A new simplified seismic stability analysis taking into account degradation of soil undrained stress-strain properties and effects of compaction

International Symposium «Qualification of dynamic analyses of dams and their equipments and of probabilistic assessment seismic hazard in Europe», 31th August 2nd September 216 Saint-Malo Session 3 : Soils properties and simplified analysis A new simplified seismic stability analysis taking into account degradation of soil undrained stress-strain properties and effects of compaction Antoine Duttine, Fumio Tatsuoka, Taiki Shinbo, Yoshiyuki Mohri Saint-Malo Yannick LE GAL

SUMMARY 1. OUTLINE AND BACKGROUND 2. MODIFIED NEWMARK SLIDING BLOCK ANALYSIS Procedure Analysis example 3. PSEUDO-STATIC FEM ANALYSIS Procedure Analysis example 4. COMBINATION OF NEWMARK-D AND PSEUDO-STATIC FEM ANALYSES 5. CONCLUSION A new simplified seismic analysis taking into account degradation of soil undrained stress-strain properties and effects of compaction, Duttine et. al. 216 2

Following the 211 Great East Japan Earthquake, in the three prefectures of Iwate, Miyagi & Fukushima, around 15% of earth fill dams were significantly damaged (1951 out of 125) Current seismic design analysis (LE, kh=.15, F s >1.2) can not explain Fujinuma dam failure and this 15% damage ratio. Besides, around 2,earth-fill dams in japan!! Need for a simplified practical seismic analysis for high seismic loads: combination of FEM & Newmark Fujinuma dam failure ( H= 18.5m L= 133.2m, top fill composed of poorly compacted weak sandy soils, i.e. prone to strength degradation & liquefaction) Residual 土構造物の残留変形 displacement 許容値 Allowable limit 3. 盛土の残留変形 Combined displacements = by : 1. 1. FEM 連続体としての残留変形 (continuum) + 2. 2. Newmark すべり変形 method 1. 1. Residual すべり変形を伴わない繰 displacement by cyclic loading w/o 返し載荷による残留変形 considering slip (FEM) (FEM) 2. Slip 2. すべり変形 displacement (Newmark) (Newmark 法 ) 1. 極限釣り合い法での最小瞬間安全率 Lowest factor of safety in LE (limit equilibrium) 3

Crucial need to take into account in a simple way the degradation of soil rigidity and strength by seismic loading i.e. undrained cyclic loading (CL) Development of modified Newmark method and pseudo-static FEM analyses in a united framework (cumulative damage and total stress concepts) along with relevant undrained cyclic+monotonic loading tests 1Linear-equiv. response analysis max. resp. acc [gal] 2Newmark-D Mobilized cyclic shear stresses 3Pseudo-static FEM Undrained LE stability analysis Accumulated damage along slip surface Accumulated damage for each element せん断応力 τ (kn/m 2 ) deterioration by damage 香川県打越下池強度低下モデル剛性低下モデル H25 新堤体 - 前刃金 4 σ c '=1kN/m 2 DA=%. 5% 3. 75% 2% 2 5% 7.5% 1% 1 damage 15% 25% 4% 6% 8% 1% 5 1 15 2 25 せん断ひずみ γ (%) Rotational displacement by Newmark Residual displacement by FEM Inertia from response accelerations deterioration by damage Response acceleration of sliding mass 4

