RELIABILITY ANALYSIS OF GEOTECHNICAL SYSTEMS. Dr. G L Sivakumar Babu Department of Civil Engineering Indian Institute of Science Bangalore, India

Transcription

1 RELIABILITY ANALYSIS OF GEOTECHNICAL SYSTEMS Dr. G L Sivakumar Babu Department of Civil Engineering Indian Institute of Science Bangalore, India

2 Contents Motivation Shallow foundations Pile foundations Unsaturated soil slopes Retaining systems Buried pipes Conclusions

3 Acknowledgments Seshagiri Rao D S N Murthy Sumanta Haldar Munwar Basha Amit Srivatsava

4 Motivation Contd.. Steel and concrete Manufactured Controlled conditions Material behaviour Soil Natural material Formed through complex processes Complicated material behaviour (nonlinear and stress dependent, numerous widely accepted transformation models)

5 Motivation Contd.. Is it appropriate to neglect such high degree of soil property variations associated with mean design parameter???

6 Motivation Contd.. Resistance factors are functions of variability in LRFD design

7 Reliability analysis ) ( 0) ) ( ( β = Φ < = X g P p f + Φ = ) ( 1 S R S R p f σ σ μ μ ) ( 0) ) ( ( β = Φ < = X g P p f Performance function is Z=R-S + = ) ( S R S R σ σ μ μ β ( ) = Φ = ) )(1 ln(1 )} ) /(1 (1 ) ln{( 1 1 S R S R S R p f δ δ δ δ μ μ β Φ = ) )(1 ln(1 ) )(1 (1 ) / ln( 1 S R S R S R p f δ δ δ δ μ μ (USACE 1999) Probability density function of safety margin (R-S)

8 Whitman (1985) f N diagram adopted by Hong Kong Planning Department for planning purposes

9 Practical implications of effect of variability on design of shallow foundations (Lacasse 001) Probability density function of FS=R/S

10 Seismic stability of slopes Contd.. CoV c,φ =10%

11 Variability Inherent variability Inhomogeneous Anisotropic Measurement uncertainty Transformation uncertainty due to use simplified mathematical correlations Under this uncertain environment, how far solutions based on deterministic approaches produce realistic estimates of safety in designs??

12 Probabilistic Analysis Reliability The probability that a system performs satisfactorily the intended function under specified operating conditions, during its design period Input parameters Moments (mean, variance, etc.) Distributions (Normal and Lognormal etc.) Auto-correlation

13 The mean of the soil property is defined as mean of the trend function fitted to the experimental data μ = t n 1 n i= 1 t( z i ) The standard deviation of variability is σ R = 1 n [ R( n 1 c. o. v. R = i= 1 σ R μ t z i )] The coefficient of variation of variability is:

14 Evaluation of spatial variability - Autocorrelation Indication of distance within which the property values show relatively strong correlation The sample autocorrelation function is = = + = n i Y i j n i Y j i Y i j Y n Y Y j n 1 1 ) ( 1 1 ) )( ( 1 1 ) ( μ μ μ τ ρ

15 Evaluation of spatial variability - Variance reduction function For theoretical triangular fit to sample autocorrelation function For theoretical exponential fit to sample autocorrelation function For theoretical double exponential fit to sample autocorrelation function a, b, d are the autocorrelation distances, and T is the averaging distance, the domain over which the soil properties are averaged

16 Inherent Soil Variability Commonly used theoretical fits to sample autocorrelation functions (vanmarcke, 1983)

17 USEFULNESS OF CPT BASED PROBABILISTIC ANALYSIS OF SOIL PROFILES WHY CPT BASED METHOD? Simple, fast and continuous. Analysis is based on well established concepts Less average cost compared to soil boring (LTRC, 1999). Laboratory tests can be avoided. Provides a format for quantifying information regarding subsurface condition of a particular site. ADVANTAGES in Reliability Based Design To quantify variability from CPT data that map into load-settlement behaviour and integrate with the design of shallow/pile foundations. The pile-soil interface parameters can be calculated from undrained shear strength values obtained from CPT data. Propose reliability based design methodologies for foundations considering Ultimate Limit State (ULS) and Serviceability Limit State (SLS).

