MÓDULO: TÉCNICAS ELEMENTALES DE PREELABORACIÓN

The response of piles to lateral loading is sensitive to soil properties near the ground surface. As the surface layers may be subject to disturbance, reasonably conservative soil parameters should be adopted in the prediction of pile deflection. An approximate assessment of the effects of soil layering can be made by reference to the work by Davisson & Gill (1963) or Pise (1982).

Poulos (1972) studied the behaviour of a laterally-loaded pile socketed in rock. He concluded that socketing of a pile has little influence on the horizontal deflection at working load unless the pile is sufficiently rigid, with a stiffness factor under lateral loading, Kr, greater than 0.01, where Kr = _E EpIp

sL4 , and Ip and L are the second moment of area and length of the pile respectively.

The effect of sloping ground in front of a laterally-loaded pile was analysed by Poulos (1976) for clayey soils, and by Nakashima et al (1985) for granular soils. It was concluded that the effect on pile deformation will not be significant if the pile is beyond a distance of about five to seven pile diameters from the slope crest.

The load-deflection and load-rotation relationships for a laterally-loaded pile are generally highly non-linear. Three approaches have been proposed for predicting the behaviour of a single pile :

(a) equivalent cantilever method, (b) subgrade reaction method, and (c) elastic continuum method.

Alternative methods include numerical methods such as the finite element and boundary element methods as discussed in Section 6.13.2.4. However, these are seldom justified for routine design problems.

A useful summary of the methods of determining the horizontal soil stiffness is given by Jamiolkowski & Garassino (1977).

It should be noted that the currently available analytical methods for assessing deformation of laterally-loaded piles do not consider the contribution of the side shear stiffness. Some allowance may be made for barrettes loaded in the direction of the long side

of the section with the use of additional springs to model the shear stiffness and capacity in the subgrade reaction approach.

Where the allowable deformation is relatively large, the effects of non-linear bending behaviour of the pile section due to progressive yielding and cracking together with its effect on the deflection and bending moment profile should be considered (Kramer & Heavey, 1988). The possible non-linear structural behaviour of the section can be determined by measuring the response of an upstand above the ground surface in a lateral loading test.

6.13.3.2 Equivalent cantilever method

The equivalent cantilever method is a gross simplification of the problem and should only be used as an approximate check on the other more rigorous methods unless the pile is subject to nominal lateral load. In this method, the pile is represented by an equivalent cantilever and the deflection is computed for either free-head or fixed-head conditions. Empirical expressions for the depths to the point of virtual fixity in different ground conditions are summarised by Tomlinson (1994).

The principal shortcoming of this approach is that the relative pile-soil stiffness is not considered in a rational framework in determining the point of fixity. Also, the method is not suited for evaluating profiles of bending moments.

6.13.3.3 Subgrade reaction method

In the subgrade reaction method, the soil is idealised as a series of discrete springs down the pile shaft. The continuum nature of the soil is not taken into account in this formulation.

The characteristic of the soil spring is expressed as follows :

p = kh δh [6.16]

Ph = Kh δh [6.17]

= kh D δh (for constant Kh)

= nh z δh (for the case of Kh varying linearly with depth) where p = soil pressure

kh = coefficient of horizontal subgrade reaction δh = lateral deflection

Ph = soil reaction per unit length of pile Kh = modulus of horizontal subgrade reaction D = width or diameter of pile

nh = constant of horizontal subgrade reaction, sometimes referred to as the constant of modulus variation in the literature

It should be noted that kh is not a fundamental soil parameter as it is influenced by the pile dimensions. In contrast, Kh is more of a fundamental property and is related to the Young's modulus of the soil, and it is not a function of pile dimensions. Soil springs determined using subgrade reaction do not consider the interaction between adjoining springs. Calibration against field test data may be necessary in order to adjust the soil modulus to derive a better estimation (Poulos et al, 2002).

Traditionally, overconsolidated clay is assumed to have a constant Kh with depth whereas normally consolidated clay and granular soil is assumed to have a Kh increasing linearly with depth, starting from zero at ground surface.

