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1 25ƒ 3C Á œ pp. 181~188 ª w MCS» w w MCS-based Reliability Analysis of Axially Loaded Single Pile Structure x Á Á» Huh, JungwonÁJeong, Sang-SeomÁKwak, Kiseok Abstract This paper deals with development of a reliability analysis algorithm to quantify the risk in axially loaded single pile structure in consideration of pile-soil interaction and uncertainties on various design variables, and its application to realistic problems. To develop the reliability analysis algorithm of axially loaded pile-soil system, (1) the finite difference method based on an equivalent soil spring model and a load transfer method (2) the Monte Carlo simulation method are integrated, and a computer program is then developed. Explicit consideration of uncertainties in vertical soil properties is accomplished by considering unit skin friction and unit end bearing resistances as random variables, leading introduction of variability in load transfer functions. Applicability of the proposed algorithm to safety assessment of axially loaded pile-soil system is verified using a realistic example. Soil resistance strength and vertical settlement of pile in the axially loaded pile-soil system appear to be more controlling failure modes than pile strength from the results of a reliability analysis. A sensitivity analysis is also conducted and sensitive random variables are identified for each performance function. Since the proposed algorithm can explicitly consider uncertainties in various design variables, and quantify failure probability of a pile foundation, it can be directly used to estimate risk, to obtain basic informations for life cycle cost analysis, and to develop code requirements for a reliability based design of pile foundation. Keywords : pile reliability analysis, MCS, risk assessment, pile-soil interaction, load transfer function - y w y š w w x yw w š w ü» w. w - w š x w (1) ƒ v w» w w w (2) le» ww» w ful v w. p w y ¾ y š w ƒ w w w y š w. ew mw w - sƒ w w. w l w w w ewƒ w - q q. w ww ƒ w w y w.» w» w w wš y y š w š q y ey w,»» w x sƒ» w w» œ ª» y w» w. w : w, MCS, x sƒ, - y, w w 1. z» x q ƒ w y, w w ƒ swwš w w y š w w. ƒ šd» w w š w w y j p y, w y š * z Á w w œw œ, œw ( ** z Á w m œw, œw ( *** z Áw», œw ( p y, ƒ» y ü wš. p w š w w p y x p w» w w w x (in-situ test) w ü x w w w yw œ l». x x s w œ (spatial variability) y 25ƒ 3C œ 181

2 , (pore water pressure) w y, x yƒ p sƒ w y (Tandjiria et al., 2000). w w y š wš» w w» m w. w š w y š w š x q» w - w œw w q w sƒw. w» w x wš w w, p, p, - p - ƒ w txwš, swwš y yw š w w (x, 2003). w» w w p w ƒ š, - y š w yw x sƒ w w. - y š w» w ƒ v (equivalent base spring model) w w w, ƒ v xk v ƒ w 1867 Winklerƒ w» w. üá w w y» w w (Eloseily, 1998; Barakat et al., 1999; Tandjiria et al., 2000)» (½, 1991;, 1995; ¼, 1996; Beker, 1996; Yoon & O'Neill, 1997) ƒ y w ù, y š w w ù w» w yw w w w w w ù w w w. y - w w, w y y š w w š z x sƒ w, (1) - y š w ƒ v w» w w w (2) w w ƒ ü wš y y š w x sƒw le» ww w š v w. w e mw - ü y m y š w w x sƒ w» w. 2. sƒ w 2.1 MCS w sƒ œw x» w k ( )w w sƒw, w w kw w w y w xkù w w š 182 xk tx. q y (1) tx (x, 2003). P f = f gx ( ) < 0 x ( x 1, x 2,, xn )dx 1, dx 2,, dxn», f x ( x 1, x 2,, x n )» y X 1, X 2, Ã, X n w y w (joint probability density function). œw w kw sw» y w y w w ƒ w, w y ƒ (1) mw w q y w. w ƒ w w» MCS(Monte Carlo simulation) t q y w Level III» ƒ y s³ š sx k w (reliability index) w p š Level II». w Level II» MVFOSM(Mean Value First Order Second Moment method), Generalized Safety Index Method, FORM(First Order Reliability Method), š SORM(Second Order Reliability Method). w» w w w š xk ƒ (implicit)w w k w w Level II» š w w (Haldar & Mahadevan, 2000b). MCS w w ww. MCS w œw. MCS w q y (1) ƒ w. P f = EIx [ ( )] = I[ x]fx ( )dx (2) D», I[x] q w (indicator function) Binary. 1if g x [ ] Ig [ ( x) ] ( ) 0 failure = = 0if g( x) > 0 safe I x (2) q y P f w I[x]» e ùkü MCS w q y w w. N s P f PMC 1 = f = Ix [ N i ] s i = 1», x i ù (random number) u i w w y w f(x) l w t e, MC MCS w e, š Ns t ùkü. m MCS» y w l t w q y w» q t y q y ew (Haldar & Mahadevan, 2000a). 2.2 w w Terzaghi m 1 ù x (1) (3) (4)

