KINEMATIC BENDING OF FIXED-HEAD PILES IN NON- HOMOGENEOUS SOIL

Transcription

1 KINEMATIC BENDING OF FIXED-HEAD PILES IN NON- HOMOGENEOUS SOIL Raffaele Di Laora Univerity of Napoli Parthenope Emmanouil Rovithi Intitute of Engineering Seimology and Earthquake Engineering EPPO-ITSAK, Thealoniki, Greece Abtract Kinematic bending of pile in inhomogeneou oil i explored in tatic and dynamic regime. The ytem under conideration conit of a fixed-head pile embedded in a continuouly inhomogeneou vicoelatic oil layer reting on a rigid bae. A generalized parabolic function i employed to decribe the variable hear modulu in the inhomogeneou tratum. The problem i treated numerically by mean of rigorou elatodynamic finiteelement analye. A deign formula for kinematic pile-head moment i derived both for tatic and dynamic loading by employing a characteritic pile wavenumber. A new normalization cheme for dynamic pile bending i propoed by mean of a ingle dimenionle frequency parameter governing kinematic pile-head moment. A numerical example i alo provided. 1. Introduction Piled foundation may be ubjected to large curvature during earthquake due to deformation developing in the urrounding oil, even in abence of force applied at the top. Thi interaction mechanim i known a kinematic interaction. Evidence on kinematically-treed pile ha been identified in pot-earthquake obervation (Tazoh et al. 1984, Mizuno 1987) in oil that have not experienced large movement uch a thoe induced by liquefaction. The above field data ha revealed the poibility of damage cloe to the pile head or near interface eparating oil layer with harply different tiffne. Pile-oil kinematic interaction ha been the ubject of ytematic reearch (e.g. Kavvada and Gazeta 1993, Mylonaki 2001, Nikolaou et al. 2001, Di Laora et al. 2012) providing implified olution for kinematic pile moment at the interface between two conecutive layer with harply differing tiffne. However, pile-head kinematic moment may be equally important for oil with mall tiffne near urface where kinematic force tend to dominate over inertial one, epecially for large-diameter pile (Di Laora and Mandolini 2011). Kinematic bending moment at the pile-head are explored in thi paper referring to a long pile embedded in a continuouly inhomogeneou layer over a rigid bae. The variation of oil tiffne with depth i decribed by a generalized parabolic function accounting both for zero and finite hear modulu at the urface. The cope of the tudy i: (a) to elucidate the role of key dimenionle parameter of the problem; (b) to propoe a new formulation for the active length of pile in continuouly inhomogeneou oil; (c) to introduce the notion of an ective oil curvature being equal to pile-head kinematic curvature both in tatic and dynamic regime and (d) to implement a unique dimenionle frequency governing dynamic pile-head kinematic bending.

2 Figure 1. Single elatic fixed-head pile embedded in a continuouly inhomogeneou layer over rigid rock. 2. Problem tatement The ytem under conideration conit of a fixed-head pile embedded in a continuouly inhomogeneou vicoelatic oil layer on a rigid bae (Fig. 1). The pile i modelled a a linearly elatic cylindrical olid beam of diameter d, length L, elatic modulu E p and ma denity ρ p. Soil ma denity, ρ, Poion ratio, v, and hyteretic damping ratio, β, are conidered contant with depth, wherea hear modulu G (z) i aumed to increae according to the generalized power law function: z G (z) = Gd a + (1 a) d n (1) where a = (G o / G d ) 1/n and n are dimenionle inhomogeneity factor, G o being the hear modulu at ground urface (z = 0) and G d referring to the hear modulu at the depth of one pile diameter (z = d). For value of the inhomogeneity factor n cloe to zero or G o / G d ratio cloe to 1, Eq. 1 decribe a homogeneou medium (i.e. G o = G d ), wherea for n = 1 a Gibon-type oil i modeled having either zero or finite tiffne at the free urface depending on the value of parameter a. The pile-oil ytem i ubjected to harmonic S-wave having different frequencie propagating in the oil ma. 3. Static behavior 3.1 One-dimenional oil repone Under contant ground acceleration (a ), equilibrium of an one-dimenional oil column with contant ma denity ρ and variable hear modulu, G(z), i decribed by the differential equation: du (z) dz d G(z) = a ρ dz (2) Upon integrating twice and impoing the boundary condition of zero hear tre at oil urface [(0) = 0] and zero oil diplacement at the bae [u (H) = 0], oil lateral diplacement i obtained a: 2 1n aρd H H 2 G d (1a) (n2)(n1) d d u(z)= a+(1 a) a+1a n1 1n z z a+(1 a) a+(1 a) n 1 d d (3) The olution in term of oil hear train and oil curvature i expreed by:

