Geotechnical Earthquake Engineering and Soil Dynamics IV
GSP 181 © 2008 ASCE
FD Solutions for Static and Dynamic Winkler Models with Lateral Spread Induced Earth Pressures on Piles D.W. Chang1, B.S. Lin2, C.H. Yeh2 and S.H. Cheng2 1
Professor, Department of Civil Engineering, Tamkang University, Tamsui, Taiwan 25137;
[email protected] 2 Research Assistant, Department of Civil Engineering, Tamkang University, Tamsui, Taiwan 25137
ABSTRACT: Finite difference (FD) solutions for static and dynamic Winkler’s foundation models are applicable in modeling pile responses. In this paper, relevant solutions on liquefaction induced lateral spreading are presented. Direct and indirect earth-pressure approximations and implementations are introduced. Pile foundation failures of 1995 Kobe earthquake were examined using these models. It was found that both the static and dynamic analyses could provide rational pile displacements in agreement with field observations. The largest pile displacements were found at the pile head from static modeling and the dynamic one using indirect earth pressures. For dynamic solutions with direct earth pressure approximations, maximum pile displacements were found at pile tip. These solutions seem reasonable to model different types of lateral spreading. The mechanism of ground motion, strongly affected by geologic and geographic site conditions as well as the soil-foundation-structure interactions, needs to be carefully verified before applying these solutions. INTRODUCTION Pile foundations subjected to liquefaction-induced lateral spreading under the earthquake have been extensively studied in the past decade. Seismic pile performance with this concern can be analyzed utilizing a rigorous FE technique, or much simpler one representing by Winkler foundation model. For static solutions of the later, the ground forces acting on pile could be obtained from two alternatives, 1. Direct earth pressure approach, available as the one suggested by Japan Road Association (1990), 2. Indirect earth pressure approach, in which the earth pressures are obtained from prescribed ground displacements and soil springs. The displacement profile could be represented by the one proposed by Tokimatsu and Asaka (1998). The static Winkler model and nonlinear moment-curvature relationships of piles are often used for analysis of nonlinear pile responses. For dynamic solutions of the Winkler model, a two-step computational procedure has been suggested. It can be found that Chang et al. (2001, 2003), Boulanger et al. (2003, 2007), Arduino et al. (2005), Lin et al. (2005) and
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Liyanapathirana and Poulos (2005) have all pointed out that free-field ground motions can be obtained first and then used to solve for the pile responses. Pseudo static pile displacements at any specific time can be obtained from the corresponding ground displacement profile. This simulation was found comparable to the static ones. To analyze the seismic pile responses under earthquake, the authors have suggested discrete FD solutions for the dynamic Winkler model. For lateral spreading effects, preliminary study (2007) was made using the seismic earth pressures suggested by Zhang et al. (1998). This model initially suggested for retaining structure was adopted ignoring the differences of geometry and structure/material rigidity. As the results, deformed shape and magnitude of the pile displacement were found compatible to the field observations of Ishihara and Cubrinovski (2004), where the piles tilted to yield large displacements at the bottom. For dynamic analysis using the indirect earth pressure approach, the permanent ground displacement profile (Tokimatsu and Asaka, 1998) is adopted and modified in this paper. Other than that, any feasible solution could be obtained from proper ground deformational analysis. Figure 1 illustrates available numerical schemes for the task problem. The discrete equations based on difference formulas of the static/dynamic modeling and the corresponding earth pressure approaches were presented next. Preliminary comparisons of these solutions were discussed for case studies on 1995 Kobe earthquake. Direct Earth Pressure Model
JRA (1990) Earth Pressure Specification
Indirect Earth Pressure Model
