Single Piles in Lateral Spreads: Field Bending Moment Evaluation
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1 Single Piles in Lateral Spreads: Field Bending Moment Evaluation Ricardo Dobry, M.ASCE 1 ; Tarek Abdoun, A.M.ASCE 2 ; Thomas D. O Rourke, M.ASCE 3 ; and S. H. Goh 4 Abstract: The results of the six centrifuge models of instrumented single pile foundations presented in a companion paper, are used to calibrate two limit equilibrium LE methods to evaluate bending response and factor of safety against bending failure of piles in the field subjected to lateral spreading. These six models simulate single reinforced concrete piles in two- and three-layer soil profiles, mostly end bearing but including also one floating pile, with and without a reinforced concrete pile cap, and one model where the liquefiable sand layer was densified locally around the pile to simulate the effect of pile driving. The measured permanent maximum bending moments in the pile, M max, invariably occurred at the boundaries between liquefied and nonliquefied soil layers, and in most cases the moments at such boundaries reached their peak M max and then decreased during shaking. These values of M max before decrease, which were associated with failure of the soil against the deep foundation, are used to calibrate the two proposed LE engineering methods. For the piles where M max was controlled by the pressure of the liquefied soil, the measured prototype M max in the centrifuge tests ranged between about 100 and 200 kn m. It is found that a lateral pressure per unit area of pile or pile cap constant with depth (p ) of 10.3 kpa, predicts M max of the single piles tested within 15%. For the cases where M max was controlled by passive failure of the shallow nonliquefied layer, the prototype M max measured at the upper and lower boundaries of the liquefied soil in the centrifuge tests ranged between 160 and 305 kn m. The M max values of kn m measured at the upper boundary were reached during the shaking, and then observed to decrease towards the end of shaking. At the lower boundary, the measured M max of 305 kn m was reached at the end of shaking. Use of passive pressure against the pile of the shallow nonliquefiable soil layer, obtained from the ultimate plateaus (p ult )ofp-y curves, in conjunction with basic pile kinematic considerations and parameters addressed herein, explains well the development of moments measured in the centrifuge at both the upper and lower boundaries of the liquefied layer. This good accord validates the simplified LE prediction of M max at the upper boundary. The two proposed simplified engineering LE methods are used to evaluate bending response and distress of end-bearing and floating piles in the Niigata Family Court House building during the 1964 Niigata earthquake, with good agreement between predicted and observed performance. DOI: / ASCE : CE Database subject headings: Pile foundations; Spread foundations; Limit equilibrium; Bending moments; Earthquakes; Liquefaction. Introduction Permanent lateral movements and damage to pile foundations caused by soil liquefaction have been reported for many earthquakes in the U.S., Japan, Costa Rica, Mexico, and other countries McCulloch and Bonilla 1970; Hamada et al. 1986; Mizuno 1987; Benuzka 1990; Yoshida and Hamada 1991; Hamada and O Rourke 1992; O Rourke and Hamada 1992; Youd et al. 1992; 1 Professor, Dept. of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY corresponding author. dobryr@rpi.edu 2 Research Asst. Professor, Dept. of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY Professor, School of Civil and Environmental Engineering, Cornell Univ., Ithaca, NY Engineer, Mueser Rutledge Consulting Engineers, New York, NY Note. Discussion open until March 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on April 25, 2001; approved on October 8, This paper is part of the Journal of Geotechnical and Geoenvironmental Engineering, Vol. 129, No. 10, October 1, ASCE, ISSN /2003/ /$ Bartlett and Youd 1992; Youd 1993; Dobry 1994; Tokimatsu et al. 1996; Swan et al. 1996; Yokoyama et al. 1997; and Tokimatsu Evaluation of this case history information reveals the significance of several factors influencing the deformation of deep foundations as well as bending moments and cracking of damaged piles. These factors include: free field permanent lateral ground displacement; thicknesses and properties of soil strata penetrated by the piles; and the geometry and properties of the pile foundation. While in some cases the top of the foundation moves laterally an amount similar to the free field, in others it moves much less, with the soil failing pseudostatically, mostly in the passive mode against the piles and pile cap or flowing around them. The observed damage and cracking to the piles is often concentrated at the upper and lower boundaries of the liquefied sand layer where there is a sudden change in soil properties e.g., Fig. 1, see also Hamada 1992; Yokoyama et al. 1997; Tokimatsu 1999, or at the connections between pile and pile cap e.g., Hamada It has also been observed that more damage occurs to single piles when the lateral movement of the foundation is forced by nonliquefied soil layers end-bearing Pile 2 in Fig. 1, as opposed to the case when the forces on the foundation are limited by the strength of the liquefied soil floating Pile 1 in Fig. 1. These observations suggest a complex, pseudostatic, kinematic soil-structure interaction phenomenon, driven by the lateral JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003 / 879