18 Characterization of uncertainty- Measurement uncertainty The soil properties are measured by physical means. This measurement process introduces variability. Measured soil property (Y m (z)) can be described as : Y m ( z) = Y ( z) + e( Z) Where Y(z) is the in-situ soil property, e(z) is the measurement uncertainty. The expanded form of above equation as : Y m ( z) = t( z) + R( z) + e( z) In the published literature the range of measurement error i.e. for ECPT is generally 5%-15%. CoV e

19 Characterization of uncertainty - Transformation Uncertainty A transformation model is required to relate the test measurement to an appropriate design property. The correlation between the undrained shear strength and tip resistance is: s u = q c σ N k vo = D K ( qc σ vo) D K = 1 N K σ vo where s u is the undrained shear strength; N K is the empirical constant, q c is the cone tip resistance, total overburden stress

20 Total variability The uncertainty associated with design soil property such as cone tip resistance is a function of inherent soil variability (w), measurement error (e) and transformation uncertainty ε. The design soil property is predicted from test measurement using the following transformation model using second-moment statistics ( ξ ε ) Design property and measurement are related by ξ d = T m, Design property, related to inherent variability, measurement and transformation is given by ξ = T ( t + w e,ε ) d + SD ξd T = SD w w T + e SD e T + ε SD ε SD ξa = T w Γ T e T ε ( L) SD + SD + SD w e ε

21 Studies on shallow foundations

22 Moments of cone tip resistance-shear failure criterion-nges data

23 Moments of design parameters-shear failure criterion-nges data φ TC = log 10 q c σ / vo p / a p a

24 Analysis of allowable pressure Deterministic approach Probabilistic approach for system reliability index of three

25 Allowable bearing pressure-keswick clay For undrained conditions (Skempton 1951) Total CoV of S u or c u For footing with Df/B=1.1 Spatial averaging CoV of S u or c u

26 Bearing capacity Keswick clay All three components of uncertainty Factor of safety Vs. Reliability index shows that lower FS can be allowed.

27 Bearing capacity of clays Power plant site - India A proposed 445 MW Konaseema EPS Oakwell gas-fired combined cycle power plant on the East coast in Indian state of Andhra Pradesh

28 Bearing capacity-power plant clay site All three components of uncertainty

29 Effect of anisotropic spatial correlation

30 Effect of anisotropic spatial correlation Variance reduction factors for -D space, Lv= m and Lh=7 m

31 Effect of anisotropic spatial correlation Coefficient of variation of bearing capacity (autocorrelation distance in the vertical direction=0.19 m) Assumption of isotropic correlation structure influences reliability

32 RELIABILITY BASED DESIGN OF PILE FOUNDATIONS

33 CONVENTIONAL DESIGN METHODOLOGY q u q us W D q su us q sb ub L VERTICALLY LOADED PILE Ultimate axial load carrying capacity: Q u = p x L x q us + A b xq ub -W D: Pile diameter L: Length of pile p: Pile perimeter = pi x D A b : Area of pile base q us : Ultimate unit skin frictional resistance q ub : Ultimate unit end bearing resistance W: Weight of pile Design load capacity: Q u /FOS; FOS varies from -3 q us and q ub are the functions of shear strength of soil INADEQUECIES IN THE PRESENT APPROACH INADEQUECIES IN THE PRESENT APPROACH It is not unique and varies significantly over a wide range In-situ behaviour of the pile foundation is considerably influenced by variability in soil properties. Handles ultimate state and serviceability states separately

34 EVIDENCE OF VARIABILITY Load (kn) Case S/6 Case S/3 Case S/7 Case S/ Settlement (m) 4 PILE LOAD TEST RESULTS (TEJCHMAN & GWIZDALA, 1977) For 4 pile load tests: Pile Diameter: 1.5 m Pile length: 1 m Load tests are from the same site If the allowable settlement is 0.0 m, the allowable load varies from 300 kn- 500 kn This variation indicates the randomness of pile-soil interface properties SOURCES OF VARIABILITY Inherent soil variability: in-situ variation in soil strength parameters depth wise Measurement error: due to the process of measurement of field data Transformation uncertainty: use of various transformation model to estimate soil parameters (say e.g. undrained shear strength from CPT data)

