Nonlinear seismic response of single piles N. Makris, D. Badoni Department of Civil Engineering, University ofnotre Dame, 7/4

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1 Nonlinear seismic response of single piles N. Makris, D. Badoni Department of Civil Engineering, University ofnotre Dame, 7/4 Abstract A macroscopic model that consists of distributed hysteretic springs and frequency dependent dashpots is utilized to model the soil-pile interaction and a practical method based on one dimensional finite element formulation is developed to compute the nonlinear seismic response of single piles. 1. Introduction Realistic modeling of the nonlinear dynamic response of the pile-soil system is very important, especially for the dynamic analysis of highwaybridges, overcrossings and ramps, several of which experienced significant damage in the recent 1994 Northridge earthquake^. While an extremely flexible pile might simply follow the seismic motion of the ground, real piles with finite bending rigidity "resist" the induced soil deformation, generating additional soil strains in their vicinity. As a result, the incident seismic waves are scattered and the seismic excitation to which the pile-foundation is effectively subjected differs from that of the free field motion. Analysis of the linear kinematic response of single pile and pile groups has been studied by many researchers (Kaynia and Kausel^, Kaynia and Novak^, Makris and Gazetast Makris^ and references therein). However, linear analysis is limited to small displacement gradients, and fails to describe the stress- and displacement-fields in the vicinity of piles when excited by strong seismic motions. 2. Macroscopic Constitutive Model The problem studied herein is that of a single pile embedded in a layered soil and subjected to strong lateral motion because of the passage of vertical shear waves. The seismic loading considered produces horizontal harmonic

2 476 Soil Dynamics and Earthquake Engineering oscillations at a "free-field" point of the deposit, i.e. at a point unaffected by the presence of the piles. Around the piles the "perturbed" wave field is a complicated combination of incident waves reflected at the surface, and waves diffracted by the piles. At the limit of linear elastodynamics a rigorous method of solution to this three dimensional dynamic boundary value problem has been presented by Kaynia and Novak^ based on the earlier work developed by Kaynia and Kausel^. In the case of strong loading, where the deformation level near the pile soil interface is large and linear elasticity fails to realistically represent the displacement and stressfields,the problem becomes even more complicated, and a rigorous three-dimensional dynamic formulation using theories of viscoplasticity becomes a formidable task. Work in the context of dynamic viscoplasticity has been conducted by Cheng et al.^, where it was shown that even for a two-dimensional plane-strain case, the computational effort is indeed very large. In contrast, the Winkler foundation model utilized herein is a versatile and economical approach, since the problem of pile-soil interaction becomes one-dimensional. Some constitutive models are more attractive than others because of their simplicity, the physical meaning of some of their parameters, their validity over large frequency-spectra and load-intensities, and their consistency at limiting cases. The above reasons motivated for the formulation of the constitutive model proposed herein, which is a continuously distributed nonlinear spring in parallel with a frequency dependent linear dashpot. This model is the extension at large deflections of the linear Kelvin model, and the extension atfinitefrequencies of the hysteretic-type nonlinear spring utilized by Trochanis et al^. The force resulting from the nonlinear spring alone is given by Fj(z) = A(z).S(z)^, for cohesive soils (1) F (z) = fjij d-- ] -.z,, for cohesionless soils (2) * * 1 smcp In equations (1) and (2) f is a hysteretic dimensionless quantity that is governed by the following equation In equation (1), S(z) is the shear strength of the soil at depth z, d is the pile diameter and A is a dimensionless quantity 9.142<A< , where the lower limit = is for a perfectly smooth pile and the upper limit = 4^/ is for a perfectly rough pile. The theoretical values for A derived by Randolph and Houlsby^ for plane-strain conditions do not differ much from the value A = 9 that was proposed by Broms^ after experimental and theoretical studies. At shallow depths, the plane-strain assumption is inappropriate because of the non-zero vertical deformation of the soil during lateral motion of the pile. Accordingly, near the surface parameter A assumes smaller values. Broms^ proposed a value of A = 2 near the surface whereas Matlock^ used

