Geodynamics in ZSoil 2012

Transcription

1 Geodynamics in ZSoil 212 Andrzej Truty ZACE Services

2 Dynamic capabilities of ZSoil 212 Transient linear/nonlinear time history analysis for 1 structures 2 single-phase soil-structure interaction 3 two-phase fully/partially saturated soil-structure interaction Eigenvalues and eigenvectors detection via Lanczos method

3 Geodynamic tools Implicit time integration schemes (Newmark, HHT α ) for single phase and two-phase problems Viscous boundaries for single and two-phase media Domain reduction for single and two-phase media Modeling of damping Rayleigh damping Hysteretic damping: creep, dissipative stress-strain laws Numerical damping (via time integration schemes) Mass discretization (lumped/consistent) Dynamic analyses run in the absolute or relative format Boundary conditions Imposed displacements, velocities and accelerations Correcting acceleration time histories via Butterworth filtering and baseline correction procedures Application of HSs model for hard soils and Densification model for loose deposits

4 Nonlinear Newmark scheme Nonlinear balance equation Ma n+1 + Cv n+1 + N (d n+1 ) = F n+1 Integration scheme in time d n+1 = d n + tv n + t2 2 [(1 2β)a n + 2βa n+1 ] v n+1 = v n + t[(1 γ)a n +γa n+1 ] Integration coefficients for zero numerical damping: β =.25, γ =.5

5 Nonlinear HHT α scheme Linear form of the algorithm is written: Ma n+1 +(1+α)Cv n+1 αcv n +(1+α)Kd n+1 αkd n = F n+α [ ] n+α =(1 + α)[ ] n+1 α[ ] n F n+α =(1 + α)f n+1 αf n Nonlinear version Integration scheme M a n+1 + C v n+α +N(d n+a ) = F n+α d n+1 = d n + t v n + t2 [(1 2β)a n + 2β a n+1 ] 2 v n+1 = v n + t [(1 γ) a n + γ a n+1 ] (1 2α) Hilber proposes:.3 <α<, γ= 2 α= corresponds to Newmark s algorithm, β= (1 α)2 4

6 Domain Reduction Method (DRM) (Bielak 21) A A A A u (), t u& (), t u&& () t P e Fault Goal: analyze computational model that concerns the structure and only a small adjacent part of subsoil Single and two-phase formulations are supported

7 DRM decomposition A A A A u (), t u& (), t u&& () t P e Fault 1 This model with a large subsoil zone and source of load P e (t) is decomposed into two models: background model reduced model

8 DRM: background model A u (), t u& (), t u&& () t P e Fault 1 In the background model the structure is removed and free field motion is analyzed 2 Displacements, velocities and accelerations induced by P e (t) are denoted by u (t), u (t), ü (t)

9 DRM: reduced model Boundary layer e e e b i ~ Ω + ~ Ω + Γ + ~ Ω + Γ Ω ~ Ω + ~ Ω + Γ + ~ Ω + Γ Ω ~ 1 Viscous dampers are added to ˆΓ + to cancel wave reflections 2 Displ. decomposition in the exterior domain: u e = u e + w e

10 DRM: 2D example 3m 12m m=75kg/m m=75kg/m m=75kg/m 2m 6m 4m 4m 6m 2m E Frame dimensions: L = 2m, H = 12m (spacing 6m) Columns:.6m.8m Horizontal beams:.4m.75m Distributed mass: 75 kg/m Concrete: E = 3 kpa, ν =.2 Subsoil: E = 1 kpa, ν =.35, ρ = 2 kg/m 3 Subsoil depth: 3m Excitation: 5 Hz harmonic imposed base displacements Damping: δ con =./.15, δ soil =./.3

11 DRM: 2D example: Large model A u A u B B 36 m To verify results a large model 36m long was analyzed Periodic BC were assumed on left and right vertical walls In the considered case v s = 136.1m/s, v R = m/s and λ R = 25.4 m

12 DRM: 2D example: reduced model A u A = u B B 3m 3m dashpots P 1m dashpots Elements in exterior domain Elements in boundary layer Boundary condition w e = Reduced model lengths for undamped vibrations L i : 2m, 15m, 1m L i for damped vibrations: 2m, 15m, 1m, 7m oraz 5m

13 DRM: 2D example: results for undamped vibrations Comparizon of u x time histories at point P for reference and 1m long DRM model ux(t) [m] Model referencyjny t [s] Model DRM 1m

14 DRM: 2D example: error in solution for undamped vibrations (ux-ux_ref)/ux_ref_max 2m t [s] (ux-ux_ref)/ux_ref_max 1m t [s] (ux-ux_ref)/ux_ref_max 15m t [s]