2 Fig. 1. Observed pile deformation and soil conditions at Niigata Family Court House NFCH building, 1964 Niigata earthquake Yoshida and Hamada 1991; Haamada 1992; Hamada, personal communication, 1992 movement of the ground Hamada et al. 1986; Yoshida and Hamada 1991; Hamada 1992; Meyersohn et al. 1992; Meyersohn 1994; Berrill et al. 1997; and Ramos Beam-on-Winkler-Springs BWS analyses of the phenomenon have been reported e.g., see Meyersohn 1994, using the p-y curves originally suggested by Reese et al corresponding to nonlinear horizontal soil springs connecting the pile to the laterally displaced free field. An alternative approach to BWS is limit equilibrium LE. The LE approach is based on the assumption that all involved soil layers apply a static lateral pressure against the pile foundation, generally in the direction of the free field lateral spread. The loads applied to the pile in the LE analysis are equivalent to the maximum values of p from the aforementioned p- y curves. This method provides upper bounds for the maximum bending moments, displacements and rotations of the pile foundation, and is especially useful as an engineering evaluation tool for design and retrofitting decisions. LE analyses have been used by Berrill et al to explain the excellent performance of a bridge foundation to lateral spreading induced by the 1987 Edgecumbe earthquake in New Zealand; and by the Japan Road Association 1996 and Yokoyama et al to evaluate the response of bridge foundations to lateral soil deformation during the 1995 Kobe earthquake. As shown clearly by experiences in the field Youd et al. 1992; Berrill et al and in the centrifuge see companion paper, if the foundation is strong enough relative to the soil around it, the soil layers will fail in the passive mode before the bending moments reach a level capable of damaging the piles and connections, with the corresponding maximum moment staying constant or decreasing afterwards irrespective of further increases in the level of free field displacement. In that case, only the effect of the deformation of the foundation on the superstructure remains of concern. On the other hand, weak piles combined with a relatively strong shallow soil layer can induce bending high enough to damage the piles Fig. 1. The companion paper Abdoun et al presents results of six centrifuge models of vertical single piles embedded in twoand three-layer soil profiles see Fig. 2. The six models were subjected to base shaking and subsequent lateral spreading due to the liquefaction of the layer of Nevada sand having a relative density (D r ) of 40%. The nonliquefiable soil layers depicted in Fig. 2 consisted of slightly cemented Nevada sand with an internal friction angle of 34.5 and a cohesion c of 5.1 kpa. The top nonliquefiable layer in the three-layer profile included vertical Fig. 2. Pile foundation centrifuge models tested using two- and three-layer soil profiles with liquefiable sand layer of D r 40% values of D p listed were measured at pile head, 0.5 m above ground surface, simultaneously with M max ; values of D H are free field ground surface displacements at end of shaking drainage holes to simulate a pervious prototype soil. The companion paper reported the permanent pile bending moments and permanent pile and soil lateral displacements, obtained from the corresponding measured time histories after filtering out the cyclic component associated with the shaking. In all models of Fig. 2, the maximum permanent bending moments (M max ) were induced in the piles at boundaries between liquefied and nonliquefied soil layers. Most but not all of these maximum moments occurred during the shaking, significantly after the Nevada sand layer had liquefied, and then the moments decreased afterwards despite the continued shaking and progressive increase of the free field ground deformation. This, plus direct observations of soil failure conducted after the tests, revealed that the strengths of the relevant liquefied and nonliquefied soil layers had been mobilized in all centrifuge models. Therefore, the measured values of M max,as well as the corresponding measurements of pile head lateral displacement (D p ) at the time of M max, provide an excellent opportunity to calibrate the LE method for a wide range of single pile conditions. The corresponding values of M max and associated D p are presented in Table 1 and Fig. 2. Such calibration of LE procedures is implemented herein. At the end of the paper, the two proposed LE methods are used to predict the observed distress of end-bearing and floating piles at the NFCH building in the 1964 Niigata earthquake Fig. 1. The six centrifuge models of Fig. 2 are classified below using the same two categories defined in the companion paper: Case I single pile with M max controlled by the pressure of the liquefied soil, models 3, 4, 5a and 5b ; and Case III single pile with M max controlled by the pressure of the shallow nonliquefied soil, models 1 and 2. Case II in the companion paper corresponded to centrifuge experiments of pile groups which are not discussed herein. Limit Equilibrium Analysis of M max in Case I Models 3, 4, 5a, and 5b As illustrated in Fig. 2, the large increases in the measured values of M max, from 113 kn m for the single pile model 3, to as much 880 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003
3 Table 1. Summary of Maximum Bending Moments and Maximum Pile Head Displacements Measured in Centrifuge Tests, and Those Predicted Using Proposed Limit Equilibrium Approach Centrifuge model number Measured max. bending moment, M max (kn m) a 0.5A p H p A c H c m 3 Predicted max. bending moment, M max (kn m) Measured maximum pile head displacement at z 0.5 m, D p (m) Predicted maximum pile head displacement at z 0.5 m, D p (m) b 162 d b 162 d c e f b e 5a 170 c e f 5b 195 c e a All bending moment measurements have a sensitivity of 8 knm. b M max recorded at a depth of 2 m see Figs. 2 and 5. In models 1 and 2, M max 305 kn m was also measured at a depth of 8mattheendofshaking. c M max recorded at a depth of 6 m see Fig. 2. d M max predicted at a depth of 2 m using Eq. 7. In model 1, M max 356 kn m was also predicted at a depth of 8matafree field displacement, D H 0.8 m, versus M max 306 kn m measured see text. e M max predicted using Eq. 1 with p 10.3 kpa; that is, M max A p H p A c H c. f See text. as M max 195 kn m when adding a cap and densifying the sand around the pile in model 5b, are consistent with the increases in areas exposed to the liquefied soil lateral pressure. The next question is what is the shape of the diagram of maximum liquefied soil pressure versus depth. No attempt is made in the paper herein to resolve this issue. Instead, the simplest possible assumption is made that this maximum liquefied soil pressure, p, is constant and independent of depth, so as to be able to backfigure a single value of p from the M max measured in the centrifuge experiments of Fig. 2. This constant pressure p (kn/m 2 ) is assumed to act in model 3 on a width