Tokimastsu and Asaka (1998) Permanent Ground Displacement Profile
Direct Earth Pressure Model
Seismic Earth Pressures Model (Zhang et al., 1998)
Indirect Earth Pressure Model
Solutions from CYCLIC-1D analysis, or modified T&A (1998) Model w/ Time Dependent Ground Displacement Profile
Static FD solution
Winkler Model
Dynamic FD solution
FIG. 1. Available numerical schemes of the task problem. STATIC AND DYNAMIC FD FORMULATIONS For static Winkler’s foundation model, the governing equation and corresponding finite difference formulation are written as follows, d 4u ( z ) d 2u ( z ) EI + Px = q(z) kh ( f ( z ) u ( z ) ) (1) dz 4 dz 2 In Equation (1), E = Young’s modulus of pile, I = pile’s moment of inertia, Px = axial
load, q ( z ) = earth pressures (units in F/L), u ( z ) = pile displacement, kh = modulus of subgrade reaction (units in F/L2; for OC clay, kh = kch D ; for NC clay or sand, kh = nz , where kch = coefficient of subgrade reaction, n = constant of subgrade reaction), = ratio for nonlinearity, f ( z ) = ground displacement, z = spatial variable along the pile, D = pile diameter. The earth pressures could be evaluated using direct and indirect
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models with the prescribed ground displacements. Corresponding FD solutions are respectively presented as follows, A1u ( i 2 ) + ( 4 A1 + B1 ) u ( i 1) + ( 6 A1 2 B1 ) u ( i ) + ( 4 A1 + B1 ) u ( i + 1) (2) + A1u ( i + 2 ) = q ( i ) A1u ( i 2 ) + ( 4 A1 + B1 ) u ( i 1) + ( 6 A1 2 B1 + kh ) u ( i ) + ( 4 A1 + B1 ) u ( i + 1)
+ A1u ( i + 2 ) = kh f ( i )
where A1 =
EI
( z)
4
; B1 =
Px
( z)
2
(3)
; i is the ith node along the pile. Boundary conditions at
pile head and pile tip would change the discrete formulation accordingly. To solve the equations representing for a single pile and the surrounding soils, one need to conduct a matrix analysis. For dynamic solution of the Winkler’s foundation model, governing equation is written as follows. 4 2 2 u ( z, t ) u ( z, t ) u ( z, t ) EI A P + + = q ( z, t ) x 4 2 (4) z t z2 kh ( f ( z , t ) u ( z , t ) )
where = mass density of the pile, A = area of pile’s cross section. Similarly, Equation (4) could be evaluated using direct and indirect earth pressures. If the seismic earth pressures were known already, then Eq. (4) could be written as u ( i + 2, j ) + ( 4 B2 ) u ( i + 1, j ) u ( i, j + 1) =
1 + ( 2 A2 + 2 B2 6 ) u ( i, j ) A2 + ( 4 B2 ) u ( i 1, j ) u ( i 2, j )
(5)
A2 u ( i, j 1) + C2
where A2 =
A( z)
Px ( z )
4
2
q ( z , t )( z )
4
; C2 = ; j is the jth time step. ; B2 = 2 EI EI EI ( t ) For prescribed ground motions applied to the indirect earth pressures, Eq. (4) can be solved as follows: u ( i + 2, j ) + ( 4 B2 ) u ( i + 1, j ) u ( i, j + 1) =
1 + ( 2 A2 + 2 B2 C3 6 ) u ( i, j ) A2 + ( 4 B2 ) u ( i 1, j ) u ( i 2, j )
(6)
A2 u ( i, j 1) + C3 f ( i, j )
where C3 =
kh ( z )
4
. It is necessary to point out that the modulus of subgrade EI reaction, kh could be found in p-y models or simply determined from SPT-N values and undrained shear strength, Cu for sands and clays, respectively. Again boundary conditions at the pile head and tip would affect these formulations. Long pile conditions were usually assumed at bottom of the pile. Modified equations
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Geotechnical Earthquake Engineering and Soil Dynamics IV
GSP 181 © 2008 ASCE
need to be derived for boundary nodes and their neighboring nodes in pile shaft. The independent equations can save considerable time of computations. Details of the derivations and all the numerical formulations can be found in Lin (2006) and Yeh (2006). Pile nonlinearities could be properly simulated using rigorous modeling or a simple treatment based on iterative analysis and moment-curvature relations of the pile (Ishihara and Cubrinovski, 2004). A recent study conducted by Rajaparthy and Hutchinson (2006) using the program LPILE (Reese and Wang, 2000) to show useful performance measures for plastic hinge and maximum moment of the pile was also founded on moment-curvature behavior of the piles. DIRECT AND INDIRECT EARTH PRESSURE MODELS
The direct earth pressure model has been suggested by Japan Road Association (1990). Earth pressures of the upper crust and the liquefied layer were suggested as follows, qNL = cs cNL K p NL z ( 0 z H NL ) (7) qL = cs cL { NL H NL + L ( z H NL )} ( H NL z H NL + H L )