.. Procedure of Newmark-D analysis 1Plane strain FEM dynamic response analysis (equivalent linear or non-linear analysis, not taking into account the degradation of stiffness by undrained CL and shear banding) response 最大 acc. 加速 [gal] 度コンター図 6. 586. 572. 558. 544. 53. 516. 52. 488. 474. 46. 446. 432. 418. 44. 39. 376. Searching for the slip circle providing the maximum displacement 2Quasi-static LE stability analysis by Fellenius slice method (using undrained strength with saturated soil) to find the critical failure plane 3Estimation of undrained strength degradation due to undrained CL by the cumulative damage concept based on uniform undrained CL & ML test results 主応力差 (σ a σ r ) (kn/m 2 ) 主応力差 15 1 5-5 圧密過程 排水条件 単調載荷 のみ 繰返し載荷過程 * εd A 土の繰返し非排水三軸試験方法 ( JGS 54 1) 非排水条件 繰返し載荷過程 単調載荷過程 土の圧密非排水 ( Cubar) 三軸圧縮試験方法 (JGS 5 23) 非排水条件 強度低下 単調載荷過程 繰返し載荷 + 単調載荷 -1-1 -5 5 1 15 2 軸ひずみ ε a (%) 時間.2 362. 348. 334. 32. (gal) 4Calculation of slip displacement for a given seismic motion by the Newmark-D method using degrading shear strength. Acc [m/s 2 ] θ [rad/s 2 ] θ [rad/s] δ=r θ [m].1..8.4..4. -.4 -.8 6. 3. 滑動開始時刻 : 13.32s. Duttine et. al. (215): Evaluation of seismic dam displacement by Newmark method taking into account soil strength reduction due to undrained -3. cyclic loading, JGS Special Issue on earth dam seismic stability, 215.3, pp.8-11, (in Japanese) Tatsuoka et al. (214). Evaluation of seismic slip displacement of slope by Newmark method taking into account soil strength reduction due to undrained cyclic loading and strain-softening, Proc. Special JGS Symp. Overcoming the Great East Japan Earthquake, pp.394-43, (in Japanese) 降伏加速度 5 1 15 2 25 3 35 4 45 経過時間 Elapsed time [s] δ max =146.5 mm 5

1 Dynamic response analysis No need for complicated constitutive model & analysis (effects of pore water pressure buildup will be taken into account later in modified Newmark) Hyperbolic model (Hardin-Drnevich, Generalized Hyperbolic Equation GHE model Tatsuoka et al., 1991) Simplified analysis: Linear equivalent Shear modulus ratio G/G 1..8.6.4.2 G/G -γ GHE model (Tatsuoka & Shibuya, 1991) h-γ. 1E-4 1E-3.1.1 1 Shear strain γ [%] Core & shell materials common for all Dc experimental data 45 4 35 3 25 2 15 1 5 Damping ratio h [%] Tatsuoka,F. and Shibuya,S. (1991), Deformation characteristics of soils and rocks from field and laboratory tests, Keynote Lecture for Session No.1, Proc. of the 9 th Asian Regional Conf. on SMFE, Bangkok, Vol.II, pp.11-17. 6

In the analysis further, it will be assumed that yielding or slip occurs when apparent mobilized stresses by linear equiv. response analysis exceeds the deteriorating soil shear strength (computed based on cumulative damage) τ w zero crossing s e apparent mobilized stress by FEM τ w actual mobilized stress τ w (= τ f ) τ i a) equivalent uniform cyclic loading until (n-1) th pulse (cumulative damage theory) n th pulse time τ γ DA s τ initial e undrained τ~γ relation before shaking at n th pulse continuous deterioration by undrained cyclic loading b) shear strain, γ c) τ undrained τ~γ relation γ DA s before shaking e τ f τ initial at n th pulse τ~γ relation upon equiv. uniform cyclic loading shear strain, γ 7

2 Initial stability analysis (Limit Equilibrium: LE) Initial slip with lowest yielding seismic coefficient (LE, slice method Fellenius&others) Slice internal forces equilibrium s mi : undrained initial shear strength s mi ' [ ] u P tan φ + c L σ tan φ + c ' i u i u u τi Fs Fs = = F s :safety factor 8

3 Random cyclic loading used in cumulative damage Simply obtained for each slice by projecting the stresses of nearest element along the slice base plane 1 1 σn = ( σ x + σy ) + ( σy σx ) cos 2α τxy sin 2α 2 2 1 τ = ( σy σx ) sin 2α + τxy cos 2α 2 τ τ xy σ N σx τ xy σ y Normal and shear stress time histories for each slice Slice No.5 σ N Slice No.5 τ 9

