ABSTRACT

A semi-analytical method is developed for predicting the load-settlement behavior of vertically loaded piles formed in weak rocks and cohesive soils. The aim is to have a simpler and more cost-effective alternative to the traditional load transfer (t-z) method, whereby the data required from various depths, are not only difficult to interpret from routine site investigations but also costly to develop from sophisticated field tests. The proposed method is based on formulating generic functions to uncouple shaft and base transfer mechanisms. The functions are developed through analysis of field performance data from instrumented pile tests. After combining the generic functions with the basic governing equations for soil-pile interaction, the full set of equations is solved iteratively with the aid of a newly developed computer program. The program computes the ratio of load sharing between the shaft and the base, at every stage of loading, leading to determination of the axial force distribution and load-settlement variation. Using the program, load-settlement predictions are carried out for 3 CFA piles formed in mudstone. Also, parallel predictions are made using the t-z approach and another Published method. In every case, it is shown that the proposed method yields the most accurate predictions.

Keywords: piles, settlement, cohesive soils, weak rocks.

INTRODUCTION

Although numerical methods such as boundary element and finite element analyses are sufficiently powerful to account for complex stress transfer mechanisms in pile-soil systems, the sophisticated parameters required are seldom obtainable from a standard site investigation. Therefore, many engineers of-ten resort to alternative methods such as the load transfer (t-z) method, which still has a drawback in that load transfer curves are not only expensive to develop from trial piles but also difficult to interpret from in-situ/laboratory soil tests.

In this paper, semi-analytical solutions for shear transfer and load-settlement response are developed in order to provide a practical and cost-effective alternative to the t-z method. Governing equations for pile-soil interaction are first developed, by uncoupling shaft and base resistances. Through analysis of a large database of instrumented pile tests formed in weak rocks/clays, relationships are formulated to represent shaft and base transfer characteristics. The method takes into consideration spatial variations in shear load transfer with depth, due to changes in the intensity of applied pile head load.

METHODOLOGY OF ANALYSIS

The governing equations for an axially loaded pile can be written as follows, after Randolph and Wroth (1978):

(1) |

(2) |

(3) |

where *P*(z) = axial force in pile at depth z; *f*_{s}(z) = local unit shaft resistance mobilised at depth z; w(z)= vertical displacement of pile at depth z; *D*_{s} = pile shaft diameter; and *E*_{p} = elastic modulus of pile material.

It is usually very difficult to model and predict the shaft resistance versus depth variations, even for piles in homogeneous soil media. This is because shaft resistance mobilization is influenced not only by the pile and soil properties but also by the following: (i) pile-soil interface geometry and slip characteristics, (ii) method of pile installation, (iii) stresses acting on the pile-soil interface and (iv) pile loading procedure and speed. Examination of a large database of pile tests shows that *f*_{s}(z) can be approximated based on the assumption that the shear modulus, Gs of the soil varies as a parabolic function of depth:

(4) |

where *A*, *B*, *C* are constants. For a particular pile head load, the mobilised shaft resistance can be related to the shaft settlement D_{s} through the function:

(5a) (5b) |

where *f*_{us}=maximum average unit shaft resistance; D_{sc} = critical shaft settlement (D_{s} value corresponding to *f*_{us}). For a given pile, D_{sc} is related to *D*_{s}, pile length, *L*, the mean shear strength of the soil around pile shaft, cu and two empirical constants *A*_{1} and *A*_{2}. Hence:

(6) |

The values of *A*_{1} and *A*_{2} are expected to be influenced by factors such as pile-soil interface properties, pile installation effects, rate of loading and correlation between shear strength and shear modulus of soil.

Corresponding to the shear modulus variation model in Equation 4, the variation, with depth, of the limiting shear stress at the pile-soil interface is assumed to be

(7) |

Equations (1)-(3) lead to the following general solutions:

(7) |

(8) |

where the basic coefficients a_{1}, a_{2}, a_{3} and a_{4} and the constant *C*_{1} are determined by iteration, by satisfying simultaneously the force equilibrium and displacement compatibility conditions of the pile-soil system. From analysis of a hypothetical pile, typical plots of *f*_{s}(z) and P(z) versus z are shown in Figures 1-2.