3 w» ew w» ful» t 3ƒ w -, k w w ƒ w y š. r z š w w -» w w» w. 1 w - ¼ w v wù v. x š w v p wù k w, x š w v p w (t z) š (q z) š xk w. x š w w (t z)š (q z)š (Mosher, 1984; Vijayvergiya, 1977; Kraft et al., 1981; API, 1993) d e w w kw w x x w» w. w xk x w tx. EA d2 z dx 2 2πRt( x, z) = 0», E k, A, x ¾ w t, z, R z, š t(x,z) e w. 1. w w w ew w, Matlock et al.(1981) w w z» (recursive method) w w -» w. 2 v w w, w ùkü. ƒ w (EA/h) ùkü v,» h ¼. w P v S (5) 2. w y ew. (z) +x w w ƒ (T) w, T i (i) (i 1) w ü ùkü. w x (Reese & Welch, 1975; Matlock et al., 1981;, 2001) -»» (beam-column method) w œ y ew ùkù w w» w. 3. w w» š w y š w x y w» w x (ASD; allowable stress design) œw» w w w (LRFD; load and resistance factor design, AISC, 2002) (performancebased design) y š. w» w x w w yw w» mw w w w. w - w ww w. 3.1» w y y w w w y w y p p y sww - w w y j w. p y wk w w» w, k (E)» (σ ck ) š (A) y (y ) š w. w y w œ x w e» w, w y š w. w y, y w ƒ. p y v ü y (inherent uncertainty) x sá w p w» w w x t w ü x w w m y (statistical uncertainty)» w e y (modeling uncertainty) sww. w w y w 25ƒ 3C œ 183

4 y w x sƒw v w. (Vanmarcke, 1977; Phoon & Kulhawy, 1999) w p œ w» w 2ƒ ƒ w. (1) d» w w mw x p dü p m ³ w. (2) p s œ w œ w û ƒ. w p w w d x x v w w š w. w dü ¾ p e 3 w ù x ƒ w š w. ƒ d p w w ƒ w (, ƒ d p e ). p w v p y š w w w w. (q z) š w w w w. w w w (t z) š (q z) š (Mosher, 1984; Vijayvergiya, 1977; Kraft et al., 1981; API, 1993), API(American Petroleum Institute) w w» w œ sƒ w w k (Driven Piles) w š w. w š ³ y w (f s ) (q p ) w w. 5 t w š xk w ƒ wš ƒ d y f s q p y š w d w w w w. 5. w t z š w y š 3. p w y š 3.2 v x x t-z š q-z š w y š x v p txw k y š w ƒ d y y š w. ù w - w x š w v w (t z) š 3.3» w w w w w w q k ³ w w k w w v w. ( )w q (1) / w ew, (2) ww w sww q, š (3) - w q w w ew w ew w w w. g z = z allow z», z allow ³ x (6) 4. API ³ w š (API, 1993) 184