1 + sin ; NL , L , H NL , H L are the unit weights and thickness of the 1 sin layers. Note that the coefficients cs , cNL , cL are to be determined according to the distance to waterfront, the liquefaction potential index, PL and the engineering judgment. Pile shaft underneath the liquefied layer could be assumed either rigid or flexural. Eqs. (2) and (7) are combined to solve for the pile deformations. One must multiply Eq. (7) with the pile diameter to obtain corresponding loads for solutions. Ishihara and Cubrinovski (2004) adopted this method to analyze pile responses. For static modeling with the indirect earth pressures, the permanent ground displacement profile proposed by Tokimatsu and Asaka (T&A, 1998) are as follows, L / H = ( 25 ~ 100 ) D0 / H (8) where K p =
1 D ( x ) / D0 = 2
5x / L
(9)
f LS ( z , x ) = D ( x ) f LS ( z , x ) = D ( x ) cos
(z ! ( z zw ) 2H
D ( x) 1
(z
zw ) H
zw )
( z " zw )
(10)
where L = length of lateral spreading zone, H = thickness of liquefied layer, D0 = maximum ground displacement at the waterfront, x = site distance to waterfront, D ( x ) = maximum ground displacement at the site, f LS ( z , x ) = ground displacement
profile, z = depth, zw = depth of water table. Note that in using this model, ground displacements are only applicable to the zone of lateral spreading. Soils underneath the zone are assumed stable, where no soil displacement would exist. For dynamic modeling using Eq. (4), it is rather difficult to find proper direct and indirect earth pressure models as those discussed in static modeling. Ignoring the
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Geotechnical Earthquake Engineering and Soil Dynamics IV
GSP 181 © 2008 ASCE
structural geometry and rigidity differences, seismic earth pressure model (Zhang et al., 1998) used for the embedded pile cap (Tokimatsu, 2003) may be used. The seismic earth pressures are simply modeled as follows. pE = pEP
pEA
pEP = K EP s h (1 ± Kv ) = s h (1 ± Kv ) f EP pEA = K EA s h (1 ± Kv ) = s h (1 ± Kv ) f EA
( , l,# , I ( , l,# , I
) ,$ )
E
= tan
1
p
Kh / (1 ± Kv ) ,$
E
= tan
1
a
K h / (1 ± Kv )
(11)
where pE = net seismic pressure, pEA = active earth pressure, pEP = passive earth pressure, K h , K v = seismic coefficients in horizontal and vertical directions, s = unit weight of soil, h = depth of soil, f EP and f EA are time-dependent potential function, = soil’s friction angle, l = pile length, # a and # p = friction angel between soil and
pile for active and passive earth pressures, I E = angle of seismic forces, $ = inclination angle of ground surface. In applying this model, the sloping ground amenable for the lateral spreading occurrence is able to monitor. For adequate loadings, Eq. (11) needs to be multiplied with the pile diameter too. One could modify the corresponding equations according to 3D pile geometry and the pile-to-pile interaction effects such as those suggested on p-y relationships (Reese and Van Impe, 2001). For dynamic modeling using indirect earth pressure model, time-dependent ground displacement profiles ought to be obtained first. One may conduct a proper seismic analysis for the ground displacements and then use them to solve for the corresponding pile displacements. Computer program CYCLIC-1D (Elgamal et al., 2002) is available for the ground displacements affected by lateral spreading. For simplicity and to elaborate the use of indirect earth pressure mode, the liquefaction-induced permanent ground displacements suggested by Tokimatsu and Asaka (1998) is considered. A time dependent normalized equation, H ( t ) is proposed herein. This function is suggested by integrating the normalized ground accelerations, a ( t ) ( = a ( t ) / amax ) twice with time. Then, normalize and multiply it with original static displacement function, f ( z , x ) as follows, a (t ) H (t ) = ' ' dt / H peak (12) amax f ( z , x, t ) = f ( z , x ) H ( t )
(13)
where H peak = peak of the integral, amax = peak ground acceleration. Displacement-time history of the soils at different depths will then have the same variations but different quantities according to this approximation. CASE STUDIES ON 1995 KOBE EARTHQUAKE
Pile foundation damages reported by Ishihara and Cubrinovski (2004) on Oil-storage tank TA72 located about 20m from the waterfront in Mikagehama island during the 1995 Kobe earthquake are studied herein. The tank, having a diameter of 14.95 m and
