LIMIT STATE ANALYSIS OF FIXED-HEAD CONCRETE PILES UNDER LATERAL LOADS

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1 1 th Wold Confeence on Eathquake Engineeing Vancouve, B.C., Canada August 1-6, 004 Pae No. 971 LIMIT STATE ANALYSIS OF FIXED-HEAD CONCRETE PILES UNDER LATERAL LOADS S. T. SONG 1, Y. H. CHAI & Tom H. HALE SUMMARY Unde seismic loads, dee foundations with fixed ile/ile-ca connection may be subjected to a lage cuvatue demand at the ile head. Damage induced by local inelastic defomation deends on the magnitude of the lateal dislacement imosed on the ile. In this ae, an analytical model elating the dislacement ductility facto to the local cuvatue ductility demand is oosed fo fixed-head iles embedded in cohesive and cohesionless soils. The model indicates that the cuvatue ductility demand deends on the stength and stiffness of the soil-ile system, as well as the location and length of the lastic hinges. The model is useful fo design of fixed-head iles since it is caable of estimating the seveity of the local damage in the ile fo a wide ange of ile and soil oeties. The vesatility of the model is illustated using an examle of a fixed-head concete ile constucted in soil tyes cuently classified in the US building codes. Seismic efomance of the ile subjected to dislacement ductility facto commensuate with the cuent design is assessed fo diffeent soil conditions. INTRODUCTION Dee foundations fo buildings and bidges often ely on the use of concete iles that ae estained fom otation at the ile head. Unde lateal seismic loads, howeve, the fixity at the ile/ile-ca connection induces a lage cuvatue demand at the ile head, with a otential fo failue in the ile. Sevee damage of ile-suoted foundations had been obseved in ecent eathquakes. As ost-eathquake insection of ile foundations is difficult, damage assessment of iles becomes imotant, aticulaly if a cetain level of efomance is to be guaanteed fo the stuctue. Fo a fixed-head ile subjected to a lage lateal load, sequential yielding of the ile occus until a lastic mechanism is fully develoed. Figue 1 shows the deflected shae and the associated bending moment distibution at vaious limit states of a lateally loaded fixed-head ile. The fist yield limit state of the ile, which is shown in Figue 1(a), is chaacteized by a maximum bending moment at the ile/ile-ca 1 Gaduate Reseach Assistant, Det. of Civil & Envion. Eng., UC Davis, Califonia, USA. ssong@ucdavis.edu Associate Pofesso, Det. of Civil & Envion. Eng., UC Davis, Califonia, USA. yhchai@ucdavis.edu Senio Stuctual Enginee, Office of Statewide Heath Planning and Develoment, Sacamento, Califonia, and Chai of the SEAOC Cental Seismology Committee, USA. THale@oshd.state.ca.us

2 connection whee the flexual stength M u of the ile is eached. A lastic hinge is then assumed to fom at the ile head with the cente of otation occuing at the gound level. Futhe dislacement beyond the fist yield limit state involves a concentated otation of the lastic hinge, which is accomanied by a edistibution of intenal foces in the ile. The edistibution inceases the bending moment in the nonyielding otion of the ile until the fomation of a second lastic hinge. Figue 1(b) shows the second yield limit state whee the second lastic hinge foms at a deth L m. Continued lateal dislacement afte the second lastic hinge fomation is facilitated by inelastic otations in both lastic hinges until the ile eaches the ultimate limit state, as shown in Figue 1(c). The ultimate limit state is assumed to be associated with a flexual failue, as dictated by a limiting cuvatue in the lastic hinge. In ode to contol the damage due to flexual yielding of the ile, the cuvatue ductility demand in the ile fom an imosed lateal dislacement must be oely assessed. A simle mechanistic model is develoed in subsequent sections fo chaacteizing the lateal esonse of fixed-head iles fo vaious limit states. The model is caable of edicting the lateal stiffness and lateal stength of the ile as well as the cuvatue ductility demand in the ile. Figue 1. Deflected shae and bending moment distibution of a lateally loaded fixed-head ile (a) fist yield limit state, (b) second yield limit state, and (c) ultimate limit state. ANALYTICAL MODEL Satisfactoy seismic efomance of fixed-head iles deends on the level of inelastic defomation imosed on the ile. Inelastic defomation, as commonly chaacteized in tems of cuvatue demand, is elated to the stiffness and stength of the soil-ile system as well as the lastic hinge length of the ile. In this section, a kinematic model, which elates the dislacement ductility facto to the cuvatue ductility facto, is deived. The model is develoed fo diffeent soil conditions. Lateal Stiffness of Soil-Pile System: Cohesive Soils A common aoach fo seismic design of ile foundations assumes that a lateally loaded soil-ile system can be analyzed as a flexual membe suoted by an elastic Winkle foundation. In this case, the soil is elaced by a seies of sings, which ovide a soil eaction that is ootional to the lateal deflection. Fo cohesive soils, the stiffness of the soil-sing is assumed to be indeendent of the deth, esulting in a constant hoizontal subgade eaction k h (in units of foce/length ) fo the Winkle foundation. Closedfom solutions fo the deflection and bending moment of an elastic ile embedded in cohesive soils ae

