ABSTRACT: Piles are normally subjected to axial loads. However, in many cases lateral loads have also to be transferred in addition to axial loads. Till now, it is not clear how important the interaction effects are, dependent on various factors such as pile stiffness, soil type and loading conditions. In this paper, results of numerical simulation of behavior of short and nearly rigid piles as well as long, flexible piles embedded in sandy soil under inclined loads are presented. For the sandy soil an elasto-plastic constitutive material law has been used. The results are shown in terms of load-deformation curves for the lateral and the axial loading portions. It is shown that for inclined compression loading the behavior under lateral load is nearly independent of a vertical load, whereas the behavior under vertical load is strongly influenced in particular for short piles. For inclined tension loading of short piles also the horizontal load deformation curve is dependent on the vertical load portion, because it significantly affects the ultimate vertical load. Interaction diagrams are presented, which characterize the pile behavior under inclined loads.

1 INTRODUCTION

In different foundation systems, especially for offshore structures, vertical or nearly vertical driven piles are often used to found structures, e.g. platforms. Normally the axial loading of the pile is predominant. But in special cases, for instance for piles supporting offshore wind energy foundations or conductors, the axial (mostly vertical) load is accompanied by a lateral (horizontal) load.

Current design practice involves separate analysis of the axial and lateral responses of piles and does not consider the effect of interaction between the different load directions. Several results of investigations on the behavior of piles subjected to inclined loading have been reported in the literature, e. g. by Yoshimi (1964), Das et al. (1976), Chari & Meyerhof (1983), Ismael (1989), Sastry & Meyerhof (1990), Shahrour & Meimon (1991), Meyerhof (1995), Amde et al. (1997) and Abdel-Rahman & Achmus (2006).

These investigations indicate that the pile response to horizontal loading is only slightly affected by a vertical load, whereas horizontal loads significantly affect the vertical pile response. However, there has as yet been no clear and comprehensible presentation of important parameters and a quantification of interaction effects.

In this paper, the results of a numerical study on the behavior of vertical piles embedded in sand under variable inclined compression and tension loads are presented.

BEHAVIOR OF PILES IN SAND SUBJECTED TO INCLINED LOADS

Martin Achmus, Khalid Abdel-Rahman & Klaus Thieken Institute of Soil Mechanics, Foundation Engineering and Waterpower Engineering, Leibniz University of Hannover, Germany

2 FINITE ELEMENT MODELING

For the investigation of the behavior of vertical piles subjected to combined loading a three-dimensional (3-D) numerical model was developed. The finite element program ABAQUS (Abaqus 2008) was used.

Solid concrete piles with a diameter of 2m were considered. Two different embedded lengths L= 10m (Case 1) and L = 30m (Case 2) were chosen for studying the effect of pile stiffness.

The most important issue in geotechnical numerical modeling is the simulation of the soil’s stress-strain behavior. An elasto-plastic material law with Mohr-Coulomb failure criterion was used to describe the behavior of medium dense fine sand. The soil stiffness is represented here by a stiffness modulus for oedometric compression ES and a Poisson’s ratio ν. To account for the non-linear soil behavior, a stress-dependency of the stiffness modulus was implemented as follows:

λ

σσσκ ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

at

matSE (1)

Here σat = 100 kN/m2 is a reference (atmospheric) stress and σm is the current mean principal stress in the considered soil element. The parameter κ determines the soil stiffness at the reference stress state, and the parameter λ governs the stress dependency of the soil stiffness.

The material parameters used in the calculations are given in Table 1.

Table 1. Material parameters used for medium dense sand

Unit weight γ 19.0 kN/m3 Oedometric stiffness parameter κ 250 Oedometric stiffness parameter λ 0.70 Poisson’s ratio ν 0.25 Internal friction angle ϕ’ 35.0° Dilation angle ψ 2.5° Cohesion c’ 1.0 kN/m2

A linear elastic material behavior of the piles was assumed with the parameters E = 3.0⋅104

MN/m2 (Young’s modulus) and ν = 0.20 (Poisson’s ratio) for concrete. A classification of the stiffness of the piles can be obtained by the ratio of the elastic length

Le and the actual embedded length L. The elastic length relates the pile and the soil stiffness and can be derived from models in which the soil reaction is idealized by elastic springs (subgrade reaction models). For sandy soils a linear increase of the spring stiffness with depth z under the soil surface is a common approach: Ks = kr z, with kr in MN/m3. With this approach, the elastic length is defined as follows:

5

r

ppe k

I.EL = (2)

Here Ep Ip is the bending stiffness of the pile. For medium dense sand, a typical value for kr is kr = 15 MN/m3 (see e.g. API 2000). Using this value, the L/Le-values given in Table 2 apply to the two cases considered. With these values the pile in Case 1 can be classified as short and relatively rigid and the pile in Case 2 as long and flexible. This means that for Case 2 the bending deformation of the pile prevails, whereas for Case 1 both bending and “rigid body” rotation dominate the behavior of the pile (Titze 1970).

