Analysis of Laterally Loaded Piles with Nonlinear Bending Behavior

Transcription

1 7 TRANSPORATON RESEARCH RECORD 1169 Analysis of Laterally Loaded Piles ith Nonlinear Bending Behavior STEVEN L. KRAMER AND EDWARD J. HEAVEY Analysis or the response of plle! fnundatlnns to lateral loads requires aurate harateriation of the behavl.or of the pile and the soil surrounding the pile. The oon assuption of pile linearity n bending ay not be valid n any ases. A odel for representing nonlinear plle bending behavior n the analysis of laterally loaded piles ls ntrodued. The fourth order differential equation governing the response of a laterally loaded pile ls solved teratively by a finite differene tehnique. The ethod onverges to a solution featuring displaeent-opatible soil resistane and urvatureopatible bending oents along the length of tl1e plle. The ability of the ethod to represent nonlinear pile bendlng behavior ls llustrated. The ability of the odel to predit observed behavior ls illustrated by appliation of the odel to lateral load test ase histories. The response of pile foundations to lateral loads depends on the interation beteen the piles and the surrounding soil. Analysis of this response requires aurate harateriation of the behavior of both the pile and the soil surrounding the pile. Coonly 'used existing ethods for analysis of the lateral load response of single piles onsider the nonlinearity of the soil resistane but treat the pile as a linear, elasti bea. The assuption of pile linearity ay not be valid in any ases. This paper onsiders the effet of this assuption on the behavior of laterally loaded piles and introdues a ethod for representing nonlinear pile bending behavior in the analysis of laterally loaded piles. PREVOUS WORK The proble of alulating the response of a single, laterally loaded pile has been studied by any investigators using different approahes inluding ethods of applied elastiity (J), finite eleent analysis (2, 3), boundary eleent ethods (4), and subgrade reation ethods (5). While eah of these ethods has advantages in partiular appliations, the sub grade reation ethods have beoe the ost oonly used for pratial analysis of routine probles. Subgrade reation ethods of lateral load analysis, in hih the soil.resistane is desribed by p-y urves, have been used for any years and have been reently suaried (6). Suh ethods for analysis of laterally loaded piles reported in the literature, hoever, alost uniforly treat the pile as a linear, elasti aterial. Nakai and Kishida (3) inorporated pile bending nonlinearity into an inreental finite eleent odel, S. L. Kraer, Departent of Civil Engineering, University of Washington, Seattle, Wash E. J. Heavey, Converse Consultants Northest, 3131 Elliot Avenue, Suite 55, Seattle, Wash hih iteratively solved for the defleted shape of the pile using a Rayleigh-Rit proedure. Their odel shoed reasonable predition of load-defletion behavior hen applied to several ase histories. For piles loaded into their nonlinear bending range, Reese (6) suggests analytial deterination of the pile flexural stiffness orresponding to the axiu bending oent and use of that stiffness for the entire pile in a linear pile analysis. ANALYSS OF LATERALLY LOADED PLES WTH NONLNEAR BENDNG BEHAVOR The response of laterally loaded piles ebedded in an elasti ediu is usually desribed by the bea-on-elasti-foundation theory of Hetenyi (7). Equilibriu of an eleent of a bea subjeted to axial and transverse loading as shon by Hetenyi to be desribed by the differential equation: here M bending oent; x position along axis of bea; Q = axial load; y = defletion noral to axis of bea; and p = transverse loading along length of bea. f the bea is assued to exhibit linear bending behavior, the bending oent ay be taken as the produt of the onstant bea flexural stiffness, E, and the indued urvature, in hih ase the governing differential equation is of the failiar for: For the analysis of laterally loaded piles in soil, this equation is oonly expressed in differene for and solved by a nuerial proedure that iterates toard a solution in hih the unit soil resistane, p, is opatible ith the lateral pile defieliuu, y. The bending oent indued in a pile exhibiting nonlinear bending behavior is not proportional to the pile urvature. Defining a seant pile flexural stiffness, (E J)s, as the ratio of the bending oent to the indued urvature, the bending oent ay be represented as Substitution of Equation 3 into Equation 1 and expansion of the first differential ter leads to the general governing differential (1) (2) (3)