3 well known [1]. Fo a fixed-head ile with an imosed lateal dislacement at the gound level, the lateal stiffness of the soil-ile system is given by V EIe K 1 = (1) Rc whee V is the lateal foce equied to oduce an elastic dislacement, EI e is the effective flexual igidity of the ile, and R c is the chaacteistic length of the ile, which is defined as R 4 c EI e / kh. At the fist yield limit state, the lateal deflection y1 at the gound level can be obtained by equating the bending moment at the ile/ile-ca connection to the ultimate moment caacity M u of the ile, assuming an elasto-lastic moment-cuvatue esonse i.e. M u R c y1 = EIe () Using the lateal stiffness K 1 of Eq. (1) and the yield dislacement y1 of Eq. (), the lateal foce to cause the fomation of the fist lastic hinge is: M u Vy = K1 y1 = () Rc Uon the fomation of the fist lastic hinge, the bounday condition of the ile effectively changes to a fee-head condition, whee closed-fom solutions ae also eadily available. The educed lateal stiffness K and the coesonding lastic otation θ at the gound level afte the fist yield limit state ae given by: V Vy EI K = fo V > V y and > y1 (4) c y1 e R c y1 θ = fo > y1 (5) R The lateal stiffness of fixed-head iles embedded in cohesive soils equies the detemination of the chaacteistic length R c of the ile, which in tun equies an estimation of the modulus of hoizontal subgade eaction k h. An exession fo k h has been oosed by Davisson [] fo estimating the modulus of hoizontal subgade eaction of cohesive soils: kh = 67 s u (6) whee s u is the undained shea stength of the cohesive soil, which may be detemined fom field tests o fom site classifications in cuent US building codes. Fo examle, NEHRP [] o ATC-40 [4] ovides a coelation between the undained shea stength and soil ofile tye, which is eoduced in Table 1 fo comleteness of this ae. Table 1. Soil ofile classifications and soil oeties (adated fom NEHRP [] and ATC-40 [4]) Soil ofile Descition Shea wave velocity (m/sec) SPT N (blows/0.05m) Cohesive soils Undained shea stength (kn/m ) Cohesionless soils Fiction angle φ (degees) S E Soft soil < 180 < 15 < 50 < S D Stiff soil S C Dense soil > 50 > 100 > 40

4 Lateal Stiffness of Soil-Pile System: Cohesionless Soils The above aoach fo detemining the lateal stiffness of fixed-head iles can be extended to iles embedded in cohesionless soils. In this case, the soil esistance is modeled by a Winkle foundation with a linealy inceasing modulus of hoizontal subgade eaction. Solutions fo estimating the defomation and bending moment of elastic iles in cohesionless soils have been oosed by Matlock and Reese [5]. Fo a fixed-head ile, the lateal stiffness of the soil-ile system is given by: V EIe K 1 = Rn (7) 5 whee R n is the chaacteistic length, which is defined as Rn EIe / nh fo cohesionless soils, and n h is the constant ate of incease of the modulus of hoizontal subgade eaction (in units of foce/length ). At the fist yield limit state, the lateal deflection and foce at the gound level, denoted as y1 and V y esectively, ae given by: n M u R y1 = (8) EIe M u Vy = K1 y1 = (9) R n Simila to the case fo cohesive soils, the educed stiffness K afte the fist yield limit state and its coesonding otation at gound level fo iles in cohesionless soils ae: V Vy EI K = fo V > V y and > y1 (10) y1 R e n y1 θ = fo > y1 (11) Rn The lateal stiffness of a ile embedded in cohesionless soils deends on the ate of incease of modulus of hoizontal subgade eaction n h. An estimation of n h and its coelation with the effective fiction angle and elative density of the soil is suggested in ATC- [6] and is eoduced in Figue in this ae. Note that Table 1, which is adoted fom ATC-40 [4], also ovides a coelation between the fiction angle and the soil ofile classifications by NEHRP []. NEHRP [] Soil Pofile Tye S S S E D C Subgade Coefficient n h (kn/m ) 8 o Fiction Angle 9 o 0 o 6 o 41 o 45 o Vey Loose Loose Medium Dense Vey Dense n h (kn/m ) 1 Above Wate Below Wate Relative Density (%) Figue. Subgade coefficient of cohesionless soils [6]

