Pile foundation analysis and design using experimental data and 3-D numerical analysis

Computers and Geotechnics 36 (2009) 819–836

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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Pile foundation analysis and design using experimental data and 3-D numerical analysis Emilios M. Comodromos a,*, Mello C. Papadopoulou b, Ioannis K. Rentzeperis c a b c

Department of Civil Engineering, University of Thessaly, Pedion Areos, 383 34 Volos, Greece GeoStatiki S.A. Consulting Engineers, 3, Sapfous Str., 54627 Thessaloniki, Greece Egnatia Odos S.A., P.O. Box 60030, 57001 Thessaloniki, Greece

a r t i c l e

i n f o

Article history: Received 29 August 2008 Received in revised form 5 December 2008 Accepted 19 January 2009 Available online 25 February 2009 Keywords: Pile test Pile group response 3-D nonlinear analysis Pile foundation design

a b s t r a c t Capacity based design of pile foundations limits the soil-structure interaction mechanism to group bearing capacity estimation, neglecting, in most cases, the contribution of the raft. On the other hand, a straightforward, nonlinear, 3-D analysis, accounting for soil and structural nonlinearities and the effects arising from pile–soil–pile interaction, would be extremely high CPU-time demanding and will necessitate the use of exceptionally powerful numerical tools. With the aim of investigating the most efficient, precise, and economical design for a bridge foundation, a hybrid method, compatible with the notion of sub-structuring is proposed. It is based on both experimental data and nonlinear 3-D analysis. The first step to achieve these targets is a back-analysis of a static pile load test, fitting values for soil shear strength, deformation modulus, and shear strength mobilization at the soil–pile interface. Subsequently, the response of 2  2 and 3  3 pile group configurations is numerically established and the distribution of the applied load to the raft and the characteristic piles is discussed. Finally, a design strategy for an optimized design of pile raft foundations subjected to non-uniform vertical loading is proposed. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The response of a pile group under axial loading is considered to be among the factors that most affect the superstructure behaviour for gravity loading. Capacity based design of structures, still used in many cases, necessitates only analysis of the determination of the bearing capacity of a pile group. However, the development of new design criteria renders a new displacement-based design concept more adequate for pile foundations. A superstructure based on pile foundation is a three-dimensional (3-D) physical system, and straightforward analysis of such a problem can be achieved only under the assumption of linearelasticity in both the foundation and the superstructure. This conventional design procedure ignores soil nonlinearity and effects from the pile group response. To consider the latter factors, it is necessary to adopt a numerical tool, which can provide the ability to model both soil and structural nonlinearities. This procedure has not yet been incorporated in design practice on account of its complexity and time demand. However, in the case of very important structures, an effective iterative procedure can be applied to readjust the stiffness of pile foundations within the notion of the sub-structuring technique. In that case * Corresponding author. Tel.: +30 24210 74143; fax: +30 2310 252840. E-mail addresses: [email protected] (E.M. Comodromos), [email protected] (M.C. Papadopoulou), [email protected] (I.K. Rentzeperis). 0266-352X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2009.01.011

the total structure is considered to be an assemblage of substructures [1]. According to the concept of sub-structuring, a superstructure based on a pile foundation can be subdivided in the superstructure and the pile foundation. The two sub-structures can be solved separately in that way that both solutions provide the same stresses and displacements at the common boundary (compatibility conditions). It is, therefore, possible to couple a nonlinear analysis of a complex system by using, independently, a nonlinear superstructure analysis code and a nonlinear foundation analysis code. As a result an accurate solution and an optimum design of the superstructure and the pile foundation may be achieved, as the foundation effect can be calculated from the pile group response and precisely introduced in the analysis. The application of this approach allows for efficient analysis of very large finite elements or finite difference systems [2–4]. There exist many procedures for estimating the response of pile groups, ranging from application of empirical relationships and simple closed form solutions to sophisticated nonlinear numerical procedures. Based on the experience gained through the research of the last decades, empirical relationships were proposed to estimate reduction factors on both the bearing capacity and the stiffness of a group because of the interaction between the piles. Specific values for these factors have been proposed in tabular or graphical form resulting from simplified analyses based on elastic continuum analysis and the principle

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of superposition [5–7]. Lee [8] used the load-transfer (t–z) method, originally proposed by O’Neill et al. [9], to estimate the response of a single pile and Mindlin’s solution [10], to assess the interaction between the piles. Another simplified approach, providing a methodology for estimating the settlement of a pile group, is the representation by an equivalent pier [11–13]. Most of these methods involve soil profile simplifications and other idealizations, rendering them computer cost-effective with the drawback, however, of limited accuracy in many cases. 3-D finite element analyses revealed a significant interaction between the piles in a group with a spacing of 3.0b (where b stands for the pile diameter), which was still notable even when the spacing was increased to 6.0b. Using 3-D nonlinear analysis Comodromos [14] demonstrated that, in the case of pile groups with fixed head conditions (no contribution of the raft), the group bearing capacity efficiency factor, defined as the ratio of the ultimate bearing capacity of a pile group to that of the single pile multiplied by the number of piles, did not deviate significantly from unity. Comodromos et al. [15] came to the same conclusion when examining free-head pile groups in which each pile was loaded with the same load. In contrast, it was found that the interaction affects the group stiffness efficiency factor considerably. It was also revealed that the stiffness efficiency factor, defined as the ratio of the deflection of the single pile and that of the pile group under the same mean load, depends not only on the pile arrangement, but on the settlement level as well. Moreover, Comodromos [14] proposed a simplified relationship, with the capability of predicting the response of pile groups, provided that the response of a single pile is known. The application of this relationship was limited to the commonly applied pile spacing of 3.0b and to soil profiles similar to the one used in the analysis. With the aim of examining the validity of the proposed relationship to different soil profiles, Comodromos and Bareka [16], carried out an extensive numerical analysis for various pile dispositions and different soil profiles, covering the range from very soft to hard clays. According to their results, the proposed relationship was able to predict the response of a fixed head pile group in clayey soils with a reasonable level of accuracy. These methods are mainly concentrated on pile resistance neglecting the contribution of the raft, as adopted in common practice by engineers in many countries and prescribed by the majority of existing codes and regulations, de Sanctis and Mandolini [17]. Such an approach is quite conservative for soils of medium resistance and compressibility. In the case of very compressible soil formation, depending on the foundation configuration, the response of the pile raft foundations for a small level of settlements is mostly due to pile resistance and therefore the stiffness efficiency factor remains unaffected by the raft contribution. On the contrary, when the settlement level increases the raft resistance increases as well and influences both the bearing capacity and the stiffness of the foundation. With the aim of investigating and quantifying the contribution of the raft, in the case of a bridge foundation, in highly compressible soil, a 3-D nonlinear analysis was carried out. The first step to achieve these targets was a back-analysis through a nonlinear numerical simulation of a static pile load test. Appropriate design values for soil shear strength, deformation modulus, and information regarding shear strength mobilization at the soil–pile interface were determined by backfiguring from the results of the pile load test. Subsequently, the response of the 2  2 and 3  3 pile group configurations was numerically established and the distribution of the applied load on the raft and the characteristic piles was calculated. Finally, interesting conclusions were drawn regarding the effect of the raft on the bearing capacity and the stiffness of pile groups in conjunction with the settlement level.

