On the Dynamics of Inclined Piles Amalia Giannakou, National Technical University of Athens, Greece Nikos Gerolymos, National Technical University of Athens, Greece George Gazetas, National Technical University of Athens, Greece Abstract This paper utilises -D finite-element modeling to study the static and dynamic harmonic response of inclined piles, in a homogenous and a non-homogenous soil, modeled as a linear hysteretic medium. The results of a comprehensive parameter study are presented in the form of dimensionless plots of: static and dynamic stiffnesses and damping ratios, and distribution of internal forces (bending moment, shear and axial force) with depth. These plots can be readily utilised in practical applications and they offer insight on the role of the inclination in the response of flexible piles. Introduction The dynamic behavior of single vertical piles has been extensively studied by several researchers. Analytical solutions to the problem have been derived in the form of: linear and non-linear Winkler-type formulations, which treat the soil surrounding the pile as a series of distributed springs and dashpots (Reese and Matlock, 9, Mylonakis and Gazetas, 999, Gerolymos and Gazetas, ). Although this assumption ignores the shear transfer between soil layers, it has proven to be effective in analysing the static and dynamic lateral pile response, semi-analytical type formulations, which utilize Mindlin s (9) closed form solution (Poulos and Davis 9, Banerjee and Davies, 97), and finite-element formulations (Angelides and Roesset, 9, Desai and Kuppusamy, 9). However, little attention has been drawn to the behavior of inclined (also called raked or batter) piles and only few studies can be found in the literature (Poulos and Davis 9, Poulos, 999) on their response to lateral and vertical loading. Raked piles find frequent use in foundations when lateral resistance is required to transmit horizontal loads, such as those arising from earth or water pressures. For years, the seismic behaviour of batter piles has been considered detrimental, and many codes require that such piles be avoided (French Seismic Code AFPS 9, Eurocode EC / Part ). The main arguments against the use of inclined piles are: the development of parasitic bending stresses on the pile due to soil settlement, the development of large forces onto the pile cap (failure of pile- pile cap connection), and the permanent rotation in asymmetric pile group configurations. However, in recent years evidence has been accumulating that inclined piles may, at least in certain cases if properly designed, be beneficial rather than detrimental both for the structure they support and the piles themselves (Gazetas and Mylonakis, 99). In this paper D finite element analyses are performed to study the behavior of inclined piles under dynamic head loading. Such loading arises from the inertia forces in the superstructure. A single pile is studied in: a homogenous soil deposit, and a nonhomogenous soil deposit, where the Young s modulus varies linearly with depth. The effects of the most important parameters of the problem are analysed and non-dimensional diagrams are derived for the dynamic response of raked piles. Problem Definition Figure shows the analyzed pile configuration and soil profile. A single floating inclined pile of Young s modulus E p, diameter d, and moment of inertia I p, with two different rake angles o and o, is analysed. For comparison, a single vertical pile is also analysed. The results of the dynamic analyses of the single vertical pile compare well with previously published results (Gazetas, 9, Mylonakis and Gazetas, 999, Makris and Gazetas 99). The soil Young s modulus E s is assumed to vary with depth according to the following equation: a z E = E s o () d where E o is the Young s modulus at depth z = d. Notice that a = leads to the homogenous soil deposit, while a = describes a soil where E s varies linearly with depth. In most cases of practical interest a takes values in between.