3 Soil Dynamics and Earthquake Engineering 477 A = 3. Between the surface and large depths the value of A increases progressively as plane-stain conditions prevail. Matlock^ assumed a simple form for the increment of A between the minimum value A^ = 3 and A^ = 9 : where J is a model parameter to be calibrated from experimental data. In equation (2), % = p^g is the specific weight of the soil, d is the pile diameter, 0 is the angle of internal friction measured from drained triaxial or direct shear test, z is vertical distance from the surface, and p is a parameter for which Broms^ assumed the value ^ - 3. In the case where the deposit is saturated the buoyant specific weight should be used. Following Broms work^, other researchers elaborated on the mechanisms that develop when the soil surrounding the pile fails. Reese et al^, using soil mechanics theory and a wedge-type failure near the surface, presented expressions more sophisticated than equation (2) to approximate the ultimate soil resistance near the ground surface and at a depth. Nevertheless, in this study the expression proposed by Broms^is preferred because of its simplicity, since it involves only one parameter compared to the expressions proposed by Reese et al.^, which involve three parameters. However, the formulation presented herein is not restricted to the use of a specific expression that approximates soil resistance, and the best possible available expression should be used. The expression for the distributed force resulting from the dashpot which accounts for radiation damping is^ F,= (Qa~^p^d]y (5) where the term within the bracket is the distributed frequency dependent damping coefficient c(a^) = Qa^p^V^d, a^ = ajd/v^ is a dimensionless frequency, and the coefficient Q is given by the expression Q = 2 34 ^ 0.75 where ^ is the Poisson's ratio of the soil. In the proposed model (equations (1) or (2) and (5)) the parameters to be determined are n and J for cohesive soils, or n and jm for cohesionless soils. The remaining parameters,?,, E,, S, <, are standard geotechnical parameters obtained from field or laboratory tests. The efficiency of the proposed model has been validated by predicting the experimentally measured single pile response from five well instrumented full-scale experiments^. Table 1 summarizes the soil properties, pile characteristics and the values of the calibrated model-parameters from the five experiments. Table 1 also provides a realistic range for the values of n, J and JJL. Parameter n controls the transition from the linear to the nonlinear range and

4 478 Soil Dynamics and Earthquake Engineering 'lest/ Year Kramer el. al Crousect. al Brown et. al Blaney& O'Neill 1986 Browner, al (Sand) Jennings et. al Ting et. al Table 1: Constitutive Model Parameters for Single Pile Soil Properties Cohesive Soils v^= 0.49 V% (tip) = 30 m/s Ps= 1.12 Mg/rrP Vs= 0.49 V,j(tip) = 115 m/s Ps=2.1 Mg/m^ Cohesionless Soils v^= 0.48 V,(tip) = 160 m/s ps= 1.6Mg/iiv* Vg= 0.49 Vg(tip)= 125 m/s p^= 1.6 Mg/m^ v^= 0.48 Vj,(tip) =70 m/s ps= 2.0 Mg/m^ Pile Characteristics L= 14.9 in d= 0.2()m t= m Eplp= 3.81x10^ kn-nr L= 13.4 m d= 0.273m [= m Eplp= 7.3x1 ohn-nr L= 13.1 in d= 0.273m t= in EpIp=7.3xl0^kN-nr L= 6.75 m d= 0.45 m t= 0.01 Om HpIp=0.8xH)5kN-nr L=9.75 m d=0.61 m t= 0.013m Eplp= 2.2x1 O^kN-m- Model Parameters parameters J and /x are associated with the strength of a cohesive and cohesionless soil respectively. It is observed that for cohesive soils n < \ and 0.15 < J < 0.5, whereas for cohesionless soils 2 < n < 3 and 3 < JLL < 5. It is interesting to note that the values of parameter J obtained by Matlock^ after fitting experimental data were: J=0.25 for the Lake Austin profile and J=0.50 for the Sabine profile. These values reported by Matlock lie within the range of values obtained from this study. Of course, more experiments are needed to establish with confidence the range of values for the model parameters. 3. Nonlinear seismic response of single pile The developed model in conjunction with a one dimensionalfiniteelement formulation is used herein to compute the nonlinear seismic response of the single pile. The free-field soil displacement is imposed as a support motion to the distributed nonlinear springs and frequency dependent dashpots; and the pile response is computed using the developed one dimensional finite element formulation. Details on the implementation of the method can be found in Badoni's thesis^t The effects of the nonlinear soil-pile kinematic interaction are portrayed in the form of two kinematic response factors ly(0)l,, N'(0) d *y = iw> * 4=^-^ (7) J ii n n