2. Ground investigation and site soil profile For the design of the foundation of a composite-girder bridge a geotechnical investigation has been carried out. The area is located between the harbour and the intersection of the main motorway and the ring road (10 km south-west of Thessaloniki city centre) in northern Greece. The bridge carries a single carriageway with an overall length of 286 m (spans of 56.3, 86.0, 80.0, and 61.55 m) and is a part of an extended bridge project approximately 2.5 km long, connecting the new installations of Thessaloniki’s port with the main motorway to Athens. A part of the project has been already constructed and is actually in use, even as a second construction phase is anticipated to complete the project. The subsoil in the area of the project is relatively compressible (Young’s modulus of the order of 30 MPa), presenting relatively low shear strength down to a depth of at least 25.0 m. The soil profile was determined based on the results of a geotechnical investigation including four boreholes and a Cone Penetration within the area of the project. Laboratory tests were carried out on Shelby tubes samples and included triaxial tests as well as direct shear tests. The soil stratigraphy consists of four main layers. The upper layer, assigned as Layer A, consists of brownish, loose, silty sand. The shear strength and the deformation modulus improve slightly with depth and hence it is subdivided in sub layers A1 and A2. At the level of 12.0 m the second main layer, Layer B, is located, consisting of soft, dark green clay with sand particles and organics. The high organic content is responsible for the high plasticity index attributed to the layer. Layer B extends down to 25.0 m, where very stiff, brownish-green sandy clay, assigned as Layer C, is detected, extending down to 35.0 m. From that level to the end of the borehole, lies Layer D, similar to Layer C, but with higher shear strength and deformation modulus. The groundwater level varies with seasons, within the range of 0.5–1.0 m below ground level. The main soil properties of each soil layer, derived by geotechnical investigation, and the evaluation of in situ and laboratory tests are presented in Fig. 1, whereas, detailed information is given in Comodromos et al. [18].

3. Foundation parameter assessment It should be mentioned that, in the particular case of pile foundations, where the applied loads provoke shear strains in the immediate vicinity of the piles, partial drainage occurs in this area even in the case of short-term loading. This is the case of pile testing or even of pile foundations carrying important variable loads. For this reason direct shear tests were carried out covering the range from undrained to drained conditions and appropriate range of values were defined for simulating the behaviour of clayey soils under partial drainage. It should also be reminded that the stress path developing around a pile does not correspond to conventional laboratory tests [7] and that specific laboratory tests, such as, constant normal stiffness (CNS) direct shear tests [19] for determining pile skin friction might be required, particularly for clayey soils of high strength. Furthermore, the adhesion between pile and soil could be taken as being equal to the soil cohesion for soft clays, whereas, for stiff and hard clays a reduction factor should be applied. A variety of such reduction factors are given by Tomlinson [20] and the German code DIN 4014 [21]. For safety reasons, these factors are rather conservative and significant economical consequence might arise from the project, particularly in the case of pile foundations in soils of limited shear strength. In such cases pile load tests might play an important role in value engineering and in geotechnical

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0

5

qc (MPa)

SM, ML

Layer A1: Brownish loose silty SAND NSPT= 4±3 γ= 20 WL= 33±2 φ= 30ο c= 3 E= 25.0

qc= 2

Layer A1: Brownish loose silty SAND NSPT= 4±3 γ= 20 WL= 33±2 φ= 35ο c= 5 E= 35.0

qc= 4

6.0 m

10

SM, ML

12.0 m

15

25

Layer B: Dark green soft CLAY of high plasticity with organics NSPT= 8±6 γ= 17 WL= 57±2 WP= 26 qu= 30-50 φ= 5±5 c= 25±10 E= 30.0

25.0 m Layer C: Brownish-green medium plastic very stiff sandy CLAY NSPT≥ 21 γ= 21 WL= 48 WP= 26 qu= 170±70 φ= 0-5 c= 110±20 E= 80.0

30

35

CL

35.0 m Layer D: Brownish-green medium plastic hard sandy CLAY NSPT≥ 40 γ= 21 WL= 39 WP= 19 qu= 250±80 φ= 0-5 c= 140±40 E= 90.0

40

70

OH

CL

70.0 m ANNOTATION : : SPT blow count NSPT W L (%) : liquid limit qu (kPa) : unconfined strength c (kPa) : cohesion E (MPa) : Young's modulus

γ (kN/m3) : unit weight W P (%) : plastic limit qc (MPa) : cone bearing pressure φ (deg) : angle of internal friction

Fig. 1. Geotechnical soil profile at the project area.

and structural optimization. A full scale test might contribute to the elimination of practically all the uncertainties arising from soil behaviour and interface parameters, which govern the mechanism of pile–soil interaction. This is the foremost reason that some pile design regulations [21,22] require that the pile design calculations must be related to results from pile load tests. Furthermore, a detailed back-analysis of a pile test can provide appropriate design values for the aforementioned parameters, whereas, by back-figuring from the results of pile load tests, further information regarding the shear strength mobilization can be attained. Given the magnitude and the importance of the project, a pile load test was proposed to be carried out, including both vertical and horizontal loading, and the aforementioned procedure was considered indispensable for adjusting the values of soil parameters resulting from the geotechnical evaluation, given in Table 1, and for achieving an effective design for bridge foundation.

Table 1 Final geotechnical model justified by the pile test back-analysis process. STRATA

A1

A2

B

C

D

Bottom elevation (m) Bulk modulus K (MPa) Shear modulus G (MPa) Angle of friction u (deg) Dilation angle w (deg) Cohesion c (kPa) Undrained shear strength Su (kPa) Unit weight c (kN/m3) Friction angle at the interface ui (deg) Adhesion along the interface ca (kPa)

6 20.8 9.6 30 0 3 – 20.0 30 3

12 38.9 12.9 33 0 5 – 20.0 33 5

25.0 33.3 11.1 5 0 25 – 17.0 3 15

35.0 266.7 27.6 – – – 110 21.0 – 45

70.0 300.0 31.0 – – – 140 21.0 – 65

4. Static load test According to a preliminary design of the bridge, a working load of 4400 kN and 6750 kN was estimated under static and seismic