**Figure 1.** Typical axial force versus depth variation, for various applied head loads (%) of ultimate head capacity) plotted in a normalised form, from analysis of a hypothetical pile

**Figure 2.** Typical shaft resistance versus depth variation, for various applied head loads (as in Fig.1)

plotted in a normalized form, from analysis of a hypothetical pile

EQUATIONS FOR IDEALISED PILE

To ensure force equilibrium and displacement compatibility for the idealized pile, the following boundary conditions must be satisfied.

(i) when z=0, P(z) equals the applied head load *P*_{h}; at pile toe level z=*L* (where *L*=pile length) and P(z)=*P*_{b}=(1-y)*P*_{h}, where y=*P*_{s}/*P*_{h} in which *P*_{s} and *P*_{b} are the mobilised shaft and base resistances respectively. It should be noted that y varies with *P*_{h}.

(ii) d*f*_{s}(z)/dz=0 when z=w*L*, where w is a constant (0 < w < 1) for a given *P*_{h} but varies with the ratio *P*_{h}/*P*_{uh}, where *P*_{uh} is the maximum pile head load. Reese et al. (1976) Published shaft resistance distribution plots for 4 instrumented piles formed in clay, for the full range of head loads from zero to the ultimate value. Based on non-linear regression analysis of the shaft resistance distribution profiles, it has been found that the best fit parabola representing *f*_{s}(z) is such that dfs(z)/dz=0 when z=w*L*, where w can be calibrated from pile data in terms of *P*_{h}/*P*_{uh}.

(iii) positive *f*_{s} values are guaranteed for all values of z as *P*_{h} increases from zero to *P*_{uh} by using the ratio fsb/fso of the unit shaft resistance at z=*L* to that at the surface, z=0 as a parameter to control the shape of the *f*_{s}(z) curve. Practically, fso is usually small but not zero hence fso may be assumed to correspond to a small value of z, say z=1x10^{-5}L. Based on examination of the pile data reported by Reese et al. (1976), *f*_{sb}/*f*_{so} can be represented by three different values, depending on the magnitude of the applied head load as a percentage of the ultimate head capacity.

Invoking the above boundary conditions, the following solutions are obtained:

(9a) |

where

(9b) |

(9c) |

B = 1x10^{-5}L |
(9d) |

Through parametric analysis using the pile data reported by Reese et al. (1976), the following values of the parameter *A* are found to be appropriate for the loading ranges specified:

(9e) |

An expression for the constant C1 in Equation (8) can be derived by satisfying the boundary condition that at z=0, w(z)=((b+ep) where ep is the compression of the pile. Therefore:

(10) |

As will be discussed later, the first step in applying the proposed method is to choose a suitable range of D_{b} values. Then the *P*_{b} values corresponding to the D_{b} values are calculated. Hence, it is imperative that a generic relationship between *P*_{b} and D_{b} must be obtained. This is discussed in a separate section ahead.

Using the condition w(z) = D_{b} when z=*L* enables *e*_{p} to be expressed as follows:

(11) |

The average shaft displacement D_{s} is the mean value of w(z) from z=0 to z=*L*, which is derived to be:

(12) |

It should be borne in mind that Equations (6)-(8) are only fully defined if, for a given *P*_{h} value, ( is known from which (1 to (4 can be calculated. To calculate(, an iterative scheme can be developed, which ensures compatibility between Equation (12) and the shear resistance versus displacement characteristics of the soil at the particular site. A general function for shear resistance versus displacement is presented in the following section

SHAFT TRANSFER RELATIONSHIP

Wright and Reese (1979) reported load tests on 4 large diameter bored, cast in-place piles formed in stiff over-consolidated clay. The piles were instrumented with Mustran cells installed at several selected levels. The results have been interpreted in order to model the relationship between mobilised shaft resistance *P*_{s} and average shaft displacement (s. It is found that a plot of *P*_{s}/((s versus ((s is reasonably linear, with correlation coefficients, R2, falling in the range R2=0.89-0.99.

