Dynamics. Linear/Nonlinear Analysis Of Soil Structure .

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Dynamics.Linear/Nonlinear analysis of soil structureinteraction problems.Constitutive aspectsc ZACE Services LtdAugust 20141 / 68

Scope of the lectureExample of a building subject to the earthquake (usingDomain Reduction Method (DRM))1Free eld motion as 1D wave propagation problem12342Basic requirements for mesh size and time stepMesh and BC for rigid base modelApplication of earthquake signal to the model (relative vsabsolute)Signal processing (linear deconvolution, baseline correction,Butterworth ltering)Analysis of reduced model12Setting reduced DRM model (exterior and boundary domainsetting)Running DRM modelPure periodic BC modelApplication of nonlinear models to the problem2 / 68

Problem: time history analysis of a building subject to theLoma Prieta earthquakeTo run this example we can use so-called Domain ReductionMethod (DRM)It is the two step analysis that consists of12Free eld motion analysisReduced model analysis (structure and small part of subsoiladjacent to it)3 / 68

Free eld motion: layered subsoil subject to Loma Prietaearthquake (1D) (single phase)4 / 68

Free eld motion: estimating mesh size and time stepTo trace wave propagation in the medium we needapproximately 10 nodes per wavelengthThe mesh size depends on the maximum frequency fmax that isto be representedFor typical seismic application fmax is limited up to 10 HzHence the maximum mesh size should be smaller thanhe λ10 vs10 fmaxHence the time step limitation can be formulated as follows (ina single time step signal should not pass through more thanone element and v is the maximum wave velocity) t hev5 / 68

Free eld motion: estimating mesh size and time stepMaterial ][kN/m30.250.250.3Sand61Clay102Bedrock 2LayerDepth···105105106γe[kN s2 010Clay101.832093622.12.00.010Bedrock 2.341281239612.812.00.009Sandρh teHence in sand layer we will use elements with h 1.5 m, and foreclay h 2.0 m6 / 68

General comments: Rigid base modelReal situationsoilEs s s sbedrock Eb b b bImpedance ratio αz ModelsoilEs s s sRigid basea(t) / v(t) / u(t)ρs vss 0 (complex for nonzero damping)ρr vsr7 / 68

General comments: Rigid base modelNodes at the bottom are xed(a) Motion is imposed by displacement/velocity/accelerationboundary conditions (absolute format)1Output: absolute displacements/velocities/accelerations(b) Motion is imposed by application of global acceleration tothe whole domain (relative format)1Output: relative displacements/velocities/accelerations (withrespect to the rigid base)8 / 68

General comments: Compliant base modelReal situationsoilModelEs s s sbedrock Eb b b bsoilEs s s sb d kbedrockCompliant base(viscous dashpots)a(t)Impedance ratio αz ρs vss 0 (complex for nonzero damping)ρr vsr9 / 68

Compliant base model: basic assumptionsNodes at the bottom are free to moveViscous dashpots are added to the baseMotion can exclusively be applied through the accelerationrecord (seismic input)1Output: absolute s are integrated to velocities via Newmark method(a(t) vsu (t))Viscous shear tractions are computed and applied to the baseThe input a(t) does not need to be compatible with a(t)computed at the base (!)L.H. Mejia, E.M. Dawson. Earthquake deconvolution for FLAC.2006 (available on the web).10 / 68

General comments: Domain Reduction Method: GeneralideaA&& A (t )u A (t ), u& A (t ), uPeFaultGoal: analyze computational model that concerns thestructure and only a small adjacent part of subsoilSingle and two-phase formulations are supportedThis model with a large subsoil zone and source of load Pe (t)is decomposed into two models:Background modelReduced model11 / 68

DRM: Background modelA&& 0 (t )u0 (t ), u& 0 (t ), uPeFault12In the background model structure is removedBackground model yields free eld motion:u0 (t), u̇0 (t), ü0 (t)12 / 68

DRM: Reduced modelBoundary layereee ib 12Viscous dampers are added to Γ̂ to cancel wave re ectionsDispl. decomposition in the exterior domain: ue u0e we13 / 68