35 Depth (m) EFFECT OF SPATIAL VARIABILITY ON PILE q c (MPa) L Typical CPT profile D r t = D e tan( tan(πφ) r b = D e φtan( tan(φ) cos(φ) For clay, r t and r b = D Failure zone SPATIAL AVERAGING The fluctuation in the soil property tends to cancel in the process of spatial averaging. Spatial averaging length, which is equal to the failure zone, needs to be considered in the reliability analysis of foundations. The larger the length over which the property is averaged, higher is the fluctuation that tends to cancel in the process of spatial averaging. This causes reduction in standard deviation. ESSENTIAL PAREMETERS Vertical scale of fluctuation: indicates the distance, within which soil property shows strong correlation. Averaging length: for pile shaft it is length of the pile (L) and for pile base it is the failure zone at the pile toe i.e. (r t + r b ).

36 RELIABILITY BASED DESIGN APPROACH STEP-1: CONE TIP RESISTANCE PROFILE AND DETERMINATION OF SOIL PARAMETERS CPT profile from Konaseema site (India) qc σ vo Soil parameters: s ( ) u = = DK qc σ vo N k where D K =1/N K is the empirical constant, σ vo is the total overburden stress, s u is the undrained shear strength of soil, q c is the cone tip resistance. Averaged s u over a length of pile considered for skin friction. Averaged s u over failure zone near pile tip considered for end bearing. Axial load (kn) Fitted by t-z model Field load-settlement data Settlement (m) STEP-: INTERFACE PARAMETERS Interface parameters, (i) average shear modulus of pile-soil interface (ii) ultimate soil-pile interface shear strength and (iii) end bearing soil elastic modulus: Obtained by fitted with load test data. Interface parameters = constant x undrained shear strength Undrained shear strength: from CPT data

37 Mean and standard deviation of constants are obtained by fitting several numbers of field pile load-settlement test data. Statistical estimates of soil shear strength are obtained from CPT data. STEP-3: EVALUATION OF VARIABILITY The spatially averaged combined COV is described as (Phoon & Kulhawy, 1999): COV ( Γ ( L) COV COV ) i m s + u a + σ vo 1 μt COV tr where σ vo is the average total overburden pressure over the averaging length L, μt is the mean value of qc over a depth L, COV i is the COV of inherent variability, COV tr is the COV of transformation uncertainty and COV m is the COV of measurement error. () Γ is the variance reduction function given by Vanmarcke (1983): δ z su Γ ( L) = L L δ z su 1+ e L / δ δ z su is the vertical scale of fluctuation z s u

38 Determination of vertical scale of fluctuation and variance reduction q c (kpa) Autocorrelation Exponential fit 0.8 Fitted autocorrelation function ρsu = exp(-τ/0.85) ; R = 0.9 hence, δz = 0.85 m 0.6 Cone tip resistance Linear trend Residual ( ) 0.4 Γ Autocorrelation, ρ su 0. COV i t= 478.5z t=trend function; z=depth Depth (m) Lag distance, τ (m)

39 STEP-4: RELIABILITY ANALYSIS Axial load (kn) Basic random variables: undrained shear strength near pile shaft, near pile tip, constants and allowable settlement / serviceable settlement. Standard deviation, scale of fluctuation and variance reduction of undrained shear strength is obtained from CPT data. COV of measurement error : COV m = 15 % (Phoon & Kulhawy, 1999) COV of transformation variability : COV tr = 9 % (Phoon & Kulhawy, 1999) COV of serviceable settlement : 58.3 % (Zhang et al., 005) Random variables follow log-normal distribution Mean load-settlement curve MCS generated curves Settlement (m) Load-settlement curves are generated by t-z method from mean and standard deviations of interface parameters using Monte Carlo simulations. For an applied load Q, number of sample realizations that exceed the ultimate load as well as allowable settlement are computed and expressed in terms of probability of failure Monte Carlo samples are used.