L L u o sinωt u o sinωt θ d d E s E s u o sinωt Figure Pile configuration and soil Young s modulus profiles The homogenous soil profile is characteristic of stiff, overconsolidated clays, while nonhomogenous profiles with E s linearly increasing with depth are a reasonable assumption for normally consolidated clays (Velez et al., 9), where the undrained Young s modulus, E u, is proportional to the undrained shear strength, S u, which is proportional to the effective stress, σ ν, which in turn is a linear function of depth. Finite Element Model Description The problem is analysed in -D, making use of the Finite Element (FE) method. Both the pile and the soil are assumed to be linear elastic materials. Figure depicts the finite element discretization for the case of a vertical pile, and for an inclined pile. Soil and pile are modeled with (-node) brick elements. Constant dashpots are placed at the side boundaries of the model in the two horizontal directions, to absorb the waves emanating from the vibrating pile. Dashpot constants perpendicular to the direction of loading are c s = ρ s V s A; parallel to the direction L L u o sinωt θ d d E s E s = E o z/d E s E s = E o z/d Figure Finite element discretizetion for the vertical and inclined pile of loading, c p = ρ s V p A, where ρ is the soil density, V s the shear wave velocity of the soil, V p the P wave velocity, and A the area that corresponds to the dashpot. The bottom boundary of the model is fixed, modeling rigid bedrock. Rayleigh damping is used to model material damping, taken equal to % for the range of frequencies between the eigenfrequency of the soil deposit and the frequency of the harmonic excitation. It is noted that the pile is also modeled with D solid elements, and not with beam elements, as the latter cannot correctly enforce compatibility at the soil-pile interface. Parametric Results We analyse the steady state response of a single fixed head pile to lateral dynamic oscillation, i.e. only inertial loading on the pile head is addressed. Our scope is the determination of the dynamic impedances, K, K RR, and K HR, associated with swaying, rocking, and coupled swaying-rocking oscillations, which are defined as the ratio between the magnitudes of excitation and the resulting steady state displacement or rotation at the pile head. For example, the swaying impedance is defined as: P exp( iωt) o K = () u exp[( iωt + φ )] o where P exp( iω t) is the exciting force, and o u exp[( iωt + φ )] is the resulting steady state o displacement of the pile head. It is noted that whereas in the case of a vertical pile there is only one coupled term, corresponding to the cross swaying-rocking oscillation, K HR, in case of an inclined pile there are three coupled terms: K, K HR HV, and K RV. The two extra, K HV, and K, correspond to the coupling of the RV horizontal with the vertical mode of vibration and the rocking with the vertical mode, respectively. This means that K in the case of an inclined pile is the ratio of the amplitude of the exciting force to the resulting displacement, when no rotation and no vertical displacement is allowed at the pile head ( fixed-head inclined pile). It is more convenient to express the dynamic impedance of the pile as an equivalent spring and dashpot:
K = K + iωc () where K is the stiffness of the spring, and C is the dashpot coefficient. Both vary with frequency. The dashpot has two components, one expressing the material (hysteretic) damping and one corresponding to the radiation (geometric) damping. The dimensionless coefficient D, which can be considered to be the effective damping ratio, is defined as: D = ωc / K () More details on the dynamic impedances of vertical piles can be found in Gazetas and Dobry, (9), and Velez et al. (9). In this paper, results are presented in terms of the three normalized stiffnesses: K /E s d, K RR /E s d, and K HR /E s d = f (a, E p /E s, L/d, a ; θ), and the corresponding effective damping ratios D, D RR, and D HR, where E p /E s is the pile-soil stiffness ratio, L/d is the slenderness ratio, a = ωd/v s the dimensionless ratio. For θ = ο (vertical pile) results have been derived by Dobry et al., (9), Gazetas and Dobry, (9), and Velez et al. (9). The ranges of these dimensionless parameters are as follows: θ = o - o, E p /E s -, L/d 7., a =... The case of E p /E s = and L/d = corresponds to a flexible pile, for which L a < L, where L a is the active pile length defined as the length beyond which pile deformations become negligible. The cases with L/d =7. correspond to a short pile (i.e. L a > L) regardless of E p /E s. Static Stiffnesses Figure depicts the variation of the static stiffnesses with E p /E s, with respect to θ, for the homogenous soil profile and L/d =. Gazetas (9) derived analytical expressions for the static stifnesses of flexible piles embedded in three different idealized soil profiles. In the case of homogenous soil, the swaying stiffness is:. K.E d s () E S We extend the above equation to take into account the inclination angle, θ:.( + tan θ ) K. E d( + tan θ ) s () Es Figure shows that the results of Eq. are indistinguishable from the FE results. Rocking and cross swaying rocking stiffnesses were found to be insensitive to θ, as shown in Figure. Therefore, the expressions available from K HR d K RR d K d θ = θ = θ = E p Figure Normalized static stiffnesses swaying, rocking, and cross swaying - rocking for batter angle θ =,, and as a function of pile-soil stiffness ratio E p /E s (L/d =, homogenous soil) K / E d s s 9 Gazetas (9) θ = Eq., Eq., FE results, FE results, θ = E p Figure Comparison of the normalized swaying static stiffness calculated with the simplified Eq. () and the FE analysis (L/d =, homogenous soil)