3 Cumulative damage concept Shear stress Damage by an individual pulse of an arbitrary loading pulse Arbitrary loading 主応力差 Cyclic stress strain amplitude DA= 1% DA=2% ; 5% DA=1% time Damage D = 1 N c time 繰返しせん断応力振幅比 stress ratio SR=σ d /2σ.6.5.4.3.2 両振幅 Strain 軸ひずみ DA= DA= 1% 5% Liquefaction 液状化強度曲線 strength by undrained CL 三軸非排水繰返し載荷試験拘束圧 σ' =1kPa experiments 実験値 2% 1% 1 1 1 Number of cycles, 繰返し載荷回数 N N c is the equivalent number of cycles of same amplitude as that of individual pulse and that will lead to a damage D=1 1

3 Cumulative damage concept Damage by an individual pulse of an arbitrary loading Shear stress τ pulse Arbitrary loading damaged=1 e.g. defined for conditions of DA=5%strain Damage 1 1 D = = N 5 c time SR= τ/2σ 繰返しせん断応力振幅比 stress ratio SR=σ d /2σ.6.5.4.3.2 両振幅 Strain 軸ひずみ DA= DA= 1% 5% 2% 1% Liquefaction 液状化強度曲線 strength N c =5 by undrained CL 三軸非排水繰返し載荷試験拘束圧 σ' =1kPa experiments 実験値 1 1 1 Number of cycles, 繰返し載荷回数 N N c is the equivalent number of cycles of same amplitude as that of individual pulse and that will lead to a damage D=1 11

3 Cumulative damage concept Time history of accumulated damge each slice, from which time history of equivalent strain amplitude is backcalculated by dichotomy Shear stress τ i τ j τ n Arbitrary loading (FEM resp.analysis) damage damage D=2. time time Strain DA strainda=2% time 12

3 Degradation of shear strength stress 主応力差 Obtained experimentally by undrained cyclic loading followed by monotonic loading tests 15 Monotonic loading 単調載荷 のみ 土の繰返し非排水三軸試験方法 (JGS 541) Undrained CL σ d 土の圧密非排水 (Cubar) Undrained ML 三軸圧縮試験方法 (JGS 523) time 時間 Strain amplitude Monotonic loading DA=1%,2%,5%,1% Deviator stress [kpa] 主応力差 (σ a σ r ) (kn/m 2 ) 1 5-5 εda 繰返し載荷過程 単調載荷過程 繰返し載荷 + 単調載荷 -1-1 -5 5 1 15 2 Axial strain [%] 軸ひずみ ε a (%) 強度低下 monotonic loading cyclic loading Cyclic+monotonic せん断応力 τ φ u φ u φ u Degradation of apparent c u,φ u (total stress) c u 固定点 σ r (σ a -σ r ) max σ ad σ ad σa 直応力 σ 13

3 Degradation of shear strength Obtained experimentally by undrained cyclic loading followed by monotonic loading tests 2 C u, φ u : initial undegraded strength by undrained ML without CL (DA=%) Converted 見掛け内部摩擦角 apparent φ u [deg] φ u 18 16 14 12 1 Φ u =15.5 Φu=A1 exp{-(da/t1) d1 }+A2 exp{-(da/t2) d2 } A1 5 ± t1 17.1257 ± d1 2.14236 ±.1398 A2 1.5 ±.6991 t2 5 ± d2 2.8551 ±.1183 Degradation 見掛け内部摩擦角 Φmob of φ u 盛土の強度低下モデル by undrained CL 動員 φ u (DA) Φ u (DA) 8-2 2 4 6 8 1 12 14 16 18 両振幅ひずみDA [%] Strain DA [%] せん断応力 τ φ u φ u φ u Degradation of apparent c u, φ u (total stress method) Degradation of tan φ u &c u proportional c c u u = tan φ tan φ u u c u 固定点 σ r (σ a -σ r ) max σ ad σ ad σa 直応力 σ 14

Numerical Example 15

Newmark: Force equilibrium in circular slip of slope: Resisting moment M = { R ( τ l )} r fi i τ fi = ci + σ n. i tanφi Disturbing moment u 2 + M RG 2 t M g r u-x= R G θ: displacement in the slip direction at the gravity center of the sliding mass relative to the supporting ground r k h M g M g Slice method Angular displacement: θ Force equilibrium when F s > 1. (not slipping) 2 u M r > M g r + M RG 2 t R G u-x= R G θ δ= R θ Critical slip plane 16