15 DRM: 2D example: error in solution for damped vibrations (ux-ux_ref)/ux_ref_max 15m (ux-ux_ref)/ux_ref_max 1m t [s] t [s] (ux-ux_ref)/ux_ref_max 7m t [s] (ux-ux_ref)/ux_ref_max 5m t [s]

16 3. Charakterystyka źródła wymuszenia: Walec typu HAMM 3518 HT Walec typu STAVOSTROJ VH 115 A Walec typu BOMAG BW 164 AD Szacowanie niepewności pomiaru DRM: another practical 3D example Problem: Protection of the 4km long piped gas borehole and restrictions put on road construction technology Punkt 2 P-4x; P-5y; P-6z Punkt 1 P-1x; P-2y; P-3z Ryc. Widok ogólny na szyb wydobywczy Ciecierzyn 3 Andrzej Truty ZACE Services Geodynamics in ZSoil 212

17 DRM: a practical 3D example Pipe may be subject to excitation caused by road construction vibratory rollers F sin(θt) X L

18 DRM: a practical 3D example Protection of the pipe 12m 4m Mata wibroizolacyjna Nasyp Ściany żelbetowe 4cm Mata wibroizolacyjna Nasyp MIN 5m Podłoże Ściany szczelinowe 6cm Zarurowany odwiert

19 DRM: a practical 3D example: method of solution DRM is a robust tool for this example The free field (background model) is solved only once as an axisymmetric problem The reduced model including soil, pipes and protection walls is generated once as a 3D Analysis of influence of the distance L can be made by shifting the 3D model along x-axis

20 DRM: a practical 3D example: free field motion Setting Rayleigh damping parameters to match attenuation curve AY[cm/s2] y = e -.417x R 2 = R [m]

21 DRM: a practical 3D example: reduced model 12m 32m 29m Płaszczyzna symetrii 3D mesh: 27 DOFS Ściana szczelinowa 1 layer of elements in exterior domain and 1 layer of elements in the interior domain

22 DRM: apłaszczyzna practicalsymetrii 3D example: reduced model Diapraghm wall and piped gas borehole Ściana szczelinowa Odwiert

23 DRM: a practical 3D example: computation Computations on 64-bit Windows Vista Profesional system With the limited output computation time was 6h on Pentium Quad Core Extreme with 8 Gb of RAM With the full output for each time step computation time was about 12h The animation will be shown for force 237 kn with 35 Hz frequency Show animation video

24 Dynamics of fully/partially saturated two-phase media Balance of the momentum: σ ij + γ b i = ρ ü i x j Balance of the mass for the fluid: S ε kk vi,i F cṗ = Effective stress principle by Bishop: σ ij = σ ij + Sδ ijp ( 1 Darcy law: vi F = k ij k r (S) γ F p,j + b j 1 ) g üj Simplified u p formulation is adopted (valid for earthquake analysis)

25 Integration schemes in time Expressions for solid displacements and velocities u n+1 = u n + u n t + t2 2 [(1 2β) ü n + 2βü n+1 ] u n+1 = u n + t [(1 γ) ü n + γü n+1 ] Expression for pore pressure: In the HHT scheme p n+1 = p n + (1 θ) ṗ n t + θ tṗ n+1 p n+α = (1 + α)p n+1 αp n ṗ n+α = (1 + α)ṗ n+1 αṗ n u n+α = (1 + α)u n+1 αu n u n+α = (1 + α) u n+1 α u n ü n+α = (1 + α)ü n+1 αü n

26 Two phase: matrix form of balance equations Balance of momentum (HHT scheme) Mü n+1 + C u n+α + F INT (u n+α) + C F p n+α = F EXT n+α Balance of the mass for fluid phase written at time step n + α ( C F ) T un+α 1 γ F HF p n+α +R F ü n+α h F M F ṗ n+α +Q F = After linearization of this system we solve for unknown δü and δṗ

27 Dynamic consolidation of a soil layer benchmark To achieve steady state harmonic solution more than 2 cycles of the loading are applied Solution of this problem depends on some parameters Π 1 and Π 2 q sin( t) p= Π 1 = 2 k T o βπ g T 2 Π 2 = π 2 ( T T o ) 2 1m Ux= Natural period of vibration T and dilatational wave velocity v c are defined as Ux=Uy= T = 2H v c E oed + K F /n v c = ρ

28 Dynamic consolidation of a soil layer benchmark The considered test cases No Π 1 Π 2 k [m/s] ω [rad/s] t Table: k and ω values

29 Dynamic consolidation of a soil layer benchmark y/h PI1=.1, PI2=.1 (closed form).4 Z_SOIL HHT(+).3 Z_SOIL HHT p/q y/h PI1=.1, PI2=1. (closed form).4 Z_SOIL HHT(+).3 Z_SOIL HHT p/q Π 1 =.1 and Π 2 =.1 Π 1 =.1 and Π 2 = 1.

