60547800 Non Linear Analysis of Piled Raft Foundation HG Poulos

Non-linear Analysis of Piled Raft Foundations

J.C. small1 and H.G. ~ o u l o s ~

ABSTRACT: A method of analysis is presented for piled raft foundations where the piles exhibit non linear load-deflection behaviour. The raft is analysed through the use of finite element methods, while the piles are treated as springs having a variable stiffness, so as to model any non-linear behaviour. The soil is treated as an elastic medium that may consist of layers of soil having different stifmesses. Interaction between the piles in the group is assumed to remain constant even though the stiffness of the piles may change with load level. The method has been incorporated into the computer program GARP (General Analysis of Rafts with Piles). This program is used to analyse a tall building that was constructed on a piled raft foundation, and a comparison is made of the calculated and observed behaviour.

JNTRODUCTION

Piled rafts are a cost-effective type of foundation in many situations as the raft and the piles can combine to transfer the applied loads from the structure to the supporting soil. Differential deflections in the raft can be controlled through the use of strategically placed piles, and for this reason, the piles are often used as settlement "controllers".

In some cases, the piles can be allowed to yield under load so that although their load capacity is exceeded, they can still apply an upward force to the raft to control settlement or more importantly, differential settlement.

Piled rafts have been used in many countries and a number of different methods have been used in their analysis. El-Mossallamy et al. (2006) give examples of tall buildings on piled rafts in Frankfurt and Berlin and provide some simple means of analysis. Yamashita et al. (1993) give an example of a structure on a piled raft in Tokyo, and present two approximate methods of analysis, one treating the soil and piles as springs and using Mindlin's equations to determine the interactions among the piles and raft, and the other treating the raft as a series of piles each with a circular rigid cap on top. Reul and Randolph (2003) have presented finite element analyses of tall buildings in Frankfurt and have provided comparisons with measured behaviour as have Katzenbach et al. (1997). A high rise building in Australia has been analysed by Badelow et al. (2006) who show that an economic pile layout can be developed using the piled raft concept.

Various authors have developed different techniques for the analysis of piled rafts in addition to those mentioned above. Kitiyodom and Matsumoto (2003) have developed a method for vertical and horizontal loads, Small and Zhang (2002) have used the finite layer method to compute the behaviour of piled rafts on layered soils under both vertical

I Senior Principal, Coffey Geosciences and Professor, University of Sydney. Senior Principal, Coffey Geosciences and Emeritus Professor, University of Sydney

GSP 158 Contemporary Issues in Deep Foundations

and horizontal loading. Poulos (1994) used a finite difference analysis of a raft combined with a boundary element approach for the piles.

In this paper, the background to the latest version of the computer program GARP (General Analysis of Rafts with Piles) is discussed, and the program is applied to a case study of a multi-storey building for which field data was collected. The non-linear behaviour of the piles is taken into account to allow for some yield of the piles carrying the higher loads.

THEORY

The raft is analysed by the use of finite element methods, and here an 8 noded element is used because the shear as well as moment can be computed in the raft. The soil is treated as an elastic layered material, and the soil response to loading calculated by integration of the relevant elastic continuum solutions. For each element in the raft, the contact pressure is assumed to be constant and therefore the problem is one of calculating the values of the contact pressures for each element and load on each pile head that will lead to compatibility of vertical displacement of the raft and the soil as well as compatibility of the displacement of the pile head and the raft.

This is done by applying a unit pressure to each rectangular area corresponding to a raft element on the surface of the soil and the deflection of the surface is found at the centre of each pressure block, and at the location of each pile. A unit load is also applied to each pile and the deflection at the centre of each pressure block and at each pile location is found. The deflections of the piles is found by using an elastic solution based on integrating Mindlin's equation for a point load over segments of the pile shaft, as described by Poulos and Davis (1980). This technique can be used to find the effect of pile deflection on the surface deflection of the soil. The effect of loading a rectangular region on the surface of the soil can be computed by integrating the Boussinesq solution for a point load over the area of the rectangle. Through this technique, the influence matrix [I,] may be found for the foundation.