δ δ dθ/dt d 2 θ/dt 2 Acc (m) (rad/s) (rad/s 2 ) (gal).15.1.5..2.1.. -.5 5-5 δ = R θ Integration Integration 1 2 3 4 5 6 7 8 θ M ( R ) = M M t 2 2 G 2 d r time(s) Obtain slip displacement x-u=r θ from: && M θ = M M ( R ) d G r 2 17

....6 θ (rad/s) δ=r θ (m).4.2..2.1. D c =9% Newmark-D:.578 m Original Newmark:.25 m θ (rad/s 2 ) acceleration (m/s 2 ).1. -.1 4. 2.. -2. -4. yield acceleration (Original Newmark) input acceleration yield acceleration (Newmark-D) 2 4 6 8 1 12 Elapsed time [sec] 18

Analysis examples: Newmark-D L=67m FWL 13.m 2 1 3 H=15m 4 5 Compacted to (D c ) 1Ec =85% ; 9% ; and 95% Dc [%] γ t [kn/m 3 ] γ sat [kn/m 3 ] c [kpa] φ [deg] c u [kpa] φ u [deg] 123 core & shell 95 9 85 18.2 17.2 16.3 19.8 19.2 18.7 1 (*) 7 (*) 5 (*) 5(35*) 4(35*) 35(35*) 45 15 4 2 25 32 45 subgrade -- 21. 21. 37 31.5 37 31.5 * residual strength values 19

Analysis examples: Newmark-D acceleration [gal] 4 2-2 data:8 dt=.2sec. -4 2 4 6 8 1 12 14 16 Elapsed time [sec] Input motion (Nankai trough) MAX: 31.3 gal (t=56.2sec.) Results of undrained cyclic & monotonic loading tests Shear stress ratio SR= τ d /σ.6 strainda=.5.4.3.2.1 2% 1% 1% 5% 2% 5% 1% 1% 2%5% 1% Dc=95% Dc=9% Dc=85%..1 1 1 1 Number of cycles, N Liquefaction strength (core, shelter) Degradation ratio tanφ u /tanφ u = c u /c u 1.2 1..8.6.4.2 Dc=85% Dc=9% Dc=95% Core & shelter. 2 4 6 8 1 12 14 16 18 2 Strain amplitude DA [%] 2

Results of analysis Critical failure plane providing the maximum δ (different among different analysis cases) Newmark-D: Due to the use of degrading undrained strength, effects of (D c ) 1Ec on the slip displacement are significant. Realistic for the submerged upper-reach slope. Shear strength, Existing Newmark approaches (O,S, u): Due to the use of drained strength, effects of Dc are small (Original, S) Due to difficult estimation of u at failure (cyclic mobility..), settlement for Dc=95% overestimated ( u) Toyoura sand A: drained peak stress ratio Number of loading cycle, N c = 5 B: undrained peak stress ratio necessary to develop 15 % shear strain Relative density, D r (%) C: undrained cyclic loading strength necessary to develop 15 % double amplitude shear strain Residual settlement at crest [cm] 5 4 3 2 6 4 2 Original Newmark Newmark-S Residual settlement at crest Newmark-Δu Newmark-D 85 9 95 Degree of compaction (1.Ec) [%] Tatsuoka et al.(1982). Cyclic undrained stress-strain behavior of dense sands TSS tests, S&F Vol22( 2):55-7. 21

SUMMARY 1. OUTLINE AND BACKGROUND 2. MODIFIED NEWMARK SLIDING BLOCK ANALYSIS Procedure Analysis example 3. PSEUDO-STATIC FEM ANALYSIS Procedure Analysis example 4. COMBINATION OF NEWMARK-D AND PSEUDO-STATIC FEM ANALYSES 5. CONCLUSION A new simplified seismic analysis taking into account degradation of soil undrained stress-strain properties and effects of compaction, Duttine et. al. 216 22

