25ƒ 3C Á 2005 5œ 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 (E-mail: jwonhuh@yosu.ac.kr) ** z Á w m œw, œw (E-mail: soj9081@yonsei.ac.kr) *** z Áw», œw (E-mail: kskwak@kict.re.kr) 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 2005 5œ 181
, (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)
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 2005 5œ 183
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), 4 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. 3.3.1 ew w ew w w w. g z = z allow z», z allow ³ x (6) 4. API ³ w š (API, 1993) 184
w ew ( x ew ) z w w mw. 3.3.2 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. 3.3.3 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 2 2462kN/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 2 1 20(deg) 12 6.7 6.571 10-3 290 0.2844 2 - p 25(deg) 20 8.1 7.944 10-3 480 0.4707 25ƒ 3C 2005 5œ 185
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%, 0.0031% š 9.31% wš, w ƒ» 2.083, 4.005, š 1.322. MCS q y w 95% w (Shooman, 1968)w, t 3 ƒ w w q y 95% t w. error (%) 200 1 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. 4.3 - 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 2 2462 0.06 Log-Normal σ ck (» ) kn/cm 2 2.746 0.1 Normal p A ( ) cm 2 706.9 0.05 Log-Normal d 1 f s ( )* kn/cm 2 6.571 10-3 0.2 Log-Normal f d s ( )* kn/cm 2 7.944 10-3 0.2 Log-Normal 2 q p ( )* kn/cm 2 0.4707 0.2 Log-Normal w P ( w ) kn 800 0.15 Type I(EVD) *API ³ m w (w œwz, 2002)
t 3.» w w kw ew w x (A) 1 (cm) 2.746 (kn/cm 2 ) 1391 (kn) y w (O) 0.415 (cm) 1.121 (kn/cm 2 ) 800 (kn) w (MCS) (S.F=A/O) SF ~ 2.41 SF ~ 2.50 SF ~ 1.74 q y P f = 0.018648 P f = 0.000031 P f = 0.093097 β ~ 2.083 β ~ 4.005 β ~ 1.322 No. of Simulation 1,000,000 1,000,000 1,000,000 95% 1.45 % 35.9 % 0.62 % q y 0.018648 Û0.000271 0.000031 Û0.000011 0.093097 Û0.000581 9. MCS w 11. 10. 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. 5. 12. w - w w, w y y š 25ƒ 3C 2005 5œ 187
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