**1.0 Introduction**

Settlement prediction of single piles under axial loading is as important as the estimation of ultimate axial capacity which is the sum of ultimate total shaft resistance and end bearing resistance. Settlement control usually receives the most intention from geotechnical designers who usually have difficulties in complying with the settlement requirements as specified in many design specifications or standards if there is no thorough understanding of the mechanism involved in the settlement analysis of single piles under axial loading.

It would be very useful to develop a powerful but simple tool that could predict the pile load settlement curve in addition to estimating axial pile capacity.

**2.0 Background**

The widely-used methods for predicting settlement under axial load for a single pile can be categorized into three different types: (1) load transfer method; (2) elastic theory-based method and (3) finite element or finite difference method.

The load transfer method was developed by Coyle and Reese (1966). In this method, the pile is divided into a number of segments. The movement of the individual pile segment at mid-height is estimated from load transfer versus pile movement curves (empirical t-z curves for shaft resistance and q-w curves for end bearing resistance) and the elastic axial deformation in an iterative manner for an assumed tip movement. The calculation proceeds up the length of the pile to obtain the load and displacement at the pile head. The pile load settlement curve can thus be constructed through a series of such calculations with different assumed tip movement values.

The elastic theory-based method is adopted by Butterfield and Banerjee (1971), Randolph and Wroth (1978), and Poulos and Davis (1980). In this method, the pile is divided into a number of uniformly loaded elements and the solution is obtained by imposing the compatibility constraints between the displacement of the soil and the pile. The displacements of the pile are obtained by considering the compressibility of the pile under axial loading and the soil displacements in most cases are obtained by using Mindlin’s equations for the displacement of a soil mass caused by a loading within the mass. The limitation of the elasticity method lies in the basic assumptions that must be made – semi-infinite, elastic, and isotropic solids. The actual ground conditions rarely satisfy the basic assumption of uniform and isotropic materials. In spite of the highly nonlinear stress-strain characteristics of soils, the only soil properties considered in the elasticity method are Young’s modulus E and the Poisson’s ratio, mu. The use of only two constants, E and mu, to represent soil characteristics is an oversimplification to allow the elasticity-based methods to work in conditions involving stratified soils with varying strengths and stiffness.

The finite element or finite difference method is considered one of the most powerful approaches for the analysis of single piles and pile groups under axial loading. However, it is not commonly used in practice due to its high computational requirements.

**3.0 Numerical Simulation of Load Transfer**

Numerical simulation of the load transfer curves such as t-z curves for shaft resistance and q-w curves for end bearing resistance is a practical approach for the conditions where layered soil profiles and/or many potential load cases and trial designs need to be undertaken.

A free-body diagram of a single pile under axial loading is shown in the figure on the right. The pile is evenly divided into a number of pile segments.

The basic form of the differential equation is presented below for calculating the settlement of single piles under axial loading. The differential equation is usually solved using the difference-equation technique.

Where **Ep x Ap** is axial stiffness of the pile, z is the relative movement of the pile with reference to the soil at point z, C is the circumference of the pile, u is the modulus from the load transfer curves for t-z and q-w relationships.

PileAXL program uses the load transfer method to estimate the settlement under axial load for a single pile due to its simplicity and practicality. It can deal with any complex composition of soil layers with any nonlinear relationship of displacement versus shaft resistance or end bearing resistance. A recursive solution is adopted to solve the equation.

**4.0 T-z curves for shaft resistance mobilization**

A typical nonlinear T-Z load transfer curve used to simulate the mobilization of shaft resistance with displacement is shown in the figure below.

The following list summarizes all the nonlinear T-Z load transfer curves implemented in the PileAXL program for pile settlement calculation. Those t-z load transfer curves are widely used to simulate the mobilization of shaft resistance with vertical displacement.

1. T-Z curve – API Clay

2. T-Z curve – API Sand

3. T-Z curve – API RP2GEO Sand

4. T-Z curve – Coyle and Reese Clay (1966)

5. T-Z curve – Mosher Sand (1984)

6. T-Z curve – Reese and O’Neill Clay (1989)

7. T-Z curve – Reese and O’Neill Sand (1989)

8. T-Z curve – O’Neill and Hassan Rock (1994)

9. T-Z curve – user-defined

**5.0 Q-Z curves for end bearing resistance mobilization **

Similar to t-z load transfer curves for modeling the mobilization of shaft resistance along the pile, q-z curves are commonly adopted to simulate the mobilization of end bearing resistance under the pile toe. A typical nonlinear q-w load transfer curve is shown in the figure below.

The following list summarizes all the nonlinear Q-Z load transfer curves implemented in the PileAXL program for pile settlement calculation. Those Q-Z load transfer curves are widely used to simulate the mobilization of bearing resistance with pile toe displacement.

1. Q-Z curve – API Clay

2. Q-Z curve – API Sand

3. Q-Z curve – Skempton (1951)

4. Q-Z curve – Vijayvergiva (1977)

5. Q-Z curve – Reese and O’Neill Clay (1989)

6. Q-Z curve – Reese and O’Neill Sand (1989)

7. Q-Z curve – Elastic-Plastic Model

8. Q-Z curve – user-defined

**6.0 A cast study for pile settlement analysis**

This example involves a 1.4 m diameter reinforced concrete bored pile installed through fill, soft marine clay, firm clay and dense clayey silt layers and socketed into weak siltstone in Singapore based on the case studies from (Leung 1996).

The subsurface profile consists of 3 m thick fill, 2 m thick soft marine clay, 3m thick firm silty clay followed by dense clayey silt of about 2.5 m thick. The bottom part of the pile is located in a 5.5 m thick deep rock socket of highly fractured, very weak to weak siltstone. The unconfined compressive strength of the upper and lower parts of the rock socket is 3.5 MPa and 6.5 MPa respectively. A maximum test load of 20 MN was applied on the pile and this is adopted in this example to calculate the load settlement curve.

The figure below shows the ground profile with the pile length and loading conditions for this example. Axial force applied at the pile head is 20 MN.

The groundwater table is adopted to be at the ground surface. The detailed soil layer input parameters for all the input soil layers are summarized in the table below.

The material of “Fill” is modelled with FHWA Sand method. Both soft marine clay and firm clay are modelled with FHWA clay method.

As for dense clay silt, SPT-N method is used where the empirical factors for ultimate shaft resistance are set as 0 and 0.95, respectively to match the maximum shaft resistance of about 120 kPa for this layer reported by Leung (1996).

Both upper and lower weak siltstone layers are modelled with Weathered Rock – Williams and Pells method. The empirical factor **Beta** is adopted to be 0.6 as per Leung (1996). The empirical factors **Alpha** are adopted to be 0.185 and 0.159, respectively for upper and lower weak siltstone layers to match the measured ultimate shaft resistances of about 390 kPa and 620 kPa. The bearing capacity factor, **N****cr** is adopted to be 2.5 for both weak rock layers. The typical inputs for the upper weak siltstone are shown in the figure below.

The figure blow shows the pile settlement curves of the pile length of 16 m under 20 MN axial loading reported from the PileAXL program.

The comparison results between the testing results reported in Leung (1996) and the calculated pile settlement data by PileAXL are presented in the figure below. It can be seen that the load settlement curve predicted by PileAXL for this case is reasonably close to the one recorded during the testing.