5 Soil Dynamics and Earthquake Engineering 479 plotted as functions of the frequency factor, ^ = wd/v,, where >>(0) is the pile head displacement and u (0) is the free-field surface displacement. When the pile-soil response is nonlinear, the number of physical and geometrical parameters that affect the system response is large, and it is more practical to study individual cases. Herein, the nonlinear kinematic response factors are computed for different levels of nonlinearity, for both free-head and fixed-head piles. The level of nonlinearity is measured through the ratio u^/d where UQ is the amplitude of the input harmonic motion. Figures 1 and 2 depict values of the kinematic response factors /^, and /^ for fixed- and a free-head piles embedded in the cohesive profiles reported by Kramer et a/.^and Grouse et al.*^, for three values of the ratio %Q/d=0.01,0.1 and 1.0. The linear seismic response of the piles is also computed with the developed method (solid line) and is compared with the "rigorous" solution using the formulation developed by Kaynia and Kausel^ (stars).herein, for the computation of the seismic free-field motion the profile of cohesive soils was assumed homogeneous, although the shear modulus near the surface assumes smaller values than its value at a few diameters depth. For a homogeneous soil deposit excited at the base with motion ^ = u^exp(ia}t}, the amplitude of the free-field motion is approximated by n=l J= BEM LINEAR U*/d= U%/d= cod Figure 1. Displacement kinematic response factors for fixed-head pile embedded in soft peat, Kramer et al., 1990.

6 480 Soil Dynamics and Earthquake Engineering U(Z) = M, 0" cos z (v. cos I r L V, In figures 1 and 2, the maximum head response of the pile surrounded by yielding soil exceeds the response of the same pile surrounded by linear elastic soil. The kinematic displacement factor tends to fluctuate with frequency and in some cases the pile head deflection exceeds the free-field surface displace- (8) 1.5 n=l J=0.5 **** BEM LINEAR LL/d=O.Ol - - U;/d=0.l u;/d= i 0.5 O cod a 0 V Figure 2. Displacement (top) and rotation (bottom) kinematic response factors for free-head pile embedded in soft peat, Kramer et al. (1990).

7 Soil Dynamics and Earthquake Engineering 481 ment. This behavior is reminiscent of the linear seismic response of piles embedded in soil deposits containing a thin, soft top layer^. This similarity in behavior should not be surprising, since the yielding portion of the soil near the surface behaves to some extend as an equivalent linear soft top layer. 4. Prediction of pile response in the Ohba-Ohashi soil profile Using the method developed in the previous section together with Table 1, the nonlinear dynamic response of a single pile embedded in the Ohba- Ohashi soil profile near Tokyo, Japan is computed. 0.8r 0.6 ****BEM LINEAR 0.4- U,/d=O.Oi - - u;/d=o.i ^0 cod V LVd=0.01 lod= Figure 3. Displacement kinematic response factors for fixed-head pile embedded in the Ohba- Ohashi soil profile.