equal to the pile diameter, d 0.6 m, and on the whole embedded pile length, H p, see Fig. 3 a. Hence, model 3 becomes a vertical column of height H p 6 m, totally or partially fixed at its base and subjected to a uniform lateral load per unit length, w 0.6p (kn/m). The degree of fixity of the pile at z 6 m depends on the value of the soil rotational spring, k r, Fig. 3. Free body diagrams for limit equilibrium evaluation of M max and D p in centrifuge models 3 and 5a sketched in Fig. 3 a. The calculated moment at the base of the pile in model 3 is then M max 0.5wH 2 p (0.5)(0.6)(6) 2 p 10.8p (kn m), determined from static equilibrium alone, and hence independent of k r. This moment occurs at point C in Fig. 3 a, z c 6 m. A similar calculation can be made for floating pile model 4, using the same numbers, and M max 10.8p is again calculated, now at the top of the liquefiable layer (z 2 m, see Fig. 2. For the measured values of M max 113 kn m and M max 125 kn m in the two models, slightly different values of p 113/ kn/m kpa and p 125/ kpa are obtained. In the expression above for M max, it is convenient to use A p 0.6H p dh p (m 2 ) for the total pile area subjected to the liquefied soil pressure, and thus M max 0.5A p H p p (knm). For models 5a and 5b, the same approach must take into account the rectangular area of the pile cap, A c, also subjected to the lateral pressure p Fig. 3 b. That is, for the four single pile models 3, 4, 5a and 5b, the following generalized version of the previous expression for M max is valid: M max 0.5A p H p A c H c p (1) where A p, H p and p are as defined before, A c area of pile cap subjected to the liquefied soil pressure, and H c height of center of the pile cap area above the boundary between liquefied and nonliquefied layers. Again, M max in Eq. 1 does not depend on the rotational flexibility of the lower nonliquefiable layer represented by the spring k r depicted in Fig. 3, as Eq. 1 was obtained from static equilibrium alone without any kinematic consideration related to the pile s displacement or rotation. For both models 5a and 5b, A c (2.5)(0.5) 1.25 m 2, H c 5.75 m, and H p 5.5 m; for model 5a, A p (0.6)(5.5) 3.3 m 2, and for model 5b, A p (1)(5.5) 5.5 m 2. In the case of model 5b, the effective prototype diameter of 1 m used in the previous expression includes the pile diameter, 0.6 m, plus a 0.2-m-wide ring of densified sand around the pile. This was obtained from the width of sand actually densified during construction of model 5b, which was about 0.4 cm m at 50 g 0.2 m in the prototype, verified by observation of the flow of colored liquefied sand around the pile after the test see Fig. 7 of companion paper. Therefore, using Eq. 1 : 1. in models 3 and 4, M max 10.8p ; 2. in model 5a, M max 16.3p ; and 3. in model 5b, M max 22.3p. That is, if p is assumed to be the same in the JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003 / 881
4 Table 2. Computation of Equivalent Liquefied Soil Lateral Pressure, p, from M max Measured in Single Pile Centrifuge Models Centrifuge model number Measured max bending moment, M max (kn m) 0.5A p H p A c H c m 3 p a kn/m a b Note: Average p a p M max / 0.5A p H p A c H c. different tests, the limit equilibrium method predicts that for models 5a and 5b, M max should be, respectively, about 50 and 100% greater than in model 3, which is in good agreement with the measurements presented in Table 1 and Fig. 2. Therefore, this simple limit equilibrium approach to evaluate M max accounts well for the measured increases of M max in the single pile, when the pile cap and local soil densification are considered as additional areas loaded by the liquefied soil pressure p. The values of 0.5A p H p A c H c 10.8, 16.3 and 22.3 m 3 have been listed in Tables 1 and 2. The last column of Table 2 lists the four values of p backfigured from the measured M max in each of these single pile tests on the basis of Eq. 1. The values of p in the table are remarkably constant, with an average p 10.3 kpa that predicts all four individual values of p within 15%. The predicted M max for models 3, 4, 5a, and 5b using Eq. 1 and p 10.3 kpa are listed in Table 1 and are compared with the measured M max in Fig. 4. The experimental ranges of M max in Fig. 4 correspond to the measurement sensitivity of 8 knm. Rotational Spring k r In the previous static analyses of measured M max for single piles, the rotational spring at the base of each pile (k r ) was not used. However, this k r, representing in Fig. 3 the flexibility of the bottom nonliquefiable layer, is necessary to explain the pile head displacement D p measured simultaneously with M max. Also, k r will be required in the next section when analyzing model 1. This value of k r can be easily backfigured from analysis of the maximum pile head deflection measured in model 3. This maximum horizontal deflection of the pile head, D p 0.27 m, was measured at 0.5 m above ground surface in model 3a at the same time when M M max 113 kn m occurred at 6 m depth. The calculation was done using the same limit equilibrium procedure utilized for M max and sketched in Fig. 3 a, except that the loading p kpa already determined for model 3, as well as the measured D p 0.27 m, were used in conjunction with the system of Fig. 3 a to compute k r. The value backfigured from this analysis of D p in model 3 is k r 5738 kn m/rad. Finally, it is now possible to predict D p for both model 3 and model 5a in a way fully consistent with the proposed limit equilibrium procedure which had given the predicted values of M max listed in Table 1. That is, the rotational spring, k r 5738 kn m/rad, and p 10.3 kpa can be used in conjunction with the sketches of Fig. 3 to predict D p at z A 0.5 m in the two centrifuge models. This calculation predicts D p m for model 3 compared with D p 0.27 m measured; the agreement is not surprising considering that model 3 was used to backfigure k r. A better test of the predictive power of the method and of k r 5738 kn m/rad is provided by model 5a, where D p 0.42 m is predicted, which is in reasonable accord with the measured value of 0.35 m Table 1. Therefore, the value of k r 5738 kn m/rad, backfigured from the measured D p in model 3 and verified with model 5a, will be used in the next section for evaluating model 