Gazetas (9) for vertical piles (Eq. 7 and ) can also be used in the case of raked piles..7 K.d E RR s (7) E s K.d E HR s () E s Figure portrays the dependence of the normalized static stiffnesses on L/d for a pile with a batter angle of o. The differences are insignificant. For the non-homogenous soil profile, the variation of the static stiffnesses with the inclination angle θ is depicted in Figure. In this case, the swaying stiffness becomes more sensitive to the increase of E p /E s, with the increase of θ, something that was not observed in the homogenous soil. K HR d K RR d K d 9. L / d = 7. L / d = L / d = 7. L / d = L / d = 7. L / d = E p Figure Normalized static stifnesses swaying, rocking, and cross swaying rocking for slenderness ratio L/d = 7. and, (θ = ο, homogenous soil) K HR d K RR d K d 7 9 θ = θ = θ = E p Figure Normalized static stiffnesses swaying, rocking, and cross swaying - rocking for batter angle θ =,, and as a function of pile-soil stiffness ratio E p (L/d =, non-homogenous soil) For the non-homogenous soil, the swaying stiffness for a vertical pile is (after Gazetas, 9):. K.dE o (9) E o where E o is the Young s modulus taken at depth equal to one pile diameter. Again, we extend Eq. (9) to take into account the inclination angle, θ:.(+. tan θ ) K. de ( + tan θ ) o () E Figure 7 shows that the results from Eq. 9 and our FE analyses compare well. Rocking and cross swaying rocking stiffnesses were found to be, as in the case of homogenous soil, o
Gazetas (9) θ= Eq., Eq., K d FE results, FE results, E p Figure 7 Comparison of the normalized swaying static stiffness calculated with the simplified Eq. (7) and the FE analysis (L/d =, non-homogenous soil) insensitive to the batter angle, θ. Hence, expressions for a vertical pile embedded in nonhomogenous soil (Eq. and ) derived by Gazetas (9) can be used for all inclination angles.. K.d E RR o () Es. K.7d E HR o () E o The influence of L/d on the static stiffnesses is portrayed in Figure. The swaying stiffness is sensitive to L/d, especially for greater values of E p /E s. This is because with increasing values of batter angles, the participation vertical stiffness of the pile to its lateral response increases, leading to larger values of active length. On the contrary, rocking stiffness is not influenced by L/d. Dynamic stiffnesses and effective damping ratios The variation of the three pairs of stiffness coefficients and damping ratios with respect to frequency, for different values of θ is portrayed in Figure 9. In general, the swaying stiffness increases with the rake angle. It is worth noting that both the swaying and rocking stiffness coefficients exhibit little sensitivity to frequency variations and can be efficiently approximated by their static values for all values of θ. Rocking effective damping ratio, D RR, is smaller than the swaying damping ratio. The latter is found to increase with θ. Figure a depicts the dependence of the dynamic swaying stiffness coefficient and damping ratio on E p /E s. K HR d K RR d K d 7 9 L/d = L/d = 7. L/d = L/d = 7. L/d = L/d = 7. E p Figure Normalized static stiffnesses swaying, rocking, and cross swaying rocking for slenderness ratio L/d = 7. and (θ = ο, non-homogenous soil) The swaying stiffness coefficient K decreases more rapidly with E p /E s and a. This can be attributed to the fact that the increase of E p /E s leads to the increase of the active length of the pile L a, making the pile short (L a > L), and leading to the decrease of the lateral stiffness. Furthermore, the increase of the excitation frequency causes additional increase of the static active length of the pile (Mylonakis and Gazetas, 999), which in turn leads to further decrease in the lateral stiffness of the pile. The influence of L/d is depicted in Figure b, for a pile inclined at o, embedded in soft homogenous soil with E p /E s =.
K HR d K RR d K d 7 7 θ = θ =...... a D θ = θ =. D RR D HR............... θ = θ =...... a Figure 9 Normalized dynamic stiffnesses and damping ratios swaying, rocking, and cross swaying rocking for batter angle θ =,, and as a function of the dimensionless frequency parameter a (L/d =, E p /E s =, homogenous soil). K d D.. Ep / Es = Ep / Es =. Ep / Es = Ep / Es =. K d L / d = L / d = 7. D... L / d = L / d = 7............. a a Figure Normalized dynamic swaying stiffness and damping ratio of a pile inclined at o, embedded in homogenous soil, for: pile-soil stiffness ratio E p = and and slenderness ratio L/d =, and slenderness ratio L/d = 7. and and pile-soil stiffness ratio E p /E s =