5 Lateal Stength of Soil-Pile System: Cohesive Soils Fo a lateally loaded fixed-head concete ile, the esonse may be assumed to be chaacteized by a deendable lateal stength with a vaying level of ductility caacity, deending on the level of confinement ovided fo the ile. The lateal stength of the ile can be detemined by assuming that a sufficiently lage deflection has occued so that an ultimate soil essue, extending to the deth of the maximum bending moment, is fully develoed. The deth to the maximum bending moment, which deends on the flexual stength of the ile and the ultimate essue of the soil, defines the location of the second lastic hinge and theefoe influences the lateal stength and ductility of the ile. The magnitude and distibution of the ultimate soil essue acting on the ile deend on the failue mechanism of the soil, the shae of the ile coss-section, the fiction between the ile suface and suounding soil, etc. Fo cohesive soils, an estimation of the ultimate soil essue distibution may be obtained fom consideation of a failue mechanism of the soil aound the ile, as suggested by Reese and Van Ime in [7]. The failue mechanism in the ue egion is contolled by a sliding soil wedge esulting in a soil essue that inceases linealy with deth, while a lastic flow occus aound the ile in the lowe egion leading to a constant ultimate lateal essue. Using such failue mechanism, the ultimate soil essue distibution of cohesive soils maybe witten as: 9 x + su fo x x ψ D = u( x ) (1) 11su fo x > x The deth delineating the two egions is defined as the citical deth x, which vaies with the undained shea stength and is given by 9 su x ψ D = D (1) γ' D + su whee ψ is the citical deth coefficient and γ ' is the effective unit weight of the soil. The ultimate soil essue distibution of Eq. (1) is lotted in Figue (a) and will be used fo calculating the lateal stength of fixed-head iles embedded in cohesive soils. V P M u s u V P M u Gound Level L m x L m M u M u D 11s u D σ' vk Reese and Van Ime [7] (a) Boms [8] (b) Figue. Ultimate soil essue distibution fo lateally loaded fixed-head iles: (a) cohesive soils, (b) cohesionless soils.

6 The deth at which the second lastic hinge foms L m and the ultimate lateal stength of the soil-ile system V u can be detemined using the equilibium condition fo lateal foce and bending moment. Fo cohesive soils with an ultimate essue distibution given by Eq. (1), the nomalized deth to the second lastic hinge, defined as L m L / D, is given by the solution of the equation: M u 1 L = 11 L 4 m m + L ψ ψ 4 m m fo L m fo L m ψ > ψ whee M u M u / ( su D ) is the nomalized flexual stength of the ile. Uon the detemination of the nomalized deth to the second lastic hinge, the nomalized lateal stength V can be obtained by V u 9 L m L m + ψ = 9 11L m ψ fo L fo L m m ψ > ψ whee the nomalized lateal stength is defined as Vu Vu / ( su D ). Lateal Stength of Soil-Pile System: Cohesionless Soils The lateal stength, as well as the deth to the second lastic hinge, of a fixed-head ile embedded in a cohesionless soil can be detemined similaly to that fo cohesive soils. Studies on the magnitude and distibution of the ultimate soil essue on iles in cohesionless soils have been made in the ast, and an ultimate lateal essue distibution that is convenient fo design has been oosed by Boms [8]. The lateal essue u on the ile is taken to be equal to times the Rankine assive essue of the soil: u ( x ) = σ v ( x) K (16) whee σ v (x) is the vetical effective ovebuden stess, which may be taken as the effective unit weight γ ' multily by the deth x, and the tem K is the coefficient of assive soil essue and is given by K = 1 ( 1+ sin φ) ( sin φ) whee φ is the fiction angle of the cohesionless soil. The ultimate essue distibution of Eq. (16), which vaies linealy with deth, is shown in Figue (b). The deth at which the second lastic hinge foms L m and the ultimate lateal stength of the soil-ile system V u can be detemined using the ultimate soil essue distibution of Eq. (16). The nomalized deth to the second lastic hinge, defined as L m Lm / D, and the nomalized ultimate stength, defined as Vu Vu / ( K γ D ), fo iles in cohesionless soils, ae given by: L m = M u (18) m V u = L (19) 4 whee the nomalized flexual stength M is defined as M u M u / ( K γ D ). u u (14) (15) (17)

7 Kinematic Relation between Dislacement and Cuvatue Ductility Factos To ensue a satisfactoy efomance, the seveity of local damage may be contolled by limiting the cuvatue ductility demand in the otential lastic hinge egion. The cuvatue ductility demand, which is diffeent fo the two lastic hinges, deends on the dislacement ductility imosed on the ile. In ode to estimate the local inelastic defomation and hence the cuvatue ductility demand in the citical egion, the lateal esonse of a fixed-head ile is aoximated by a ti-linea foce-dislacement esonse, as shown in Figue 4, with an initial stiffness K 1 followed by a educed stiffness K. The fist and second yield limit states ae defined by the lateal dislacement y1 and y, esectively. The lateal dislacement beyond y is chaacteized by a constant lateal foce signifying a fully lastic esonse. The ultimate limit state is defined by the lateal dislacement u, which deends on the ductility caacity of the lastic hinges. The lateal foce-dislacement esonse of the fixed-head ile can be futhe idealized by a bilinea elastolastic esonse, which is also shown in Figue 4, with the equivalent elasto-lastic yield dislacement y. Heein the dislacement ductility facto µ is defined as u y µ = y y + y whee is the inceased lastic dislacement fom the stage of second lastic hinge fomation to the ultimate limit state, as indicated in Figue 4. The elation between the dislacement and cuvatue ductility factos can be established by fomulating Eq. (0) as a function of the yield and ultimate cuvatues of the ile section. (0) Lateal Foce V V u V y Elasto-Plastic Resonse K Limit State of 1 st Hinge Limit State of nd Hinge Ti-Linea Resonse Actual Resonse Ultimate V L m µ P 1 st Hinge u K 1 = y nd Hinge y1 y y u Lateal Dislacement Figue 4. Idealized lateal foce-dislacement esonse of fixed-head iles The equivalent elasto-lastic yield dislacement y in Eq. (0) may be elated to the ultimate lateal stength V u using the initial stiffness K 1 of the bilinea cuve: V u y = K 1 (1) while the lateal dislacement at the second yield limit state y in Eq. (0) may be detemined fom the idealized ti-linea esonse, i.e.: Vy Vu Vy y = + () K1 K whee V y is the lateal foce equied to develo the fist lastic hinge. The next ste of the fomulation involves the detemination of the lastic otation fo both hinges. The lateal dislacement of the ile fom y to u esults in a otation of θ in both lastic hinges. The lastic hinge otation θ is elated to the lastic dislacement, as shown in Figue 1(c), by