1. Limit Equilibrium Analysis of M max in Case III Models 1 and 2 Fig. 4. Comparison between maximum pile bending moments measured in centrifuge tests and those calculated using limit equilibrium method, for individual piles embedded in two-layer soil profile. In all calculations, lateral pressure p 10.3 kpa was used for liquefied sand. As shown in Fig. 2, the largest bending moments of the whole series of centrifuge tests were measured in models 1 and 2, due to the largest strength of the top and bottom nonliquefiable soil layers, as compared to the liquefied soil strength controlling M max in Case I. In Case I, at any given time the maximum moment always occurred at one liquefied nonliquefied soil boundary. On the other hand, in models 1 and 2, at any given time, maximum negative and positive moments occurred simultaneously at the top (z 2 m) and bottom (z 8 m) boundaries of the liquefied layer, with the bending moment being zero at some elevation within this liquefied layer see Fig. 10 of companion paper. This indicates that the displaced pile shape had a double curvature, qualitatively similar to that sketched in the inset of Fig. 5 herein. The measured bending moment profiles in models 1 and 2 were almost exactly straight lines within the liquefied Nevada sand layer, indicating that the effect of the liquefied soil pressure was negligible compared with the forces and moments imposed on the pile by the nonliquefied layers above and below Fig. 10 of companion paper. This simplifies considerably the analysis of models 1 and 2, as the presence of the liquefied soil between z 2 and 8 m can be neglected. Finally, the measured increase and then decrease during shaking of the bending moment at z 2 m in both models 1 and 2, as well as observation of the model conditions during posttest dissection see companion paper, indicate that the 882 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003
5 amount of rotational flexibility existed at both supports A and B, defined by two identical soil rotational springs k r. The corresponding expression for M at both supports is D H M L L/6EI 1/k r (2) Fig. 5. Bending moments measured at upper (M A ) and lower (M B ) interfaces in models 1 and 2 data points ; and calculated for rigid (k r ) and flexible (k r 5738 kn m/rad) supports at A and B lines slightly cemented shallow sand layer did reach passive failure against the pile foundation, while the bottom layer below z 8 m did not fail. In the rest of this section, an elastic analysis is first presented of the bending response of centrifuge models 1 and 2 for small to moderate free field lateral deformation D H. This is followed by development of a limit equilibrium LE, simplified engineering prediction method validated by the maximum bending moment at z 2 m measured in model 1, M A (M A ) max 160 kn m Figs. 2 and 5. This LE method assumes plastic failure of the top layer at the time of (M A ) max. Then, the elastic and plastic analyses are combined into a more detailed elastic perfectly plastic procedure for model 1 which predicts well the evolution of both M A at z 2 m) and M B at z 8 m), as D H increases. This combined elastoplastic analysis explains both the values and occurrence of (M A ) max and (M B ) max during shaking and at the end of shaking, respectively. Finally, model 2 is briefly discussed. Elastic Analysis of M A and M B Fig. 5 presents the permanent bending moments M A and M B versus permanent lateral ground displacement in the free field of the top nonliquefiable layer (D H ), recorded in models 1 and 2 throughout the shaking, including the time when M A (M A ) max. Points A and B are defined in the inset of Fig. 5. This inset sketches the distorted shape of the pile for a given D H, as well as the predicted bending moments M M A M B (6EI/L 2 )D H and shear forces H H A H B (12EI/L 3 )D H, calculated neglecting the liquefied soil and assuming that both top and bottom layers, represented by the supports at A and B, are infinitely rigid and do not allow for any rotation of the pile. In these expressions for moment and shear, static equilibrium is combined with kinematic considerations reflected in the presence of D H in the equations as well as in the assumption of zero rotation at points A and B. This is in contrast with Eq. 1 for the two-layer system, where the computation of M max was based only on static equilibrium. In models 1 and 2, EI 8000 kn m 2 and L 6 m, and hence M 1,333D H (kn m), with D H in meters. This equation has also been plotted in Fig. 5 dashed straight line labeled k r ; it greatly overpredicts all measured bending moments by a factor of at least 2 or 3. An additional analysis was performed of the same system in the inset of Fig. 5, but now assuming that the same Again with EI 8,000 and L 6, and using k r 5738 kn m/rad previously obtained for the bottom slightly cemented sand layer from the results of model 3, Eq. 2 predicts M 557D H. This expression has been plotted as a solid straight line in Fig. 5, and it provides a fair estimate of the measured values of M A and M B in both centrifuge models at low values of D H. It is important to note that both straight lines in Fig. 5 assume that pile and soil in the top and bottom layers remain linear as D H increases. This assumed linear response was certainly true for the pile models throughout the shaking, and it seems a reasonable first order approximation for the soil during the first part of shaking. Comparison Between Elastic Analysis and Measurements Inspection of the measured data points for centrifuge models 1 and 2, and of their relation to the two analytical lines in Fig. 5 indicates the following. 