DRM: Consequences12345Continuum in the exterior and boundary layer must work in theelastic modeZSoil freezes sti ness moduli in these zones at the begining ofthe analysisThe interior domain can be analyzed with a suitable nonlinearsoil model (HSs, Densi cation (DNS), HB)For HSs, DNS, HB small strain sti ness is automaticaly takeninto accountFor MC we have to adjust sti ness moduli (after staticanalysis) by hands or by using load time function14 / 68

DRM: Dimensions of background and reducedmodelsSpace dimension for background model can be lower than forthe reduced modelREDUCED MODEL (2D)BACKGROUND MODEL (1D)15 / 68

DRM: Background (2D) Reduced (3D)BACKGROUND MODEL(2D) or (1D)REDUCED MODEL (3D)16 / 68

DRM: Background (2D-axisymmetric) Reduced(3D)F A sin(wt)REDUCEDMODEL (3D)BACKGROUND MODEL(AXISYMMETRIC 2D)17 / 68

DRM: Background (3D) Reduced (3D)BACKGROUND MODEL ((3D))REDUCEDMODEL (3D)18 / 68

Coming back to our problemWe will use DRM methodRigid base model is analyzed19 / 68

Free eld motion: Model generation, step by step1234File/New - select Plane strain modelFill Project preselection form (set ON option Dynamics )Under (Control/Units) set units system (use [s] for time unit)File/Save As : shear-layer-RB-rel.inp20 / 68

Free eld motion: Setting material data1Add three continuum materials for sand, clay and bedrock viaAssembly/Materials (material data is given in tabular formin slide 6)21 / 68

Free eld motion: geometrical model1234Use option draw rectangle from ribbon menu DrawDraw 2 rectangles for sand and clay layers (so far we do notknow how big the reduced model should be; hence let usassume 100m from the axis of the building)Create 2 subdomains using Macro-model/ Subdomains/Create continuum inside contourApply materials to subdomains using ribbon menuParameters/Update par.22 / 68

Free eld motion: model discretization12Create virtual meshes(4 elements in sand layer, 5 elements in clay layer)(use options from ribbon menu Mesh or right menuSubdomain/Mesh/Create virtual meshThen convert virtual mesh to the real one23 / 68

Free eld motion: boundary conditions for rigid basemodel1234Select 2 nodes at the base and x themCreate an auxiliary vertical symmetry plane (at x 0.0)Select nodes at the left wall (except one at the base)Set periodic BC (use FE model/Boundary conditions/PeriodicBC/Nodes & Plane option)24 / 68

Free eld motion: Let us de ne seismic input (relativeformat)123Go to Assembly/Seismic input optionBy using this option our results will be output as relative withrespect to the rigid baseSet scaling factor to 1.0 (or another), de ne if a given signal isnormalized (or not) by g value and edit the load time functiona(t)/g (or a(t)/1 sm2 ) (click on Edit. in the combo boxcontaining list of load time functions)25 / 68

Free eld motion: Signal processingBefore we apply given earthquake record as a seismic input in ourmodel the following operations must be carried out1 Baseline correction and Butterworth ltering on the targetearthquake record2 Deconvolution of a given signal to the base (only lineardeconvolution is handled in ZSoil)26 / 68

Signal processing: baseline correctionE ect of baseline correction is well visible when earthquake signal isapplied as an imposed acceleration (at the base)(absolute format)Baseline correction is inactive27 / 68

Signal processing: baseline correctionGiven set of points: ai (ti )Goal: remove trend line from ai (ti ) (qubic polynomial)PMethod: solve optimization problem: ni {ai ã(ti )}2 MINTrend line equation: ã(ti ) ao a1 ti a2 ti2 a3 ti328 / 68

Signal processing: baseline correctionBaseline correction OFFBaseline correction ON0.6u(t) [m]0.40.20.00.20246810t [s]1214161829 / 68