Further, it is found that the numerical values of the gradient ms and the intercept I on the *P*_{s}/((s axis are such that the quantity {-I2/(4ms)} represents the maximum shaft resistance, Pus whereas {I/(2ms)}2 is the value of (s corresponding to Pus. This (s value will be referred to as the “critical shaft settlement”, (sc. Therefore, for (s((sc, the variation of *P*_{s} with (s can be represented as follows:

For (s>(sc, it is assumed that *P*_{s}=Pus.

Based on studies with model piles, Bea (1975) found that (sc is proportional to shaft diameter *D*_{s}. Whitaker and Cooke (1966) reached the same conclusion, based on a large number of full-scale load tests on bored piles formed in clay. BS 8004 (1986) states that (sc is proportional to *D*_{s} and recommends that (sc should be taken as 1-5% *D*_{s}, for piles in clay. It is suggested that (sc should be expressed as (sc=nDs, where n is a constant, which is determined by the properties of the soil and of the pile-soil interface. To determine the constant n, plots of *f*_{us}Ds/(sc (where *f*_{us}=maximum average unit shaft resistance) versus average undrained strength cu, along the pile shaft, were plotted using Wright and Reese (1979) data. The graphs are reasonably linear with a strong correlation coefficient R2=0.985 on average. Therefore n can be expressed as follows:

where the units of (sc and *D*_{s} are metres, *f*_{us} and are in MN/m2. Hence combining Equations (12)-(13), it is possible to relate *P*_{s} to (s as follows:

(14) |

For a given site, cu can be measured using conventional methods based on laboratory soil tests and/or in-situ tests such as SPT (standard penetration tests) or CPT (cone penetration tests). To determine *f*_{us}, where data from static load tests are available, Eurocode 7 (1995) and BS 8004 (1986) recommend that use should be made of calculation formulae, which are based on established correlation between the load test results and field/laboratory test findings.

BASE TRANSFER RELATIONSHIP

Different base transfer functions, relating unit base resistance fb and base movement (b, have been proposed. Vijayvergiya (1977) suggested that fb varies as a power function of (b. Hirayama (1990) and Carrubba (1997) suggested a two-constant hyperbolic relation between fb and (b, for bored piles and rock-socket piles. For clays and weak rocks, examination of pile database shows that base transfer response can be modeled realistically in (a) two phases for “clean” bases (b) three phases for situations where highly compressible soil debris or rock fragments is present immediately beneath the piles base (“unclean” bases).

Figure 3 shows normalised base transfer curves for some instrumented piles in Mercia mudstone (TP3, TP4 and TP5) and clay (SA, HBT and PR3). The plots illustrate the variation *P*_{b}/*P*_{ub} with (b/(mDb), where *P*_{ub} is ultimate base resistance; Db is pile base diameter and m is a constant. The product mDb is the critical base displacement, which is the value of (b corresponding to *P*_{ub}. From the load test results, where ultimate base capacity was very closely approached (piles TP5 and HBT), it has been estimated that m=0.20-0.22 for the piles formed in weathered Mercia mudstone and m=0.06-0.11 for the piles in clay. The shapes of the curves for piles TP3-TP5 (Fig. 3) strongly suggest unclean base conditions. The remaining piles are regarded as having “clean” bases.