2 Kraer and Heavey 71 equation for a laterally loaded pile ith nonlinear bending behavior: (El)s(fydx 4 ) + 2[d(El),jdx] (d3yjdx3) + [Q + d 2 (El)di2)] (d 2 ydi2) - p = (4) This governing equation ay be solved nuerially by disretiing the pile into segents of length, h, and expressing the differentials in differene for, hih yields a banded syste of siultaneous equations expressed, letting R = (El),, in the for: (a)..; u.. Oi., " "' ::i Pile Defletion, y (5) here Ai = Ri + (Ri+t - Ri-1)2 Bi = 2Ri-l - 6Ri + Qh 2 Ci = lori - 2R;+ 1-2R'_ 1-2Qlf- + E,h 4 Di = 2Ri+t - 6R; + Qh E; = R; + (Ri-t - R;+ 1 )2 (b) ::E., E ::E.., "'., SOLUTON OF PLE RESPONSE EQUATON An iterative proedure as developed to solve the governing equation in differene for aounting separately for both the nonlinearity of the soil resistane and lhe bending nonlinearity of the pile. The proposed proedure retains the onventional iteration toard displaeent-opatible soil resistane, used in progras like COM624 (5), herein the soil resistane is taken as the negative produt of the pile displaeent and a seant soil odulus. The proedure iterates until the oputed displaeent is ithin soe sall toler.ane of the displaeent fro lhe previous iteration, as shon in Figure la. The proposed proedure also iterates siultaneously toard a urvature-opatible bending oent. fu this proess, the atuaj bending oe at soe point along lhe pile is taken as the produt of t.oe urvature of lhe pile at tha.t point and a seant flexural stiffness. The seant flexural stiffness is varied until the urvature oputed by a partiular iteration is ithin soe sall tolerane of lhe urvature used in the previous iteration, as shon in Figure lb. By this siultaneous iteration proedure, the solution onverges to one in hih the soil resistane is opatible ith lhe pile displaeent and the bending oents are opatible ith the indued pile urvature. This siultaneous iteration proedure has a nuber of attrative features. t is a relatively siple proedure, both oneptually and oputationally. t an be inorporated into existing progras for lateral load analysis ith a oderate aount of prograing effort. The bending harateristis of lhe pile are desribed by one or ore oent-urvature diagras and ay be varied along the length to aoodate tapered, segented, or oposite piles. While it does not provide a rigorous ethod for representation of nonlinear pile behavior, the ethod is onsistent ith thal onventionally used to represe the soil resis1ane. Considering that the oent-urvature behavior of a pile is likely to be uh ore reliably knon than the unit soil resistane-displaeent behavior of the soil, the unertainty assoiated ith this nuerial ethod of representation of pile nonlinear bending behavior is sall. Pile Curvature, d 2 ytdx 2 FGURE 1 Sheati diagra of teration proedure used to onverge to (a) deoetlonopatlble soil stiffness and (b) urvatureopatlble pile Oexural stiffness. MOMENT-CURVATURE BEHAVOR OF PLES Piles generally exhibit nonlinear bending behavior as a result of one or bolh of to ehaniss. First, yielding of the pile aterial(s) ay lead to a nonlinear oent-urvature relationship. Nonlinearity as a result of aterial yielding an our in steel, onrete, and tiber piles hen bending oentindued stresses in the ou1er fibers of the pile setion exeed the elasti range of the pile aterial. Seond, the flexural stiffness of the pile ay vary as a result of hanges in geoetry of the oent-resisting pile setion. This effet is usually the result of raking of the pile setion and is typially observed only in onrete piles. Steel piles are generally not subjet to raking under noral onditions. Large pile urvatures, hoever, ay indue opressive and tensile stresses that ay exeed the yield strength of the steel, partiularly in portions of the pile setion in hih residual stresses are high. f additional loads ause further urvature, lhe assoiated bending oent ill not inrease proportionajly. The theoretial axiu bending oent for hih a seel pile ill have onstant flexural stiffness ay be deterined fro siple bea theory. The oent-urvature relationship for steel piles stressed beyond the yield stress of the slee1 ay be alulated based on the ross-setional shape of the pile and an assuption of elasti, perfetly plasti steel behavior (8). Conrete is used in pile foundations in the for of reinfored onrete, as in drilled, ast in-plae piles, or in preast, prestressed piles. Conrete piles, hoever, ay develop tensile raking even at oderate urvatures. Sine tensile stresses annot be transitted aross a rak, the oent-urvature