3 a ρ z (z) = a + (1 a) G d d z d 1 dγ (z) aρ z z (z) = a (1a)(n 1) a + (1a) R dz G d d n n1 (4) (5) Figure 2. Ditribution of free-field and pile curvature along depth and definition of oil ective curvature. 3.2 Effective oil curvature (1/R), a a meaure of pile-head curvature Toward the definition of a phyically-baed interaction factor to be applied in any uboil condition, one could think that pile-head curvature (1/R) p hould depend on the whole ditribution of free-field deformation along the pile (Fig. 2). Hence, in thi tudy the notion of an ective oil curvature (1/R), i introduced a: γ (z ) (1/R), = (6) z where z tand for an ective depth of oil contributing to kinematic pile-head bending and γ (z ) i the correponding hear train of oil computed at z = z by mean of Eq. 4. (1/R), in Eq. 6 may be viewed a an average oil curvature along z which ha a finite value and reflect the phyic of the phenomenon (Fig. 2). The ratio of pile-head curvature to ective oil curvature [(1/R) p / (1/R), ] i employed herein to decribe pile-head kinematic bending in inhomogeneou oil. The ective depth z introduced in Eq. 6 may be derived by conidering that the ective portion of oil controlling pile head bending i proportional to a characteritic wavelength of the pile-oil ytem. To thi end, an average wavenumber μ may be introduced a (Mylonaki 1995): μ = 1 L a λ(z)dz L (7) 0 a where the active pile length L a may be taken equal to 10 pile diameter, a a firt approximation, for typical value of pile-to-oil tiffne ratio, baed on the correponding expreion propoed by Randolph (1981), andzi the well-known (tatic) Winkler wavenumber. By conidering μ and λ a real-valued function implying low-frequency loading and auming that the Winkler pring modulu k x (z) varie with depth with the ame law a oil Young modulu E (z) doe, the olution of the integral in Eq. 7 may be expreed a:

4 4 4n 4n d μ 4 4 n ad ad La ala 4 d La 4na1 (8) In Fig. 3, finite-element reult in term of pile-head curvature (Di Laora and Rovithi 2014) are compared to the ective oil curvature computed by mean of Eq. 6, where z i aumed a: z = 1.25 μ (9) Figure 3. Correlation between pile-head curvature (FE) and oil ective curvature (Eq 6 and 9) Table 1. Parameter value employed in the parametric tudy. ρ p / ρ E p / E d a n ν A et of 136 oil-pile configuration were analyzed in total baed on the parameter ummarized in Table 1. Each circle point in Fig. 3 correpond to a different oil-pile configuration. The excellent correlation between the above parameter i evident. 3.3 Active pile length and deign formula Active pile length L a i defined in thi tudy a the length beyond which a further increae in pile length doe not exert any ignificant ect on pile-head bending. A et of finite element analye wa performed to derive a imple expreion of L a that may be readily employed in deign procedure. For thi reaon, a cylindrical pile of increaing length embedded in the inhomogeneou oil under tudy wa employed. (1/R) p /(1/R), ratio are plotted in Fig. 4a againt the dimenionle pile length μl. It i oberved that for μl larger than 2.5, the curve converge leading to (1/R) p /(1/R), ratio equal to 1. The above value may be interpreted a an active dimenionle pile length, leading directly to a imple expreion for L a : L a =2.5 μ (10)