**Figure 3.** Normalised base transfer curves for bored piles installed in Mercia mudstone and clay

It can also be seen that, for the intervals 0(*P*_{b}/*P*_{ub}(( (clean bases) and ((*P*_{b}/*P*_{ub}(( (unclean bases), the plots are reasonably linear for values ((0.2 and ((0.4. Typically, the slope Sg of the linear segment can be expressed as Sg=(Nb, where is the average SPT “N” value at the pile base level. By back analysis of pile test results and borehole data, it has been established that, for weathered Mercia mudstone, ((0.03. Details of the formulation of the general base transfer functions, based on satisfying the *P*_{b}-(b boundary conditions, is described by Delpak et al (2000). The derived functions for the three phases are:

for ((P_{b}/P_{ub}(1 |
(15c) |

where *A*_{2} is defined by the following transcendental function, which can be solved using Newton’s (or bisection) method:

(16) |

The remaining constants are then calculated as follows:

(17) |

The ultimate unit base resistance, *f*_{ub}, through which *P*_{ub} is calculated, can also be determined using laboratory soil tests and/or in-situ tests. It is again emphasized that, where possible, *f*_{ub} should be derived from calculation rules based on established correlation between the results of static load tests and the results of field/laboratory tests.

ALGORITHM FOR PILE ANALYSIS

The pile dimensions, pile elastic modulus and the required soil parameters must be known. Values (N in number) of (b, ranging from zero to a maximum of 10%Db, are then generated. Equal (b increments of say, 10%Db/N are used for a pile whereby the load resistance is more or less uniformly shared between the shaft and base, for most of the loading range. In other cases, there can be a great difference between the rates of increase of shaft and base resistance with increasing settlement. In such a situation, a reasonable spread of data points in the calculated load-settlement curve can be obtained by taking (b increments as follows:

of 10%Db/N for the first N/4 values, then

1/3 of 10%Db/N for the next N/4 values, next

of 10%Db/N for the subsequent N/4 values, then

of 10%Db/N for the following N/8 values, finally

10%Db/N for the last N/8 values.

The authors have produced a new interactive computer program, in graphical user interface format, for calculating load-settlement and axial force distribution. The program determines the correct ( value for each (b value as follows: First, Pus and *P*_{ub} are computed from *f*_{us} and *f*_{ub} respectively. Then *P*_{uh} is evaluated from *P*_{uh}= Pus+*P*_{ub}; n is calculated using Equation (13), hence (sc is found from (sc=nDs. For every (b value:

Calculate *P*_{b} from Equations 15(a)-(c), appropriate for the regime of (b, after re-arranging as necessary

(i) Assume (s=(b

(ii) If (s<(sc then carry out ste*P*_{s} 4-16, otherwise skip to step 17

(iii) Calculate *P*_{s} from Equation (14)

(iv) Obtain *P*_{h} from *P*_{h}=*P*_{s}+*P*_{b}

(v) Calculate ( from (=*P*_{s}/*P*_{h}

(vi) Evaluate the constant *A* from Equation 9(e), knowing *P*_{h}/*P*_{uh}

(vii) Compute the current value of ( knowing *P*_{h}/*P*_{uh}

(viii) Determine the constant *C* from Equation 9(b)

(ix) Calculate (4, (1, (2 and (3 (in that order) from Equation 9(a)

(x) Determine ep from Equation (11)

(xi) Evaluate the constant C1 from Equation (10)

(xii) Compute (s using Equation (12)

(xiii) If the absolute value of the difference between the assumed and calculated (s values (from ste*P*_{s} 2 and 13 respectively is greater than, say 0.01mm then repeat ste*P*_{s} 3-14, using the last calculated (s value as the assumed value, until convergence is achieved.

(xiv) Obtain (h from (h=ep+(b

(xv) Compute the axial force distribution from Equation (7) using, say 20 equal increments of depth, z. This is the final step for (b values for which (s<(sc.

(xvi) For all (b values, *P*_{s}=Pus

(xvii) Carry out ste*P*_{s} 5-13 and 15-16.