8 θ () = Lm whee L m = L m D is the deth to the second lastic hinge. In ode to estimate the cuvatue ductility, the lastic otation is taken to be unifomly distibuted ove the lastic hinge. Fo the fist lastic hinge, the lastic otation θ can be witten as ( φu1 φi ) 1 θ = L fo φu 1 φi φ y (4) whee φ i is the cuvatue in the fist lastic hinge at the lateal dislacement y, φ u1 is the ultimate cuvatue in the fist lastic hinge, and L 1 is the equivalent lastic hinge length fo the fist hinge. By nomalizing the fist lastic hinge length to λ 1 L 1 / D, the combination of Eqs. () and (4) gives the lastic dislacement as ( φu1 φi ) 1 L m D = λ (5) Eqs. (1), () and (5) can be substituted into Eq. (0) to give a elation between the dislacement ductility facto and the cuvatue demand in the fist lastic hinge: ( Vu Vy ) Vy K1 λ 1 L m D ( φu1 φi ) µ = + + Vu K Vu y By defining the coefficients α Vy / Vu = y1 / y and β y / ( φ y Lm ), the cuvatue ductility facto µ φ1 in the fist lastic hinge is elated to the dislacement ductility facto µ by µ K 1 ( α) + ( µ φ1 i ) = α µ φ K β L m λ whee µ φ1 is defined as µ φ1 φ u 1 / φ y, and µ φi is the cuvatue ductility demand in the fist lastic hinge at the lateal dislacement y, which is defined as µ φi φ i / φ. The kinematic elation in Eq. (7) equies the detemination of the cuvatue ductility µ φi, which involves the lastic otation of the fist lastic hinge at the lateal dislacement y. The lastic dislacement, which occus fom the fist yield limit state to the second yield limit state, indicated as y = y1 in Figue 4, is associated with a lastic otation θ in the fist hinge. Assuming a unifom distibution of lastic otation in the lastic hinge, the cuvatue ductility facto µ φi is elated to the lastic otation θ by: φi θ µ φ i = = 1+ (8) φ y φ y L1 The lastic otation θ in Eq. (8) can be obtained using the exession fo ile head otation, as given in Eq. (5) fo cohesive soils and Eq. (11) fo cohesionless soils. By elacing the numeato y 1 by in Eq. (5) and (11), the lastic otation θ can be witten as: θ (9) = ηlm whee the coefficient η is defined as η Rc / Lm fo cohesive soils and η 1.5Rn / Lm fo cohesionless soils. Fom the idealized ti-linea esonse shown in Figue 4, the lastic dislacement in Eq. (9) is elated to the lateal stength and educed stiffness of soil-ile system, i.e. Vu Vy K1 Vu = = K K K1 Vy (0) By substituting V y = αv and V u u / K 1 y = βφ y Lm y = into Eq. (0), can be e-witten as (6) (7)

9 K1 = βφ y Lm ( 1 α) K The combination of Eqs. (8), (9) and (1) allows the cuvatue ductility µ φi to be detemined: K1 β L µ φ i = 1+ m 1 K ηλ 1 ( α) whee L m is the nomalized deth to the second lastic hinge and λ 1 is the nomalized lastic hinge length of the fist hinge. Uon the detemination of the intemediate cuvatue ductility facto, i.e. µ φi in Eq. (), the ultimate cuvatue ductility demand µ φ1 in the fist lastic hinge can be detemined using Eq. (7) fo a given dislacement ductility facto µ. Damage assessment of fixed-head iles also equies an estimation of the cuvatue ductility demand in the second lastic hinge, even though the cuvatue ductility demand would likely be smalle than that of the fist lastic hinge. Simila to Eq. (4) fo the fist lastic hinge, the otation θ due to the lastic dislacement of may be witten in tems of the ultimate cuvatue demand φ u in the second hinge: ( φu φ y ) θ = L φu φ y fo () whee L is the equivalent lastic hinge length of the second hinge. The combination of Eqs. () and () gives the lastic dislacement as ( φu φ y ) L m D = λ (4) whee λ L / D is the nomalized lastic hinge length of the second hinge. Following the same aoach fo the fist lastic hinge, Eqs. (0), (1), () and (4) can be solved simultaneously to obtain the elation between the dislacement ductility facto µ and the cuvatue ductility demand µ φ : K ( 1 α) + ( µ 1) 1 µ = α + φ K β L m λ whee µ φ φ u / φ y is the cuvatue ductility demand in the second lastic hinge. Note that Eq. (5) is simila to the kinematic elation fo the fist lastic hinge in Eq. (7), excet that the lastic hinge length is diffeent and the cuvatue ductility demand at lateal dislacement y is equal to unity fo the second lastic hinge. In ode to ensue a good efomance of a ile-suoted foundation, the ultimate dislacement imosed on the ile may be limited to a design dislacement. If the design dislacement is sufficiently lage to cause inelastic defomation in both lastic hinges, the cuvatue ductility demand can be edicted using the kinematic elation of Eqs. (7) and (5). Howeve, in the case of a small lateal dislacement whee the limiting design dislacement u is less than the dislacement at the fomation of the second lastic hinge y (but lage than y1 ), only one lastic hinge will fom at the ile head. In this case, the kinematic elation will be diffeent fom that given by Eq. (7). In ode to deive the kinematic elation fo this condition, the dislacement ductility facto µ may be witten as: u y1 ' µ = + (6) y y y whee y is the elasto-lastic yield dislacement as befoe, and ' is the lastic dislacement, which is less than o equal to. Simila to Eq. (9), the lastic dislacement ' can be elated to the lastic otation of the fist hinge θ by m ' = θ ηl (7) (1) () (5)