3 72 TRANSPORTATON RESEARCH RECORD 1169 relationship beoes nonlinear as soon as raking develops. The nonlinear portion of the oent-urvature relationship for a onrete pile annot be alulated as easily as that of steel pile sine the bending behavior depends on the onstitutive behavior of both the onrete and the reinforing steel or prstressing strands. Analytial odels are available (6, 9), hoever, that are apable of prediting the oent-urvature behavior of reinfored onrete and prestressed onrete setions. The oent-urvature behavior of tiber piles ay have signifiant nonlinearity at oderate urvatures beause of the aterial nonlinearity of ood. Ultiate bending oents and initial flxurnl stiffnesses appear to vary fro pile to pile (1). The nonlinear bending odel is ell suited for inorporation into a probabilisti analysis of laterally loaded tiber pile response. o Ī.D... ::: ::: (!) LLUSTRATON OF EFFECTS OF PLE NONLNEARTY The nonlinear bending odel as applied to to reported ase histories of laterally loaded piles. The ase histories ere both drilled shafts, one onstruted in stiff lay and the other in loose sand, or soft lay. n eah ase, the response of the pile as analyed by assuing linear briding behavior and then assuing nonlinear bending behavior. The results are then opared ith the observed response of the piles. Johnson et al. Test Results of a full-sale field lateral load test on an aged drilled shaft ere presented by Johnson et al. (Jl). The shaft as 18 in. (.46 ) in diaeter by 34.5 ft (1.5 ) long inluding a 36-in. (.9 ) diaeter underrea. The shaft extended through expansive lays, hih had fratured the shaft at the nek of underrea by the lie it as tested 18 yr after onstrution. The shaft as loaded to failure ith one interediate unloadreload yle. Yielding of the shaft as observed at failure. Johnson et al. analyed the response of the shaft ith a onstant oputed flexural stiffness and the Reese and Welh (12) stiff lay p-y riteria and alulated load-displaeent response stiffer than observed. The authors suggested that the atual flexural stiffness as less than oputed. Analysis of the load test ith a saller, onstant flexural stiffness as indiated in Figure 2 gave reasonable agreeent at lateral loads belo about 28 kips (125 kn). At lateral loads above 28 kips (125 kn), hoever, linear pile analyses are unable to predit the large defletions observed in the field, as shon ithout the unload-reload yle in Figure 3. Assuing that the axiu bending oent of the shaft as reahed at a lateral load of 28 kips (125 kn), the bilinear oenturvature relationship arked "Nonlinear Pile" in Figure 2 ay be taken to represent the bending behavior of the shaft. Solution ith this nonlinear oent-urvature relationship predits the load-displaeent response over the entire range of loading shon in Figure 4. The superior ability of the FGURE 2 Moent-urvature diagras for linear and nonlinear bending behavior of drllled shaft (11). () Q. 6 5 Linear Pi e Bend in gy :.: 4 Observed ,, PLE HEAD DEFLECTON (N) FGURE 3 Coparison of observed linear pile bending odel (11). nonlinear odel to predit the observed load-displaeent response for this ase history is apparent. Tsuji et al. Test A lateral load test originally reported by Tsuji et al. in Japanese as desribed by Toinaga et al. (13). The test as perfored on a 47.2-in. (12 ) diaeter, ast-in-plae bored pile, installed in loose sand, or soft lay, ith standard penetration resistane of 4 to 5.,,

4 Kraer and Heavey Observed : 5 Vl. - : :.: 1 5. P 1 -L 1 near 1 e Bending , PLE HEAD DEFLECTON (N) FGURE 4 Coparison of observed nonlinear bending odel (11) , PLE HEAD DEFLECTON (N) FGURE 6 Coparison of observed linear plje bending odel (JJ). The unraked pile as reported to have a ftexural stiffness (El) of x 1 12 lb-in. 2 Toinaga et al. developed an expression for Lhe ultiate unit soil resistane as P ult = l in pounds per inh here = depth in inhes. Using these values and assuing bilinear p-y behavior, Toinaga et al. oputed the response of the pile by varying the initial soil stiffness until reasonable agreeent ith the observed behavior as obtained. By this proedure, good agreeent as obtained at lateral loads belo approxiately 132 kips. Above this level, hoever, the linear odel is unable LO predit the rapidly inreasing defonnations attributed by Toinaga et al. to raking of