5 w ew ( x ew ) z w w mw w w q w w. T c g σ = σ ck σ c = σ ck ---- A», T c w w w mw, σ ck A ƒ» gj p» y š w w w w - w w w kw w w ww q w tx. g Q = Q ult Q req = ( Aq p + LA s f s ) ( Q p + Q s )», A, L A s ƒƒ, ¼ ¼, q p f s (kn/cm ) 2 (kn/cm ) y 2 š, Q p Q s w w mw (kn) (kn). (7) (8) 4. w e 4.1 y m p e e 6 l 15m ¾ ¾ k w 30cm x gj p» š w. gj p s³» 2.746kN/cm 2 (280kgf/cm ) š s³ k kN/cm 2. t l 5m¾ d w md 5m 15m¾ d w - p md, t 1 API³ w s³ w ƒ p tw 2 t z š q z š 7 8 w. w w - w w 1.74( w w q y w ) w 800kN w š w. w, w w y w w t 2 y s³ w y w ww t 3 w š, q v APILE ew (, 2001; x,, 2004). 3.4 w MCS w w š ƒ w k w - w y yw š w w x sƒw» w w w w MCS» ww š wš v w. w MCS w w kw w» w w w w xk w kw ƒ š w w ƒ w. w x(simulate)w» w y s ƒ ù w (a set of pseudo-random numbers) k z wù y w w y w ( w w ) w w kw w q k sƒw. w z j w x m q y ù m p e w. t 1. API ³ m w d - ƒ (N q ) 6. w w 4.2 w kw w w w ƒ w š w. (6) tx ew w w x ew x tonf/m 2 kn/cm 2 tonf/m 2 kn/cm (deg) p 25(deg) ƒ 3C œ 185

6 7. API³ t-zš ww q w. w kw w w w š x(bogard & Matlock, 1980; Eloseily, 1998) e (in-situ-test) (Sparks & Rollins, 1997)» w y m p e( sxk) w, t 2 t w. w v w 100,000z 1,000,000z¾ w w ww, ƒ w k 1,000,000z w q y t 3 š q y 9 w. t 3 9 q y z ƒ 1,000,000z ƒw q y ew, w w w ƒƒ 1.86%, % š 9.31% wš, w ƒ» 2.083, 4.005, š MCS q y w 95% w (Shooman, 1968)w, t 3 ƒ w w q y 95% t w. error (%) P f = N P f (10) š x yw m», 1cm ƒ w. w kw. g Z = z allow z = 1 cm z 8. API³ q-z š w w w gj p» w q ³ w (7) tx, w w (8) - w w (9) 186», N z P f q y. t 3 œ y w w t w w y w š ƒ. w w q y» ƒ š y y š w, ƒ w q w e œw. q w ew w q y w q y ùkü, w ewƒ w - q q. w ew w š w x ew w š yw» v w š q q y w š w y w ƒ w q y q w» w, 0.05~0.5 y jš w ù 6 y t 2 s³, sx t 2. y m p e y s³ sxk E (k ) kn/cm gj p p Log-Normal σ ck (» ) kn/cm Normal p A ( ) cm Log-Normal d 1 f s ( )* kn/cm Log-Normal f d s ( )* kn/cm Log-Normal 2 q p ( )* kn/cm Log-Normal w P ( w ) kn Type I(EVD) *API ³ m w (w œwz, 2002)

7 t 3.» w w kw ew w x (A) 1 (cm) (kn/cm 2 ) 1391 (kn) y w (O) (cm) (kn/cm 2 ) 800 (kn) w (MCS) (S.F=A/O) SF ~ 2.41 SF ~ 2.50 SF ~ 1.74 q y P f = P f = P f = β ~ β ~ β ~ No. of Simulation 1,000,000 1,000,000 1,000,000 95% 1.45 % 35.9 % 0.62 % q y Û Û Û MCS w ew k w ww 10~12 ùkü. q y 500,000z w. 10 y y ew w q y ùküš, w y w P, d 2 f s2, A k E. w f ew w q y f, d w j» w y y w. y y q y w 11 ùküš, q y w e y gj p» σ ck A w P q š ù q y y x w e ùkû. 12 y y w w q y ùkü w d 2 f s2, w P d 1 f s1 - w y q. ù ù w w w q y y w w x w e. w yw d p e w Á y k ü x x x w w p esƒ w y š w»» (reliability-based design) œ w ƒ v w w - w w, w y y š 25ƒ 3C œ 187