Table 2. Pile dimensions and stiffnesses

Case Diameter (D) Length (L) Elastic Length (Le) L/Le Class 1 2.0 m 10.0 m 4.36 m 2.30 Nearly rigid (short) 2 2.0 m 30.0 m 4.36 m 6.88 Flexible (long)

Due to the symmetric loading only a half-cylinder representing the sub-soil and the pile was considered. 8-node continuum elements were used for the soil and the pile. The diameter of the discretized model area was 30 times that of the pile diameter. The bottom edge of the model was extended by 15 times the pile diameter below the base of the pile. With these model dimensions the calculated behavior of the pile is not influenced by the limits (Fig. 1).

Fig. 1. Finite element mesh

For the contact behavior of the surface between pile and soil an elasto-plastic model was used. The maximum frictional shear stress τfric,max is dependent on the normal stress σn and a coefficient of friction μ. In the numerical simulations presented here μ = 0.5 was assumed. In order to describe the frictional behavior between the pile and the surrounding soil realistically, the maximum frictional shear stress was limited to a value of 80 kN/m2, which is approximately the value for medium dense sand recommended in API (2000). The following equation holds for the maximum frictional shear stress:

τfric,max = μ σn ≤ τlimit = 80 kN/m2.

For full mobilization of the limit frictional stress the relative displacement (elastic slip) between the pile and the surrounding soil was set to Δuel,slip = 0.005 m. The stiffness of the contact (kτ) before reaching the maximum frictional stress is thus defined (Fig. 2).

Fig. 2. Modeling of contact behavior

The finite element calculation is carried out in steps. First, for the generation of the initial stress state the whole model area is discretized using soil elements only. Subsequently, the pile installation process is modeled by replacing the soil elements located at the pile position by pile elements and activating the contact conditions between the pile and the soil.

Finally, the vertical load and the horizontal load are applied simultaneously and increased gradually until the required maximum loads are reached.

The piles were subjected to variously inclined loads (α = 0.0, ±30.0°, ±60.0, ±90.0°), with α measured from the horizontal direction, whereby the positive sign of α stands for compression loads and the negative sign for tension load (Fig. 3).

Fig. 3. System and denominations

3 RESULTS OF NUMERICAL MODELING

3.1 General remarks In order to investigate the influence of combined axial and lateral loading, the load inclination α was varied from -90° (pure axial tension) to 0° (pure horizontal loading) to 90° (pure axial compression). In this study, the vertical (u) and horizontal (w) pile head displacements are calculated and presented with the vertical (V) and horizontal (H) loading components. A comparison of the H-w and V-u curves (cf. to Fig. 3) clearly shows the effect of the vertical load on the horizontal load-deformation behavior and vice versa.

In order to identify the range of the loads, the vertical pile capacities were calculated with the usual design approach for driven offshore piles of the API regulation (API 2000). The design values recommended for medium dense sand were used. The results are shown in Table 3. In the numerical modeling, the ultimate loads were defined at deformations (vertical or horizontal) of 20 cm (i.e. 10% of the pile diameter). This means that a simulation was concluded if either the vertical or the horizontal displacement of the pile head reached 20 cm.

Table 3. Ultimate loads derived from the analytical approach in API (2000)

Pile dimension

Vertical pile capacity (Compression)

Vertical pile capacity (Tension)

D=2.0m, L=10.0m

14.17 MN 2.23 MN

D=2.0m, L=30.0m

27.32 MN 12.24 MN

3.2 Results for piles under inclined compression loading The horizontal load deformation behavior of the pile with a diameter of 2m and an embedded length of 10m is shown in Fig. 4. These results show the horizontal displacement (w) to be almost independent of the load inclination angle and thus independent of a vertical load component acting together with the horizontal load. Similar results were found in experiments by Sastry & Meyerhof (1990) and also in a numerical study of the behavior of steel pipe piles under inclined loading by two of the authors (Abdel-Rahman & Achmus 2006).