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conditions, respectively. The length and diameter of bored piles to carry the above load, required piles that could be longer than 45.0 m and have a diameter larger than 1.50 m in many piers, when designed according to the German code DIN 4014 [21], even without taking into account the group effect. Codes and regulations are conservative in many cases and, therefore, it was decided to carry out a load test to appropriately define the soil shear strength and deformation modulus. The pile load test was entrusted to Geostatiki S.A. Consulting Engineers by Egnatia Odos S.A., the authority in charge of the design, construction, maintenance, and exploitation of the Egnatia Motorway traversing the northern part of Greece. To minimize the requirements in the reaction piles it was decided to use a pile with a 0.8 m diameter. The pile load test arrangement comprised of the test pile and two reaction piles. The length and the reinforcement of the reaction piles were designed appropriately to allow full mobilization of shaft and tip resistance of the test pile. For this reason the piles used for the test were not part of the foundation piles. 5. Experimental setup The compressive load was applied using four hydraulic jacks having a capacity of 2500 kN each, which were placed between the pile head and a reaction frame consisting of a spreader steel beam, as shown in Fig. 2. The reaction was transferred through the steel beam to the tension piles, which were placed at a centre-to-centre distance of 3.0 m from the test pile, equivalent to 3.75 diameters. An appropriate reinforced concrete square cap was cast on the head of the test pile, which enabled transfer of the applied load without any concrete cracking. Furthermore, 20 mm thick steel plates were installed on both the pile head and the spreader beam to distribute the load on the pile head and the reaction frame. Particular attention was paid to minimize load eccentricity, and spherical bearings were used on the top of the jacks to minimize eccentricity of load application. The pile was instrumented with 13 fiber-optic sensors (FOS) of very high resolution (smart bar long-gauge). On the upper part of the pile the FOS (SA-1 to SA-5 and SB-1 to SB-5) were placed in pairs so that influence of bending could be accounted for in the test. Three more sensors (SA-6 to SA-8) were installed at a deeper elevation to infer the axial strain variation along the pile. The sensors had been placed during the disposition of the reinforcing cage, Fig. 3. Because of instrumental failure, associated with cable damage during tremie pipe mounting for concreting purposes, no data were obtained from FOS SA-7 and SA-8. The application of stresses on a FOS provokes the change of its wavelength. The induced wavelength variation is calculated through the following equation:

Fig. 3. Installation of fiber-optic sensors during the disposition of reinforcing cage.

Dk ¼ K e de

ð1Þ

where Dk is the wavelength variation, de the FOS strain and Ke the constant correlating wavelength to strain. As provided by the manufacturer the value of Ke is 1.2 pm/lstrain, the FOS resolution is 1/ 1.2 lstrain and its range ±20,000 lstrain. A COTS W3/4250 series optical interrogator manufactured by Smart Fibres has been used for recording wavelength variation every 30 s. The pile head settlement was monitored by a data acquisition system including four electronic displacement transducers (LVDT-1 to LVDT-4), having a resolution of 0.001 mm. The transducers were fixed on a rigid steel frame, which was supported at points placed at a distance of 3.5 m far from the test pile and the reaction piles, to eliminate any interference effects. The hydraulic pressure in the jacking system was measured by a pressure cell. The complete pile load test layout is presented in Fig. 4. The test procedure was to load up the pile in increments of 1.0 MN. After four increments, the pile was unloaded in decrements of 2.0 MN (Cycle 1). It was then reloaded to a maximum load of 5.6 MN and unloaded (Cycle 2). Each load level was held constant until the rate of movement was 0.25 mm/h before adding the next increment to the pile head. The procedure is described by Fleming et al. [23] and Tomlinson [20]. It is also important that the holding time at each load increment is the same, so as to lead to the same degree of soil consolidation, as it has been found that the slower the rate the smaller the ultimate failure load is. 6. Pile load results – evaluation

Fig. 2. Illustrartaion of vertical pile test setup.

Fig. 5 shows the results of the loading test together with the load-movement curve derived according to the German code DIN 4014. The code DIN 4014 proposes the application of the following equation for estimating the pile resistance:

E.M. Comodromos et al. / Computers and Geotechnics 36 (2009) 819–836

Fig. 4. Pile test setup arrangement.

Q ¼ AF r s þ

i X 1

in which

Asi smi

ð2Þ

Q: pile resistance; AF: pile base area; rs: point resistance as a function of head settlement;

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Asi: the surface area of the pile shaft along the embedded length in stratum i; smi: the skin friction resistance as a function of head displacement; i: the number assigned to a particular stratum. Using the geotechnical data given in Fig. 1, the values for smi and rs have been defined from DIN 4014 tables and the pile resistance has been estimated for settlements of 2, 2.5, 3, and 10%b, as shown in Fig. 5. It can be seen that both the ultimate resistance and the pile stiffness obtained from the test are significantly higher than those derived by DIN 4014. The closeness of the reaction piles may affect the response of the tested pile by overestimating the pile head stiffness. According to Fleming et al. [23], Poulos and Davis [6], and Kitiyodom and Matsumoto [24], the overestimation may be very significant when the reaction piles are close to the test pile. As pointed out by Comodromos et al. [15], the effect is due to the earlier mobilization of the shear strength of the soil between the pile under test and the reaction piles. Eventually, a simultaneous downward movement of the tested pile and upward of the reaction pile produces a higher level of shear strain, and, as a result, shear strength mobilization is developed earlier than in the corresponding case of single pile. It is also stated that the level of overestimation significantly depends on the pile test layout, the soil behaviour, and the level of loading, and that contrary to the observed significant effect on the stiffness of the tested pile, the impact on its bearing capacity was found to be negligible. For the current pile test, where only two tension piles are used at axial distance of 3.75b, the effect of the interaction was assumed to be limited. The effect can be estimated using the chart proposed by Poulos and Davis [6] for the applied pile test configuration and considering end-bearing pile. By interpolating the specific values of L/b  50 and K = Ep/Es  400 with those of the chart, the correction factor Fc, by which the measured settlement should be multiplied to estimate the true settlement, is of the order of 15–20%. This value represents also the increase in stiffness due to the presence of the reaction piles. It should be noted that the effect is valid for the initial loading part of the curve (linear-elastic part) and that the effect becomes less important as yielding of the surrounding soil occurs [15]. In the back-analysis of the pile test, mentioned below, this assumption is addressed.

The increment in pile axial force during the pile test is given by Eq. (3).

dNt ¼

pb2 4

Ep de

ð3Þ

where dNt is the pile axial force increment, de is the strain increment calculated using Eq. (1) and Ep the pile modulus of elasticity. The maximum difference of wavelength increment between the upper pair of sensors (FOS SB-1 and SA-1) during loading increments has also been used to estimate the curvature u using the following equation:

max u ¼

maxðeSB-1  eSA1 Þ maxðDkSB-1  DkSA1 Þ ¼ h hK e

The maximum difference in wavelength was 0.011 nm (11 pm), observed when the applied load was 5.0 MN. Taking into account that the distance h between the sensors was 0.66 m and the value of Ke is equal to 1.2 pm/lstrain, Eq. (4) provides a maximum curvature of 13.9  106 m1, which, using Eq. (5) results to a bending moment of 9.5 kN m. These very low values demonstrated the limited effect of applied load eccentricity, which must be attributed to the existence of spherical bearings between the jacks and the reaction beam.