the pile : 5 Vl. - : :.: Nonlinear Pile... + Ī D - - ::; ::; Cl Pile Bending \Nonlinear Pile Bending......J...,. J FGURE S Moent-urvature diagras for linear and nonlinear bending behavior of drilled shaft (13) PLE HEAD DEFLECTON (N) FGURE 7 Coparison of observed nonlinear pile bending odel (JJ). A siilar proedure as used in the present investigation. Using p-y urves of the hyperboli tangent for (14, 15), hih is onsidered to be ore representative of the atual nonlinear soil behavior than the siple bilinear odel, the response of the pile as oputed assuing linear pile bending behavior. The oent-urvature relationship as that indiated for linear pile bending in Figure 5. This analysis also indiates good agreeent beteen oputed and observed behavior at lateral loads belo 132 kips as shon in Figure 6. Assuing that the ultiate bending oent as reahed at a lateral load of 132 kips, the bilinear oent-urvature relationship arked "Nonlinear Pile Bending" in Figure 5 ay be taken to represent the bending behavior of the pile. Solution of

5 74 lhe pile response ilh this nonlinear oent-urvature relationship gives the load-displaeent urve shon in Figure 7. Again, lhe ability of the nonlinear odel to predit Lhe large pile defletions assoiated ilh nonlinear pile bending behavior an easily be seen. SUMMARY A elhodology for analysis of lhe lateral load response of piles ilh nonl.inear bending behavior has been developed. Appliation to ase histories of lateral load tests on piles and drilled shafts loaded into lheir nonlinear bending ranges indiates lhat the odel is apable of prediting lhe large defletions often assoiated ilh the lateral load response of suh piles. The odel is expeted to have partiular appliation to reinfored onrete drilled shafts and to tiber piles. REFERENCES 1. H. G. Poulos. Behavior of Laterally Loaded Piles: -Single Piles. Pro.. Aerian Soiety of Civil Engineers, Vol. 97, No. SM5, May 1971, pp M.. Faruque and C. S. Desai. 3-D Material and Geoetri Nonlinear Analysis of Piles. Pro., 2nd Jn1ernational Conferene on Nue-ria Methods in Offshore Piling, The University of Texas at Austin, S. Nakai and R. Kishida. Nonlinear Analysis of a Laterally Loaded Pile. Pro.. 4th nternational Conferene on Geoehanis, Edonton, Alberta, Canada, 1982, pp P. K. Banerjee and T. G. Davies. Analysis of Soe Reported Case Histories of Laterally Loaded Pile GToups. Nuerial Methods in Offshore Piling. nstitute of Civil Engineers, London, England, 198, pp L. C. Reese and W. R. Sullivan. Douentation of Copuer Progra COM624. Geotehnial Engineering Softare GSB-1. TRANSPOKTATON RESEARCH RECORD 1169 Geotehnial Engineering Center, The University of Texas at Auslin, L. C. Reese. Behavior of Piles and Pile Groups Under Laleral Load. Report GHWA-RD PHWA, U.S. Departent of Transportation, 1986, 275 pp. 7. M. Hetenyi. Beas on Ela.sti FoundaJion. University of Mihigan Press, Ann Arbor, 1946, 235 pp. 8. E. P. Popov. ntrodution to Mehanis of Solids. Prentie Hall, Engleood Cliffs, NJ., 1986, 571 pp. 9. X. Tao, J. F. Stanton, and N. M. Hakins. A Coputer Progra for the Cyli Moent-Curvature Response of Reinfored, Pie. stressed, and Partially Prestressed Conrete Setions. Strutures and Mehanis Report SM84-2. University of Washington, Seattle, 1984, 4 pp. 1. G. E. Phillips, T. Bodig, and J. R. Goodan. Wood Pole Properties, Vol. 1: Ba!groun.d and Sou.thern Pine DaJa. nteri Report HL-419. Eleooial Poer Researh nsritute, July L. D. Johnson, J.-L. Briaud, and W. R. Stroan. Lateral-Load Test of an Aged Drilled Shaft. Lalerally ljaded Deep Foundations: Analysis and Perforane (1. A. Langer, E. T. Mosley, and C. D. Thopson, eds.). ASTM STP 835. Aerian Soiety for Testi.ng and Materials, 1984, pp L. C. Reese end R. C. Welh. Lateral Loading of Deep Foundations in Stiff Clay. Journal of the Geotehnial Engineering Division, ASCE, Vol. 11, No. GT, July 1975, pp K. Toinage, K. Yaagata, and H. Kishlla. Horiontal Displae.. ent of Soil in Front of Laterally Loaded Piles. Oils and Foundations, Vol 23, No. 3, Sept. 1983, pp. 8o F. Parker, Jr. and L. C. Reese. ExperienJal and Analytial Study of Behavior of Single Piles in Sand Under Lateral and Axial LuJings. Researh Report Center for Highay Researh, The University of Texas at Austin, Nov M. W. O' Neill and J. M. Murhison. An EvaluaJion of p-y Relationships in Sands. Researh Report GT-DF2-83. The University of Texas at Houston, 1983, 174 pp. Publiation of this paper sponsored by Coitlee on Foundations of Bridges and Other Strutures.