For a uniform pile with a given bending stiffness (EpIp), there is a critical length (Lc) beyond which the pile behaves under lateral load as if it were infinitely long and can be termed a 'flexible' pile.

The expressions for the critical lengths are given in the following Lc = 4

4 _{Ep Ip}

Kh [6.18]

= 4 R for soils with a constant Kh Lc = 4

5 _E p Ip

nh [6.19]

= 4 T for soils with a Kh increasing linearly with depth

The terms 'R' and 'T' are referred to as the characteristic lengths by Matlock & Reese (1960) for homogeneous soils and non-homogeneous soils, respectively. They derived generalised solutions for piles in granular soils and clayey soils. The solutions for granular soils as summarized in Figures 6.18 and 6.19 have been widely used in Hong Kong.

A slightly different approach has been proposed by Broms (1964a & b) in which the pile response is related to the parameter L/R for clays, and to the parameter L/T for granular soils. The solutions provide the deflection and rotation at the head of rigid and flexible piles.

In general, the subgrade reaction method can give satisfactory predictions of the deflection of a single pile provided that the subgrade reaction parameters are derived from established correlations or calibrated against similar case histories or loading test results.

Typical ranges of values of nh, together with recommendations for design approach, are given in Table 6.11.

The parameter kh can be related to results of pressuremeter tests (CGS, 1992). The effects of pile width and shape on the deformation parameters are discussed by Siu (1992).

Table 6.11 – Typical Values of Coefficient of Horizontal Subgrade Reaction

Consistency _{(N value 4-10)} Loose _{(N value 11-30)} Medium Dense _{(N value 31-50)} Dense

nh for dry or moist sand

(MN/m3₎ 2.2 6.6 17.6

nh for submerged sand

(MN/m3₎ 1.3 4.4 10.7

Notes : (1) The above nh values are based on Terzaghi (1955) and are valid for stresses up to about

half the ultimate bearing capacity with allowance made for long-term movements. (2) For sands, Elson (1984) suggested that Terzaghi's values should be used as a lower limit

and the following relationship as the upper limits : nh = 0.19 Dr1.16 (MN/m3)

where Dr is the relative density of sand in percent. Dr can be related to SPT N values and

effective overburden pressure (see Figure 6 of Geoguide 1 : Guide to Retaining Wall Design (GEO, 1993)). The above equation is intended for sands and should be used with caution for saprolites. If this equation is used as a first approximation, it would be prudent to determine the design value of Dr involving the use of insitu and laboratory

density tests. In critical cases where the design is likely to be dominated by the behaviour under lateral loading, it is advisable to carry out full-scale loading tests in view of the design uncertainties.

(3) Limited available loading test results on piles in saprolitic soils in Hong Kong suggest that the nh values can be bracketed by the recommendations by Terzaghi and the above

equation by Elson.

(4) Other observed values of nh, which include an allowance for long-term movement, are

as follows (Tomlinson, 1994) :

Soft normally consolidated clays : 350 to 700 kN/m3

Soft organic silts : 150 kN/m3

(5) For sands, nh may be related to the drained horizontal Young modulus (Eh') in MPa as

follows (Yoshida & Yoshinaka, 1972; Parry, 1972) :

nh =

0.8Eh' to 1.8Eh'

z

where z is depth below ground surface in metres.

(6) It should be noted that empirical relationships developed for transported soils between N value and relative density are not generally valid for weathered rocks. Corestones, for example, can give misleading high values that are unrepresentative of the soil mass.

The solutions by Matlock & Reese (1960) apply for idealised, single layer soil. The subgrade reaction method can be extended to include non-linear effects by defining the complete load transfer curves or 'p-y' curves. This formulation is more complex and a non- linear analysis generally requires the use of computer models similar to those described by Bowles (1992), which can be used to take into account variation of deformation

characteristics with depth. In this approach, the pile is represented by a number of segments each supported by a spring, and the spring stiffness can be related to the deformation parameters by empirical correlations (e.g. SPT N values). Due allowance should be made for the strength of the upper, and often weaker, soils whose strength may be fully mobilised even at working load condition.