We can therefore write

where

I, is the influence matrix for the foundation 6, are the deflections at the centres of each rectangular pressure block, and at the pile

locations P is a vector containing the unknown contact stresses q acting over each block, and

the unknown loads P acting on the piles i.e.p = (ql, 92, ...... q,,, PI, Pz, ...... p,,JT if there are n elements and m piles.


The same can be done for the raft to form an influence matrix. A unit uniform load is applied to each raft element in turn and the deflection that this causes at the centre of all raft elements and at the location of each of the piles is found. Unit point loads are applied at pile locations and the deflections at the centres of raft elements and at pile locations are found. The deflections so determined form the columns of the influence matrix [I,] for the raft.

However, the unit loads cannot be applied to the raft unless it is fixed in some way to avoid rigid body rotations and translations. The raft is therefore fixed at one node against rotation in the x and y directions and against vertical movement.

We can then write the deflection of the raft as

where

1, is the influence matrix of the raft 6, are the deflections at the centres of each raft element, and at the pile locations A is the unknown rigid body translation of the raft 0, is the unknown rigid body rotation of the raft about the x-axis 0 is the unknown rigid body rotation of the raft about the y-axis iW is the deflection caused at the element and pile locations due to the applied loads.

The raft must also be in equilibrium, so vertical force and moment equilibrium about both thex and y axes must be satisfied.

We can therefore write

aT P =

P T p = M ,

Y I ' P = M y

......... ......... where 6= (Al, Aa A,, I , I , I ) ~ T ......... fi = (Alx1, AzxZ, A&,,, X I , ~ 2 , ......... x,JT T y = (Alyl, Am, ......... A+m yl, y2, ......... y d T

and A, are the areas of each raft element of which there are n, m is the number of piles, P T ~ ~ is the total vertical load applied to the raft, M, is the total moment about the pin in the x-direction due to loads applied to the raft and M, is the total moment in the y- direction if x and y are the axis directions in the plane of the raft.

Combining equations 1, 2 and 3 gives the final set of (n+m+3) equations to be solved (since the influence matrices of equations 1 and 2 are of size (n+m) by (n+m)).


By solving equation 4, we have the contact pressures under each element, the pile loads, and the rigid body translations and rotations of the raft. By applying the loads back to the pinned raft, the moments, deflections and shears in the raft can be found.

Interactions between the piles, and the piles and raft, as well as the pile stiffness are computed using a boundary element approach as mentioned previously. An approximate non-linear behaviour can be introduced for the piles by allowing the tangent stiffness of the pile k to vary with load level according to a hyperbolic law

where ko is the initial stiffness of the pile, Rfis a modifying factor a n d p ~ , pull are the load carried by the pile and the ultimate pile load respectively. It may be seen from the equation that the stiffness of the pile will decrease as the ultimate load is approached. The interaction factors acting between the piles (and the piles and raft) are not varied during non-linear loading but are held at the original values. This provides a more conservative result (i.e. larger settlement) than the approach adopted by Mandolini and Viggiani (1997) in which the pile-pile interaction factor is applied only to the elastic component of the pile settlement.

EXAMPLE

The Messe-Torhaus is a 130 m high building that was constructed in Frankfurt between 1983 and 1986. The structure is supported on a double piled raft foundation as shown in Figure 1 with each of the separate rafts supported by 42 piles in a rectangular 7x6 configuration. The piles are 20 m long and have a diameter of 0.9m. Each raft is 17.5 m x 24.5 m in plan, 2.5 m thick and is founded at 3m below the ground surface. The two rafts are separated by a distance of 10m.

The piled raft was monitored during construction by the installation of strain gauges on 6 piles and a load cell at the pile base, 11 earth pressure cells beneath the raft and 3 extensometers installed down to a depth of 40.5 m below raft level.