Development of modified Newmark method and pseudo-static FEM analyses in a united framework (cumulative damage and total stress concepts) along with relevant undrained cyclic+monotonic loading tests 1Linear-equiv. response analysis max. resp. acc [gal] 2Newmark-D Mobilized cyclic shear stresses 3Pseudo-static FEM Undrained LE stability analysis Accumulated damage along slip surface Accumulated damage for each element せん断応力 τ (kn/m 2 ) deterioration by damage 香川県打越下池強度低下モデル剛性低下モデル H25 新堤体 - 前刃金 4 σ c '=1kN/m 2 DA=%. 5% 3. 75% 2% 2 5% 7.5% 1% 1 damage 15% 25% 4% 6% 8% 1% 5 1 15 2 2 5 せん断ひずみ γ (%) Rotational displacement by Newmark Residual displacement by FEM Inertia from response accelerations deterioration by damage Response acceleration of sliding mass 23

Degradation of stress strain properties by undrained CL obtained from same CL+ML test results as for Newmark (no additional experiments) 主応力差 σ d 時間 y =τ/τ peak 1..8.6.4.2 DA= % 1% 3% Undrained CL followed by ML Normalized relations 5% 7% 1% Undrained CL+ML loading Dc=9% ρ d =1.528g/cm 3 ; σ' r =σ' v =1kPa experimental data (Fig. 5) fitting by Eqs. 9~11 土の繰返し非排水三軸試験方法 (JGS 541) Undrained CL 土の圧密非排水 (Cubar) Undrained 三軸圧縮試験方法 ML (JGS 523)...2.4.6.8 1. x = γ/γ peak c x x = + 1+ x 2e 1 2 1 y f (x) x a e m + (m m ) x b x b y= τ/τ peak x= γ/γ peak :normalized peak stress strain f (x):ghemodel (backbone curve DA=%: 5 parameters) m1,m2,a,b,c: 5 parameters that may depend on DA τ peak -DA:degradation of peak strength used in NewmarkD Shear stress [kpa] 3 25 2 15 1 5 Dc=9% σ c '=25kN/m 2 DA= % 2% 1% 15% 3% 5~1% 5 1 15 2 25 4% 6% Shear strain [%] 7% damage strainda increasing 24

pseudo-static FEM analysis constitutive relation (nonlinear elas.) ( e) τ σ = ε 1+ e K v 2 e volumetric deviatoric Shear stress [kpa] 3 25 2 15 1 5 Dc=9% σ c '=25kN/m 2 DA= % 2% 4% 6% 7% damage strainda increasing 1% 15% 3% 5~1% K = Ki + Kw n total bulk modulus 2( 1+ ν ) Ki = G5 3( 1 2ν ) : water bulk modulus(=2.15gpa) K w : bulk modulus of granular skeleton (constant) n: porosity 5 1 15 2 25 Shear strain [%] Taking into account inertia derived from nodes response accelerations T ( ) ( ) int = Ω 内力 F d B σ d d Ω ext T T 外力 F = ρn u&& tdω ρn gdω Ω Ω Inertia by resp.acc gravity constitutive relation d : node displacement (unknown) N, B : shape function and its spatial differentiation g : gravity acceleration u&& = Nd && t t node response acceleration obtained from response analysis. Volumetric locking countermeasure: Selective Reduced Integration or2nd order element(serendipity elem.) 25

Analysis examples : effects of Dc Shear stress [kpa] Shear stress [kpa] 25 2 15 1 5 Dc=85% σ c '=25kN/m 2 DA= % 5% 7.5% 4% 6~1% 5 1 15 2 25 3 25 2 15 1 5 Dc=9% σ c '=25kN/m 2 Dc=85% DA=.5% 2% Shear strain [%] Dc=9% % 2% 4% 6% 7% 1% damage strainda increasing 2% damage strainda increasing 1% 15% 3% 5~1% Shear stress ratio SR= τ d /σ Shear stress [kpa].6 strainda=.5.4.3.2.1..1 1 1 1 Number of cycles, N 6 5 4 3 2 1 1% 1% 5% 2% 2% Dc=95% σ c '=25kN/m 2 5% DA= % 1% 1% 2%5% 1% Dc=95% 1% 5% Liquefaction strength (core, shelter) Dc=95% Dc=9% Dc=85% 7.5% 1% 2% 3% 4% 5% 6% 8% 1% damage strainda increasing 5 1 15 2 25 Shear strain [%] 5 1 15 2 25 Shear strain [%] 26

