Macroelement Modelling of Laterally Loaded Piles and Pile-groups

Transcription

1 1 st Intenational Confeene on Natual Hazads & Infastutue 8-30 June, 016, Chania, Geee Maoelement Modelling of Lateally Loaded Piles and Pile-goups Nikos Geolymos 1 National Tehnial Univesity of Athens Oestis Papakyiakopoulos Tehnial Univesity of Munih ABSTRACT A mao-element model is pesented fo analysing the esponse of single piles and pile-goups unde lateal loading. Fomulated within the famewok of lassial elasto-plastiity, the mao-element ombines a hadening ule fo loadingunloading-eloading of the Bou-Wen type oupled with an assoiative plasti flow ule. The pape aims at detemining two fundamental omponents of the model, namely the failue sufae and the flow ule at post failue esponse, based on esults fom 3D finite element analysis. Compaisons ae also given with peditions fom uppe bound limit equilibium solutions based on evisions of Bom s theoy fo the lateal apaity of piles Keywods: Piles, Finite Element Analysis, Mao-element, Constitutive Model INTRODUCTION Reent eseah has shown that the use of laws and equations povided by the theoy of elastoplastiity an be dietly appliable to the analysis of foundations in ohesive soil unde undained loading onditions (Matin and Houlsby, 000). It has been demonstated that this appoah povides bette esults in ompaison to Winkle based model, as it is apable of ealistially epesenting the oupling between the vaious degees of feedom. Within the famewok of elastoplastiity, the "global" esponse of the pile-soil system is teated in a manne simila to that fo the "loal" esponse of an infinitesimally small soil element. The stesses and stains fo the soil element ae substituted by the genealized foes (in 3-dimensional M-Q-N spae) and the oesponding displaements (θ u v) espetively. In this pape a mathematial famewok fo maoelement modeling of piles and pile-goups is biefly pesented, emphasizing the nonlinea behaviou of both the soil and the pile. The alibation of the model paametes is ahieved though ompaisons with 3D finite element analyses with the use of ode PLAXIS. Given that the ultimate lateal apaity of a pile is dietly elated to its diamete and bending moment esistane, whih in tun is a funtion of the axial foe imposed (o developed) on the pile, a simple uniaxial stess-stain model based on the Moh-Coulomb yield iteion is also developed apable of epoduing the oss-setional behaviou of iula einfoed onete piles in tems of bending moment uvatue elationship and bending moment axial foe failue envelopes. 1 Coesponding Autho: N. Geolymos, National Tehnial Univesity of Athens, geolymos@gmail.om

2 PHENOMENOLOGICAL MODEL FOR RC PILE SECTION BEHAVIOUR A Moh-Coulomb based uniaxial stess-stain onstitutive law is developed fo modeling the maosopi behaviou of a RC iula pile setion subjeted to a ombined bending moment and axial foe loading. Consideing foe equilibium at failue in the axial dietion, one obtains: Axial Foe (kn) As=% Fibe Analysis Poposed Fit FEM σt Figue 1. Compaison of M-N failue envelopes fo RC pile setions omputed fom fibe analysis and fom PLAXIS ode, and pedited by the poposed optimization poedue. The examined pile diametes ae: (a) D = 0.8 m, (b) D = 1 m, () D = 1.5 m, with a longitudinal einfoement atio of A s = % 4 N 4N t 0 dd t 0 0 dd 0 D D (1) in whih σ and σ t ae the ompessive and tensile stength of the omposite (einfoed onete) setion, espetively: os os, t 1 sin 1 sin () whee, φ ae the stength paametes of the Moh-Coulomb model, namely the ohesion and the intenal fition angle. In Eq (1), and t ae the adii of the segments in ompession and in tension, espetively, defined as: x0 D os os D 4x0 D os 4 D x0 4 x0 os 4 D x0 os os (3)

3 Moment (knm) and t by substituting x 0 in Eq (3) with (D x 0 ). D is the pile diamete and x 0 is the absissa that defines the bounday between the zones of the setion unde ompession and tension, espetively. By applying moment limit equilibium with espet to the ente of the pile setion, the following equation is deived: t D M dd t dd x N (4) 0 Shea Foe (kn) F sv = (tension) Shea Foe (kn) Figue. Compaison of the failue envelopes fo a pile in ohesive soil, alulated by PLAXIS and pedited by the limit equilibium method, fo 4 diffeent fatos of safety to vetial loading F sv. The pile has a diamete of D = 1 m and a longitudinal einfoement atio of A s = 1.5 %. Tension (extation of the pile) is denoted with negative values Eqs (1) and (4) fom a nonlinea algebai system. Fo a given pile diamete and a known ombination of bending moment axial foe at stutual failue onditions, thee ae thee unknown vaiables:, φ and x 0. The afoementioned system is solved with the use of a geneti algoithm based optimization poedue, implemented in MATLAB. The pefomed optimization tagets to a best fit on a pedefined M N failue envelope by minimizing the oot mean squaed eo (RMSE) of the bending moment at failue (the fitness funtion): n i T i i 1 1 RMSE( M ) M M n, (5)