M ¼ Ep I p u

ð5Þ 2

6.1. Pile test back-analysis 6.1.1. Numerical simulation procedure The numerical simulation of the pile load test was performed through the finite difference code FLAC3D [25]. The bottom elevation and the lateral sides of the computational domain were taken far enough from the group to avoid any significant boundary effect. More specifically, the distance between the piles’ tip and the bottom of the mesh was 32.0 m and the lateral side of the domain was taken at a distance of 30.0 m from the outermost side of the

6

Axial Load N (MN)

5 4 3 2 Loading Cycle 1 Loading Cycle 2 DIN 4014

1

0

10

4

where Ip is the pile moment of inertia (2.0  10 m ). The pile axial force distribution arisen from the FOS has been used to adjust the interface shear strength parameters, which in most cases differ from those determined by laboratory tests due to the fact that the stress path being developed around a pile does not correspond to conventional laboratory tests [7].

7

0

ð4Þ

20

30

40

Settlement S (mm) Fig. 5. Comparison between pile test load-settlement curve and that provided from the application of code DIN 4014.

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piles. At the bottom level of the computational domain all movements were restrained, whereas, at the lateral external sides, lateral movements perpendicular to the boundary were prohibited. The number of nodes and elements was 21840 and 21458, respectively. In the particular case of pile foundations in clayey soils, where the applied loads provoke shear strains in the immediate vicinity of the piles, partial drainage occurs in this area even in the case of short-term loading. This is the case in pile testing or even in pile foundations carrying important variable loads. For this reason, the use of values of Poisson’s ratio approaching drained conditions may be considered appropriate depending on the pressure parameter B as pointed by Bishop and Hight [26]. On the other hand, even if partial drainage is anticipated in clayey layers and pile resistance can be elevated because of friction between soil and pile, the Tresca model is considered rather appropriate. This can be attributed to the fact that the applied load does not change the overall stress field in the ground significantly, therefore the strength mobilized on the pile shaft is akin to that existed prior to loading. The approach mentioned earlier is conservative in estimating the bearing capacity of piles and is applied by many codes [21,22,27] even in the case of permanent loads (drained conditions). Therefore, the elastic perfectly-plastic Tresca constitutive model was used to simulate the behaviour of clayey layers, whereas, the elastic perfectly-plastic Mohr–Coulomb constitutive model was used to simulate the behaviour of the silty and sandy layers. Table 1 summarizes the final values of soil properties as defined through the back-analysis iterative process. Pile behaviour was considered as linear-elastic with a Young’s modulus of 34 GPa, correlated to the strength of grade C20/25 concrete and allowing for stiffening of piles because of steel bar reinforcement. A ‘ramp loading’ procedure was used to avoid numerical instabilities due to the high load value. Between the piles and the surrounding soil, interface elements were used to simulate the pile–soil interaction. The constitutive model of the interface elements was defined by a linear Coulomb shear-strength criterion that limits the shear force acting at an interface node and is given by the following equation:

F smax ¼ ci A þ tan ui F n

ð6Þ

where Fsmax is the limiting shear force at the pile–soil interface, ci the adhesion between pile and soil, Fn the normal force at the interface, and A the contact area between pile and soil.

The normal and shear forces at the interface nodes are determined by the following equations:

F nðtþDtÞ ¼ kn un A þ rn A ðtþDtÞ

F si

ðtþ0:5DtÞ

ðtÞ

¼ F si þ ks Dusi

ð7Þ A þ rsi A

ð8Þ

where Fn and Fsi: normal and shear force, respectively; kn and ks: normal and shear stiffness, respectively; A: area associated with an interface node; Dusi: incremental relative shear displacement vector; un: absolute normal penetration of the interface node into the target face; rn: additional normal stress due to pre-existing interface stress condition; rsi: additional shear stress due to pre-existing interface stress condition. In many cases, particularly when linear-elastic analysis is performed, values for interface stiffness are assigned to simulate the nonlinear behaviour of a physical system. In the present study, where nonlinear analysis is carried out, the value for the interface stiffness must be high enough, in comparison with the surrounding soil, in order to minimize the contribution of those elements to the accumulated displacements. To satisfy this requirement, the guidelines of the FLAC3D manual propose values for kn and ks of the order of ten times the equivalent stiffness of the stiffest neighbouring zone. The use of considerably higher values is tempting as it can be considered as more appropriate, but in that case the solution convergence will be very slow. Based on these observations, a value of 10 GPa/m has been taken for both kn and ks. According to the results of numerical analyses this value is sufficient to ensure that no additional settlements are attributed to the pile because of the deformation of the springs representing the interface. Thus, the interface elements behave practically as a slider (St. Venant model), with a rigid/plastic behaviour [28]. The simulation sequence included an initial step in which the initial stress condition was established, followed by seven loading steps. More specifically, compression loads of 1.0, 2.0, 3.0, 4.0, 5.0, 5.5, 5.7, and 6.0 MN were applied on the tested pile, and simultaneously, tension loads equal to one half of the compression load were applied to each reaction pile.

7

Axial Load N (MN)

6 5 4 Loading Cycle 1

3

Loading Cycle 2 2 Single Pile 3-D Simulation 1

Pile Test 3-D Simulation

0 0

10

20

30

40

50

Settlement S (mm) Fig. 6. Measured and numerically established load-settlement curves for the pile test and single pile.

60

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7. Results – comparison – evaluation The predicted load-settlement curve is presented in Fig. 6, together with the test results demonstrating remarkable agreement. On the contrary, a comparison between the load-settlement relationship resulting from the application of the code DIN 4014 and that of the pile test, clearly illustrates an underestimation of both the ultimate bearing capacity and the pile head stiffness. Fig. 7 demonstrates the variation of the numerically predicted axial load Ns along the test pile for applied loads of 1.0, 2.0,3.0, and 4.0 MN, resulting from the pile test simulation, whereas, values assigned by Nt stand for axial force distribution calculated from the strains on the FOS. The remarkable agreement of both the pile head re-

sponse and the axial load distribution observed between the data of the pile test and the results of the 3-D analysis justifies the assessment of the geotechnical model and the pile test simulation procedure. To determine the effect of the tension piles on the behaviour of the test pile, the same loading sequence was applied to establish numerically the load-settlement relationship in the case of a single pile. The analysis was performed using the same finite difference mesh in which only the tested pile was introduced in the mesh. This pile was loaded with gradual compression loads as previously stated. The results of the analysis are given in the load-displacement curve of Fig. 6 together with those of the pile test and pile test simulation. A very small difference between the load-displace-

Axial Force Nt, Ns (MN) 0

1

2

3

4

5

0

Depth (m)

10

20

Nt= 1 MN FLAC3D Ns= 1 MN Nt= 2 MN FLAC3D Ns=2 MN Nt= 3 MN FLAC3D Ns= 3 MN Nt= 4 MN FLAC3D Ns= 4 MN

30

40

Fig. 7. Measured and numerically established axial force distribution for the test pile.