Signal processing: Butterworth lterNB. In most cases we will apply low pass lter to cancel highfrequencies (in earthquake engineering we usually use 10 Hz lowpass BF lter)30 / 68

Free eld motion: Signal processing: deconvolution1Assembly/Load functions - Add a function and label it asLoma Prieta23Press button Acceleration time histories toolbox and thenin the foregoing dialog box Import from local resourcesrecord LomaPrieta-18-10-1989-Station-Corralitos.earSet ON 4-th order, 10 Hz low pass Butterworth ltering ,set ON Baseline correction and Update the signal31 / 68

Free eld motion: Signal processing: deconvolution1234select tab Linear deconvolution and ll the form with datadeclare that the signal is given on top of layer 1 (sand) and is to betransfered to top of layer 3 (bedrock)use Rayleigh damping (for f1 2 Hz ξ1 0.05 and for f1 8 Hzξ1 0.05) (use button . next to the combo box)add quiet zone ( Add zeros left ) to the record at the beginningmmwith t 6m/150 10m/208 0.1 ss5sRun deconvolution and press OK to accept modi cations32 / 68

Free eld motion: shifting load time function in timeIn order to be able to reuse this project as a free eld motionfor the reduced one we need to shift the dynamic driver in time(by 1s for instance)(will be explained later on why.?)33 / 68

Free eld motion: Drivers12Set drivers (Control/Drivers)Note that Default control for each driver can be di erent(2014)34 / 68

RAYLEIGH DAMPING .b)Speci c settings: Rayleigh dampingα (or ao ) parameter applies to the mass (low frequencydamping, may represent viscous damping)β (or a1 ) parameter applies to the sti ness (high frequencydamping, may represent hysteretic damping)NB. If all materials have the same damping coe cients we can setthem at the global level in settings for all dynamic drivers (will bedone later on). We can do it also at the material level.35 / 68

Free eld motion: Dynamic settings12Set dynamic settings in Control/Dynamics menu or bypressing button . in column Dynamic analysis settingsdirectly in the list of driversset up global Rayleigh damping coe cients, set ON } HHTintegration scheme36 / 68

Free eld motion: Computation and postprocessing1Displacements, velocities and accelerations are output asrelative to the rigid base37 / 68

Reduced model: Geometry of the concrete frame123Concrete columns 60 60cm (spacing in Z direction: 6m)Concrete beams 60 100cm (spacing in Z direction: 6m)Foundation raft 70 cm38 / 68

Reduced model: Creating new project12345File/NewSelect plane-strainIn Project preselection set ON DynamicsFile/Save As reduced-RB-relImport data from free eld motion project (in main ZSoilmenu use File/Import data from another *.inp le)39 / 68

Reduced model: importing meaningful data from freeeld motion projectAdd 3 new materials for the raft (material 4), beams (5) andcolumns (6) (assume E 30000000 kPa, ν 0.2, γ 24kN/m3 ); in group Main select} Flexibility based formulation (single beam per member issu cient once mass lumping only at connections betweenbeams and columns is a correct assumption)NB. In } Flexibility based formulation beam elements have 2nodal points (as in standard) but stress resultants are output in 5integration points (two of them are placed exactly at theendpoints); mass is lumped at two nodal points; these beamelements are mainly dedicated to structural aplications.40 / 68

Reduced model: Creating macro-model12Let us go to the preprocessorFirst we will create construction axes (X { 50, 9, 0, 9, 50}, Y {0, 10, 13, 16, 21, 26}(here we have to be coherent with the free eld motiongeometry !) and then draw the geometry41 / 68

Reduced model: meshing (at the macro-level)123The optimal situation is when at the left and right walls wehave mesh that is compatible with the one used for free eldmotionTherefore in the vertical direction we will have 4 elements insand layer and 5 elements in the clay layerAfter meshing continuum select all beam macro-elements andcreate virtual mesh with split parameter equal to 142 / 68

Reduced model: DRM setup (1)1234Here interior domain is extended until the bedrockBoundary and exterior layers are located at the left rightvertical walls (single column of elements in each)On the exterior domain we have to add viscous dashpotsIn our case setup (1) is optimal43 / 68