The piled raft was constructed in the Frankfurt clay which is a stiff, overconsolidated clay and according to Reul and Randolph (2003) has an elastic modulus E (in MPa) that increases with depth z (in m) according to Equation 6.


This gives a variation of modulus that is almost constant for the first 20 m depth and then increases almost linearly with depth. It is assumed here that this modulus is applicable during construction and is an "undrained" modulus, as this is not stated by Reul and Randolph.

The finite element mesh for the twin rafts was made up of 8-noded isoparametric elements and as can be seen in Figure 1, the two rafts were joined by a mesh of thin shell elements. This is so the whole raft mesh can be analysed separately, and the connecting elements were made very thin (0.05 m) so as to have negligible bending stiffness. They therefore have very little effect on the solution.

Figure 1: Finite element mesh for the twin rafts of the Torhaus showing instnunented pile locations P1 to P6.

PLOT OF FINITE ELEMENT MESH

To allow for the fact that the raft was constructed 3 m deep, the load applied to the foundation was taken as the applied load of 466.5 kPa minus the overburden pressure of 62.5 kPa (i.e. 404 kPa). This approach implies that during recompression of the foundation, the clay will be very stiff and so its movement will be negligible.

350

- 300

. 250

2 0 0

-50

100

- 5 0

Sectn I

0 0 Sectn I1

- 4 0

303,

0 0 80 0 ma no 400

G m P A m h a

AUWS 2 m

MU= TU~IMUS rlilrMurL


Pile 6 Pile 1 Pile 2

Figure 2: Pile loads for elastic analysis - Cross-section I.

Pile 5 Pile 4 Pile 3

Figure 3: Pile loads for elastic analysis - Cross-section I1

Shown in Figures 2 and 3 are the measured and calculated loads that were carried by the piles. These are shown for two cross-sections; section I is through piles 6, 1 and 2 and section I1 through piles 5 ,4 and 3 as shown in Figure 1. Pile loads are shown for a linear elastic pile response, and for a non-linear pile response. For an elastic pile response, the pile loads predicted are reasonably close to the measured values, except for piles 4 and 5 that are predicted to cany more load than was measured.

For the non-linear pile response, the ultimate capacity of the piles must be calculated. The ultimate pile loads have been estimated to be about 9 MN as mobilised pile skin


frictions of the order of 140-150 kPa have been measured on piles in the Frankfurt clay, and as piles 3 and 2 were carrying loads of about 6 7 MN, the ultimate capacity of the piles must have been at least this value. The pile head stiffness at the start of loading was computed to be 539 MN/m and the Rf value was chosen to be 0.5.

The effect of the non-linear pile response is to shed load from the more heavily loaded piles to the more lightly loaded piles as they will have a stiffer response. However, the non-linear analysis does not predict the high loads on the edge piles 2 and 3 as well as the elastic analysis but gives an improved prediction of the load on pile 5. This may be due to the actual distribution of loading, as here the load was assumed to be uniformly applied.

Deflection of the rafts is shown in Figure 4 where it may be seen that the two foundations rotate towards each other. The mean settlement of each raft calculated for an elastic response was about 80 mm whereas for the non-linear case it is slightly higher at 85 mm. The measured settlements at the end of construction have been reported as 65 and 80 mm at the centre of the northern and southern rafts respectively (Hemsley 2000). Unlike the calculated settlements, the measured settlements are not the same for each raft, but the calculated settlements are of the same order of magnitude as the mean settlement of the southern raft. Table 1 shows the mean measured and calculated settlements. In this case, nonlinear effects appear to have only a relatively small effect on the computed behaviour of the foundation system.

Table 1: Measured and calculated settlements at end of construction.