Fig. 4. Horizontal displacement at the pile top dependent on horizontal load

(inclined compression, D=2m, L=10m)

Since the same result is obtained for the long pile with an embedded length of 30m (Fig. 5), this finding seems to be independent of the pile length and the relative stiffness. For the long pile, the differences in the load-deformation curves are even smaller than for the short pile.

Fig. 5. Horizontal displacement at the pile top dependent on horizontal load

(inclined compression, D=2m, L=30m)

In contrast, for the short pile the vertical displacement (settlement) of the pile is strongly

affected by a horizontal loading component. In Fig. 6 the vertical settlement and the vertical load for variously inclined loads is given for the short pile (D=2m, L=10m). Before reaching a certain “critical” load level, the horizontal load has a favourable effect, since it leads to a stiffer behavior in the vertical direction. However, exceeding this load level the behavior of the pile tends to be less rigid and the vertical settlement increases with the load inclination angle compared to the pure vertical loading case. The magnitude of the stiffness increase before reaching the critical load level and the critical load level itself strongly depends on the load inclination angle (α).

In Fig. 7 the vertical load-settlement curve for the long pile with an embedded length of 30m is given. Tendentially, a stiffness increase before reaching a critical load level and a decrease after exceeding this load level can also be observed due to a horizontal load acting simultaneously. However, these effects are much less pronounced than in the case of a short pile (Fig. 6).

It should be noted that the curves for α less than 90° in Figs. 6 and 7 stop at certain points due to the fact that at that stage the corresponding horizontal deformation reaches the limit displacement of 20cm.

The reason for the favourable effect of a horizontal load in the case of inclined compression has already been discussed by Abdel-Rahman & Achmus (2006). Due to the large horizontal bedding stresses induced by the horizontal load particularly in the upper part of the pile the mobilized skin friction is increased. It seems obvious that this effect is the more pronounced the shorter the pile is. However, if the horizontal load becomes large and approaches the ultimate horizontal load, further inclined loading causes large horizontal and vertical deformations, which leads to a stiffness decrease and to a reduction of the ultimate vertical load.

Fig. 6. Vertical settlement of the pile top dependent on vertical load (inclined compression, D=2.0m, L=10.0m)

Fig. 7. Vertical settlement of the pile top dependent on vertical load (inclined compression, D=2.0m, L=30.0m)

3.3 Results for piles under inclined tension loading In Fig. 8 the horizontal load-displacement behavior under variable tensile inclined loads is presented for the pile with an embedded length of 10m. At first, for relatively small loads, there is no significant influence of the vertical load on the H-w curve. But, starting from a certain load level which is dependent on the load inclination, the curves for inclined loads deviate from the curve for pure horizontal loading. A vertical load acting simultaneously significantly decreases the horizontal stiffness of the pile-soil system.

The reason for this deviation in the H-w curves is that a strong vertical heave causes an upwardly directed force acting on the passive soil wedge induced by the horizontal load, and so to a decrease in the ultimate horizontal load. When the vertical heave is smaller than the heave of the soil wedge, a downwardly directed force results, which means that the passive wall friction angle is negative and a larger ultimate passive pressure applies.

For the case of the long pile only a very small stiffness decrease in a horizontal direction due to a vertical load is obtained (Fig. 9). The reason for this is that the vertical loads corresponding to the maximum horizontal deformation of 20cm are much smaller than the ultimate tension load and so the vertical deformations are too small to change the direction of the resultant passive wall friction angle considerably. Thus, the ultimate horizontal load remains almost unchanged.

Fig. 8. Horizontal displacement at the pile top dependent on horizontal load (inclined tension, D=2m, L=10m)

Fig. 9. Horizontal displacement at the pile top dependent on horizontal load (inclined tension, D=2m, L=30m)

The corresponding vertical load-displacement curves for the cases considered are shown in

Figs. 10 and 11. Here again, as for the inclined compression load, a significant influence of the horizontal load is found. The vertical pile stiffness is distinctly reduced when compared to the case with pure axial tension. But, on the other hand, a horizontal load component up to a certain loading angle (α) increases the ultimate vertical pile capacity. For the short pile (D=2m, L=10m), the pile capacity is increased up to an inclination angle of -45°, after which there is no increase in the pile capacity.

The observed behavior can be explained by the interaction of two effects. A horizontal load induces bedding stresses around the pile, but also a soil heave around the upper part of the pile due to the formation of a passive soil wedge. Due to the soil heave, negative skin friction on the pile arises, which leads to additional loading and thus a stiffness decrease in vertical direction. However, if the vertical deformations become relatively large, i.e. the ultimate vertical load is approached, large pile heave occurs and the negative skin friction is reduced and can finally be turned into positive skin friction. Thus, with large deformations, even an increase in the ultimate tension load compared with the purely axial loaded pile can occur. An increase is observed only if the vertical load portion is above a certain limit, i.e. the load inclination angle exceeds a certain limit. This limit differs for the short and the long pile and is thus dependent on the pile-soil system.