1. For values of lateral spreading D H at least up to 0.2 or 0.3 m, M A M B in both centrifuge models, and this value is reasonably well represented by Eq. 2 with k r 5738 kn m/rad (M 557D H ). This suggests that for this range of low D H the soil in both top and bottom nonliquefiable layers can be represented by linear rotational springs. The fact that the values of M A and M B are similar in models 1 and 2, indicates that for this range of D H the bending response was not much affected by the pile cap. 2. The top nonliquefiable layer failed and M A (M A ) max for values of D H of the order of m in models 1 and 2, with (M A ) max being significantly greater in model 2 due to the presence of the pile cap, and with M A decreasing in both models at greater D H. 3. The value of M B continued to increase monotonically after failure of the top soil layer, with the maximum value of M B happening at the end of shaking in both models. Limit Equilibrium Prediction of M A max in Model 1 It is possible to use limit equilibrium to evaluate the measured (M A ) max of 160 kn m in model 1, as this value was controlled by the strength of the slightly cemented 2 m shallow sand layer above point A. However, as already suggested by the elastic analysis above, this use of limit equilibrium in model 1 is somewhat more complicated than what was done in the previous section for the single piles in the two-layer profiles, where M max at z 6 m could be calculated from static equilibrium alone. Additional considerations beyond statics become necessary when applying limit equilibrium to evaluate both M A and M B in model 1. Fig. 6 shows the free body diagrams used for the limit equilibrium analysis needed to predict the maximum bending moment (M A ) max in model 1 at a depth of 2 m. Three free body diagrams FBD are included, all corresponding to the time when M A (M A ) max at z 2 m. They are: FBD1 for the pile in the slightly cemented shallow sand layer between z 0 and 2 m; FBD2 corresponding to the pile in the liquefied sand layer between z 2 and 8 m, where the presence of the liquefied soil is ignored, similar to the analyses generating the straight lines in Fig. 5; and JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003 / 883
6 Table 3. Relation between L/h and M max /(p 0 h 3 ) from Eq. 7 L/h M max /(p 0 h 3 ) z ps /h Fig. 6. Free body diagrams for limit equilibrium evaluation of bending moments, M A (M A ) max and simultaneous M B in centrifuge model 1 (EI 8,000 kn m 2 ) FBD3 indicating the interaction between the pile and the rotational spring k r at point B in the bottom slightly cemented sand layer (z 8 m). This FBD3 allows predicting the rotation B M B /k r of the pile at point B. The direction of lateral spread D H has been indicated at the top of Fig. 6. Despite this being a limit equilibrium analysis, the value of M A (M A ) max cannot be solved exclusively by statics. The reason for this is the change in the direction of the soil passive pressure against the pile in FBD1 at an unknown depth z ps. This change in direction is required by the signs of bending moment and shear force at point C, which indicate that while the soil pushes the pile in the direction of D H near the bottom of the slightly cemented sand layer, the pile pushes the soil, also in the direction of D H, near the top of the layer. This was confirmed by the observed failure of the pile near the ground surface in centrifuge model 1, which consisted of snapping of the pile, that is, rotation of the upper part of the pile and penetration into the soil toward the downslope, D H direction see companion paper. Therefore, FBD1 provides only two equations of static equilibrium for three unknowns (z ps, M A and H A ), hence requiring additional FBDs. An important aspect of FBD1 in the figure is the linear variation of the ultimate soil force per unit length on the pile, p p 0 z, defined by the slope of the line, p 0, which has been assigned the value p kn/m 2 for model 1. A linear variation of p with depth is to be expected for a sand without any cohesion; this is reasonable for the slightly cemented sand layer, for which 34.5 with a small cohesion was measured in triaxial tests Abdoun The selected value of p kn/m 2 was obtained from an in-flight centrifuge pile lateral load test in this soil and corresponding derivation of p- y curves at different depths using program LPILE Reese and Wang A plot of the ultimate plateaus of these p-y curves versus depth gave the slope p kn/m 2 used here Abdoun The three static equilibrium expressions derived from Fig. 6 are listed below as Eqs Equations 3 and 4 correspond to force and moment equilibrium of FBD1, while Eq. 5 was obtained from moment equilibrium of FBD2. In Fig. 6 and the equations below, it is understood that M A (M A ) max z ps /h M A /p 0 h 3 2 2/3 3 1/6 (3) H A /p 0 h (4) M B /p 0 h 3 H A /p 0 h 2 L/h M A /p 0 h 3 (5) These three simultaneous static equilibrium Eqs. 3 5 are not sufficient to calculate the four unknowns (M A, M B, H A, and z ps ); a fourth equation is needed. This additional equation is obtained from the observation, earlier in this section, that M A M B for moderate values of D H, as illustrated by Fig. 5 and by elastic Eq. 2. If the assumption is made that at the time of failure of the upper layer, when M A (M A ) max, this is still true, the desired fourth equation is M A M B (6) Now the four equations 3 6 suffice to compute the four unknowns. This use of elastic Eq. 6 is equivalent to assuming an elastic perfectly plastic behavior for the upper sand layer, where the elastic response characterized by Eq. 6 and valid for a range of small and moderate values of ground displacement D H, turns suddenly and without any transition to plastic failure of the whole layer, at which point the passive pressure distribution of FBD1 in Fig. 6 is valid, and M A (M A ) max M B. On this basis, Eqs. 3 6 can be reduced to the following simple relation between (M A )/(p 0 h 3 ) (M A ) max /(p 0 h 3 ) and the geometric parameter L/h 3 1 2/ L/h M A max / p 0 h / L/h M A max / p 0 h (7) The prediction of (M A ) max in Eq. 7 requires only knowledge of two geometric parameters h and L and one passive resistance parameter for the upper layer (p 0 ). From an engineering viewpoint, it is significant that Eq. 7 does not require knowledge, neither of the stiffnesses of pile and soil as Eq. 2 did, nor of the free field displacement D H at which (M A ) max occurs. Table 3 lists values of normalized (M A ) max calculated for different L/h using Eq. 7 : the values of (M A ) max /(p 0 h 3 ) range between and For L/h 2, the following approximation to Eq. 7 can be used which provides the same (M A ) max of Table 3 with an error not exceeding 2%: 884 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003