CONCLUSIONS

Measured mean settlement (mm)

Northern raft Southern raft

The program GARP has been used to analyse a building that was monitored during construction and the predicted performance has been compared with field measurements. It has been noted that if the piles are allowed to behave in a non-linear fashion, the piles carrying more load shed some load to the more lightly loaded piles that have a stiffer response.

In the case of the Torhaus building, the use of non-linear analysis improves load prediction on some of the piles, but not for two of the edge piles. This may be due to a poor prediction of the ultimate load that the piles could carry as this was not known. The computed settlements at the end of construction are consistent with the measured values for the southern raft, but overestimate the mean settlement of the northern raft. In the case of the the Torhaus building, the piles appear to have had reserve load carrying capacity

Calculated mean settlement (elastic)

65 80

Calculated mean settlement (non-

mm

80

linear) mm

85


and were not acting as settlement-reducing piles that would have shed more load than observed.

Figure 4: Vertical deflection contours in the two piled rafts of the Messe-Torhaus.

CONTOURS OF VERTICAL DISPLACEMENT

Messe Torhaus Frankfurt

REFERENCES

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Badelow, F., Poulos, H.G., Small, J.C. and Moyes, P. (2006) "Economic foundation design for tall buildings", Proceedings of lofh International Conference on Piling and Deep Foz~ndations (DFIIEFFC - Amsterdam, 31 May-2 June, 2006), Eds. J.Lindenberg, M.Bottiau, A.F. van Tol, 200-209.

El-Mossallamy, Y., Lutz, B. and Richter, T. (2006) "Innovative application of piled raft foundation to optimize the design of high-rise buildings and bridge foundations", Proceedings of lorh International Confrence on Piling and Deep Fozrndations (DFIIEFFC - Amsterdam, 31 May-2 June, 2006), Eds. J. Lindberg, M. Bottiau & A.F. VanTol, 269-278.

Hemsley, J.A. (Ed.) (2000) Design Applications of Rafi Fotmndations, Thomas Telford.

Katzenbach, R., Arslan, U., Gutwald, J., Holzhauser, J. and Quick, H. (1997) "Soil- structure interaction of the 300m high Commerzbank tower in Frankfurt am Main - Measurements and numerical studies", Proceedings of the 141h Int. Con$ on Soil Mechs. and Fo~ind Eng., Hamburg 2, 1081-1084.


Kitiyodom, P. and Matsumoto, T. (2003) "A simplified analysis method for piled raft foundations in non-homogeneous soils", Int. JZ. For Numerical Mefh. in Geomechs., 27,

Mandolini, A. and Viggiani, C. (1997) "Settlement of piled foundations", Giotechniqrre, XLVII(4), 791-8 16.

Poulos, H.G. (1994) "An approximate numerical analysis of pile-raft interaction", Znt. J1. for Nr~merical and Analytical Meth0d.r in Geomechanics, 18(2), 73-92.

Poulos, H.G. & Davis, E.H. (1980) Pile foundation analysis anddesign. John Wiley and Sons, New York.

Reul, 0. & Randolph, M. F. 2003. "Piled rafts in overconsolidated clay: comparison of in-situ measurements and numerical analyses", Giotechniqr~e, 53(3), 301-31 5 .

Reul, 0. & Randolph, M. F. 2004. "Design strategies for piled rafts subjected to nonuniform vertical loading", Jl. of Geotechnical and Geoenvironmental Eng., ASCE, 130(1), 1-11

Small, J.C. and Zhang, H.H (2002) "Behaviour of piled raft foundations under lateral and vertical loading", International Journal of Geomechanics, 2(1), 29-45.

Yamashita, K., Kakurai, M. and Yamada, T. (1993) "Settlement behaviour of five-storey building on a piled raft foundation", Proceedings of the znd Int. Geotech. Seminar on Deep Foundations on Bored and Auger Piles - BAP II, Ghenf Belgium, W. Van Impe (ed.), Balkema, Rotterdam, 35 1-356.


60547800 Non Linear Analysis of Piled Raft Foundation HG Poulos

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