Axial Force Nsn, Ns (MN) 0

1

2

3

4

5

6

0

Depth (m)

10

20

30

40 Nsn= 1 MN

Ns= 1 MN

Nsn=2 MN

Ns=2 MN

Nsn= 3 MN

Ns= 3 MN

Nsn= 4 MN

Ns= 4 MN

Ns= 5 MN

Nsn= 5 MN

Ns=5.5 MN

Nsn=5.5 MN

Fig. 8. Numerically established axial force versus depth relationship for the test and the single pile.

E.M. Comodromos et al. / Computers and Geotechnics 36 (2009) 819–836

ment curves of the test pile and the single pile could be observed, rendering the tested pile slightly stiffer than the single pile, whereas, both piles exhibited the same ultimate capacity. Its maximum effect corresponds to the load of 3.0 MN where the pile test exhibits a settlement of 4.18 mm and the single pile 5.17 mm, rendering the pile of the test 19% stiffer than the single pile. This value is in close agreement with that previously estimated from the chart proposed by Poulos and Davis [6]. The limited effect should be attributed to the layout of the reaction piles, given that for cross tension piles configuration in the same spacing as above, the stiffness overestimation of the test pile head exceeds the level of 200% [15,24]. A reduction of the interaction effect between the tested pile and the reaction piles starts when yielding of the soil surrounding the pile begins to take place and the soil is unable to follow the stress field induced by the tension piles. The result of the interaction between the test pile and the reaction piles can also be seen in Fig. 8, where axial forces Ns and Nsn correspond to the pile test and single pile simulation of the applied loading sequence on the pile head. Because of the uplift forces applied by the tension piles, the contribution of the shaft resistance is always larger in the test pile than in the single pile, until plastification occurs and the two piles have the same shaft resistance and end bearing. Fig. 9 shows the predicted vertical displacement field at the vertical section y = 0 for the test pile corresponding to an applied axial load of 4.0 MN. The uplift forces applied on the tension piles produce an upward movement, whereas, shear stresses are also transferred to the interface of the tested pile through the surrounding soil. 8. Numerical analyses of pile groups A parametric numerical analysis using the finite difference code FLAC3D was carried out to investigate the consequences of the interaction between the piles and the pile cap, on both the ultimate bearing capacity and the stiffness of individual piles and that of the entire pile group. Moreover, quantitative and qualitative evaluation of the contribution of the raft to the ultimate resistance and pile head stiffness is also an interesting point for investigation, as the existing codes do not refer to that. As a consequence, and bearing in mind difficulties arising from such a goal, engineers com-

827

monly neglect raft contribution, rendering the approach of solely concentrating on pile resistance quite conservative and, in most cases, economically aggravating. Pile groups of similar configurations to those applied in the foundation of piers where examined. More specifically 2  2 and 3  3 pile group configurations with constant spacing of 3.0b were examined. On the contrary, pile diameters of 1.00, 1.20 and 1.50 m were considered, whereas, pile lengths were taken as 38.0 and 42.0 m. The geometry of the mesh was parametrically defined to give the possibility of geometrical variations when needed. A mesh generator subroutine was implemented using the FISH built-in programming language, providing the possibility of mesh refinement and geometry variation. Based on the experience gained through this and previous numerical works, a mesh refinement around the piles leads faster to a solution despite the fact that the number of linear equations to be solved becomes higher. This is valid when soil yielding occurs and must be attributed to the better stress distribution which helps iterative nonlinear algorithms to achieve the required resolution in less iterations. The bottom elevation and the lateral sides of the computational domain were taken far enough from the group to avoid any significant boundary effect. The same constitutive models and simulation process as in the previous paragraph were adopted. The load was applied at the centre of the pile cap using a ‘ramp loading’ procedure. 9. Load-settlement response Fig. 10 illustrates the finite difference mesh utilized in the analysis of the 3  3 pile group configuration with 3.0b spacing and pile length L = 42 m, which consisted of 21840 elements, 23624 nodes, and 360 structural elements, simulating also the 3.0 m thick concrete pile cap. Fig. 11 shows the predicted vertical displacement field at the vertical section y = 0 for 3  3 group, L = 42.0 m and pile diameter b = 1.50 m under a total vertical load of 90 MN (mean load Nm = 10 MN, defined as the applied group load divided by the number of piles). It can be observed that the surrounding soil settles together with the pile group, indicating that most of the load is carried by the shaft resistance and that no slip occurred at the pile soil interface. When the applied load increases to 171 MN (mean load Nm = 19 MN) the piles clearly manifest larger settlements, while pile tip settlements are larger than that of the overlying soil (Fig. 12). This is a clear indication of slip occurrence at the pile–soil interface. Figs. 13 and 14 illustrate the load-settlement response of the pile groups with cap obtained from all numerical analyses examined. For comparison purposes curves are plotted as the mean axial load Nm versus Sns (settlement normalized to pile diameter, Sns = S/b). It can be observed that both 2  2 and 3  3 group configurations demonstrate the same mean ultimate bearing capacity, defined as the ultimate capacity of the group divided by the number of piles in the group. On the contrary the group mean stiffness, defined as the stiffness of the group divided by the number of piles, clearly depends on the number of piles, justifying the findings of previous researchers [7,14,16], who state that when the pile number increases the group mean stiffness decreases as a result of pile– soil–pile interaction. It must be underlined that this is valid for low level settlements, since, as soil yielding occurs the effect of the interaction decreases and finally leads to the coincidence of loadsettlement curves, as can be observed in Figs. 13 and 14. 10. Raft contribution

Fig. 9. Vertical displacement field around test pile for N = 4.0 MN.

Fig. 15 illustrates the numerically derived load-displacement curves for the 3  3 pile group with pile cap and a 3  3 pile group

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Fig. 10. Finite difference mesh simulating a 3  3 raft pile group, b = 1.00 m.

with no pile cap, but their heads restricted to exhibit the same settlement. All piles have a diameter of 1.20 m and pile length L = 38.0 m. The bold line with circle markers shows the response of the 3  3 group with pile cap, which can be separated to the contribution of the piles (thin line with diamond markers) and that of the raft (thin line with square markers). Separate analyses have also been carried out for the group without cap and for the raft without piles. The bold line with asterisk markers in Fig. 15 corresponds to the response of the 3  3 group without cap, whereas, the bold line with triangles stands for the raft load divided by the number of piles. Valuable conclusions can be drawn when comparing the load-settlement response of the above-mentioned configurations. A notable conclusion is that the stiffness of the pile group remains largely unaffected by the cap. This should be attributed to the fact that, for low level settlements the contribution of the raft is practically negligible in this particular case of very com-

pressible surface soil material. However, it is clearly demonstrated that as the load increases, higher values of settlement are attained, and the soil under the pile cap exhibits higher resistance, rendering the raft contribution more notable. When comparing the load-settlement curves, it can be stated that the piles’ ultimate resistance remains unaffected by the pile group configuration and the ‘‘existence or not” of the pile cap, justifying the widely accepted approach that for pile groups resting on clayey soils the group efficiency factor does not significantly deviate from unity. On the contrary the contribution of the raft increases with the level of settlement and its resistance is always less than that resulting from the analysis of the unpiled raft. Both conclusions are in agreement with the findings of de Sanctis and Mandolini [17]. It must finally be noted that the bearing capacity of the overall foundation, corresponding to a settlement value of at least 10%b, increases, and therefore the allowable load, defined as 50% of the

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829

11. Load distribution to piles and the raft

Fig. 11. Vertical displacement field contours of a 3  3 pile group, L = 42.0 m, b = 1.50 m, and mean load Nm = 10 MN at cross section y = 0.0.