Fig. 10. Vertical heave at the pile top dependent on horizontal load (inclined tension, D=2m, L=10m)

Fig. 11. Vertical heave at the pile top dependent on horizontal load (inclined tension, D=2m, L=30m)

3.4 Interaction Diagrams To characterize the mutual effects of horizontal and vertical loads on the pile behavior, the concept of interaction diagrams is applied. Such diagrams represent the deformations of a pile under all possible combinations of horizontal and vertical loads. The interaction diagrams derived from the results of this study are shown in Fig. 12 for the short pile and in Fig. 13 for the long pile. To determine the curves with sufficient accuracy, several numerical simulations for inclined loads from -90° to 90° in 2.5° to 10° steps were carried out.

To give a more general representation, normalized vertical and horizontal loads are depicted (V/Vult and H/Hult). The ultimate loads were derived from the numerical results for purely vertical and purely horizontal loading. For compression and horizontal load, the ultimate state was defined at vertical and horizontal displacements of 10% of the pile diameter, i.e. 20 cm. With this approach, the ultimate loads for pure compression, pure tension and pure horizontal loading given in Table 4 were obtained.

Table 4. Ultimate loads derived from the numerical results

Pile dimension

Vertical pile capacity (Compression, Vc,ult)

Vertical pile capacity (Tension, Vt,ult)

Horizontal pile capacity (Hult)

D=2.0m, L=10.0m

9.96 MN 1.89 MN 3.11 MN

D=2.0m, L=30.0m

27.61 MN 11.80 MN 7.23 MN

The mainly horizontal lines in Figures 12 and 13 connect the points with the same vertical pile displacement, whereas the mainly vertical ones connect the points with the same horizontal displacement. From these interaction diagrams, the horizontal and vertical pile displacement and the resultant displacement of the pile can be obtained for any loading combination. The upper halves of the diagrams present the compression loading case and the lower ones the tension loading case.

Fig. 12. Interaction diagram for the short pile (D=2m, L=10m)

Fig. 13. Interaction diagram for the long pile (D=2m, L=30m)

The interaction diagrams give a good insight into the effect of interaction for the respective

piles. For the long and relatively flexible pile, the lines of constant deformations are only slightly inclined, especially in the range of admissible vertical loads (i.e. V/Vult ≤ 0.5 to 0.6). This means that in most cases H-V interaction might be of minor importance. However, for the short and relatively stiff pile, the lines are significantly inclined also in the range of admissible loads. Thus, for such piles the consideration of interaction could be of significance.

4 CONCLUSIONS

Regarding the behavior of piles under inclined compression loads, the horizontal pile stiffness is almost unaffected by the vertical compression load acting simultaneously. The design of piles under horizontal loading using standard methods (e.g. p-y method) without taking into account vertical loads therefore seems generally justified. Regarding the compression pile stiffness in the vertical direction, first an increase in pile stiffness is induced by the horizontal loading. After exceeding a certain loading level, the behavior becomes less rigid and the pile stiffness decreases compared to the case without horizontal loading. This stiffness decrease is more pronounced for short piles.

Different results were obtained for combined axial tension and horizontal loading. Under simultaneously horizontal and vertical tension load the horizontal stiffness of the short pile-soil system decreases significantly. For long piles this effect is comparably small even under larger horizontal loading. The vertical pile stiffness in both cases (short and long pile) is reduced by the horizontal force, which should be taken into account in the pile design procedure under working loads. Under larger vertical displacement a minor increase in the vertical pile capacity (Vt,ult) was obtained.

The main features of the behavior of a pile under combined loading can be depicted in an interaction diagram. From the diagrams presented for a short and a long pile, the load combinations for which interaction significantly affects the pile behavior can be clearly identified.

As part of an ongoing research project, interaction diagrams for various pile geometries and soil conditions will be determined in order to elaborate general statements regarding the importance of interaction effects dependent on the pile-soil system features.

5 ACKNOWLEDGEMENT

The results presented in this paper were obtained in a research project entitled “Piles under combined loading” funded by the German Research Fund (DFG), Germany. The support is gratefully acknowledged. REFERENCES Abaqus 2008. User’s Manual, Version 6.8. Abdel-Rahman, K. & Achmus , M. (2006). Numerical modeling of combined axial and lateral

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