Alternatively, the load-transfer curves can be determined based on instrumented pile loading tests, in which a series of 'p-y' curves are derived for various types of soils. Nip & Ng (2005) presented a simple method to back-analyse results of laterally loaded piles for deriving the 'p-y' curves for superficial deposits. Reese & Van Impe (2001) discussed factors that should be considered when formulating the 'p-y' curves. These include pile types and flexural stiffness, duration of loading, pile geometry and layout, effect of pile installation and ground conditions. Despite the complexities in developing the 'p-y' curves, the analytical method is simple once the non-linear behaviours of the soils are modelled by the 'p-y' curves. This method is particularly suitable for layered soils.

6.13.3.4 Elastic continuum methods

Solutions for deflection and rotation based on elastic continuum assumptions are summarised by Poulos & Davis (1980). Design charts are given for different slenderness ratios (L/D) and the dimensionless pile stiffness factors under lateral loading (Kr) for both friction and end-bearing piles. The concept of critical length is however not considered in this formulation as pointed out by Elson (1984).

A comparison of these simplified elastic continuum solutions with those of the rigorous boundary element analyses has been carried out by Elson (1984). The comparison suggests that the solutions by Poulos & Davis (1980) generally give higher deflections and rotations at ground surface, particularly for piles in a soil with increasing stiffness with depth.

The elastic analysis has been extended by Poulos & Davis (1980) to account for plastic yielding of soil near ground surface. In this approximate method, the limiting ultimate stress criteria as proposed by Broms (1965) have been adopted to determine factors for correction of the basic solution.

An alternative approach is proposed by Randolph (1981b) who fitted empirical algebraic expressions to the results of finite element analyses for homogeneous and non- homogeneous linear elastic soils. In this formulation, the critical pile length, Lc (beyond which the pile plays no part in the behaviour of the upper part) is defined as follows :

Lc = 2 ro (E_Gpe c )

2/7 _[6.20]

where G* = G(1+ 0.75 νs)

Gc = mean value of G* over the critical length, Lc, in a flexible pile G = shear modulus of soil

ro = radius of an equivalent circular pile νs = Poisson's ratio of soil

Epe = equivalent Young’s modulus of the pile = 4E_πrpIp o4

For a given problem, iterations will be necessary to evaluate the values of Lc and Gc. Expressions for deflection and rotation at ground level given by Randolph's elastic continuum formulation are summarised in Figure 6.28.

Results of horizontal plate loading tests carried out from within a hand-dug caisson in completely weathered granite (Whiteside, 1986) indicate the following range of correlation :

Eh' = 0.6 N to 1.9 N (MPa) [6.21] where Eh' is the drained horizontal Young's modulus of the soil.

The modulus may be nearer the lower bound if disturbance due to pile excavation and stress relief is excessive. The reloading modulus was however found to be two to three times the above values.

Plumbridge et al (2000b) carried out lateral loading tests on large-diameter bored piles and barrettes in fill and alluvial deposits. Testing arrangement on five sites included a 100 cycle bi-directional loading stage followed by a five-stage maintained lateral loading test. The cyclic loading indicated only a negligible degradation in pile-soil stiffness after the 100 cycle bi-direction loading. The deflection behaviour for piles in push or pull directions was generally similar. Based on the deflection profile of the single pile in maintained-load tests, the correlation between horizontal Young's modulus, Eh' and SPT N value was found to range between 3 N and 4 N (MPa).

Lam et al (1991) reported results of horizontal Goodman Jack tests carried out from within a caisson in moderately to slightly (grade III/II) weathered granite. The interpreted rock mass modulus was in the range of 3.1 to 8.2 GPa.

In the absence of site-specific field data, the above range of values may be used in preliminary design of piles subject to lateral loads.