40 p = f 1 Ultimate Limit State (ULS): When the applied load is greater or equal to pile ultimate load carrying capacity, the probability of failure due to applied load is estimated by Monte Carlo Simulation (MCS): Number of samples exceeding the ultiamte load ( corresponds to settlement Total number of samples 0.05d) under load Q ( ) 1 Reliability index corresponding to ultimate limit state criteria: βuls = Φ 1 p f 1 Serviceability Limit State (SLS): When the settlement is greater or equal to serviceable limit (S SER ), the probability of failure due to serviceable criteria is estimated at any axial load by MCS: p f = Number of samples exceeding the serviceable allowable Total number of samples settlement Reliability index due to serviceable limit state criteria: System reliability: β S SER under ( 1 ) 1 SLS = Φ p f load Q p f ( Q Qu S S SER ) = p f ( Q Qu ) + p f ( S S SER ) p f ( Q Qu S S SER ) = p f ( Q Q u ) + p f ( S S SER ) p f ( Q Q u S System reliability index: = Φ 1 β 1 p ( Q Q S S ) SYS S SER ) p ( ) f u SER f ( S S ) SER

41 DESIGN LOAD Required pile diameter (m) Conventional FOS design gives design load, Q = 413 kn For β SYS =, Q = 350 kn (S SER = m) m Q = 410 kn (S SER = 0.05 m) m Q = 45 kn (S SER = m) m For β SYS =.5, Q = 30 kn (S SER = m) m Q = 375 kn (S SER = 0.05 m) m Q = 4 kn (S SER = m) m 1 0 Serviceable settlement = 0.015m Serviceable settlement = 0.05m Serviceable settlement = 0.030m Target reliability index, β System reliability, βsys D P = 0.8 m L P = 15 m COV = 38 % SSER =0.015 m Conventional FACTOR OF SAFETY Approach SSER =0.05 m Design load (kn) SSER =0.03 m CHOICE OF PILE DIAMETER If the pile is designed for the target reliability indices of.0,.5, and 3.0, required diameters are 0.8 m, m 1.m, 1.7m for S SER = 0.015m. For the same reliability, the required pile diameters are 0.7 m, m 1.0 m, m 1.4 m for S SER = 0.05 m and 0.5 m, m 0.7 m and 1.0 m respectively if S SER = m. m

42 P 0 δ LATERALLY LOADED PILES DESIGN OF LATERALLY LOADED PILE Maximum lateral displacement at pile head. Maximum bending moment Load Maximum lateral displacement relation: δ = 0.707( E p I * 0.66 P δ p ) 0.5 o * δ k 3 / 4 h 0.3P o Load Maximum bending moment relation: M max M max ( E I ) = 0.5 ( δ ) 0.0 p p * 0.9 ( P ( k o h ) ) k h is the coefficient of lateral subgrade reaction d is the pile diameter E p I p is the uniform flexural rigidity of the pile δ* is the yield displacement of soil Ref: Hsuing and Chen (1997) The coefficient of lateral subgrade reaction: k h = κ s u /d κ is the correlation parameter

43 EVALUATION OF VARIABILITY RANDOM VARIABLES Undrained shear strength of soil, s u (mean value: average over the pile length) Correlation constant, κ Coefficient of lateral subgrade reaction, k h COV of k h : COV k = COVs + COV κ 10 % h From CPT data MEAN AND VARIANCE OF * RESPONSE 0.66 Po δ δ = 0.5 * 4 Mean maximum lateral displacement : 0.707( E pi p ) δ ( kh ) 3/ 0.3Po u δ k h Variance of maximum lateral displacement : = σ σ δ = 0.35P0 ( δ ) 0.5 ( E I ) δ ( k ) 3/ 4 ( P P h 0.3P0 ) σ δ at k = k h ( E I ) ( k ) 3/ ( COV. k ) P P h h k kh h h Mean of k h Mean maximum bending moment : Variance of maximum bending moment : M σ max M max ( E I ) = 0.5 ( δ ) 0.5 = 0.0 p p * 0.9 ( Po ) ( k ) 0.0 ( E I ) ( P ) P P 0.9 ( δ ) ( k ) k h ( COV. k ) kh h