7 M A max p 0 h 3 / / L/h (8) For the parameters of model 1 (L 6 m, h 2 m, and p kn/m 2 ), Eqs. 7 and 8 predict (M A ) max 162 kn m, in excellent agreement with the measured (M A ) max 160 kn m; the predicted value has been included in Table 1. This excellent agreement between prediction and measurement of (M A ) max in centrifuge model 1 suggests that Eqs. 7 and 8 can be used with confidence for engineering field evaluations of (M A ) max ; this is done later herein in the Niigata lateral spreading pile case history analyzed by the authors. It is important to note that the (M A ) max predicted with Eqs. 7 and 8 is sensitive to the value of p 0 used in the calculations. For example, a different value, p kn/m 2 would have been estimated from the manual of program LPILE Reese and Wang 1993 for the strength parameters of the slightly cemented soil in the top layer measured in triaxial tests: c 5.1 kpa and If the calculation with Eqs. 7 and 8 had been done with p kn/m 2 instead of the correct 250 kn/m 2, (M A ) max 110 kn m would have been predicted, much smaller than the measured 160 kn m. Bending Moment Development Versus Ground Displacement The limit equilibrium analytical model of Eqs. 3 5, based exclusively on static equilibrium and assuming plastic failure of the top slightly cemented sand layer as reflected in the soil load distribution on the pile p 0 in the FBD1 of Fig. 6, can also be used to predict the evolution of moments M A and M B versus ground displacement D H, after (M A ) max occurs. This assumes that once the top layer has failed plastically, the load distribution given by p 0 in Fig. 6 remains constant at large D H irrespective of the value of D H. This prediction from Eqs. 3 5 at large D H can then be combined with elastic equation M A M B 557D H at low D H before (M A ) max occurs, see Fig. 5, to provide a complete picture of the development of M A and M B for the whole range of free field ground displacement D H. However, the prediction of M A and M B at large D H from Eqs. 3 5 requires a fourth equation, to allow for calculation of the four unknowns (M A, M B, H A, and z ps ) as functions of D H. This additional expression is Eq. 9 EI/p 0 h 3 L 2 D H 1/3 H A /p 0 h 2 L/h 1/2 M A /p 0 h 3 EI/p 0 h 3 L M B /k r (9) Equation 9 was obtained from calculation of the horizontal displacement D pa D H of point A of the pile relative to point B, using both FB2 and FBD3 in Fig. 6. Strictly speaking, the free field displacement D H is equal to the lateral displacement of the pile at z z ps, that is D pc D H, rather than D pa D H as assumed in Eq. 9. However, D pa D pc, and the use of D H in Eq. 9 simplifies considerably the calculations. Once D H is specified, the four simultaneous Eqs. 3 5 plus Eq. 9 can be used to calculate the four unknowns (M A, M B, H A, and z ps ). Fig. 7 illustrates the bending moment development at points A and B for centrifuge pile model 1, as well as of the shear force H A H B, predicted by Eqs. 3 5 and 9. The point at which M A M B at the onset of plastic equilibrium in the upper layer is shown by the intersection of three lines: 1 the elastic line, M A M B 557D H, which corresponds to Eq. 2 and is the same line of Fig. 5, up to D H 0.29 m; 2 a plastic line for M A versus D H, for D H 0.29 m, obtained by solving Eqs. 3 5 and 9, which intersect the elastic line at D H 0.29 m; and 3 a second plastic line for M B versus D H obtained from the same Fig. 7. Elastoplastic analytical model solid line segments, and measured values of M A, M B, and H A in model 1 data points. Parameters used to solve Eqs. 3 5 and 9 are those indicated in Fig. 6. solution of Eqs. 3 5 and 9, which also intersects the elastic line at D H 0.29 m. The three lines in Fig. 7 were calculated using the same parameters used before for model 1 and shown in Fig. 6. The data points in Fig. 7 are the measured M A and M B in model 1 for various D H. The curve of H A versus D H calculated in both the elastic (D H 0.29 m) and plastic (D H 0.29 m) ranges has also been included at the top of Fig. 7. Data points corresponding to measured values of H A are also plotted in the same graph, obtained in all cases from the measured M A and M B and H A (M A M B )/6. The complete elastoplastic analytical model, defined by the three solid lines in Fig. 7, assumes that the behavior of the upper soil layer is elastic for low values of D H, for which the elastic demand M A 557D H is smaller than the plastic resistance of this upper layer corresponding to the value of M A calculated with the free body diagrams of Fig. 6 using Eqs. 3 5 and 9. Plastic failure of the upper layer occurs as soon as the demand is equal to the resistance, that is when M A 557D H predicted by the elastic model becomes equal to M A calculated with Eqs. 3 5 and 9. The point at which the elastic response of the upper layer turns into plastic response occurs in Fig. 7 at D H 0.29 m. At this point, and only at this point, all expressions used in the analysis Eqs. 2 9 are valid and the predicted M A (M A ) max 162 kn m as calculated earlier herein. Beyond this point that is for D H 0.29 m), the upper soil layer behaves plastically and Eqs. 2 and 6 and 8 are not valid, with both M A and M B predicted with the plastic LE Eqs. 3 5 and 9.AsM A predicted by Eqs. 3 5 and 9 decreases when D H increases, the elastoplastic analytical model of Fig. 7 predicts that failure, and thus (M A ) max, occurs at the intersection of the lines, which happens at D H 0.29 m in the case of Model 1. This prediction of (M A ) max can JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003 / 885