8 482 Soil Dynamics and Earthquake Engineering The Ohba bridge is located in Fujisawa City near Tokyo. It is supported by eleven piers and is485m long and 10.8m wide. The bridge piers are supported on pile foundations. For instance, pier 6 is supported on an 8X8=64 group with steel piles (32 batter and 32 vertical). The piles have diameter=0.6m, length=22m and wall thickness=0.009m (for the vertical piles) and 0.012m (for the batter piles). The river runs between pier 6 and pier 7. The top layers through which the piles penetrate consist of soft alluvial strata of humus and silty clay with shear wave velocity ranging from 50m/s to 60m/s. The total thickness of the alluvium is about 22 meters. The underlying substratum of alluvial deposits consists of stiff clay and sand with shear wave velocity of about 400 m/s. The undrained shear strength, Su, assumes values of the order of lokn/rn- from the surface down to ten pile diameters and from there gradually increases to 2>5kN/m~. Parameter n was taken equal to one, since the soft deposit of the Ohba Ohashi area resembles the Mercer Slough peat^, for which the calibration of the model resulted: n=l and J=0.5. Figure 3 plots the nonlinear kinematic response factors of a fixed-head pile embedded in the Ohba profile for three values of the ratio M</d=0.01,0.1 and 1.0, and two values of J=0.1 and 0.6. The computed linear response of the pile (solid line) is compared with the rigorous solution by Kaynia and Kausel^ (stars). The general trend in Figure 3 is that nonlinear behavior amplifies the values of kinematic response factors at moderate frequencies. 5. Conclusions A macroscopic viscoelostoplastic model which consists of distributed hysteretic springs and frequency dependent dashpots in conjunction with onedimensional finite element formulation was used to predict the seismic response of the single piles. The model is physically motivated and involves standard geotechnical parameters. Only two parameters have to be calibrated by fitting experimental data. Nonlinear behavior of the pile soil interface amplifies the kinematic response factors at moderate frequencies. Acknowledgments Partial financial support for this project has been provided by Shimizu Corporation, Japan (Grant No. NCEER/RF A to NCEER), and by the Department of Civil Engineering and Geological Sciences, University of Notre Dame. References 1. Moehle, J. P. "Preliminary report on the seismological and engineering aspects of the January 17, 1994 Northridge earthquake", Report No UCB/ EERI-94/01, Kaynia, A. M. and Kausel, E. "Dynamic stiffness and seismic response of pile groups." Research Report, Dept. of Civil Engrg., MIT, Cambridge, MA, 1982.

9 Soil Dynamics and Earthquake Engineering Kaynia, A. M. and Novak, M. "Response of pile foundations to Rayleigh waves and to obliquely incident body waves", Earthquake Engrg. Struct. DvM. 21, , Makris, N. and Gazetas, G. "Dynamic pile-soil-pile interaction. Part II: Lateral and seismic response", Earthquake, eng. struct, dyn. 21, , Makris, N. "Soil-pile interaction during the passage of Rayleigh waves: An analytical solution," Earthquake, eng. struct, dyn. 23, , Cheng, F-R, Roesset, J. M. and Tassoulas, J. L. "Dynamic response of circular foundations in an elastoplastic medium." Geotech, Engrg. Report, GR 86-3, Dept. Civil Engrg. Univ. Texas Austin, Austin, TX, Trochanis, A., Bielak, J. and Christiano, R "Simplified model for analysis of one or two piles, 'V. gfofec/% engrg, ASCE, 120, , Randolph, M. F. and Houlsby, G. T. "The limiting pressure on a circular pile loaded laterally in cohesive soil," Geotechnique, 34, , Broms, B. B., "Lateral resistance of piles in cohesive soils," J. soil mech EdMj Dh/. ASCE, 90, SM2, Matlock, H. "Correlations for design of laterally loaded piles in soft clay," Proc., Offshore Technology Conf., Paper No. OTC 1204, , Broms, B. B. "Lateral resistance of piles in cohesionless soils," J. soil mzca Fdnj D;'v. ASCE,90, SM3, Reese, L. C, Cox, W. R., and Koop, F. D. "Analysis of laterally loaded piles in sand," Proc., Offshore Technology Conf., Paper No. OTC 2080, , Gazetas, G and Dobry, R. "Horizontal response of piles in layered soils", 7. ggofgc/i. zng. ASCE 110, 20-40, Badoni, D. "Nonlinear analysis of piles under inertia and seismic loading", Master thesis, University of Notre Dame, IN, Kramer, S. L. Satari, R. and Kilian, A. P. "Evaluation of in situ strength of a peat deposit from laterally loaded pile test results," Transportation Research Record, No 1278, Transp. res. board, Washington, D. C., , Crouse, C. B., Kramer, S. L., Michell, R. and Hushmand, B. "Dynamic test of pipe in saturated peat," J. geotech. engrg, ASCE, 119, (1993). 17. Gazetas, G., Fan, K.,Tazoh, T., Shimizu, K., Kavvadas, M. and Makris, N. "Seismic pile-group structure interaction," Piles under Dynamic Loads, S. Prakash ed., Geotechnical Special Publication No 34, 56-93, 1992.