10 whee the coefficient η has been defined eviously fo cohesive and cohesionless soils. By witing the lastic otation as θ = ( φu1 φ y ) L 1, whee φ u1 is the ultimate cuvatue in the fist lastic hinge, the dislacement ductility facto µ in Eq. (6) can be witten as η L L y1 1 m u1 µ = + y y ( φ φ ) y Substituting α = y 1 / y and y = βφ y Lm into Eq. (8), the elation between the dislacement ductility facto µ and cuvatue ductility facto µ φ1 fo y1 u y is: ηλ 1 µ = α + ( µ φ1 1) (9) β L m whee µ φ1 φ u 1 / φ y. The set of equations, namely Eqs. (7), (5) and (9), allows a full ange of cuvatue ductility demand fo fixed-head iles to be estimated. Plastic Hinge Length of Fixed-Head Concete Piles The cuvatue demand in the yielding egion of a ile is elated to the equivalent lastic hinge length of the ile. Studies of bidge columns o extended ile-shafts have esulted in emiical exessions fo the equivalent lastic hinge length. Fo the case of fixed-head iles, it is easonable to assume that the length of the fist lastic hinge is simila to the lastic hinge length of a fixed-based bidge column, since the fist lastic hinge of the ile foms against a suoting membe. In this case, the equivalent lastic hinge length L 1 of the ile is assumed to be the same as that of a fixed-based column excet that the height of the column is elaced by one-half of the distance to the second hinge. This aoach is based on the assumtion that the bending moment in the ue egion of fixed-head iles is simila to the evesed moment distibution in a lateally loaded column with full fixity at both ends. Moe secifically, the equivalent lastic hinge length fo the fist hinge of the fixed-head ile is taken fom that oosed by Piestley et al. [9]: L = f d (40) Lm f ye dbl ye bl whee f ye is the exected yield stength of the einfocing steel (in MPa units) and d bl is the diamete of the longitudinal einfocement of the ile. The equivalent lastic hinge length of the fist lastic hinge, howeve, should not be taken as geate than the ile diamete. Fo the second lastic hinge, the sead of cuvatue will be moe significant than that of the fist lastic hinge. In this ae, the equivalent lastic hinge length fo the second lastic hinge is taken fom the lastic hinge length fo extended ile-shafts with a zeo above-gound height, as oosed by Chai [10]. In this case, a lastic hinge length of L = D, o a nomalized lastic hinge length of λ = 1.0, is aoiate fo the second lastic hinge. ILLUSTRATIVE EXAMPLE To illustate the use of the analytical model, conside the following einfoced concete fixed-head ile embedded in two diffeent soils, which ae classified accoding to NEHRP [] as: (1) class E site with cohesive soils, and () class C site with cohesionless soils. The ile has a diamete of D = 0.61 m and an embedded length of m, as shown in Figue 5. The following aametes ae assumed fo the ile: (i) the longitudinal einfocement is ovided by 1 # 5 bas, esulting in a longitudinal steel atio of 0.00; (ii) the tansvese einfocement is ovided by #1 sial at a itch of 75 mm, esulting in a confining steel atio of 0.015; (iii) a concete cove of 75 mm is used fo the tansvese sial of the ile; (iv) the exected concete comessive stength, as suggested by ATC- [6], is f ' ce = 1. f ' c = 44.8 MPa; and (v) the longitudinal and tansvese steel ae ovided by gade A706 steel with a exected yield stength f ye = 475MPa. The ile is subjected to an axial comession of 000 kn o 0. f c Ag. (8)