8 be done with either Eq. 7 or Eq. 8, and as mentioned before does not require knowledge of D H at which failure occurs. In the analytical model of Fig. 7, all three lines intersect at the point of failure, where M A M B. There is excellent overall agreement between the moments predicted by the three line segments and the measured data points in Fig. 7, indicating that this elastoplastic analytical model captures well the mechanics of development of M A and M B in centrifuge model 1. There is also very good agreement between predicted and measured values of the shear force H A in the same figure. The analytical model defined by the three lines in Fig. 7 predicts well M A M B at low values of D H, it predicts very well (M A ) max 162 kn m compared with 160 kn m measured, itexplains both the decrease after failure of M A and the simultaneous increase in M B, predicting reasonably well the values of both M A and M B (M B ) max 356 kn m versus 305 kn m measured at the end of shaking (D H 0.80 m). However, it predicts that (M A ) max should occur at D H 0.29 m versus 0.46 m measured, and also that at the time of failure, M B M A, which is not the case in model 1. The continued increase of M B as D H increases beyond plastic failure of the upper soil layer, measured in model 1 and captured by the analytical model of Fig. 7, is explained by the reduction in the value of z ps Fig. 6. This reduction in z ps decreases the bending moment M A but simultaneously increases the shear H A that must be transmitted by the pile at point A see Fig. 7. As the model piles used in the centrifuge experiments were very strong and remained elastic, clearly the pile in model 1 was able to take this increased shear force. On the other hand, if the pile cracks and ultimately fails at point A at an intermediate value of D H,it may not be able to transmit this larger shear force as D H continues to increase; in that case the actual subsequent values of M B may be significantly smaller than predicted by the analytical model. The significance of this is discussed again later herein when analyzing the Niigata Family Court House NFCH case history. The excellent agreement in Fig. 7 between the measured data points of M A, M B and H A versus D H in centrifuge model 1, on the one hand, and the three lines in the same figure predicted using the elastoplastic model defined by Eqs. 2 5 and 9, on the other, further increases confidence in the simple LE engineering procedure developed before herein in Eqs. 7 and 8 for evaluation of (M A ) max. Model 2 The only difference between centrifuge models 1 and 2 Fig. 2 was the existence of a pile cap in model 2. This pile cap increases the moment resistance of the pile foundation to lateral spreading and adds significant complexity to the LE analysis. In terms of the free body diagrams of Fig. 6, the cap would be represented by two actions of the shallow layer on the pile: 1. an additional lateral force corresponding to the passive soil resistance to the cap acting on the pile in the FBD1 of Fig. 6 at a very shallow depth of the order of 0.3 m; and 2. a resisting rocking moment, also acting on the pile at a shallow depth in FBD1, representing the resistance of the upper layer to rotation of the cap. As calculation of this rocking moment is well beyond the scope of the simplified limit equilibrium approach proposed by the authors, no further analysis of model 2 is conducted herein. Table 4. Evaluation of Floating and End-bearing Piles at NFCH Building in 1964 Niigata Earthquake Using Proposed Limit Equilibrium Approach Parameter Estimated axial load on pile Meyersohn 1994 Cracking moment of pile, M c Meyersohn 1994 Ultimate moment of pile, M u Meyersohn 1994 M max at z 2.75 m calculated using limit equilibrium Eq. 1, with p 10 kpa M max at z 2.75 m calculated using limit equilibrium Eqs. 7 and 8 presence of pile cap neglected Pile 1 Floating Comparison with NFCH Case History in 1964 Niigata Earthquake Pile 2 End bearing 29 kn 290 kn 2.65 kn m 18.2 kn m 56.6 kn m 86.2 kn m kn m kn m Factor of safety against cracking, (FS) c M c /M max Factor of safety against ultimate bending failure, (FS) u M u /M max Observed distress of pile at z 2.75m after 1964 Niigata earthquake Fig Bending cracks Shear dislocation of about 0.1 m The observed damage to reinforced concrete single piles in the NFCH building caused by the 1964 Niigata earthquake Fig. 1, was selected as a case history to verify the two limit equilibrium procedures proposed in this paper for evaluating M max at the top of the liquefied layer. Comparison of Figs. 1 and 2 reveal that the three centrifuge models tested in a three-layer soil profile approximate this case history. Centrifuge models 1 and 2 Fig. 2 roughly correspond to end-bearing Pile 2 in the field Fig. 1, while centrifuge model 4 corresponds to floating Pile 1. The NFCH case history has been documented by Yoshida and Hamada 1991, Hamada 1992, 2000; Hamada, personal communication, 2001, Meyersohn 1994 and Goh While there are some differences between the published data for this case history with regard to depth of groundwater at the time of the earthquake, depths of conspicuous bending deformation, and lengths of piles, the final information consolidated in Fig. 1 and used in the calculations below, is consistent with best estimates kindly provided to the writers by Professor Hamada in Both piles at NFCH were driven, hollow concrete piles of 0.35 m outside diameter and 75 mm wall thickness; with longitudinal reinforcement consisting of mm-diam steel bars, and transverse reinforcement provided by a spiral hoop 3 mm in diameter and a pitch of 800 mm. While the two piles had the same section, they were subjected to different vertical loads, and thus had somewhat different ultimate capacities. Meyersohn 1994 has provided estimated cracking bending moments for Piles 1 and 2 of M c 2.65 and 18.2 kn m, respectively; the corresponding ultimate 886 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003
9 bending moments are M u 56.6 and 86.2 kn m. These values of cracking and ultimate moments are based on the estimated axial loads acting on the piles shown in Fig. 1; this information is also included in Table 4. As sketched in Fig. 1, the groundwater level in 1964, z w, was somewhere between 2.5 and 3 m; this range of values will be used in the calculations herein for the depth of the upper boundary of the liquefied layer. The lateral ground displacement due to lateral spreading in the vicinity of the NFCH building was between 0.5 and 2 m. As indicated in Fig. 1 and Table 4, the damage to the NFCH piles at a depth of m consisted of bending cracks for Pile 1 and of a shear dislocation of approximately 0.1 m for Pile 2. That is, the damage was greater to end-bearing Pile 2 as compared with floating Pile 1. This is consistent with the greater values of M max at a similar location measured in the centrifuge in end-bearing pile models 1 and and 270 kn m, as compared with floating pile model kn m ; see Fig. 2. This is in turn due to this M max being controlled by the passive strength of the shallow nonliquefied soil in one case, and by the strength of the weaker liquefied sand in the