4 in whih M i is the bending moment omputed by Eq (4), M T,i is the taget bending moment, and n, the numbe of M T.i N i pais that define the failue envelope. It is inteesting to obseve that the poposed simplified Moh- Coulomb based onstitutive model an be easily edued to a Tesa with tension ut-off based one, by equating the ompessive stength in Eq () with σ = and setting the tensile stength equal to the tension ut-off. Fig 1 shows the M-N failue envelopes fo a pile oss-setion with a longitudinal einfoement atio of A s = % and fo thee diffeent diametes (D = 0.8 m, 1 m and 1.5 m). Compaison is given between the peditions of: (a) the poposed optimization poedue, (b) the 3D FE analysis with PLAXIS, and () the fibe analysis with the ompute ode USC-RC (Esmaeily 001). M / M y h p (m) Μ ω Q / Q y N Q h p, F sv = 1.5 h p, F sv = inf u pl /φ pl, F sv = inf Nomality ω (deg) h p Figue 3. Finite element veifiation of the assoiative plasti flow ule fo a wide ange of load ombinations. Obseve that the depth to the plasti hinge h p ineases fo ineasing negative load angles [M / Q = ω, ω* = (Q y / M y ) ω] eahing a maximum value at the vetex of the failue envelope (at appoximately ω = -50 o ). Fo load angles geate than ω > 50 o, the plasti hinge moves violently fom its deepest loation to the head of the pile. FINITE ELEMENT ANALYSIS Single Pile A 16 m long pile with a diamete of 1 m is analyzed. Both the pile and the soil ae modeled with 10-node tetahedal elements. The plan view dimensions of the finite element model is 1.3L x 1.3L in width with a depth of 1.5 L (whee L is the length of the pile), aefully weighting the effet of the boundaies on the esponse of the pile and the omputational time. Zeo-displaement bounday onditions pevent the out of

5 plane defomation at the vetial sides of the model, while the base is fixed in all thee dietions. Speial intefae elements wee plaed between the pile and the soil, thus allowing slippage and gapping to ou. Fo the total stess analysis unde undained onditions, soil behaviou is desibed by the Moh-Coulomb model with S u = 50 kpa, and speifi weight of γ = 18 kν/m 3, elastiity modulus of E s = 5000 kpa and Poisson s atio of v = Based on the maosopi onstitutive law fo einfoed onete iula pile setions, the behaviou of the pile was modelled via a Moh-Coulomb failue iteion with = 156 kpa, φ = 0 ο, tension ut-off σ t = 7534 kpa and Elastiity modulus E = 30 GPa, oesponding to a longitudinal einfoement atio of A s = 1.5 %. Fig ompaes the failue envelopes in moment shea foe spae as fo 4 diffeent fatos of safety against vetial loading, as it is pedited by an uppe bound limit equilibium solution based on Bom s theoy (1964) fo the lateal apaity of piles and alulated by the FE models. The hypothesis of plasti flow ule fo the inemental displaement to otation atio at the pile head is veified in Fig 3. 3 m 5 m 1.5 m 3 m 5 m Dietion of lateal loading 5 m Figue 4. The two studied pile goup onfiguations and thei oesponding FE models. The piles ae 16 m in length with a diamete of 1 m and longitudinal einfoement atio of A s = 1.5 % Pile Goup The Bom s theoy fo the ultimate lateal apaity of single piles is evisited and is extended in the analysis of pile-goups with due onsideation to pile-to-pile inteation. Due to lak of spae, only the esults fo: (1) a 1 x and () a x goup of piles with a minimum spaing fom ente to ente of s = 3D ae pesented (Fig 4). The piles, whih ae igidly onneted to the ap, ae initially subjeted to a vetial load equal to 1/5 of the ultimate axial beaing apaity of the whole goup (stati fato of safety to vetial loading F sv = 5). Then, they ae loaded to a ombined M-Q load until omplete failue. The geometial and mehanial popeties of the piles as well as those of the soil ae as those fo the single pile. Fig 5a ompaes the failue envelopes in dimensionless M-Q spae as pedited by the limit equilibium analysis and alulated by the FE models. M y, Q y and N y ae the apaities of the pile in pue ovetuning moment, shea foe and axial foe loading, espetively. A 3D epesentation of the failue sufae fo the 1 x pilegoup is also depited in figue 5b. Finally, Fig 6 identifies all the possible failue mehanisms of lateally loaded pilegoups: (1) axial