11 V P = 000 kn 0.61 m (1) #5 # 1 75 mm itch m Fixed-Head Concete Pile D = 0.61 m (Metic Ba Size Shown) Soil Conditions (1): Site Class E Soft Clay (): Site Class C Dense Sand Figue 5. Details of a fixed-head ile The moment-cuvatue esonse of the ile section may be idealized by an elasto-lastic esonse. In this 5 case, the effective flexual igidity of the ile is EI e = kn m, and the equivalent elastolastic yield cuvatue is φ y = ad/m. The ultimate bending moment of the ile, based on the elastolastic idealization, is M u = kn-m. As a means fo ensuing a good seismic efomance of a stuctue, studies have suggested that damage can be contolled by limiting the stain values in the citical egions. Fo examle, Kowalsky [11] suggested a damage-contol stain of fo the exteme comessive fibe of the confined concete coe, o fo the exteme tension fibe of longitudinal steel. Following this suggestion fo damage-contol of fixed-head iles, the limiting cuvatue of the ile in this examle is φ u = 0.16 ad/m, which will be taken as the ultimate cuvatue of the section. Thus fo the level of confining steel ovided fo the ile, the cuvatue ductility caacity is ( µ φ ) ca = To demonstate the alicability of the model, the cuvatue ductility demand will be estimated fo a ange of imosed dislacement ductility facto u to 4. Examle 1: Soft Clay in Site Class E The lateal esonse of the fixed-head ile in this examle will be assessed fo a cohesive soil, classified as ofile tye S E o soft clay e NEHRP []. The effective unit weight of the soft clay is taken as γ ' =17.5 kn/m and the undained shea stength is taken as s u = 5 kpa. Fom Eq. (6), the modulus of hoizontal subgade eaction is k h = 45 kn/m. It should be noted that the soil stiffness estimated by Eq. (6) is intended fo analyses at the woking load level. Fo assessment of cuvatue ductility demand, howeve, the soil stiffness should coesond to the fist yield limit state of the ile. Thus the soil stiffness that is aoiate fo cuvatue ductility assessment should stictly be educed since softening of the soil would have occued uon fist yielding of the ile. Cuently no ecommendation exists fo the aoiate level of modification, and as such, the lateal stiffness of the soil edicted by Eq. (6) will be used in this examle without eduction to illustate the ocedue. Fo k h = 45 kn/m and 5 EI = kn m e, the chaacteistic length of the ile is Rc 4 EI e / kh =. 57 m. The citical deth of the ile, beyond which the ultimate lateal esistance of soil emains constant, is x = 1.75 m, o coesonding to a citical deth coefficient of ψ =. 87, as estimated fom Eq. (1). The initial lateal stiffness of the soil-ile system, as calculated fom Eq. (1), is K 1 = 856 kn/m, wheeas the educed lateal stiffness, due to the fist lastic hinge fomation, is K = 468 kn/m, as calculated fom Eq. (4). The atio of the two lateal stiffness coefficients is K 1 / K =.0. Fom Eqs. () and (), the lateal dislacement and foce at the fist yield limit state ae y1 = 0.05 m and V y = 445 kn, esectively.

12 Using a nomalized flexual stength of M M / ( s D ) u = u u = 10.0 and a citical deth coefficient of ψ =.87, the nomalized deth to the second lastic hinge may be obtained by solving Eq. (14), which gives L m = 6.7, o an actual deth of L m =.8 m. The coesonding nomalized lateal stength may be estimated by Eq. (15), which gives V u = 56. 1, o an actual lateal stength of V u = 70 kn. Fom Eqs. (1) and (), the elasto-lastic yield dislacement is y = m and the lateal dislacement at the fomation of the second lastic hinge is y = m. The cuvatue ductility demand deends on the atio between V y and V u, which is eesented by the coefficient of α = Vy / Vu = Using an elastolastic yield dislacement of y = m, an elasto-lastic yield cuvatue of φ y = ad/m and the deth to the second lastic hinge of L m =.8 m, the coefficient is β = y /(φ y L m ) = Fo R c =.57 m and L m =.8m, the coefficient η, which is defined as η = Rc / Lm fo cohesive soils, is equal to Fo this examle, the equivalent lastic hinge length of the fist lastic hinge of the ile is taken as L 1 = 0. 5 m fom Eq. (40), which coesonds to a nomalized length of λ 1 = 0. 86, while the lastic hinge length fo the second lastic hinge is taken as L = D = m, which coesonds to a nomalized length of λ = 1. The cuvatue ductility demand µ φi in the fist lastic hinge at the lateal dislacement y is µ φ i = 5. 46, as calculated fom Eq. (). The substitution of α = 0. 61, β = 0. 74, η = 0.95, λ 1 = and L m = 6. 7 into Eq. (9) gives the kinematic elation fo small lateal dislacements whee only one lastic hinge foms. The same set of values lus K 1 / K =.0, λ = 1 and µ φ i = can be substituted into Eqs. (7) and (5) fo the case of lage lateal dislacement whee both lastic hinges fom. The esulting kinematic elations fo the fist and second lastic hinges ae lotted in Figue 6(a). It can be seen that the cuvatue ductility facto inceases linealy with the dislacement ductility facto fo both lastic hinges. The sloe of the staight line fo the fist lastic hinge is geate than that fo the second lastic hinge due to the shote length of the fist lastic hinge. In the small dislacement ange whee only one lastic hinge foms, i.e. µ < 1. 9, the sloe of the line is also slightly diffeent fom the sloe whee two lastic hinges fom. Fo a cuvatue ductility caacity of 16.0 as estimated fo the ile section, the esult in Figue 6(a) indicates that the fixed-head ile can toleate a dislacement ductility facto of.. Note that fo a dislacement ductility facto of µ = 1, the cuvatue ductility demand in the fist lastic hinge is µ φ1 =.. The eason fo the cuvatue ductility facto geate than unity is due to the definition of the elasto-lastic yield dislacement y, which is lage than the lateal dislacement to cause fist yield of the ile y1. (a) (b) Figue 6. Kinematic elation fo a fixed-head ile embedded in the (a) S E cohesive soil (b) S C cohesionless soil