LOAD TRANSFER ANALYSIS

Shaft t-z relation

Hyperbolic t-z curves are widely used and the most appropriate representation, which takes into account the roughness of the pile-soil interface, is that proposed by Kim et al. (1999). The mobilised unit shaft resistance *f*_{s}(z) at depth z is related to the local pile displacement w(z) at that depth through the following expression:

(18) |

where *f*_{us}(z)=ultimate unit shaft resistance at depth z; *D*_{s}=pile shaft diameter; K and ( are numerical constants determined by the roughness of the pile-soil interface. Based on test results from instrumented piles subjected to uplift loading in weathered rocks, Kim et al. (1999) established the following K and ( values: (a) Rough interface: K=3.86 (mm)-0.5, (=1.0 (non-dimensional) (b) Smooth interface: K=6.26 (mm)-0.5, (=1.35. For the continuous flight auger (CFA) piles analysed in this paper, it is appropriate to adopt the parameters relating to a rough pile-soil interface.

Base p-y relation

Hyperbolic functions are also widely used to model base p-y curves for cohesive soils and soft rocks. Analysis of base transfer data from the pile tests reported by Wright and Reese (1979) shows that a suitable p-y model is of the form:

(19) |

where Eb=deformation modulus of the soil within the influence zone beneath the pile base level. For weathered rocks, BS 8004 (1986) recommends that Eb should be estimated from Eb= jMr(c (in MPa) where j=mass factor (j=0.2 from Table 4 of BS 8004 for group 2/3 rocks); (c=uni-axial strength (MPa); Mr=250. Using the correlation cu=5-6 times SPT “N” (kN/m2) suggested by Stroud (1989) and, by convention, (c =2cu hence (c=12Nb kN/m2.

The authors have developed a second computer program, based on the t-z method (Coyle and Reese, 1966), in windows format. The pile to be analysed is divided into 10-12 segments, limiting segment lengths to 1.2m. Correspondingly, the soil around the pile shaft is divided into 10-12 layers. The layers can have different values of K,(, and SPT “N” or CPT-qc. These parameters must be input into the program. The program includes empirical correlations for calculating the *f*_{us}(z) values for the specified soil layers.

THREE METHODS APPLIED TO CFA PILES

Three continuous flight auger (CFA) piles, formed in weathered mudstone, were analyzed using the proposed method (UoG program), t-z analysis and the method suggested by Fleming (1992). The piles, which are 0.6 m in diameter and 23 m in length, were installed and tested in the year 2000 as part of a large scale building development in South Wales, UK. The calculated load-settlement curves for the piles are compared with the measured data in Figures 4-6.

**Figure 4.** Measured and predicted load-settlement curves for CFA pile P1 in Mercia mudstone

**Figure 5.** Measured and predicted load-settlement curves for CFA pile P2 in Mercia mudstone

**Figure 6.** Measured and predicted load-settlement curves for CFA pile P3 in Mercia mudstone

It is seen that the proposed method yields the most accurate predictions, especially at the anticipated working load of 0.4*P*_{uh}. Since the piles had no shaft instrumentation, the calculated load distributions (using the proposed method and t-z method) have not been reported here. It should be noted that Fleming’s (1992) method does not include a capability to predict axial forces in piles.

CONCLUSIONS

The semi-analytical method presented in this paper provides a reasonably accurate and practical alternative to the load transfer method. The method has the advantage over other methods in that only routinely available soil parameters are required. The method predicts accurately the axial load transfer and settlement of a loaded pile. Important factors such as pile-soil interface properties, pile installation effects, rate of loading and correlation between shear strength and shear modulus of soil are automatically accounted for in modeling shaft load transfer. Owing to its simplicity, the proposed method is readily adjustable to cater for a variety of soil types, and hence it is likely to be of significant benefit to structural/foundation engineers.

ACKNOWLEDGEMENTS

Special appreciation is due to The Royal Society, London, UK, for generously providing an Industry Fellowship grant (No. 533002.K5170) for 2 years to support the lead author while carrying out a project at Lankelma Piling & Engineering Ltd, East Sussex, UK. Other thanks go to Messrs E Zon and J Brouwer of Lankelma for their collaboration and financial contribution. The University of Glamorgan Directorate is thanked for making various facilities available.

REFERENCES

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