Reduced model: DRM setup (2)123Here interior domain is far from the bedrockBoundary and exterior layers (still single layers of elements) arelocated around the structureOn the exterior domain we have to add viscous dashpots44 / 68

Reduced model: real mesh, DRM domains1Once we have FE mesh we can de ne DRM exterior andboundary domains (FE model/DRM domains/Create.)45 / 68

Reduced model: existence function and unloadingfunction for horizontal xities12Before starting dynamic analysis horizontal xities must bereleased but static equilibrium must be preservedTo do that we need to de ne one existence function for thesexities in X-direction and one additonal load function that willbe used as an unloading one for reactions in released xities46 / 68

Reduced model: setting variable in time solid BC12First we apply standard BC (FE model/Boundaryconditions/Solid BC/On box)Then we can select nodes at both vertical walls (except thoseat the base) and de ne BC using option (in same menu Onnode)47 / 68

Reduced model: added masses12Remaining part of the structure add some mass to the systemHere we add 6000 kg at each nodal point in structure locatedon axis of symmetry and 3000 kg in the remaining one48 / 68

Reduced model: associate free eld motion project withcurrent one49 / 68

Reduced model: computation and analysis of resultsTracing time histories of relative displacements andaccelerations of selected pointsMaking envelopes in structuresObserve parasitic e ect of high displacement amplitude at endof shaking visible in displacement/acceleration time history plotfor top frame node (this is an e ect of rigid base assumption)50 / 68

Reduced model: what about nonlinearities ?Main sources of nonlinearities12In subsoil (plasticity, pore water pressure generation (in fullysaturated media) that may lead to liquefaction in case of loosedeposits)In structure (cracking)NB. Once we use nonlinear models for subsoil deconvolution is nomore unique (sti ness and damping are stress/strain dependent)51 / 68

Reduced model: nonlinear constitutive models for subsoil1M-C, H-B modelshere we have to de ne small strain E modulus that is largerthan static one; to use di erent sti ness moduli within onecomputation we can apply a load time function to thisparameterthese models have built in hysteresis e ect but for larger strainvalues; therefore we have to be careful when setting dampingparameters; using low frequency damping (to mass) is correctbut sti ness proportional damping (if de ned) is alreadyincluded (partially or fully) in the elasto-plastic law; in thiscase we could de ne mass proportional damping parameterα 4πf ξ (at f 2Hz for instance) while keeping β 0.02HSs, Densi cation modelsHSs and Densi cation models can be used for dynamicanalyses but we need to activate Small strain sti nessoption in bothIn both cases mass proportional damping is correct whilehysteretic damping is fully embedded in the constitutive theory52 / 68

Let us try M-C model for subsoilSave data le for current reduced model asreduced-RB-rel-MC.inp (File/Save As)Switch elastic model for sand to M-C (φ 30o , c 1 kPa,ψ 0o ), activate group Damping and set α 1.256 (5%damping at f 2 Hz), β 0.0Switch elastic model for clay to M-C (φ 25o , c 10 kPa,ψ 0o ), activate group Damping and set α 1.256 (5%damping at f 2 Hz), β 0.0In the preprocessor add initial stresses to sand and clay layers(use option FE model/Initial conditions/Initial stresses)In the preprocessor add contact interface between wall andsubsoil53 / 68

DRM with M-C model: contact interfaceSelect continuum element edges adjacent to the concrete wall andthen use option FE model/Interface/On continuum elementedges54 / 68

DRM with M-C model: analysis and resultsHere we can see that in some periods of shaking plastic zonesare active near boundary layerWhat about periodic BC ?55 / 68

M-C model with periodic BCSave data le for current reduced model asperiodic-RB-rel-MC.inp (File/Save As)Go to preproceesor1234567remove DRM domainsdelete horizontal xities at left and right wallselect nodes at the left wall (except one at the base)create auxiliary symmetry plane at x 0.0 (use option fromribbon menu Draw)use option FE model/Boundary conditions/PeriodicBC/Nodes & Planedashpots will not play any role (so we can leave them)quit preprocessorDelete free eld motion project from project descriptionRun analysisNB. Periodic BC can be used only for at con gurations56 / 68