13 Examle : Dense Sand in Site Class C The same fixed-head ile is analyzed fo a cohesionless soil classified as ofile tye S C e NEHRP (001). The sand is assumed to be dy with an effective unit weight of γ ' = 0. 5 kn/m and an intenal fiction angle of φ = 4. Fo the selected fiction angle, the assive soil essue coefficient is K = ( 1 + sin φ) / ( 1 sin φ ) = The ate of incease of modulus of hoizontal subgade eaction, which is extaolated fom the above-wate cuve of Figue, is n h = 7000 kn/m. Although the soil stiffness should similaly be modified fo assessment of cuvatue ductility demand, as discussed eviously fo cohesive soils, the soil stiffness estimated fom Figue will also be used without modification to 5 illustate the ocedue. Using the same flexual igidity of EI e = kn-m and a soil stiffness of n h = 7000 kn/m, the chaacteistic length of the soil-ile system is R 5 EI / n 1. 1m. n e h = The lateal stiffness of the soil-ile system, as estimated fom Eqs. (7) and (10), is K 1 = kn/m and K = kn/m, giving a stiffness atio of K 1 / K =.6. The lateal dislacement and foce equied to develo the fist lastic hinge ae y1 = 0.01 m and V y = 669. kn, as calculated fom Eqs. (8) and (9), 4 esectively. Using the nomalized flexual stength of M u = M u / ( K γ' D ) = , the nomalized deth to the second lastic hinge is L m = fom Eq. (18), which coesonds to an actual deth of L m =.95 m. The nomalized lateal stength of the soil-ile system is V u = fom Eq. (19), giving an actual lateal stength of V u = 8.6 kn. The elasto-lastic yield dislacement may be calculated fom Eq. (1), which is y = m. The lateal dislacement at the second yield limit state is y = 0.0 m, as calculated fom Eq. (). Using V y = 669. kn and V u = 8.6 kn, the coefficient α = Vy / Vu is equal to Fo y = m, L m =.95 m, and φ y = ad/m, the coefficient is β = y / ( φ y Lm ) = Using R n = 1.1m and L m =.95 m, the coefficient η fo cohesionless soils is η = 1. 5Rn / Lm = The nomalized lastic hinge lengths ae the same as befoe fo the cohesive soil, i.e. λ 1 = and λ = 1. The cuvatue ductility µ φi in the fist lastic hinge at the lateal dislacement y is µ φ i =. 01, as estimated fom Eq. (). Substituting α = 0. 81, β = 0. 4, η = 0. 66, λ 1 = 0. 86, and L m = into Eq. (9) gives the kinematic elation fo the case of small lateal dislacement whee only one lastic hinge foms. The same set of values lus K 1 / K =.6, λ = 1 and µ φ i =. 01 can be substituted into Eqs. (7) and (5) fo the case whee both lastic hinges fom. The esulting kinematic elations ae lotted in Figue 6(b) fo comaison with the case of soft clay. The cuvatue ductility demand fo the ile in dense sand follows the same tend as the soft clay with linealy inceasing cuvatue ductility facto fo inceased dislacement ductility facto. Fo a given dislacement ductility facto, howeve, the cuvatue ductility demand in dense sand is significantly smalle than the cuvatue ductility demand in soft clay. Fo examle, at a dislacement ductility facto of µ = 4, the cuvatue ductility demand is µ φ 1 = fo the case of dense sand comaed to the cuvatue ductility demand of µ φ 1 = fo soft clay. Although not lotted in Figue 6(b), the estimated cuvatue ductility caacity of 16.0 fo the ile section would coesond to a dislacement ductility facto of µ = 11.5, which is significantly lage than the dislacement ductility nomally adoted fo design. PRELIMINARY RESULTS FOR DAMAGE ASSESSMENT Unde lateal seismic loads, damage to iles is often elated to the cuvatue ductility demand in the citical egions of the ile. Consequently, the cuvatue ductility demand is used as an indicato of ile damage in this ae. The cuvatue ductility demand howeve deends not only on the dislacement ductility imosed on the ile, but also on the oeties of the soil. Since a wide ange of soil conditions