44 RELIABILITY ANALYSIS OF LATERALLY LOADED PILE For an applied lateral load pile foundation is considered to be satisfactory: Lateral displacement at pile head does not exceed allowable displacement (1% diameter of pile) Maximum bending moment does not exceed moment capacity of pile section. Performance functions: Mean of δ G 1 ( δ ) = δ a δ Allowable lateral displacement δ = ( 1 + α COV ) δ δ a Resisting moment of pile section δ G ( M max ) = M R max M Μ R = F Z Y P Probability density function M max M R ασ δ δ a

45 DESIGN APPROACH Depth (m) CPT data: Konaseema area (SCPT-9) [Clay site] 0 qc (kpa) Cone tip resistance Residual Linear trend t= 478.5z t=trend function; z=depth Undrianed shear strength(kpa) Residual Linear trend Undrained shear strength t = 3.01z t=trend function; z=depth 800 D = 0.8m L=10 m PARAMETERS OBTAINED FROM CPT PROFILE Mean value of subgrade reaction k h = kn/m 3 Inherent variability, COV i = 37% Scale of fluctuation : 0.85 m Spatial COV of undrained shear strength, COV su = 38 % ASSUMED PARAMETERS Yield displacement of soil, δ * = m Allowable lateral displacement, δ a = m Pile resisting moment, M R = 08 knm COV m = 15 % COV tr = 9 %

46 CONVENTIONAL DESIGN δ a = m M R = 08 knm Lateral load = 44 kn Lateral load = 4 kn Min Lateral load = 4 kn FOS =.75 Design Lateral load = 8 kn RELIABILITY BASED DESIGN COV kh = 39% Obtain mean (δ) & variance (σ δ ) of displacement Assume α Plot (δ + α.σ δ ) δ a = Design load & β δ Final Design load YES β system =β target NO (a) β system β mom Deterministic loaddisplacement curve M R = 08 knm (b) Obtain mean (M max ) & Variance of maximum moment for the design load Deterministic loadmaximum moment curve Reliability based design lateral load obtained 150 kn > 8 kn (from conventional FOS design) Lateral load, P0 (kn) P det = 44 kn P all = 48 kn P all = 150 kn (1+α.COV δ a )δ lines β δ =.8 β δ = 4.3 α = 4 α = a = m Lateral load, P0 (kn) P 0 = 48 kn βmom = 0.1 P det = 4 kn P all = 150 kn βmom =.5 System reliability index β =.5 for α = 8 Mmax = 97 knm MR = 08 knm System reliability index β = 0.1 for α = 4 Mmax = 48 knm Lateral displacement (m) Maximum bending moment (knm)

47 CONCLUDING REMARKS The study shows that the probabilistic analysis of soil profile provides a format for quantifying the information about the subsurface condition of the site and it also provides the basis for predicting the reliability of the pile foundations. Depending upon the uncertainty level and spatial variability of soil, allowable load can be suggested. The study shows that, it is useful to choose a suitable value for serviceability limit, so that the combined reliability index is ensured from both the considerations of ULS and SLS.