Let us use HSs and densi cation modelsFor loose sand layer we will use Densi cation model by SawickiFor clay layer HSs model will be used57 / 68

Densi cation theory by Sawicki (1989)Concept of common compaction curve (in simple shear)εacc eo C1 ln (1 C2 z)1z N γo24Only the 2 parameters: C1 i C2q0.040DNS g0 3.871e-04ZSOIL g0 3.871e-04DNS g0 7.741e-04ZSOIL g0 7.741e-04DNS g0 1.548e-03ZSOIL g0 1.548e-03DNS g0 3.871e-03ZSOIL g0 3.871e-03DNS g0 7.741e-03ZSOIL g0 7.741e-030.0350.0300.025EPS-V o sin( 0058 / 68

Global construction of densi cation modelModel is derived directly from HSs (Schanz, Vermeer)Deviatoric plastic ow rule is adopted (ψ 0o )Model is equiped with a current shear yield surface controlledin addition by an external M-C surfacerefEur EurEo Eoref p mref σp mσ refElastic sti ness modulus E varies from Eo EurAll volume changes are due to explicit densi cation mechanism59 / 68

Global construction of densi cation modelGsHardin-Drnevich700Go600M-Cq [kPa]500shearmechanism400Current position ofinternal shear mechanism300200M-CStress cyclein drained triaxialtest100Gur0 0100200300400500600p [kPa]806040TAU [kPa] c200-20-40-60-80-0.0006 -0.0004 -0.000200.0002 0.0004 0.0006GAMMA [-]60 / 68

Customization of the modelModel is very exible and can be used in di erent contexts(not only for loose deposits)Standard Mohr-Coulomb(E (p) small-strain densi cation )M-C hardening(E (p) small-strain densi cation )61 / 68

Simple shear test (static) (drained case)100 kPaBAuxA uxB 10 3 sin(2 π t)t 0.100s, step 0.25s1m {‐100,‐100,0,‐100}uyA uyBCC1 8.7 10 3 ,C2 2 105D1mmij - 0020.00020.0010.0000510z * 1.0e61520250.00000.00.20.40.60.8z * 1.0e61.01.21.462 / 68

Monotonic undrained simple shear testInternal shear mechanism is inactive (no notion of E50 is included)300250100 kPa200 kPa300 kPaMC250100 kPa200 kPa300 kPa200q [kPa]q [kPa]20015015010010050500050100150p [kPa]20025030000.000.010.020.03Gamma XY [-]0.040.0563 / 68

Monotonic undrained simple shear testInternal shear mechanism is active (e ect of E50 is included)300140100 kPa200 kPa300 kPaMC250100q [kPa]q [kPa]2001501008060405000100 kPa200 kPa300 kPa1202050100150p [kPa]20025030000.000.010.020.03Gamma XY [-]0.040.0564 / 68

Cyclic undrained simple shear testInternal shear mechanism is active (e ect of E50 is included)120120q 30 kPaMCMonotonic100q [kPa]806060404020202040p [kPa]6080001001202040p [kPa]6080100q 50 kPaMCMonotonic10080q [kPa]q [kPa]8000q 40 kPaMCMonotonic100604020002040p [kPa]608010065 / 68

So let us use HSs and densi cation modelsHere we will use periodic BC so let us reuse data prepared forM-C model and make corrections for materials (save data asperiodic-RB-rel-HSs-DNS.inp )For sand layer Densi cation model is usedEoref 200000 kPa, ν 0.25, m 0.5, γ0.7 1

Linear/Nonlinear analysis of soil structure interaction problems. Constitutive aspects c ZACE Services Ltd August 2014 . them at the global level in settings for all dynamic drivers (will be done later on). . Linear/Nonlinear analysis of soil stru

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