5 Upon combining Eq 1, 4, 6, 9 and 10, the following deign formula for kinematic pile-head bending i obtained: a ρ 1/R = p G z G L a /2 a ρ (11) The above expreion imply that the ective depth z ha the following propertie: (a) it i half of the active pile length; (b) the oil hear modulu G(z ) repreent the tiffne of an equivalent homogeneou oil that lead to equal kinematic pile-head bending a in the inhomogeneou cae. With reference to hort pile (i.e. μl < 2.5), Fig. 4a allow alo a imple expreion for the kinematic head curvature to be employed for practical purpoe: (1/R) (1/R) p,. 2 =0.3 μl (12) (1/R) p,dynamic / (1/R) p,tatic (b) (a) Figure 4. (a) (1/R) p /(1/R), ratio a function of pile mechanical lenderne μl (b) Dynamic reduction of pilehead curvature a function of the dimenionle frequency parameter a. 4. Dynamic behavior Toward an identification of the key parameter involved in kinematic pile-head bending under dynamic action, an average hear wave velocity V,av. i defined a: V,av z d 1-n /2 dz 1-n /2 z 0 z z (a-1)(n-2) V V(z) 2d -a a (1- a) d (13) providing equal travel time between a homogeneou oil with V =V,av and an inhomogeneou oil. In thi manner, a new normalization cheme for dynamic pile bending i propoed by mean of the dimenionle frequency parameter a : ω a = (14) μv,av

6 (1/R) dyn. /(1/R) t ratio for the complete et of inhomogeneou oil under invetigation are plotted in Fig. 5b againt a.. The evident imilarity of the curve indicate that kinematic pile-head moment in inhomogeneou oil are eentially governed by the ingle dimenionle frequency parameter given in Eq. 15. Approximate mean and upper-bound curve are alo uggeted a a ueful manner to determine dynamic ect in kinematic pile-head bending. 5. Application example The cae of a fixed-head olid cylindrical concrete pile embedded in normally-conolidated clay i employed a an application example of the propoed analyi. The pile ha diameter d = 0.8 m and Young modulu E p = 30 GPa. The evaluation of kinematic demand i performed under the conervative aumption of low frequency excitation. Soil hear modulu varie linearly with depth according to the law G(z) = z, where G(z) i expreed in kpa and z in meter, correponding to a = 0.6 and n = 1 in Eq. 1. Shear modulu G d at a depth of one pile diameter i, therefore, equal to 1700 kpa. Poion coicient and ma denity were et at 0.5 and 1.8 Mg/m 3, repectively, correponding to undrained condition. The deign acceleration at oil urface i a = 0.3g; g being the acceleration of gravity. Under the reaonable aumption δ = 2, the avearge wavenumber μ i obtained from Eq. 8 a: μ m (15) by taking L a = 10d, which i ufficiently accurate a a firt approximation. The ective depth and the correponding hear modulu of oil may then be computed a z = L a /2 = 1.25/μ = m and G(z )= = 5963kPa. Accordingly, the kinematic bending moment M kin at the pile head i: a (16) 4 M kin = EpIp / knm G z 5963 Note that if a homogeneou oil of equal hear modulu with the inhomogeneou oil at urface had been adopted, the correponding value of kinematic bending moment would have been equal to 6391 knm reulting evidently in a conervative deign of pile reinforcement (10 time larger). Reference Tazoh T, Dewa K, Shimizu K and Shimada M (1984) Obervation of earthquake repone behaviour of foundation pile for road bridge Proceeding of the 8th World Conference on Earthquake Engineering, San Francico, Mizuno H (1987) Pile Damage during Earthquake in Japan In: Nogami T, editor. Dynamic Repone of Pile Foundation, ASCE Special Publication. Kavvada M and Gazeta G (1993) Kinematic eimic repone and bending of free-head pile in layered oil Geotechnique, 43(2): Mylonaki G (2001) Simplified model for eimic pile bending at oil layer interface Soil and Foundation, 41(4): Nikolaou A, Mylonaki G, Gazeta G and Tazoh T (2001) Kinematic pile bending during earthquake: Analyi and field meaurement Geotechnique, 51(5): Di Laora R, Mandolini A and Mylonaki G (2012) Inight on kinematic bending of flexible pile in layered oil Soil Dynamic and Earthquake Engineering, 43: Di Laora R and Mandolini A (2011) Some Remark about Eurocode and Italian code about piled foundation in eimic area Proceeding of ERTC-12 Workhop on Evaluation of EC8, Athen, Greece. Randolph MF (1981) The repone of flexible pile to lateral loading Géotechnique, 31(2): Di Laora R and Rovithi Emm (2014) Kinematic bending of fixed-head pile in non-homogeneou oil Accepted in 2ECEES Conference Proceeding, Augut, Itanbul, Turkey.