48 Analysis of unsaturated slopes Shear strength of unsaturated soils (Fredlund and Rahardjo 1993) τ= c' + (u a -u w ) tanφ b +(σ n -u a ) tanφ' Where c' is effective cohesion (u a -u w ) is matric suction, u a is pore-air pressure, u w is pore water pressure, φ b is the angle indicating the rate of increase in shear strength relative to the increase in matric suction, σ n is the total stress normal to the sloping surface, and φ' is effective friction angle Surficial stability of unsaturated infinite slope model (Cho and Lee 00) β=slope angle z=depth of failure plane

49 Stability of unsaturated slopes Contd..

50 Stability of unsaturated slopes Contd.. Suction variation with depth

51 Stability of unsaturated slopes Contd.. Variation of FS with depth of failure plane for different elapsed periods

52 Stability of unsaturated slopes Variation of reliability index with depth of failure plane for different elapsed periods

53 Influence of saturated hydraulic conductivity Variation of reliability index with depth of failure plane for elapsed time = 5 days Variation of reliability index with depth of failure plane for elapsed time = 10 days

54 Influence of saturated hydraulic conductivity Variation of reliability index with depth of failure plane for elapsed time = 15 days Variation of reliability index with depth of failure plane for elapsed time = 0 days

55 ANALYSIS OF GRAVITY RETAINING WALLS BY RELIABILITY BASED DESIGN OPTIMIZATION

56 STABILITY ASSESSMENT OF GRAVITY WALLS OBJECTIVE Optimum wall proportions for gravity retaining structures by targeting various system reliability indices needs to be computed STABILITY ASSESSMENT The stability assessment of gravity retaining walls is characterized by many sources of uncertainty and variability The retaining wall system is modeled as a series-parallel combination of failure modes. The first order reliability method (FORM) is applied to estimate the component reliability indices of each failure mode and to assess the effect of uncertainties in design parameters. The analysis is performed by treating back fill and foundation soil properties, geometric properties of wall, reinforcement properties and concrete properties as random variables.

57 FAILURE MODES CONSIDERED S Q R U V η w 5 P a sin( 90 η + δ ) γ 1 φ 1. Overturning failure w 6 w 1 w P a w 4 δ ( ) 90 η P a cos( 90 η + δ ) ) ( 90 η + δ H. Sliding failure h P L t O bf w 3 N M b η L L h K t H 3 γ φ1 c 3. Eccentricity failure F G H I J ( L + b + S + b L ) B = + t f h 4. Bearing failure

58 FAILURE MODES CONSIDERED contd 5. Toe Shear failure 6. Toe moment failure 7. Heel shear failure 8. Heel moment failure

59 Performance functions 1. Overturning Failure mode g ( x) M = M R 1 1. Sliding Failure mode g ( x) O F = F R 1 3. Eccentricity Failure mode g 3 x ( ) 4. Bearing Failure mode g 4 x ( ) D ( B /6) = 1 e q u = 1 q max 5. Toe shear Failure mode g τ ( ) c 5 x = 1 τ vtoe 6. Toe moment Failure mode g x MR ( ) toe 6 = 1 M utoe 7. Heel shear Failure mode g τ ( ) c 7 x = 1 τ vheel 8. Heel moment Failure mode g x MR ( ) heel 8 = 1 M uheel

60 γ 1 φ ( ) ( γφγφ,,,,,,,,,, ) t h f g1,,3,4 x = f 1 1 cl L Sb bt g5,6,7,8 x f 1 1 cl L Sb bt f f A pt A pt ( ) = ( γφγφ,,,,,,,,,,,,,, /( ), /( )) t h f γc ck y stoe sheel = unit weight of backfill soil = friction angle of backfill soil Parameters to optimize S = width of stem at top of wall γ = unit weight of foundation soil b f = batter width of front face of wall φ 1 = friction angle of foundation soil b = batter width of back face of wall c = cohesion of foundation soil Lt = length of toe slab γ c f ck = unit weight of concrete = compressive strength of concrete L h t = length of heel slab = Width of stem at top of wall f y = yield strength of HYSD bars Astoe /( pt) = steel reinforcement ratio in the toe slab Asheel /( pt) = steel reinforcement ratio in the heel slab