It is widely accepted that in the case of fixed head piles, for the same settlement, the piles within the group carry different proportions of the applied load [6,7,14,16]. More specifically, for the same settlement the central pile carries the lowest load, whereas, the external piles carry the highest. At a certain level of settlement, where the surrounding soil yields considerably and the influence of interaction vanishes, all the piles behave the same. Fig. 16 illustrates the variation of the response of the characteristic piles with the level of settlements for the 3  3 fixed head group with no cap (piles with no cap, but restricted to have the same pile head settlement), L = 38.0 m and b = 1.20 m. It can be seen that the central pile P3 carries 65% of the mean load. This proportion increases with the level of settlement and finally rises to 100% at a settlement level of 3%b, where all the piles of the group share the same mean load. At this point no effect of pile–soil–pile interaction is observed, as the surrounding soil has completely yielded. In contrast to the central pile, the corner pile P1 initially carries 115% of the mean load. The response of the perimetric pile, pile P, is less affected by the interaction. It should be mentioned that the level of settlement required to achieve an equalisation of the loads for all piles in a group depends on many factors [16]. Higher values are necessary for soils exhibiting their ultimate strength at high level strain, for very close relative spacing d (d < 2.0b) and for high values of relative depth (L/ b > 50). In case of co-existence of the above conditions the equalisation is practically unfeasible. Fig. 16 demonstrates an equalisation of axial forces carried by all characteristic piles at a relatively low level of settlements (3%). This should be mainly attributed to the existence of the clayey layers C and D simulated by the Tresca constitutive model (very stiff to hard clayey layers developing their ultimate shear strength at very low strain level). When examining the case of the 3  3 group with pile cap it is realized that the contribution of the raft alters the behaviour of the characteristic piles. Fig. 17 shows the response of these piles together with the load carried by the raft. The modes of the characteristic piles remain the same as those without the cap. Initially, when the raft contribution is almost negligible the response remains unaffected. However as the settlement level increases the proportion of the load carried by the piles decreases accordingly because of the raft’s contribution. It is worth noticing that the effect of the raft continues after soil yielding and the proportion of the load undertaken by the characteristic piles decreases at a constant rate. Fig. 18 illustrates the load proportion carried out by the characteristic piles normalized to the mean total load of the piles Np given by Eq. (9). It is understood that the modes are very similar to those of the pile group with no cap.

Np ¼

Fig. 12. Vertical displacement field contours of a 3  3 pile group, L = 42.0 m, b = 1.50 m, and mean load Nm = 19 MN at cross section y = 0.0.

group bearing capacity, increases. The contribution of the raft leads to an increase in the allowable load of at least 50% with reference to pile groups where the resistance of the raft is neglected.

total applied load  load carried by the raft number of piles

ð9Þ

It should be highlighted that the rigidity of the pile cap may significantly affect the response of the characteristic piles, particularly when the load is not uniformly applied. Eventually when a point load is applied and the cap is very flexible, the load is mainly distributed to the vicinity of the point where it is applied. To investigate this effect the 3  3 pile group, b = 1.00 m and L = 38.0 m was examined with different pile cap thickness and with no cap (the restriction of common settlement on the pile heads is always valid). A load corresponding to the allowable load (settlement level of the order of 2%b) was applied at the centre of the pile group. The results are plotted in Fig. 19, where a very high diversification can be observed when pile cap thickness is less than the pile diameter. In this case, the applied load is mainly distributed to the piles in the vicinity of the loaded region. When the pile cap rigidity increases the external piles start to carry a higher proportion of the load because

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25 22_D150_L38 33_D150_L38 22_D120_L38 33_D120_L38 22_D100_L38 33_D100_L38

Mean Axial Load Nm (MN)

20

15

10

5

0 0%

2%

4%

6%

8%

10%

Normalised Settlement Sns Fig. 13. Numerically established load-settlement curves for pile groups with tip at 38.0 m, 2  2 and 3  3 layouts and diameters of 1.00, 1.20, and 1.50 m.

25

Mean Axial Load Nm (MN)

20

22_D150_L42 33_D150_L42 22_D120_L42 33_D120_L42 22_D100_L42 33_D100_L42

15

10

5

0 0%

2%

4%

6%

8%

10%

Normalised Settlement Sns Fig. 14. Numerically established load-settlement curves for pile groups with tip at 42.0 m, 2  2 and 3  3 layouts and diameters of 1.00, 1.20, and 1.50 m.

of their position and the interaction effect. When the cap thickness is three times higher than the pile diameter the rigidity renders the cap practically a rigid body, and therefore the location and the form of the applied load does not affect the distribution to the piles of a group. In that case the pile–soil–pile interaction remains the main factor affecting the load distribution, depending on the pile group configuration and the settlement level, as precisely presented by Comodromos [14] and Comodromos and Bareka [16]. The piles of the group with no cap always present a higher proportion of load than those with a cap. The difference between bold and dash lines corresponds to the load carried by the raft. Eventually when the load carried by the characteristic piles is normalized to the sum of the load carried by the piles and not to the total load applied, the corresponding curves coincide (Fig. 20), demonstrating that the effect of the raft is the same no matter what the position of the pile in a group is. It should be noted that normalizing the raft thickness to the pile diameter might be useful for small range of pile group geometry. However, in the case of large range the effect of raft span

length should be taken into account and the use of raft-soil stiffness as defined by Eq. (10) in Horikoshi and Randolph [13]:

K rs ¼ 5:57

Er ð1  v 2s Þ ðB=LÞ0:5 ðt r =LÞ3 Es ð1  v 2r Þ

ð10Þ

where Krs: raft-soil stiffness; Er: raft modulus of elasticity (32.5 GPa); Es: soil modulus of elasticity (70.0 MPa); vs: Poisson’s ratio of soil (0.35); vr: Poisson’s ratio of the raft (0.20); L: raft length (10.6 m); B: raft breadth (10.6 m); tr: raft thickness. The results of Fig. 20 are replotted with regards to raft-soil stiffness and are given in Fig. 21. It can be observed that raft-soil stiffness

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14.0 33_D120_L38 Piles Pile Cap 3-D, Group No Cap 3-D, Raft only

Mean Axial Load Nm (MN)

12.0 10.0 8.0 6.0 4.0 2.0 0.0 0%

2%

4%

6%

8%

10%

12%

14%

Normalised Settlement Sns Fig. 15. Comparison of load-settlement response of 3  3 pile group, b = 1.20 m and L = 38.0 m, with and without cap.