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GSP 181 © 2008 ASCE
storage capacity about 2450 kl , is supported on 69 precast concrete piles with length of 23~24 m and diameter of 45 cm . The water table is estimated at the depths of 2~3 m . Sand compaction pile has been conducted to increase the SPT-N values for the Masado layer around Tank TA72. Figure 2 shows the reported relations for the bending moment ( M ) and curvature ( ( ) of the piles where D0 is pile diameter and N is axial load on pile. The cracking moment ( M cr ), the yield moment ( M y ) and the ultimate moment ( M u ) are known as 105, 200 and 234 kN m respectively. The ultimate shear strength is 232 kN according to ACI specifications (1998). Initial EI of pile is 58330 kN/m2. Soil properties in use are listed in Table 1. Bore-hole cameras and inclinometers were used to inspect the damages of the piles. Deteriorating of No. 2 and No. 9 piles are shown in Figure 3. The main cracks along the shaft were found at the depths of 8~14m. Pile No. 2 have scraped wounds, and pile No. 9 was sheared off at the depth about 10.5 m . Both piles were found damaged by lateral spreading of the soils at the foundation site.
FIG. 2. Relationships of bending moment and curvature of pile (from Ishihara and Cubrinovski, 2004). Table 1. Fundamental properties of soils used in case studies. Soil layers
Z (m)
Masado soil Silty sand Silty sand Fine sand Gravel
0~13.5 13.5~14 14~20 20~23.5 23.5~26
(kN/m3) 18 18 19 20 20
(deg) 33 36 36 39 40
In the simulations, seismic record of the NS-component of 1995 Kobe Earthquake is used. Table 2 depicts all the parameters used for the comparative analyses. Note that iterative technique is used to simulate the pile nonlinearities based on moment-curvature relationships shown in Figure 2. Figures 4 shows the pile displacements from the static modeling using direct and indirect earth pressures for tank TA72. It can be seen that the maximum pile displacement from direct solution is lager than the one from indirect solution. The deviations between these solutions and the one made by Ishihara and Cubrinovski are caused by material parameters and boundary conditions. The maximum pile displacements are found at pile head rather than the
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Geotechnical Earthquake Engineering and Soil Dynamics IV
GSP 181 © 2008 ASCE
bottom. Nevertheless, their magnitudes are similar to those shown in Figure 3. On the other hand, Figure 5 depicts the ultimate pile displacements from the dynamic modeling where the direct pressure model was used. Note that in Figure 5 the difference of the peak displacements between pile head and pile tip is about the same order of field displacements. Unlike the static modeling, maximum pile displacements from the dynamic analysis are found at the bottom of piles. Although the pile displacement profiles from the static and dynamic solutions are quite different, their magnitudes are about the same. Furthermore, Figure 6 depicts the results from dynamic analysis using indirect earth pressure approach where modified Tokimatsu and Asaka model was adopted. The maximum pile displacement appearing at the pile head is found greater than the static ones. It can be seen that all these modeling can provide rational solutions for pile displacements with liquefaction-induced lateral spreading concerns. The static analysis would predict largest displacements at pile head because the earth pressures/ground motions applied are decreasing with the depth. Similarly, the dynamic modeling using modified indirect earth pressures would give largest pile displacements at pile head. Pile deformations at arbitrary time could also be reviewed. If the site was amplified by earthquake shaking and the surface ground motions are pronounced to cause shallow spreading, then one can use JRA and T&A models to analyze the pile foundation. In the contrast, dynamic modeling using seismic earth pressures would yield largest pile displacements at the bottom according to the depth-increased earth pressures. This approach implies that the lateral spreading can be modeled as a massive motion of the layered soils, in which the earth pressures could be increased with the depth. If the lateral spreading could affect the deep soils, then one should use the seismic earth pressure model for the modeling. As a result, the mechanics of lateral spreading and the soil-foundation-structure interactions should be examined prior to the analysis. Table 2. Parameters used in various numerical solutions for the case studies. Method Static / Direct Static and Dynamic / Indirect Dynamic / Direct