30 Dynamic consolidation of a soil layer benchmark y/h PI1=1., PI2=.1 (closed form).4 Z_SOIL HHT(+).3 Z_SOIL HHT p/q y/h PI1=1, PI2=.1 (closed form).2 Z_SOIL HHT(+) Z_SOIL HHT p/q Π 1 = 1. and Π 2 =.1 Π 1 = 1 and Π 2 =.1

31 Dynamics: two-phase stabilization procedures Why do we need stabilization? Nonstabilized solution Stabilized solution video video

32 Viscous boundaries for single phase Lysmer dashpots are the simplest to cancel wave reflections outgoing from the domain (exact only in 1D) Damping force vector F v = N T σ s dγ σ s is a viscous stress defined as { 1 σ = (λ s + 2µ s ) nn T + µ ( )} s t 1 t T T 1 + t 2 t 2 v s c p c s The corresponding shear and dilatational wave velocities are denoted by G λ + 2 G c s = c p = ρ ρ NB. The normalized normal and tangential vectors are denoted by n and t 1, t 2 Γ

33 Viscous boundaries: example 1m A a y (t) B a x (t) ax/g (t), ay/g (t) Material data: E = 2 ν =.25 γ = 9.81 [kn/m 3 ] ρ = γ -.1 g = 1 [kg/m3 ] t [s] G C D c 1m s = = 2 m/s ρ λ + 2 G c Viscous damper is added at p = = ρ C-D boundary m/s NB. Viscous dampers may inherit their properties from adjacent continuum elements (to be set at material level)

34 Viscous boundaries: example horizontal velocity at points C,D should vanish after time t = =.45 s Vx [m/s] t [s] Vx [m/s] t [s]

35 Viscous boundaries: example vertical velocity should vanish after time t = =.429 s Vy [m/s] t [s] Vy [m/s] t [s]

36 Viscous boundaries for two-phase (H. Modaressi) F v u = F v p = Γ Γ N T σ s dγ N T ΦdΓ { σ s = ρ c2 p (λ s + 2µ s ) nn T ( + ρ c s t1 t T T 1 + t 2 t ) } 2 v s +n (S p S o p o ) V p1 Φ = k [ ρ ( 1 c2 p V 2 p1 ) ρ F ] n T a s The approximate first dilatational wave velocity for saturated medium is defined as ( ) Q V p1 = c p 1 + λ + 2 G 1 Q = n S K F + n ds dp

37 Two-phase viscous boundaries: benchmark q q q A B p= 1 m A B p= Ux= 1 m A B p= Ux= 3 m Ux= q(t) C D C D Paraxial element 1 kn/m2 C D Ux=Uy= 1 t [s]

38 Two-phase viscous boundaries: benchmark Solution for 3m model p [kpa] t [s] 25m 5m 1m uy [m] t [s] 25m 5m 1m

39 Two-phase viscous boundaries: benchmark Solution for 1m model without paraxial elements p [kpa] t [s] 25m 5m 1m uy [m] t [s] 25m 5m 1m

40 Two-phase viscous boundaries: benchmark Solution for 1m model with paraxial elements p [kpa] t [s] 25m 5m 1m uy [m] t [s] 25m 5m 1m

41 Viscous boundaries in practice How to handle transient dynamics preceeded by an initial state or other static drivers? Relaxed fixities LTF (t=) t=1 s 1 t Viscous dampers a x (t) Initial state Dynamic driver

42 Baseline correction and filtering seismic input Import earthquake records in 3 formats: (US) Local/frameset.htm (EU) free format (ti,ai)

43 Baseline correction and filtering seismic input Butterworth filtering (low-pass,high-pass,band-pass,band-stop) Baseline correction

44 Export options For both corrected/uncorrected acceleration time history Export to excel Fourier spectra Export to excel elastic response spectra for different damping coefficients starting from ξ =.5 up to ξ =.25 Fourier spectra a [[m/s^2]] Serie1 Serie f [Hz]

45 Soil constitutive models HSs model for hard soils with strain dependent stiffness moduli Densification model for loose deposits to describe liquefaction phenomenon (2 versions: by Zienkiewicz and by Sawicki)

46 Modeling liquefaction with DNS model Soil layer subject to horizontal shaking (N-S El Centro scaled to.1g)