Seismic Mechanical Response of rigid shallow footings Part I - Some experimental/theoretical and numerical issues: monotonic and cyclic loading

Transcription

1 Seismic Mechanical Response of rigid shallow footings Part I - Some experimental/theoretical and numerical issues: monotonic and cyclic loading Departement of Structural Engineerig Claudio.diprisco@polimi.it

2 2 Soil structure interaction as a link between seismology and structural dynamics Between a 3D domain of infinite extension and a finite geometrical determined spatial domain (the structure) It s so difficulty managing heterogeneities and differences! Seismologists against structural engineers

3 3 Outlook of the presentation a) Introduction b) Cyclic liquefaction c) Failure mechanisms and punching d) The interaction domain e) Cyclic soil-structure interaction f) The visco-elastic approach

4 Definition of shallow foundations 4 GEOMETRIES Shallow (B/D>4) and deep foundations (B/D<4) Plinth Strip footings (deformable beam) Mat foundations Grid foundations pillars Stubby foundations the case of the medieval Ghirlandina bell tower (Modena, Italy) Retrofitting of historical building Inertial and kinematic effects

5 The seismic soil-structure interaction 5 What are the consequences? Irreversible differential settlements, damage to the structure energy dissipation 3 Dynamic structural response 4 Soil- foundation interaction Δτ 2 1 site amplification Depending on topography and stratigraphy As usual it is a question of coupling: uncoupled it is easier

6 Subdivide the domain into substructures 6 Performing numerical analyses of the entire domain ( the geotechnical and the structural may be quite complex) Dynamic centrifuge tests can be also performed

7 Local site amplification: topographic factor 7 The site effect: foundations placed along natural slopes 1 Δτ Symmetric cycles? xt & () Frequency and time domain

8 Local site amplification: a coupled problem due to the non-linear mechanical behaviour of soils 8 Liquefaction phenomena and pore pressure buildup 1 Is there any coupling?

9 Liquefaction phenomenon and undrained stable locus 9 1 Concept of static liquefaction: Relative density Drainage conditions Grain size distribution Initiial state of stress

10 1 cyclic liquefaction 10 Cycle asymmetry

11 1 11 What are the strategies necessary to numerically capture the cyclic liquefaction of the stratum? 1. Semi-empirical stress-path approaches: evaluating the stress path by means of visco-elastic numerical analysis and employing empirical/ numerical methods for accounting for, at least locally, the non-linearity of the mechanical response under undrained conditions 2. By employing one-dimensional numerical code in which multimechanism simplified models are implemented (in this case the soilstructure interaction is absent) 3. Fully coupled numerical analyses (by considering the drainage and the ratcheting phenomenon)

12 1 The ratcheting phenomenon and coupling 12 q Elasto-plastic/viscoplastic q Elastic-response Shake down Case a + ε Ratcheting ε q q Case b Plastic adaptative response no ratcheting ε Progressive Stabilization ε

13 1 Cyclic fatigue 13 q q q c b a Case c In case of no coupling between the principal plastic mechanism and the third one, case a is obtained and the accumulation of irreversible strains is characterised by a constant rate ε ε& coupling c a p Under undrained conditions the process is self feeding because p progressively reduces and the stress level increases b N cycles

14 1 II Multiple mechanisms 14 ( f ) g & n * n ij = φn n ' φ * n σ ij ε = α* < f > n f = β I + J n n * n * n 1 2 gn = J * n 2 I Limit locus for the cone axis s * n ' 1 = 3 ij = σ σ χ 1 n ij * n ' * n n ij ij 1 ij 3 I χ n n * n ij w e& ij & χ = J J = s s * n * n * n 2 ij ij 1 *n n * n n ij = & εij J1 χij e& * n n n 1 = 3 & εij χij Problem of the cone shape: model of Prevost [1985], [1990], [1993] 3

15 1 15 q Problem of the cone shape: Prevost model C A B p elastic zone During the unloading the progressive activation of the diffferent plastic mechanisms allows to always obtain a plastic adaptative response Owing to the chosen plastic potential a perfect close loop is always obtained and no ratcheting is simulated

16 1 Calibration on torsional shear tests (Toyoura sand specimens) 16 w n n w β β n 160 TEST TOY 0.01 Hz 25 G [MPa] G at the last cycle G(9) numerical D at the last cycle D(9) numerical D [%] p'=200 kpa γ SA [%] Experimental torsional shear test results

17 17

18 E E E E E E E E E-03 TEST TOY 0.1 Hz τ [kpa] γ Numerical simulations

19 1 cyclic liquefaction 19

20 1 CPT and SPT in situ tests 20

21 1 - Prevention 21 What kind of prevention can be done: Sand columns To improve the drainage Dynamic densification, vibro-compaction To increase the relative density Piles?