The variation with frequency of the stiffness coefficients and damping ratios for different values of θ in non-homogenous soil are portrayed in Figure. The E p /E s ratio is equal to at depth equal to one pile diameter. Again, the swaying stiffnesses are rather insensitive to frequency and, hence, the static values calculated through Eq. can be used to readily estimate the lateral stiffness of an inclined pile at any frequency. Observe that all stiffness coefficients are greater than the ones computed for the homogenous soil, for E p /E s =. On the other hand, damping ratios are significantly lower in the non-homogenous soil. No major differences are observed in the variation of the stiffness and damping ratios with respect to θ. The influence of E p /E s and L/d on the swaying stiffness and damping ratio is portrayed in Figures a and b, for o. In contrast to the homogenous soil, the effect of L/d is less pronounced in this case (E p /E s =, at z=d), while the effect of E p /E s ratio on the swaying stiffness is more pronounced here. Distribution of internal forces with depth The bending moment and shear force distributions with depth for all values of θ studied in this paper are presented in Figure. The internal forces are solely due to a horizontal force at the pile head (fixed). Only the case L/d = at E p /E s = is studied, which corresponds to the case of flexible piles. Randolph s (9) dimensionless plots for bending moments are used to study the effect of pile inclination. The inclination angle θ does not appear to have any effect on bending moments. K HR d K RR d K d θ = θ = θ =...... D D HR RR D........... θ = θ = θ =...... a a Figure Normalized dynamic stiffnesses and damping ratios swaying, rocking, and cross swaying rocking for batter angle θ =,, and as a function of the dimensionless frequency parameter a (L/d =, E p = at z = d, non-homogenous soil)
. Ep / Eο = Ep / Eο =. K d K d 9 L/d = L/d = 7. D D D D..... Ep / Eο = Ep / Eο =. L/d = L/d = 7....... a...... a Figure Normalized dynamic swaying stiffness and damping ratio of a pile inclined at o, embedded in non-homogenous soil, for: pile-soil stiffness ratio E p = and and slenderness ratio L/d =, and slenderness ratio L/d = 7. and and pile-soil stiffness ratio E p = at z = d For the homogenous case it was observed that both the bending moments and the shear forces are insensitive to frequency variations. As depicted in Figure, in the case of nonhomogenous soil, the bending moment is sensitive the inclination angle θ. The axial force remains practically insensitive to θ. M / [P cos(θ ) L a ] -. -. -.. Q / [P cos(θ )] -..... θ = z / La.. z / La.. θ =.. Figure Normalized bending moment, and shear force distributions with depth for batter angle θ =,, and, for lateral force loading at the pile head (a =., E p /E s =, L/d =, homogenous soil)
M / [P cos(θ ) L a ] -. -. -. -... N / [P sin(θ )]...... z / La. z / d. θ =. Figure Normalized bending moment, and shear force distributions for batter angle θ =,, and, for lateral force loading at the pile head (a =., E p =, L/d =, νον-homogenous soil) Conclusions The results of a parameter study on the static and dynamic behavior of inclined piles using the Finite Element method are presented in this paper. Dimensionless plots of the static and dynamic stiffnesses as well as the effective damping ratios are produced for piles with different batter angles embedded in a homogenous and a non-homogenous soil where E s is proportional to depth. The swaying static stiffness of an inclined pile was found to be different from the stiffness of a vertical pile and new expressions have been derived that take into account the batter angle θ. All dynamic stiffnesses were found to be rather insensitive to frequency. The influence of the slenderness ratio on the dynamic stiffnessess was found to be more pronounced with increasing values of batter angle. Moreover, dimensionless plots of internal forces developed for flexible inclined piles are presented for both idealized soil profiles. The effect of batter angle on bending moment distribution with depth is insignificant for the case of homogenous soil, whereas for non-homogenous case pile inclination becomes an important parameter. References Angelides, D.C. and Roesset, J.M., 9.Nonlinear lateral dynamic stiffness of piles. Journal of Geotechnical Engineering Division, ASCE, Vol. 7, No GT, pp - Banarjee, P.K. and Davies T.G., 97. The linear behanior of axially and laterally loaded single piles embedded in non-homogenous soils. Geotechnique, Vol., No, pp 9- Berril, J. B., Christensen, S.A., Keenan R.P., Okada, W., and Pettinga, J. R.,. Case study of lateral spreading forces on a piled foundation. Geotechnique, Vol., No, pp -7 Desai, C. S. and Kuppusamy, T., 9. Applications of a numerical procedure for laterally loaded structures. Numerical Methods in Offshore Piling, Institution of Civil Engineer, London, England, pp 9-99 Dobry, R., Vicente, E., O Rourke, M.J., and Roesset, J. M., 9. Horizontal stiffness and damping ratio of single piles. Journal of Geotechnical Engineering Division, ASCE, Vol, No GT, pp 9-9 Gazetas, G.,9. Seismic response of endbearing piles. Soil Dynamics and Earthquake Engineering, Vol., pp 9 9. Gazetas, G. and Mylonakis, G., 99. Seismic soil-structure interaction: new evidence and emerging issues. Geotechnical Earthquake Engineering and Soil Dynamics III, ASCE, Geotechnical Special Publication 7, pp 9-7 Gazetas, G. and Dobry, R., 9. Horizontal response of piles in layered soil. Journal of the
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