3x3_D120_L38

P1 P3

120% 100% 80 % 60 % P1

40 %

P2 20 % 0% 0%

P3

P2

P3 d

P2 Raft

100%

P2

P3 d

P1

80%

60%

40%

20%

P1

0% 2%

4%

6%

8%

0%

10%

Fig. 16. Variation of normalised axial load with normalised settlement for characteristic piles P1, P2, and P3, 3  3 layout with no cap, b = 1.20 m and L = 38.0 m.

values less than 1.0 correspond to a rather flexible raft, while for values higher than 10 the raft behaves as a rigid body. The results of the 3-D analysis are also compared to those arising from the application of the simplified method proposed by Clancy and Randolph [29] for rigid rafts (piles and raft exhibit the same settlement). Within the notion of reciprocal theorem and making use of the principle of superposition, they proposed Eqs. (11)–(13) to estimate the overall stiffness of the pile–raft system.

ðPp þ Pr Þ ½kp þ kr ð1  2arp Þ ¼ wpr ½1  ðkr =kp Þa2rp 

ð11Þ

4%

8%

12%

Normalised Settlement Sns

Normalised Settlement Sns

kpr ¼

3x3_D120_L38

120%

Normalised Mean Load N/Nm (%)

Normalised Mean Load N/Nm (%)

140%

Fig. 17. Variation of normalised axial load with normalised settlement for characteristic piles P1, P2 and P3, 3  3 layout with cap, b = 1.20 m and L = 38.0 m (pile and raft load normalised to the total applied load).

Pp ¼

Pr ¼

½1  kr ðarp =kp Þwpr ð1=kp Þ  kr ðarp =kp Þ2 ½ðkr =kp Þ  kr ðarp =kp Þwpr ð1=kp Þ  kr ðarp =kp Þ2

ð12Þ

ð13Þ

in which kpr: overall stiffness of the pile–raft system; Pp: total load carried by pile group in combined foundation; Pr: total load carried by raft in combined foundation; kp: overall stiffness (Pp/wp) of pile group in isolation; kr: overall stiffness (Pr/wr) of raft in isolation; wpr: overall pile–raft settlement (for rigid raft wpr = wp = wr); arp: interaction factor of pile group on raft.

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3x3_D120_L38

Normalised Mean Load N/Np (%)

120% 100%

P1 P2 P3 Raft

80% 60% 40% P2

P3

20% 0% 0%

d P1 2%

4%

6%

8%

10%

12%

Normalised Settlement Sns Fig. 18. Variation of normalised axial load with normalised settlement for characteristic piles P1, P2, and P3, 3  3 layout with cap, b = 1.20 m and L = 38.0 m (pile and raft load normalised to the load carried by the piles).

and substituted to Eq. (11) can lead to kpr and wpr. Eqs. (10) and (13) can then be used to calculate Pp and Pr. The aforementioned process has been initially applied for the mean applied load of 5 MN (Nm = 5 MN, total load 45 MN), which is approximately the upper limit of linear-elastic behaviour of the combined pile–raft system. Table 2 summarizes the results of both the 3-D and the simplified analysis. It can be observed that the results estimated by the simplified method are in a very good agreement with those of the full 3-D analysis. Despite the fact that the basic assumptions of the method (reciprocal theorem and the principle of superposition) limit its applicability to linear-elastic behaviour an attempt has been made to examine the prediction error when nonlinear effects are introduced. Within this framework two further cases were examined. The first corresponds to a total applied load of 90 MN (mean load Nm = 10 MN in Fig. 15) and the second to 120 MN (mean load Nm = 13.33 MN). From the results summarized in Table 3, it can be seen that the prediction just after yield of pile resistance is quite satisfactory. On the contrary, the results given in Table 4 verify that the applicability of the method is not recommended when the effects of nonlinearities become significant. 12. Use of the results to the design process

As it can be seen in Fig. 15, the 3  3 pile group layout with pile diameter of 1.20 m and a pile length of 38.0 m has been solved separately (pile group with no cap), the raft has been also solved in isolation (raft only) and a combined analysis has also been carried out (33_D120_L38). The contribution of the piles in that combined analysis is given by line with diamond markers (Piles), while the contribution of the raft by the line with square markers (Pile Cap). Hence the values of kpr, Pp, Pr, kp, and kr can be calculated, while the interaction factor arp can also be calculated from the following equation:

arp ¼

  kp Pr wpr  Pp kr

ð14Þ

The effect of the pile–raft interaction can also be estimated using the above simplified method and the values arising for Pp, Pr, and wpr can be compared. More specifically, the interaction factor arp can be estimated from charts given by Clancy and Randolph [29]

Despite the development and continuous advances in numerical methods and powerful computer manufacturing, a 3-D nonlinear analysis remains computationally demanding in both geotechnical modeling and CPU-time. For this reason a straightforward analysis of a superstructure, based on a pile foundation, cannot practically be applied, when soil nonlinearity and effects from pile group response cannot be ignored. However, hybrid methods or even the fine notion of the sub-structuring technique can facilitate this goal. More specifically, for the design of superstructures, such as bridge projects, a performance-based design approach is adopted, including many load case combinations together with partial factors of safety. From that approach various load envelopes arise and therefore every pier foundation has to be solved for various loads, rendering the analysis extremely demanding, computationally, if a 3-D nonlinear analysis is to be carried out. An alternative approach can be applied to facilitate the calculation process. Assuming that a

180%

Normalised axial load N/Nm (%)

160%

P3

P2

d

140%

P1

120% 100%

P2

80%

P1

d

60% 40% 20% 0.0

P3

1.0

P1, no raft contrib.

P1

P2, no raft contrib.

P2

P3, no raft contrib.

P3

2.0 3.0 Pile cap thickness normalised to pile diameter

4.0

5.0

Fig. 19. Variation of normalised pile axial load with cap thickness for characteristic piles P1, P2, and P3, 3  3 layout with cap, b = 1.00 m and L = 38.0 m for the allowable load (pile load normalised to the total applied load).

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180 %

Normalised axial load, Nm (%)

160 %

P3

P2

d

140 %

P1

120 % 100 % 80 % 60 % 40 % 20 %

0.0

P1, no raft contrib.

P1

P2, no raft contrib.

P2

P3, no raft contrib.

P3

1.0

2.0

3.0

4.0

5.0

Pile cap thickness normalised to pile diameter Fig. 20. Variation of normalised pile axial load with cap thickness for characteristic piles P1, P2, and P3, 3  3 layout with cap, b = 1.00 m and L = 38.0 m for the allowable load (pile load normalised to load carried by the piles).

200% P1, no raft contrib.

P1

P2, no raft contrib.

P2

P3, no raft contrib.