other. Therefore, the two different limit equilibrium LE procedures proposed in this paper are used below to analyze NFCH Piles 1 and 2, respectively. An important difference between the centrifuge models and the NFCH piles is that the model piles were very strong and remained elastic at all times, with the soil failing rather than the pile, while clearly both piles in the field cracked, and Pile 2 failed before the lateral spread in the free field reached its final value of D H 1 m. Therefore, in this as well as other engineering applications, the main objective of the limit equilibrium procedures proposed herein should be evaluation of factors of safety against cracking and ultimate bending failure, rather than actual prediction of M max. The rest of this section evaluates those factors of safety for M max at a depth z m, for both Piles 1 and 2 of the NFCH building during the 1964 Niigata earthquake, with this depth corresponding to point A discussed previously herein e.g., Fig. 6. Fig. 1 shows the presence of a pile cap in both Piles 1 and 2, down to a depth of about 1.6 m. While this cap does not affect the predicted M max for Pile 1, it may tend to increase M max of Pile 2, as shown by the larger maximum bending moment measured in centrifuge model 2 as compared to model 1 Table 1 and Fig. 2. The results of the calculations of factors of safety presented below are summarized in Table 4. Evaluation of Floating Pile 1 Eq. 1 is applicable to Pile 1, with A c 0, A p ( ) (5.5 to 6.5) 1.93 to 3.58 m 2, H p 5.5 to 6.5 m, and with p kpa used after the centrifuge results summarized in Table 2. This assumes that this value of p measured in the centrifuge is applicable to NFCH s Pile 1, despite the difference in pile diameters 0.35 versus 0.60 m, as well as different liquefied sands. Also, the effective pile diameter is assumed to be somewhere between 0.35 m actual diameter of concrete pile, and 0.55 m due to the local densification effect of the pile driving, as discussed in the companion paper in relation to centrifuge model 5b. That is, after incorporating the uncertainties about the exact depth of groundwater, length of pile subjected to liquefied soil pressure, effective pile diameter, and value of p, the predicted maximum bending moment at z m ranges from M max (0.5)(1.93)(5.5)(8.5) 45.1 kn m to M max (0.5)(3.58)(6.5)(11.5) kn m; this range has been included in Table 4. This range of predicted M max gives very small factors of safety against cracking, between (FS) c 2.65/ and (FS) c 2.65/ , hence clearly predicting cracking for this pile at z 2.75 m. The factor of safety against ultimate bending failure is in the range between (FS) u 56.6/ and (FS) u 56.6/ , predicting that in Pile 1 at this depth the pile should crack but perhaps not fail completely, consistent with the observed damage Fig. 1 and Table 4. That is, both prediction and observation indicate that Pile 1 pushed against the liquefied soil below and cracked under that pressure. While the analysis is more uncertain in relation to ultimate bending failure, in conjunction with the observation it suggests that Pile 1 was close to but did not experience ultimate failure, with the liquefied soil perhaps eventually flowing around the pile. Evaluation of End-Bearing Pile 2 For Pile 2, the LE procedure defined on the basis of Fig. 6 and Eqs. 7 and 8 is applicable if the presence of the pile cap is neglected. This should provide a lower bound value for (M A ) max at z 2.75 m, and corresponding upper bound factors of safety derived from (M A ) max. For the conditions of Pile 2 in Niigata, and for the lower boundary of the liquefied layer at a depth of 9 m, L 6 to 6.5 m and h 3 to 2.5 m Fig. 1 ; therefore, L/h 2 to 2.6. Based on standard penetration tests, Meyersohn 1994 estimated friction angles, 32 for the upper layer between z 0 and about z 2.75 m, and 40 for the denser bottom layer below z 9 m. The value 32 in conjunction with the procedure recommended in the manual of program LPILE Reese and Wang 1993 was used by the authors to estimate a soil pressure parameter, p kn/m 2 for the top nonliquefied layer and the NFCH pile. As this LPILE procedure had provided a value of p 0 which was too low for centrifuge model 1, a best estimate range between p and 200 kn/m 2 was defined for the calculations. For L/h 2 to 2.6, (M A ) max /(p 0 h 3 ) to 0.079, respectively, from Table 3 and Eqs. 7 and 8. Therefore, the maximum predicted moment at z 2.75 m ranges between (M A ) max (0.079)(160)(2.5 3 ) kn m and (M A ) max (0.074)(200)(3 3 ) kn m for the range of p to 200 assumed. This calculated range for (M A ) max 198 to 400 kn m has been included in Table 4. The corresponding factors of safety predicted for Pile 2 are also listed in the same table. They are: a very low factor of safety against cracking, (FS) c , and also a low factor of safety significantly below unity against ultimate bending failure, (FS) u As already mentioned, neglecting the presence of the pile cap means that the actual factors of safety may have been even lower. Therefore, this limit equilibrium calculation clearly predicts both cracking and ultimate bending failure of the pile, in good agreement with the observed damage to Pile 2. No attempt was made here to evaluate (M B ) max at z 9 m in Pile 2, despite the fact that a solution was available from centrifuge model 1 see Eqs. 3 5 and 9 and Figs. 6 and 7. As discussed in the previous section herein, after NFCH Pile 2 failed in the field at z 2.75 m, its capacity to transmit shear at that depth was impaired, and hence the assumption of an elastic pile included in the analytical model of Fig. 7 would most probably lead to an overprediction of the actual field bending moments at lower elevations and large ground displacements. The calculations above for NFCH Piles 1 and 2 in the 1964 Niigata earthquake, summarized in Table 4, illustrate the validity of the two proposed limit equilibrium methods to evaluate bending response of floating and end-bearing pile foundations subjected to actual lateral spreads in the field, for cases in which this response is controlled by the strength of either liquefied or shallow nonliquefied soil layers. JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / OCTOBER 2003 / 887
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