22 4 The soil-structure interaction 22 Statically determinate interaction Redundantly constrained interaction Importance of differential settlements induced by cyclic loading (ratcheting DBD approaches)

23 The pseudo-static approach (ULS): the rigid-plastic approach 23 4 Static equivalent horizontal load: step 2 is disregarded whereas step 4 is abruptly simplified. A design pseudo-static distribution of forces is applied to the structure, additional loads H and M are applied on the foundation and new limit conditions have to be accounted for From micro to macro In this perspective the design of the shallow foundation under inclined and eccentric loads becomes essential

24 Failure mechanisms: the interaction domain in quasi static conditions 24 4 Failure mechanisms: small scale 2D experimental test results (drained and undrained conditions, cohesive and granular soils) Nova e Montrasio, 1988

25 Punching mechanisms 25 Punching mechanisms and 2nd order effects 4 Lancellotta e Calavera, 1999 Lancellotta, 1993

26 The interaction domain for rigid shallow footings 26 f 2 2 2β M H 2 V = + V 1 0 ψ B µ Vc 4 m = M/ψBV MAX, h = H/µV MAX = V/V MAX MONOTONOUSLY INCREASING LOADING 1. To each point belonging to the failure locus a distinct failure mechanism corresponds 2. Difficulty in defining the failure locus when loose sand strata are concerned 3. Extension to rectangular footings 4. Extension for D/B>0 H/M V

27 Elasto-plastic finite element numerical analyses 27 Centered vertical load Centered inclined load Tochnog Difficulty of capturing localisation within the continuum, Necessary in case of either complex stratigraphies or topographies 4

28 Shape of the interaction domain in the M-H plane 28 FEM analyses under undrained conditions D/B Normalised failure M-H envelopes for ξ = 0,0.5,0.75,1: (a) D/B = 0 (surface foundations); (b) D/ B = 0.25 ; (c) D/ B = 0.5 ; (d) D/ B = 1 [after Gouvernec 2008]

29 29 Topographic effect small scale experimental test results 2009

30 DEM NUMERICAL SIMULATIONS 30

31 numerical versus experimental data 31

32 Footings along slopes 32 4 H And what about complex soil profiles? And what about saturated soils? m = M/ψBV MAX, h = H/µV MAX Cyclic loading V V = V/V MAX V M

33 0 0 Footing improvement Inclined LOADS 33 GENERALISED STRESS PATHS H v V M H u θ Q Rigid strip footing LOAD CONTROLLED TESTS q V INTERACTION DIAGRAMS THE EXPERIMENTAL TEST SERIES 0.5B 0.5B 0.5B Dense sand B

34 DENSE SAND, vertical loading 34 Failure mechanism in unreinforced dense sand layer Failure mechanism in unfastened reinforced dense sand layer

35 35 6 Unreinforced loose sand Sabbia sciolta non rinforzata Unfastened reinforced loose sand Sabbia sciolta rinforzata con geosintetici non allacciati 10 H [kpa] 4 2 H [kpa] V [kpa] V [kpa] Fastened reinforced loose sand Sabbia sciolta rinforzata con geosintetici allacciati INTERACTION DIAGRAMS H [kpa] EXPERIMETNAL DATA and NUMERICAL INTERPOLATION V [kpa]

36 Numerical simulations Tochnnog finite element code elasto-perfectly plastic constitutive model ,8 1,6 1,4 V / V MAX 1,2 1 0,8 0,6 Legge associata Legge non associata 0,4 0, ,05 0,1 0,15 0,2 0,25 0,3 v / B Truss elements 0,7 0,6 0,5 v [m] 0,4 0,3 0,2 mohr-coulomb associata mohr-coulomb non associata 0, ,2 0,4 0,6 0,8 1 1,2 1,4 u [m] Non associated flow rule (ψ = 0)

37 Soil-pipelines interaction 37

38 Role of the footing embedment δ1 δ2 δ V 3 m2 2γ Vm2 2β FQ ( ) = n + h + m ( ξ ) (1 ξ + ) = 0 V V V V m1 m2 m1 m2 4 Soil-pipe interaction

39 Fem analyses 39 Displacement controlled numerical tests

40 Pipes along slopes 40 4 In case of pipelines along inclined slopes

41 The inertial effects at the foundation level 41 2 The interaction domain may shrink and evolve because inertial forces develop The pseudo-static approach Importance of the λ/b parameter where λ = v/f, if λ/b is small the motion cannot be assumed to be synchronous

42 The dynamic effect 42 Even the coupling between the dynamic response of the structure and of the soil could be a priori accounted for: the former can be visualized by means of a time evolution of generalised loads variables within the interaction domain, the latter by means of the shrinkage of the interaction domain itself H/M V

43 Embankments foundations: the pseudostatic approach 43 Fs = F FF * = s s s F s F safety factor in case of no enbankment s In this case the inertial effect cannot be uncoupled with respect to the kinematic one

