Comparison of the Two Methods

The two methods discussed in the previous section are conceptually the same. The differences between the two methods are summarised in Table 6.1.

Method I Method II (APRILS) Separation of

Foundation

(1) Isolated raft, (2) Pile group and layered soil

(1) Isolated piled raft, (2) layered soil

Pile Connection Attached to the raft element Attached to the raft at the node Load applied at the Pile

Head Uniform pressures Concentrated loads Table 6.1 Differences between Methods I and II

Chapter 6 – Analysis of Piled Rafts

Method II has several advantages over method I:

(1) The piles are attached to the raft at the nodes which reduces the size of the mesh in terms of the total number of nodes and elements which then reduces the number of equations that need to be solved.

(2) The raft and piles are analysed as a whole structure such that the applied loads and moments are directly transferred from the raft to the pile. Therefore, the transformation of the applied moments into equivalent forces acting along the edges of the raft elements is not necessary.

(3) The influence matrix of the piled raft is generated by applying unit uniform load to the raft elements or unit load to the nodes of the piles in turn without the need to apply unit moment which reduces the computation time for matrix formation. (4) Better simulation of the loads at the pile heads can be obtained for the analysis of

large piled rafts as the loads behave as concentrated loads instead of uniform pressures at the pile head.

In the following example, a piled raft was analysed by both methods and results were compared. A square piled raft is supported by 16 piles of identical size and length. The raft has a size of 8 x 8m and a thickness of 1m. The piles are 10m long with a diameter of 0.5m. The overall depth of the soil is assumed to be 2 x pile length (i.e. 20m). The configuration and pile arrangement of the piled raft is shown in Figure 6.7. The modulus and Poisson’s ratio of the soil are taken as 10MPa and 0.3 respectively. The raft and piles have the same modulus of 25,000MPa and Poisson’s ratio of 0.3. For both analysis, each pile was divided into 10 elements and the soil was divided into 20 layers. Figures 6.8a and b show the finite element meshes for the piled raft used in the Method I and II analyses. In the Method I analysis, the raft was divided into 144 elements and the element which contains the pile has to have the same area as the pile cross-sectional area. In the Method II analysis, the raft was divided into 64 elements which was approximately half of the number of elements used in the Method I analysis. The use of fewer elements in the analysis means that the number of equations that need to be solved is reduced thus requiring less time for computation.

Chapter 6 – Analysis of Piled Rafts

Figures 6.9a and b show the contours of settlement obtained from Method I and Method II analyses. Figure 6.10a shows the comparison between the settlement along the centre line of the raft for both methods. The settlement obtained by the Method II analysis was about 6% less than that obtained from the Method I analysis. The axial load distributions along the centre (P1) and corner (P2) piles are shown in Figure 6.10b. The load distribution for the corner piles from both analyses are in good agreement except at the pile head. For the pile at the centre, the load distribution obtained from Method II was smaller than that obtained from Method I. However, the differences in the load diminished with depth along the pile. For both corner and centre piles, the differences of load at the top of the piles are larger than at the base of the piles, this implies that the influence of the type of the load applied at the pile head would become less significant as the embedment depth of the pile increases. As the depth between the load at the pile head and the pile node of interest increases, the effect of the uniform load or concentrated load at the pile head on the load in the pile will approach a similar value. The contours of bending moment in the x-direction for Methods I and II are shown in Figures 6.11a and b respectively. The contours are slightly different especially at the pile heads where different pile-raft connection exists. Figure 6.11c shows the comparison between the two methods of the moments in the x-direction at the Gauss points close to the centre line of the raft and it may be seen that the two results are in good agreement.

The difference in the results obtained by both analyses could be due to (i) different integration schemes used in the analyses as the size of the integration blocks is determined from the average size of the raft elements, (ii) the load applied at the pile head is represented by a concentrated load in the Method II analysis, (iii) the piled raft is analysed as a single structure in Method II so that forces and moments acting on the raft are directly transmitted to the piles while in the Method I analysis, the raft is separated from the pile group and the moments acting on the raft have to be transformed into equivalent forces.

Chapter 6 – Analysis of Piled Rafts

6.2.4 Non-Linear Analysis of Piled Raft

The typical load-settlement behaviour of a piled raft is shown in Figure 6.12. When the piled raft is loaded below Pe, both the piles and raft are behaving elastically. When the

piled raft is loaded between Pe and Pu, the interface between the pile and soil starts to slip.

When the pile capacity is fully mobilised, the load in excess of Pe is carried by the raft

only. As the load reaches Pu, both the piles and raft have reached their ultimate capacities

and fail to carry additional loads. Such non-linear behaviour of the foundation can be simulated by various approaches: (i) assumption of a hyperbolic load-settlement curve (Russo, 1998; Mandolini and Viggiani, 1997) (ii) limiting the load carried by the raft and piles (Horikoshi and Randolph, 1998) (iii) a soil model consisting of yield surface segments, eg. Drucker-Prager yield criterion (Reul and Randolph, 2003; Maharaj and Gandhi, 2004).

Program APRILS has the capability to take into account the non-linear response of the pile by limiting the loads acting on the pile-soil interface to simulate the slip that occurs at the interface. The analysis is implemented through an incremental-iterative process. The load is applied in increments and the forces acting on the raft-soil and pile-soil interfaces are computed for each increment. These interface forces are then compared with the limiting contact pressures acting on the raft or the limiting ring loads along the pile shaft and base load at the pile base. These limiting loads are computed from the shear strength of the soil, su. The limiting ring loads along the pile shaft for pile element i (for a

pile in clay), LRi, is

LRi = ca C δz (6.7)

and the limiting base load at the pile base, LB, is

LB = Nc su A (6.8)

where ca = pile-soil adhesion = su x α C = circumference of pile

δz = length of pile element i

α = adhesion factor and is a function of su Nc = bearing capacity factor (~ 9 for piles in clay)

Chapter 6 – Analysis of Piled Rafts