P3

P3

P2

160%

d P1

140% 120% 100% 80%

20%

0.0

2.00

1.50

40%

1.00

60%

0.50 0.75

Normalised axial load, Nm (%)

180%

5.0

10.0

15.0 Raft-soil stiffness Krs

20.0

25.0

Fig. 21. Variation of normalised pile axial load with raft-soil thickness Krs for characteristic piles P1, P2, and P3, 3  3 layout with cap, b = 1.00 m and L = 38.0 m for the allowable load (pile load normalised to load carried by the piles).

Table 2 Pile–raft interaction: comparison of 3-D analysis with the simplified method by Clancy and Randolph [29]; pile group 3  3, L = 38.0 m total load 45 MN.

arp wrp (mm) Pp (MN) Pr (MN)

Full piled raft 3-D nonlinear analysis

Clancy and Randolph [29]

Prediction error (%)

0.58 14.0 41.16 3.84

0.65 13.9 41.83 3.17

12 0.7 1.62 17

3-D nonlinear analysis has been carried out and the response of the piles and the raft is established, the stiffness of the pile head and

the raft subjected to vertical loading equal to the allowable load can be defined as a linear spring. Then the pier can be solved using linear-elasticity and simulating the cap with plate or shell elements and the piles with springs. All loading cases can be incorporated in such an analysis and the envelope of stresses, moments, and reinforcements can then be provided. Table 5 summarizes the spring values simulating the response of piles and soil under the raft as defined within the process of parametric analysis. They can then be used in a simplified numerical analysis, provided that the applied load always remains less than the level of the allowable load. Next to the absolute spring value, the ratio of the spring stiffnesses of the piles in a piled raft to

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Table 3 Pile–raft interaction: comparison of 3-D analysis with the simplified method by Clancy and Randolph [29]; pile group 3  3, L = 38.0 m total load 90 MN. Full piled raft 3-D nonlinear analysis

arp wrp (mm) Pp (MN) Pr (MN)

0.48 50.5 72.46 17.54

Clancy and Randolph [29] 0.65 58.6 73.04 16.96

Table 5 Spring values simulating pile and soil response under vertical loading. Pile layout

Prediction error (%) 35 16 0.8 3.3

3x3 P2

Pile length (nm)

Pile diameter

Spring stiffness P1

P2

P3

Soil

38.0

1.00

312 (64%) 382 (45%) 432 (40%) 342 (53%) 406 (44%) 476 (43%)

259 (53%) 316 (37%) 367 (34%) 291 (45%) 343 (37%) 417 (38%)

206 (42%) 264 (31%) 324 (30%) 246 (38%) 292 (31%) 363 (33%)

2.94

1.20

P3

1.50

d

P1

42.0

1.20

Table 4 Pile–raft interaction: comparison of 3-D analysis with the simplified method by Clancy and Randolph [29]; pile group 3  3, L = 38.0 m total load 120 MN.

arp wrp (mm) Pp (MN) Pr (MN)

Full piled raft 3-D nonlinear analysis

Clancy and Randolph [29]

Prediction error (%)

0.46 156.1 79.2 40.8

0.65 197.3 63.3 56.7

41 26.3 19.7 39.0

1.50

2x2

38.0

13. Summary and conclusions A methodology allowing an efficient and economical design of pile foundations, taking into account the contribution of the raft, has been presented. The combination of experimental data with

1.00 1.20

d

1.50

P1

42.0

the spring stiffness of a single in the same subsoil and the same applied mean load is given in parenthesis. An example of a simplified application is given a little further in the text, corresponding to a 2  2 pile group, b = 1.50 m and L = 38.0 m. The applied load of 16.5 MN corresponds to that of the outermost pier of the bridge and is transferred to the pile cap through two circular columns located between the piles, Fig. 22. The Finite Element code Sofistik [30] was used and the solution was achieved in less than a minute of CPU-time. Figs. 22 and 23 illustrate the isovalues of settlements and bending moments resulting from the above-mentioned analysis. Apparently, the combination of the aforementioned approach with the results of a 3-D nonlinear analysis, providing the precise response of characteristic piles in a group, is quiet efficient for an appropriate design of a pile foundation, in which the benefits from the contribution of the raft are taken into account.

1.00

1.00 1.20 1.50

397 (81%) 440 (51%) 463 (43%) 422 (65%) 500 (54%) 514 (47%)

2.37 1.90 3.15 2.56 1.94

1.77 1.37 1.00 1.81 1.22 0.98

Pile spring stiffness is given in MN/m, while soil spring in MN/m3.

nonlinear 3-D analysis, within the notion of sub-structuring, allowed for a precise design with reference to capacity based design methods. In addition, the proposed strategy accounts for soil and structural nonlinearities as well as for pile–soil–pile interaction, with no excessive numerical complexities and CPU-time demands, which are required in the case of a straightforward nonlinear 3-D analysis. The method is based on both experimental data and nonlinear 3-D analysis, which can be used within the notion of substructuring. For the particular soil profile it was found that results arising from the application of regulations and codes are overconservative. Appropriate design values for soil shear strength, deformation modulus, and information regarding shear strength mobilization at the soil–pile interface were determined by backfiguring from the results of the pile load test.

Fig. 22. Vertical displacement contours of a 2  2 pile cap under the load of the outmost pier of the bridge.

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Fig. 23. Bending moment contours of a 2  2 pile cap under the load of the outmost pier of the bridge.

Subsequently, the response of pile groups was numerically established and the distribution of the applied load to the raft and the characteristic piles was calculated. According to the results of the numerical analysis, the existence of the raft does not affect the pile head vertical stiffness, for loadings less than the allowable load. This should be attributed to the fact that for low level settlements, the contribution of the raft is practically negligible in this particular case of very compressible surface soil material. On the contrary, the ultimate resistance of the group (combined resistance of piles and raft) significantly increases, leading to higher values of allowable load. This has a significantly beneficial effect on the design of pile raft foundations. The mode of distribution of the applied load to the characteristic piles of a group with and without cap remains almost the same with two main differences. The piles of the group with no cap always present higher proportion of load than those with a cap. Contrary to fixed head groups with no cap, where, from a certain level of loading, all piles behave the same carrying 100% of the mean load, in the case of pile raft foundations the proportion is always less than 100% and is decreasing as the applied load increases. In addition, interesting conclusions are also drawn regarding the effect of the thickness of the pile cap in the case of non-uniform vertical loading. When the pile cap thickness is less than the diameter of the piles, the applied load is mainly distributed to the piles in the vicinity of the loaded region. For conventional pile spacings of five pile diameters or less, when the cap thickness is higher than three times the pile diameter, the rigidity renders the cap practically a rigid body and therefore the location and the form of the applied load does not affect the distribution to the piles of a group. In that case the pile–soil–pile interaction remains the main factor affecting the load distribution, depending on the pile group configuration and the settlement level. Acknowledgment The permission of Egnatia Odos S.A. to publish this article is gratefully acknowledged. References [1] Bathe KJ, Wilson EL. Numerical methods in finite element analysis. New Jersey: Prentice-Hall; 1976.

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