61 Reliability indices satisfying all the constraints in the form of o performance functions as given below g ( ) ( ) ( ) ( ) 1 x 0; g x 0; g3 x 0; g4 x 0 g ( ) ( ) ( ) ( ) 5 x 0; g6 x 0; g7 x 0; g8 x 0 in the standard normal space U as β k n = i = = i= 1 Minimize u ; k 1to 8and i 1to n Reliability index corresponding to each limit state equation can be obtained using non-linear constrained optimization technique such as the method of Lagrange multipliers and is given by Lagrange function n Lk = ui + λkgk( u); k = 1to 8 and i = 1to n i= 1 The stationary points can be found by solving the following equations ( L u ) = 0 ( L λ ) 0 k i = where k = 1 to 8and i = 1 to n k i

62 Statistics of input parameters Random variable Mean μ ) ( i Statistics Coefficient of variation COV ) ( i Distribution γ 1 18 kn/m 3 7% Normal φ 30 o 5% and10% Log-Normal γ 19 kn/m 3 7% Normal φ 1 0 o 5% Log-Normal c 30 kn/m 5% to 0% Log-Normal γ c 4 kn/m 3 5% Normal f 0 kn/m ck (M 0 concrete is assumed for the present study) 10% Normal f 415 kn/m (Fe 415 steel HYSD bars) y (Fe 415 steel is assumed for the present study) 5% Normal A stoe pt 0.5% Normal A sheel pt Mean values of wall proportions and area of 0.5% Normal L reinforcement in toe and heel slab should be t 0.5% Normal obtained from the optimizion for target system L h reliability indices 0.5% Normal S 0.5% Normal b f 0.5% Normal b 0.5% Normal t 0.5% Normal

63 β Identification of MPP in FORM In the standard normal space, the point on the first order limit state function at which the distance from the origin is minimum is the Most Probable Point of failure (MPP) and the shortest distance corresponding to MPP is called as reliability index ( β )

64 System reliability based optimization Overall stability of gravity retaining wall system is influenced by overturning, sliding, eccentricity, bearing, toe shear, toe moment, heel shear and heel moment failure modes. Series-Parallel Combination Model Considered Toe slab failure sequence is a parallel system of its toe shear and moment failure events as shown in above Figure. Probability of failure of toe slab is given by ( ( ) ) ( ) { }{ ( ( ) ) } ( ) ( ) ( ( ) ) { }{ ( )} Pf _ toe = P g5 u 0 g6 u 0 P g5 u 0 P g6 u 0 βtsh β < < = < < = Φ Φ tm probability of failure of heel slab is given by ( ( ) ) ( ) { }{ ( ( ) ) } ( ) ( ) ( ( ) ) { }{ ( )} Pf _ heel = P g7 u 0 g8 u 0 P g7 u 0 P g8 u 0 βhsh β < < = < < = Φ Φ hm

65 Assuming that the overturning, sliding, eccentricity, bearing, toe slab and heel slab failure modes are statistically independent, Probability of failure of the wall system having series-parallel combination model can be computed as follows P P f f _ system _ system ( ( ) ) ( ) < 0 ( < 0) ( ( ) < 0) ( ( ) < 0) ( g5( u) 0) ( g6( u) 0) ( g7( u) 0) ( g8( u) 0) g1 u g u g3 u g4 u = P < < < < { 1 Φ( β )}{ ( )}{ ( )}{ ( )} ot 1 Φ βsli 1 Φ βe 1 Φ βb { 1 Φ ( β )}{ 1 Φ ( β )} = 1 toe heel System reliability index of gravity retaining wall is 1 ( 1 ) _ β =Φ P sys f system

66 COMPONENT RELIABILITY VS SYSTEM RELIABILITY Fig. 5. Variation of component reliability indices ( β i ) and system reliability index ( β sys ) with batter width of back face ( b/ H ) of gravity retaining wall for COV of φ, c and φ 1 = 5%, COV of γ 1 and γ = 7 % and Astoe / pt = 0.10% and Asheel / pt = 0.6%

67 Variation of batter width of back face ( b/ H) and front face ( bf / H) of gravity retaining wall with target system reliability index ( β sys ) for COV of φ = 5% & 10% and COV of c = 5%

68 OPTIMUM WALL PROPORTIONS S / H = 0.05, / Lt H = 0.07, Lh / H = 0.07, t/ H = 0.07 A pt ) = 0.10 % A pt ) = 0.6 % Area of HYSD steel bars in the toe slab ( /( ) stoe Area of HYSD steel bars in the heel slab ( /( ) sheel β sys _ t arget bf / H b/ H bf / H b/ H bf / H b/ H bf / H b/ H bf / H b/ H The areas of cross section from optimized sections are lesser than those obtained from the specifications.