L=67m Analysis examples : effects of Dc FWL 13.m 2 1 3 H=15m 4. 4 Settlement at dam crest [m] 3.5 3. 2.5 2. 1.5 1..5 D c =85% D c =9% D c =95% Settlement at dam crest Shear strain e (-) 5 Dc=85%. 2 4 6 8 1 12 14 16 18 Elapsed time [s] Shear strain e (-) Dc=95% Shear strain e (-) Dc=9% Similarly as Newmark-D, analysis realistically reproduces strong effects of compaction on the crest settlement 27

Settlement at dam crest [m] Analysis examples : effects of Dc 4. 3.5 3. 2.5 2. 1.5 1..5. τ-γ moderately deteriorated, inertia=large D c =85% D c =9% D c =95% Settlement at dam crest FWL 13.m τ-γ most deteriorated Inertia= Shear strain e (-) 2 L=67m 1 4 5 3 H=15m Dc=85% 2 4 6 8 1 12 14 16 18 Elapsed time [s] Shear strain e (-) Dc=95% Shear strain e (-) Dc=9% Large influence of inertia, max. settlement may be underestimated by static analyses taking into account only rigidities before/after earthquake 28

Combination of Newmark and Pseudo-static FEM Pseudostatic FEM until start of sliding t=53.12 sec. Residual 土構造物の残留変形 displacement Allowable 許容値 limit 3. 盛土の残留変形 Combined displacements = by : 1. 1. FEM 連続体としての残留変形 (continuum) + 2. 2. Newmark すべり変形 method 1. 1. Residual すべり変形を伴わない繰 displacement by cyclic loading w/o 返し載荷による残留変形 considering slip (FEM) (FEM) 2. Slip 2. すべり変形 displacement (Newmark) (Newmark 法 ) 1. 極限釣り合い法での最小瞬間安全率 Lowest factor of safety in LE (limit equilibrium Rotational displacement by Newmark-D Dc=9% Total settlement at crest: s=57.1cm Newmark-D : s=47.7cm PS-FEM : s=9.4cm 29

Combination of Newmark and Pseudo-static FEM Pseudostatic FEM (max. disp.) Dc=9% Increment from t=53.12 sec slip boundaries before settlement slip boundaries after settlement Combined deformation Total settlement at crest: s=147.8cm Residual settlement at crest [cm] 6 5 4 3 2 1 Newmark-D Final residual settlement at crest Newmark-D+ pseudo-static FEM 85 9 95 Degree of compaction Dc [%] 3

CONCLUSIONS Proposed simplified dynamic analysis: Combination of Newmark & pseudo-static FEM analyses Consistent framework: cumulative damage & total stress concepts allowing to model directly degradation of rigidity and strength Results show that analysis realistically reproduces effects of degree of compaction on the seismic stability of earth fill dams Results for real case study (Fujinuma dam) show good agreement with observations, constitutes first validation of the proposed analysis Residual settlement at crest [cm] 6 5 4 3 2 1 Newmark-D Final residual settlement at crest Newmark-D+ pseudo-static FEM 85 9 95 Degree of compaction Dc [%] Scale 1 m for model dimensions 1 m for deformation s= 1.63 m: total settlement at crest obtained by Newmark-D and pseudo-static FEM s/h= 9.2 Settlement % at crest s=1.63m (s/h=9.2% Slip C2 Scale s 1 m for model dimensions 1 m for deformation Slip C1 s= 4.4 m: total settlement at the left end of crest obtained by Newmark-D and pseudo-static FEM s/h= 25 % s A new simplified seismic analysis taking into account degradation of soil undrained stress-strain properties and effects of compaction, Duttine et. al. 216 31

THANK YOU FOR YOUR ATTENTION Title 216 32