14 exist in actice, seismic efomance of a fixed-head ile may vay significantly deending on the site condition. In this section, the efomance of a fixed-head is assessed fo a wide ange of soils, fom ofile tye S E to ofile tye S C e NEHRP [] o ATC-40 [4]. The assessment is made using the same ile details as esented in the evious section. The vaiation in soil ofile tyes is achieved by vaying the undained shea stength fom s u = 0 to 50 kn/m fo cohesive soils and by vaying the elative density fom D =15% to 90% fo cohesionless soils. The cuvatue ductility demand in the fixed-head ile is calculated fo dislacement ductility factos of µ =.5 and µ =. The esulting cuvatue ductility demand fo the fist and second lastic hinges of the fixed-head ile is lotted in Figue 7. Figue 7(a) shows the cuvatue ductility demand vesus the undained shea stength of cohesive soils. Note that the soil ofile tye is also labeled on the hoizontal axis in the figue. It can be seen that the cuvatue ductility demand is elatively constant fo a given dislacement ductility facto. Fo an imosed dislacement ductility facto of µ =, fo examle, the cuvatue ductility demand in the fist lastic hinge inceases slightly fom µ φ1 = 1.4 at an undained shea stength of s u = 0 kn/m to µ φ1 = 14.9 at an undained shea stength of s u = 100 kn/m, and then deceases slightly to µ φ1 = 14.6 when the undained shea stength inceases to s u = 50 kn/m. Fo soft cohesive soils (soil tye S E ), the vaiation of cuvatue ductility demand in the second lastic hinge is less comaed to that of the fist lastic hinge. Fo the ile analyzed in this examle, the cuvatue ductility caacity of ( µ φ ) ca = 16.0 is adequate fo a lage ange of cohesive soils. (a) Figue 7. Cuvatue ductility demand of a fixed-head ile embedded in ofile tye S C to S E (a) cohesive soils (b) cohesionless soils The cuvatue ductility demand fo the same ile embedded in cohesionless soils is shown in Figue 7(b). The cuvatue ductility demand deceases with inceased soil stiffness, as signified by the incease in elative density, esecially fo the fist lastic hinge. The decease in cuvatue ductility demand is moe significant fo iles embedded in S E cohesionless soils. Simila to the tend obseved fo cohesive soils, the decease in cuvatue ductility demand in the second lastic hinge is smalle than that of the fist lastic hinge. The cuvatue ductility caacity of ( µ φ ) ca = 16.0 is adequate fo a lage ange of cohesionless soils fo an imosed dislacement ductility facto of µ =. It is imotant to note that the esults esented in Figue 7 ae eliminay since they have been calculated without a modification in the soil stiffness fo both cohesive and cohesionless soils. An adjustment of the soil stiffness howeve may (b)

15 lead to an inceased cuvatue ductility demand in the ile. Futhe eseach into the aoiate level of soil stiffness modification fo cuvatue ductility demand estimation is waanted. CONCLUSIONS Seismic design of dee foundations should include an assessment of the cuvatue ductility demand in the otential lastic hinge egion of the ile. In this ae, an analytical model is develoed fo assessing the cuvatue ductility demand of fixed-head iles embedded in cohesive and cohesionless soils. The model is useful fo efomance-based design since local damage can be contolled by limiting the cuvatue ductility demand in the lastic hinges of the ile. Fo the oosed model, the lateal esonse is chaacteized by a linea elastic esonse, followed by fist yielding of the ile at the ile head, and then by a full lastic mechanism afte the fomation of the second lastic hinge. The elastic esonse of the ile and its fist yield limit state ae detemined using classical solutions of a flexual element suoted by an elastic Winkle foundation. The ultimate lateal stength, as defined by a fully lastic mechanism, is detemined using the flexual stength of the ile and the ultimate essue distibution of the soil. A kinematic elation between the global dislacement ductility facto and local cuvatue ductility demand is develoed by assuming a concentated lastic otation in both lastic hinges. The kinematic elation indicates that the cuvatue ductility demand deends on the atio of the fist yield lateal foce to ultimate lateal foce, the initial stiffness to ost fist yield stiffness atio, the deth to the second lastic hinge, and the lastic hinge length of the ile. The vesatility of the oosed model is illustated using a fixed-head einfoced concete ile embedded in two diffeent soil tyes, as classified in cuent US building codes. Although esults esented in this ae ae eliminay, the model is nonetheless shown to be caable of edicting the local ductility demand fo a wide ange of ile and soil oeties. REFERENCES 1. Poulos HG, Davis EH. Pile foundation analysis and design. New Yok: Wiley-Intescience, Davisson MT. Lateal load caacity of iles. Highway Reseach Recod 1970; : NEHRP. NEHRP ecommended ovisions fo seismic egulations fo new buildings and othe stuctues, FEMA 68. Washington, D.C., Alied Technology Council ATC-40. Seismic evaluation and etofit of concete buildings. Redwood City, Califonia, Matlock H, Reese LC. Genealized solutions fo lateally loaded iles. Jounal of Soil Mechanics and Foundations Division (ASCE) 1960; 86(SM5): Alied Technology Council ATC-. Imoved seismic design citeia fo Califonia bidges: ovisional ecommendations. Redwood City, Califonia, Reese LC, Van Ime WF. Single iles and ile gous unde lateal loading. Rottedam: Balkema, Boms BB. Lateal esistance of iles in cohesionless soils. Jounal of Soil Mechanics and Foundations Division (ASCE) 1964; 90(SM): Piestley MJN, Seible F, Calvi GM. Seismic design and etofit of bidges. New Yok: Wiley- Intescience, Chai YH. Flexual stength and ductility of extended ile-shafts. I: Analytical model. Jounal of Stuctual Engineeing (ASCE) 00; 18(5): Kowalsky MJ. Defomation limit states fo cicula einfoced concete bidge columns. Jounal of Stuctual Engineeing (ASCE) 000; 16(8):