69 Stability Assessment of Buried pipes OBJECTIVE Optimum diameter to thickness ratio and thickness of steel pipe for buried flexible pipes by targeting various reliability indices considering four failure criteria buckling, crushing, deflection and handling flexibility (FWHA 001). STABILITY ASSESSMENT Owing to the uncertainties in soil friction angle and unit weight of the backfill, modulus elasticity of soil, modulus of elasticity and yield strength of steel pipe, the assessment of stability of buried flexible pipes needs to be on rational basis considering variability in design parameters.

70 Limit states Considered 1. Limit state for Buckling failure. Limit state for Crushing failure

71 Limit States Considered 3. Limit state for Deflection failure 4. Limit state for Handling flexibility failure

72 g Performance functions 1. Buckling Failure mode g g 1 ( x) Allowable buckling pressure ( Pa ) = 1 External pressure due to Marston's load ( P ). Crushing Failure mode ( x) b ( f ) y = Yield stress of the pipe material 1 Ring compressive strength or Bending stress 3. Deflection Failure mode 3 x ( ) 4. Handling flexibility Failure mode ( f or f ) Allowable deflection (5% of diameter of pipe) ( Δa) = 1 Horizontal deflection of pipe Δx ( ) A ( ) Maximum permissible flexibility factor FF ( ) ( ) max g4 x = flexibility factor FF e 1

73 Methodology for optimization 1. Assume a trial thickness of the steel pipe and find the diameter to thickness ratio of steel pipe for desired target reliability index against buckling failure using the formulation given below minimize subjected to g buck T uu = u ( ). Thickness of the steel pipe is needed to evaluate for the computed value of diameter to thickness ratio in the step 1 for the desired target reliability index against crushing failure using the formulation given below β buck minimize subjected to g crush T uu = u ( ) β crush

74 Methodology for optimization Verify whether the thickness computed in step is equal to the assumed trial thickness value, if not then again modify the thickness of the steel pipe and then go back to step 1 to evaluate the diameter to thickness ratio and iterate the process. 3. Reliability indices against deflection failure and handling flexibility failure are needed for the established diameter to thickness ratio and thickness of the steel pipe in steps 1 and for the desired target reliability indices using the formulations given below T minimize uu= βdef subjected to g ( u) ( ) flex def T minimize uu= β flex subjected to g u Verify whether the reliability indices and computed in step 3 are equal to the desired target reliability indices, if not then iterate the entire process (starting from the computation of diameter to thickness ratio in step (1) until the criterion is met.

75 Statistics of input parameters Random variable Statistics Coefficient of Mean variation ( μ i ) Distribution ( COV i ) γ 18 kn/m 3 7% Gaussian φ 30 o 10% Log-Normal E soil kn/m 5%, 10%, 15% and 0% Gaussian E kn/m 5% Gaussian f y 8000 kn/m 5% Gaussian ν H 5.0 m 0.5% Gaussian B d.0 m 0.5% Gaussian q 7000 kn/m 30% Gaussian D/ t ratio t Mean value of pipe diameter to thickness ratio ( D/ t) and thickness of steel pipe 0.5% Gaussian should be obtained from the Target reliability based design optimization (TRBDO) for the target component reliability indices 0.5% Gaussian

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77 Conclusions The probabilistic analysis of the soil data and soil profiles provides a format for quantifying the information about the subsurface condition of the site. it also provides the basis for obtaining the response statistics which are useful in the the reliability analysis of geotechnical structures. Reliability based optimization is useful in the design of geotechnical structures

78 Thank you for your attention G. L. Sivakumar Babu Department of Civil Engineering Indian Institute of Science Bangalore