Parameters in use
H NL = 2.5m , H L = 11m , cs = 1.0 , cNL = 0.133 , cL = 0.3 D0 = 1.75m , L = 76m , H = 11m , x = 28m , zw = 2m , = 1.0 K h = 0.83 , K v = 0 , ) = 90o , $ = 5o , # a = 0.6 , # p = 0.8 , # moba = # a , # mobp = # p
CONCLUDING REMARKS
This paper discusses the static and dynamic FD solutions for Winkler’s models on piles under lateral spreading. Direct and indirect earth-pressure models are both presented. Nonlinear pile responses could be obtained using iterative analysis with prescribed pile moment-curvature relationships. It was shown that the lateral spreading could be monitored through these solutions with careful calibrations for model parameters. Static modeling with direct and indirect earth pressures would result in maximum pile displacements at pile head. Dynamic modeling using modified indirect earth pressures can provide similar results and the pile deformations at a specified time. On the other
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GSP 181 © 2008 ASCE
hand, dynamic analysis using direct earth pressures would yield largest pile displacement at the tip. Nevertheless, all the solutions can provide similar pile displacements to those observed in the field. The differences are mainly caused by the earth pressures and ground displacements in use. For routine design applications, one must understand that these simplified solutions are good for first approximation. They may lead significant errors by neglecting the complexities of the physical mechanism. For proper use of these solutions, the geological and geographic site conditions as well as the soil-foundation-structure interactions must be evaluated carefully prior to the analysis.
(a) Pile No.2
(b) Pile No.9
FIG. 3. Lateral displacements and observed cracks of Pile No. 2 and Pile No. 9 (from Ishihara and Cubrinovski, 2004). 0
0
2 4
6
Liquefiable Layer
6
D epth (m )
D epth(m )
8
12
18
10 12 14 16 18
24 Predicted (Ishihara and C ubrinovski,2004)
20
D irectEarth Pressure (Fixed H ead) IndirectEarth Pressure (Fixed H ead)
30 -40
0
40
22
80
D isplacem ent(cm )
FIG. 4. Pile displacements from the static modeling using direct and
46.25 cm
24 0 30 60 M axim um D isplacem ent(cm )
FIG. 5. Peak displacements from dynamic modeling using direct
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Geotechnical Earthquake Engineering and Soil Dynamics IV
GSP 181 © 2008 ASCE
indirect earth pressures.
seismic earth pressures.
0
D epth(m )
6 Liquefiable Layer
12
18
24
30 -90
D ynam ic M odelU sing IndirectEarth Pressure attim e = 10 sec attim e = 20 sec attim e = 30 sec attim e = 40 sec
-45
Study C ase :TA N K TA 72 1995 K obe Earthquake
0
45
90
D isplacem ent(cm )
FIG. 6. Pile displacements from the dynamic modeling using indirect earth pressures of modified T&A displacement profile. ACKNOWLEDGEMENTS
This work is a partial result of the studies through research contract NSC95-2211-E032-043 from National Science Council in ROC. The authors express their sincere gratitude for the support. REFERENCES
American Concrete Institute (1998). "Building Code Requirements for Structural Concrete." ASCE, U. S. A. Arduino, P., Kramer, S.L., Li, P. and Home, J. (2005), "Stiffness of Piles in Liquefiable Soils." Procds., ASCE Conf. on Seismic Performance and Simulation of Pile Foundations in Liquefied and Laterally Spreading Ground, Davis, University of California, U.S.A., pp. 135-148. Brangenberg, S.J., Boulanger, R.W., Kutter B.L., Chang, D. (2007). "Liquefaction-induced Softening of Load Transfer Between Pile Groups and Laterally Spreading Crusts." J. Geotechnical & Geoenvironmental Engrg, Vol. 133 (1), pp. 91-103. Boulanger, R.W., Kutter B.L., Brangenberg, S.J., Singh, P., Chang, D. (2003). "Pile Foundations in Liquefied and Laterally Spreading Ground: Centrifuge Experiments and Analyses." Report UCD/CGM-03/01, Center for Geotechnical Modeling, University of California, Davis, 205 pp.