6 beaing apaity failue aompanied with in-depth plasti hinges, () plasti hinges at the onnetion with the ap and at a etain depth, (3) axial beaing apaity failue in onjuntion with pile-to-ap plasti hinges M (knm) 0000 N/Ny* Q (kn) Pile-goup: 1 x (a) Pile-goup: x (b) Figue 5. (a) Compaison of the failue envelopes fo a pile in ohesive soil alulated by PLAXIS and pedited by the limit equilibium method fo a fato of safety to vetial loading F sv = 5, and (b) 3D epesentation (in M-Q-N spae) of the failue sufae fo the 1x pilegoup. The pile has a diamete od D = 1 m and a longitudinal einfoement atio of A s = 1.5 %. M / M y 1 3 Q / Q y 1 3 Figue 6. Identifiation of failue mehanisms fo lateally loaded pilegoups (analysis with PLAXIS). Thee aeas ae distinguished eah oesponding to a speifi failue mode: Mode 1: Axial beaing apaity failue with in-depth plasti hinges, Mode Plasti hinges at the onnetion with the ap and at a etain depth, Mode 3: Axial beaing apaity failue with pile-to-ap plasti hinges. MACROELEMENT MODELING The model is fomulated in the famewok of lassial elastoplastiity, and ombines featues of: (a) the bounding sufae plastiity, (b) the itial state onept, and () a hadening evolution law and unloading-

7 eloading ule of the modified Bou-Wen type. Aoding to this fomulation the tangent elastoplasti stiffness matix that elates the inemental foe veto to the inemental displaement veto, is given in the following matix fom: 1 1 ep Τ e T e T e Κ Χ Κ I Φ g Φ f Κ Φg Φ f Κ Η Η Χ (6) in whih K e is the elasti stiffness matix of the piles with due onsideation to pile-to-pile inteation (in the elasti egime), Φ f and Φ g aount fo the failue sufae and plasti flow ule, espetively, and H 1 and H ontol the hadening law and unloading-eloading ule. The tems in maties H 1 and H ae funtions of the dimensionless hadening paamete ζ, whih is of the Bou-Wen type (Bou 1971, Wen 1976, Geolymos and Gazetas, 005; Geolymos et al. 009, Dosos et al. 01; Tasiopoulou and Geolymos 010). Finally, X is the geometi tansfomation matix that aounts fo the kinemati onstaints that the pile-ap (in the ase of pile-goups) imposes on the piles. CONCLUSION A maoelement model was pesented fo analysing the esponse of piles and pile-goups unde lateal loading. Two of the most fundamental omponents of the poposed maoelement, the failue envelope and the flow ule at post-failue (o at failue) onditions, ae detemined by exploiting the esults fom a ompehensive 3D finite element analysis with ode PLAXIS. The maosopi, in tems of bending moment uvatue elation, nonlinea esponse of pile unde ombined axial-moment loading was desibed by an appopiately alibated Moh-Coulomb based stess-stain model. The esults fom the numeial analysis ompae well with those fom a limit equilibium solution and veify the hypothesis of assoiativity of the plasti flow ule. REFERENCES Bou R. Modele mathematique d hysteesis. Austia. 1971, 4(1): 16-5 (in Fenh) Boms, B Lateal esistane of piles in ohesive soils. Jounal of Soil Mehanis and Foundation Division, ASCE, 90(3): Dosos V., Geolymos N., Gazetas G. Constitutive Model fo Soil Amplifiation of Gound Shaking: Paamete Calibation, Compaisons, Validation. Soil Dynamis and Eathquake Engineeing.01, 4: Geolymos N. and Gazetas G Phenomenologial model applied to inelasti esponse of soil pile inteation systems. Soils and Foundations, 45(4): Geolymos N., Dosos V., Gazetas G. Seismi Response of Single-Column Bent on Pile: Evidene of Benefiial Role of Pile and Soil Inelastiity. Bulletin of Eathquake Engineeing. 009, 7(): Esmaeily A USC-RC. Softwae fo analyzing behavio of a single einfoed onete membe. Vesion 1.0., USC Civil Engineeing Depatment. Beaty M, Byne P. UBCSAND onstitutive model, Vesion 904aR. Doumentation Repot: UBCSAND onstitutive model on Itasa UDM Web Site, 011, pp. 69. Matin, C. M., Houlsby, G. T Combined loading of spudan foundations on lay: laboatoy tests, Geotehnique, 50(4): Tasiopoulou P., Geolymos N. Constitutive Modeling of Sand: Fomulation of a New Plastiity Appoah, Soil Dynamis and Eathquake Engineeing. 016, 8: Wen Y.K. Method fo andom vibation of hysteeti systems. J. Engng Meh., ASCE. 1976, 10():