8 w š z w» x w, (1) - y š w ƒ v w» w w w (2) y Á y š w x sƒw le» ww w - w v w, e mw w - sƒ w w. e ew, w w w q y sƒw. q w ewƒ w - q ùkû, ew w w x ew š yw» v w q. w y l w w d 2 t z š w tx š w w q ù, w w w w w mw w w.» w» w w wš y y š w š q y ey w,»» w x sƒ» (LCC; Life Cycle Cost) w w» œ ª» y w» ( w w ) w q. w 2002 w w. š x ½, y, (1991) SPLT w Meyerhof œ, w wz w t, 11«, 2y, pp ¼ (1996) CPT w sƒ, 1996 w œwz ƒ w tz, pp , y, w(1995) w sƒ, w œwz, 11«, 1y, pp ,, x (2001) -e š w» w, w œwz 2001 w tz, pp x (2003) w w w w š, wm wz, 23«6Ay, pp x, (2004) y w, 2004 wm wz»w z, pp w œwz(2002) œw 4; ¾», American Institute of Steel Construction (2002) Manual of Steel Construction: Load and Resistance Factor Design, AISC, Chicago, Illinois, U.S.A. American Petroleum Institute (1993) API Recommended Practice for Planning, Designing, and Constructing Fixed Offshore Platforms-working Stress Design, Report RP-2A. Barakat S. A., Malkawi A. I. H., & Tahat R. H. (1999) Reliabilitybased optimization of laterally loaded piles, Structural Safety, Vol. 21, pp Becker, D.E. (1996) Limit state design for foundations. Part I. An overview of the foundation design process. Canadian Geotechnical Journal, Vol. 33, No. 6, pp Bogard, D. & Matlock, H. (1980) Simplified Calculation of p-y Curve for Laterally Loaded Piles in Sand, In House Report, The Earth Technology Corporation. Eloseily, K. (1998) Reliability Assessment for Pile Groups under Lateral Loads, Thesis submitted for partial fulfillment of Ph.D. degree, University of Maryland College Park, Maryland, U.S.A. Haldar, A. & Mahadevan, S. (2000a) Probability, Reliability and Statistical Methods in Engineering Design, John Wiley & Sons, New York, N.Y. Haldar, A. & Mahadevan, S. (2000b) Reliability assessment using stochastic finite element analysis, John Wiley & Sons, New York, N.Y. (2000b). Kraft, L. M., Ray, R. P., & Kagawa, T. (1981) Theoretical t-z Curves, Proceedings paper 16653, Journal Geotechnical Engineering Division, ASCE, 107(GT11) Matlock, H., Borgard, D., & Lam, I. P. (1981) BMCOL76: A Computer Program for the Analysis of Beam-Columns under Static Axial and Lateral Loading, Ertec, Inc. Mosher, R. L. (1984) Load Transfer Criteria for Numerical Analysis of Axially Loaded Piles in Sand; Part 1: Load Transfer Criteria, Technical Report K-84-1, USACE, Vicksburg, Miss. Phoon, K. K. & Kulhawy, F. H. (1999) Characterization of geotechnical variability, Canadian Geotechnical Journal, Vol. 36, pp Reese, L. C. & Welch, R. C. (1975) Lateral Loading of Deep Foundation in Stiff Clay, J. of the Geot. Eng. Div., ASCE, Vol. 101, No. GT7, pp Shooman ML (1968) Probabilistic reliability: an engineering approach, McGraw-Hill Book Co., New York, NY. Sparks, A. E. & Rollins, K.M. (1997) Passive Resistance and Lateral Load Capacity of a Full-Scale Fixed-Head Pile Group in Clay, Civil Engineering Dept. Research Report CEG , Brigham Young University, Provo, Utah. Tandjiria, V., Teh, C. I. & Low, B. K. (2000) Reliability analysis of laterally loaded piles using response surface methods, Structural Safety, Vol. 22, pp Vanmarcke, E. H. (1977) Probabilistic modeling of soil profiles, Journal of Geotechnical Engineering Division, ASCE, 103 (GT11), pp Vijayvergiya, V. N. (1977) Load-Movement Characteristics of Piles, 4th Symposium of Waterways, Port, Coastal and Ocean Division, ASCE, Long Beach, California, Vol. 2, pp Yoon, G.L. M O'Neill, M.W. (1997) Resistance factors for single driven piles from experiments, Research Record 1569, Transportation Research Board, Washington, pp ( : / : / : ) 188

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