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Chang, D.W. and Lin, B.S. (2003). "Wave Equation Analyses on Seismic Responses of Grouped Piles." Procds., 12th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering, Singapore, pp. 581-586. Chang, D.W. and Lin, B.S. (2006). "EQWEAP~a Simplified Procedure to Analyze Dynamic Pile-Soil Interaction with Soil Liquefaction Concerns." Procds., Second Taiwan-Japan Joint Workshop on Geotechnical Hazards from Large Earthquake and Heavy Rainfall, Nagaoka, Japan, pp. 155-162. Chang, D.W., Lin, B.S. and Cheng, S.H. (2007). "Seismic Earth Pressure Induced Pile Response from Discrete Wave Equation Analysis." Procds., 16th Southeast Asian Geotechnical Conference, Malaysia, PP. 663-668. Elgamal, A., Yang, Z. and Parra, E. (2002). "Computational Modeling of Cyclic Mobility and Post-Liquefaction Site Response." Soil dynamic and Earthquake Engineering, Vol. 22 (4), pp. 259-271. Ishihara, K. and Cubrinovski, M. (2004). "Case Studies on Pile Foundations undergoing Lateral Spreading in Liquefied Deposits." Procds., 5th International Conference on Case Histories in Geotechnical Engineering, New York, Paper SOAP 5. Japan Road Association (JRA) (1990). - Specification for Highway Bridges, Part V, Seismic Design. Lin, B.S. (2006) "Structural Analysis for Pile Foundations Subjected to Soil Liquefaction and Lateral Spreading." PhD Thesis, Dept. of Civil Engr., Tamkang University. Lin, S.S., Tseng, Y.J., Chiang, C.C. and Hung, C.L. (2005). "Damage of Piles Caused by Laterally Spreading – Back Study of Three Cases." Procds., ASCE Conf. on Seismic Performance and Simulation of Pile Foundations in Liquefied and Laterally Spreading Ground, Davis, University of California, U.S.A., pp. 121-133. Liyanapathirana, P.S. and Poulos, H.G. (2005). "Pseudostatic Approach for Seismic Analysis of Piles in Liquefying Soil." J. Geotechnical & Geoenvironmental Engrg, Vol. 131(12), pp. 1480-1487. Rajaparthy, S.R. and Hutchinson, T.C. (2006). "Evaluation of Methods for Analyzing the Seismic Response of Piles Subjected to Liquefaction-Induced Loads." Procds., 5th International Conference on Bridges and Highways, San Francisco. Reese, L.C. and Wang, S.T. (2000). Documentation of Computer Program LPILE PLUS 4.0. Ensoft, Inc., Austin, Texas. Reese, L.C. and Van Impe, W.F. (2001). Single Piles and Pile Groups under Lateral Loading. A. A. Balkema, Rotterdam, Brookfield. Tokimatsu, K. and Asaka, Y. (1998). "Effects of Liquefaction-Induced Ground Displacement on Pile Performance in the 1995 Hyogoken-Nambu Earthquake." Soils and Foundations, Special Issue, No. 2, pp. 163-178. Tokimatsu, K. (2003). "Behavior and Design of Pile Foundations Subjected to Earthquakes." Procds., 12th Asian Regional Conf. on Soil Mechanics and Geotechnical Engineering, pp 1065-1096. Yeh, C.H. (2006). "Study on Static Modeling of Pile Foundation in Liquefied Soils." Master Thesis, Dept. of Civil Engr., Tamkang University. Zhang, J. M., Shamoto, Y., and Tokimatsu, K. (1998). "Seismic Earth Pressure for Retaining Walls under Any Lateral Displacement." Soils and Foundations, Vol. 38(2), pp. 143-163.
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