described.
2-1-2-3 Rigid Pile and Plastic Soil
Broms (1964a, 1964b, 1965) employed the model shown in Figure 2-1(c) to derive equations for the loading that causes a failure, either because of excessive stresses in the soil or because of a plastic hinge, or hinges, in the pile. The pile is assumed to be rigid, and a solution is found by use of the equations of statics for the distribution of ultimate resistance of the soil that puts the pile in equilibrium. The soil resistance shown hatched in the Figure 2-1(c) is for cohesive soil, and a solution was developed for cohesionless soil as well. After the ultimate loading is computed for a pile of particular dimensions, Broms suggests that the deflection at the working load may be computed by the use of the model shown in Figure 2-1(a).
Broms’ method makes use of several simplifying assumptions but is useful for the initial selection of a pile for a given soil and for a given set of loads.
2-1-2-4 Rigid Pile and Four-Spring Model for Soil
The model shown in Figure 2-1 (d) was developed for the design of piles that support transmission towers (DiGioia, et al., 1989). The loading shown at the top of the pile includes an axial load. As shown in the sketch, the four springs are: a spring at the pile tip that responds to the rotation of the tip, a spring at the pile tip that responds to the axial movement of the tip, a set of springs parallel to the wall that respond to vertical movement of the pile, and a set of springs normal to the wall that respond to lateral deflection.
The model was developed by analytical techniques and tested against a series of experiments performed on short piles. However, the experimental procedures did not allow the independent determination of the curves that give the forces as a function of the four different types of movement. Therefore, the relative importance of the four types of soil resistance has not
2-1-2-5 Nonlinear Pile and p-y Model for Soil
The model shown in Figure 2-2 represents the one utilized by the LPile software. The loading on the pile is general for the two-dimensional case (no torsion or out-of-plane bending).
The horizontal lines across the pile are meant to show that it is made up of different sections; for example, a steel pipe could be used with changes in wall thickness. The difference-equation method is employed for the solution of the beam-column equation to allow the different values of bending stiffness to be addressed. Also, it is possible to vary the bending stiffness with respect to the bending moment that is computed during iteration.
An axial load is indicated and is considered in the solution with respect to its effect on bending and not in respect to axial settlement. However, as shown later in this manual, the computational procedure is such that it allows for the determination of the axial load at which a pile will buckle.
The soil around the pile is replaced by a set of mechanisms that merely indicate that the soil resistance p is a nonlinear function of pile deflection y. The mechanisms, and the corresponding curves that represent their behavior, are widely spaced in the sketch but are considered to be close together in the analysis. As may be seen, the p-y curves are fully nonlinear with respect to distance x along the pile and pile deflection y. The curve for x = x1 is drawn to indicate that the pile may deflect a finite distance with no soil resistance. The curve at x = x2 is
The soil around the pile is replaced by a set of mechanisms that merely indicate that the soil resistance p is a nonlinear function of pile deflection y. The mechanisms, and the corresponding curves that represent their behavior, are widely spaced in the sketch but are considered to be close together in the analysis. As may be seen, the p-y curves are fully nonlinear with respect to distance x along the pile and pile deflection y. The curve for x = x1 is drawn to indicate that the pile may deflect a finite distance with no soil resistance. The curve at x = x2 is