44 44 Materiali Caratteristiche fisiche, meccaniche e sismiche Peso di volume argilla 20 kn/m 3 Peso di volume misto stabilizzato 16 kn/m 3 Peso di volume LECA (pacchetto standard) 8.9 kn/m 3 Coesione non drenata argilla 20 kpa Angolo di attrito interno mezzo granulare 40 Modulo elastico E e modulo di Poisson ν 60 MPa, 0.30 Fattore relativo al profilo stratigrafico di fondazione S (suolo tipo E) 1.15 Rilevato Caratteristiche geometriche Angolo di scarpa rilevato 30 Angolo di scarpa strato compensato 60

45 The Newmark method 45 8 Gemona del Friuli 15/09/1976, h (Mw = 6.00) 6 accelerazione (m/s2) a MAX tempo (s)

46 The Newmark method velocità di rotazione (rad/s) velocità di rotazione (rad/s) velocità di rotazione (rad/s) tempo (s) tempo (s) tempo (s) rotazione (rad) rotazione (rad) rotazione (rad) tempo (s) tempo (s) tempo (s)

47 From the elastic to the visco-elastic simplified approach Δτ 1 K =k(θ)? ξ = ξ ( θ)? Can the soil-structure interaction taken into consideration to define the mechanical responseof the structure?

48 The uplift 48 Shirato et al. 2007

49 49

50 M - θ curves 50 PWRI cyclic quasi-static experimental test results, 2005 Dense sand stratum M (kn.m) θ (rad) FOOTING UPLIFT Loose sand stratum M (kn.m) θ (rad)

51 51 Interface elements:: ELASTIC ELASTO-PLASTICi CONSTRAINTS: BILATERAL UNILATERAL APPROACH SMALL DSPLACEMENTS LARGE DISPLACEMENTS

52 52 ELASTIC SOIL STRATUM UNILATERAL CONSTRAINT θ = θ V = V0 M = M 0 M ϑ

53 53 Large displacements v = + ψ tanθ i v g 1 χ = v senθ i 1 tanθ Dipendenza da K (modello elastico, secondo percorso, grandi spostamenti) e confronto con il modello rigido Non linearity due to large displacements M[kNm] K[kNm^(-3)]=1;11 molle K[kNm^(-3)]=10; 11 molle K[kNm^(- 3)]=100;11 molle K[kNm^(- 3)]=1000;11 molle rigido θ[ ]

54 M[kNm/m] θ[ ] V=-5 kn/m V=-7 kn/m V=-9.6 kn/m V=-10 kn/m V=-19kN/m Curves obtained for different values of V 0: elasto-plastic unilateral springs

55 The uplift of rigid shallow foundations 55 Comparison between the numerical results and the Nova- Confronto tra il Dominio di Interazione fornito dal modello Nova- Montrasio e i valori puntuali valutati con il modello numerico Montrasio model failure locus M [knm/m] V[kN/m] Modello Elasto-Plastico Modello Nova-Montrasio Poli. (Modello Nova- Montrasio)

56 Ispra Laboratory Elsa (Pedretti, 1998) 56 Cross section Plan view Quasi static V constant tests, M/H = constant Ispra Laboratory Elsa (Pedretti, 1998)

57 57 Loose sand dense sand

58 58 Loose sand Dense sand

59 59 Loose sand Dense sand

60 60 Loose sand Dense sand

61 61 EXPERIMENTAL TESTS Hmax = 0.2*V H kn] t [sec] v [mm] Settlements t [sec] M [kn*m] M [kn*m] θ [rad] θ [rad] Loose sand Dense sand

62 Symmetric generalised stress-paths 62 Spring-hysteretic damping model c ω 1 D = cost (LIN 1996)\ D h WD 1 πωγ c cω h = = = = 4πW 4π Gγ 2G 2G s A simplified viscplastic approach- superimposition of two effects: the dissipation due to radiation damping and to local irreversibilities

63 63 (- 0 K / K (- ) η R o t a t i o n a l S e c a n t S t i f f n e s s 1 E r o c k i n g a n g l e ( r a d ) D a m p i n g F a c t o r r o c k i n g a n g l e ( r a d ) (- ) 0 K / K T r a n s l a t i o n a l S e c a n t S t i f f n e s s 1 E d i s p l a c e m e n t ( m ) H i g h r e l a t i v e d e n s i t y D R = 9 0 % I S P R A p h a s e 1 I S P R A p h a s e 2 I S P R A p h a s e 3 n u m e r i c a l p h a s e 1 n u m e r i c a l p h a s e 2 P W R I t e s t n. 5 P W R I t e s t n. 8

64 64 In the light of modern seismic standards, the k ӨӨ - Ө -relationship also affects the expected seismic action F H : indeed, F H can be derived from the so-called response spectrum of the tower, as long as its k ӨӨ -dependent natural period is known. But are curves kөө - Ө path dependent? Are they dependent on the soil profile etc?