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NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd

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Ground motion Far-field BC needed 2-phase medium

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For time being in Z_Soil: limited structural dynamics t a, or d

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analysis by geomod with some extensions

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STATICS RECALL

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STATIC EQUILIBRIUM STATEMENT, 1-PHASE traction imposed on displacement imposed on u ( 11 / x 1) dx ( 12 / x 2) dx 2 12 f1f1 direction 1: ( 11 / x 1) dx 1 dx 2 +( 12 / x 2) dx 1 dx 2 + f 1 dx 1 dx 2 =0 L(u)= ij / x j + f i =0 x1x1 x2x2 dx 1 Equilibrium Boundary value problem (Differential equation of equilibrium)

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FORMAL DIFFERENTIAL PROBLEM STATEMENT 1-phase,linear or nonlinear) Incremental elasto-plastic constitutive equation: (equilibrium) (displ.boundary cond.) (traction bound. cond.) NB: Time is steps

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MATRIX FORM -DISCRETIZATION LEADS TO THE MATRIX FORM…. FOR LINEAR STATICS Kd=F ( K=stiffness matrix, F=vector of nodal forces d=vector of nodal displacements)

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DYNAMICS

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DYNAMIC EQUILIBRIUM STATEMENT, 1-PHASE ( 11 / x 1) dx ( 12 / x 2) dx 2 12 f1f1 direction 1: ( 11 / x 1) dx 1 dx 2 +( 12 / x 2) dx 1 dx 2 + f 1 dx 1 dx 2 =0 L(u)= ij / x j + f i =0 x1x1 x2x2 dx 1 Boundary value problem Equilibrium displacement imposed on u traction imposed on

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FORMAL DIFFERENTIAL PROBLEM STATEMENT Deformation(1-phase): Incremental elasto-plastic constitutive equation: (equilibrium) (displ.boundary cond.) (traction bound. cond.) (initial conditions) NB: Time is real

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STATICS (linear case) Kd=F We obtain (Linear system size: Ndofs=Nnodes x NspaceDim, -d=nodal displacements -F=nodal forces) DYNAMICS (linear case) Ma(t)+[Cv(t)]+Kd(t) =F(t) where We obtain (Linear system size: Ndofs=Nnodes x NspaceDim, But 3xNdofs unknowns) COMPARING MATRIX FORMS optional

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SOLUTION TECHNIQUES -MODAL ANALYSIS -FREQUENCY DOMAIN ANALYSIS both essentially restricted to linear problems -DIRECT TIME INTEGRATION appropriate for a fully nonlinear analysis

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DIRECT TIME INTEGRATION (linear case)…a) Using Newmarks algorithm : At each time step, solve:

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DIRECT TIME INTEGRATION (linear case)…b)

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STATICS (linear case) Kd=F DYNAMICS (linear case) Ma(t)+Cv(t)+Kd(t)=F(t) >>>> at any t n+1 we have an equivalent static problem K*d n+1 =F* n+1 a n+1 =………… v n+1 =………… MATRIX FORMS

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NEWMARK IS A 1-STEP ALGORITHM All information to compute solution at time t n+1, is in solution at time t n, restart is easy n n+1

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NUMERICAL ( ALGORITHMIC) DAMPING CAN EXIST Newmark(0.5,0.25) HHT IT MAY BE WANTED OR NOT and varies with parameters ( γ,β) Newmark(0.6,0.3025)

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DISCRETIZATION APPROXIMATES HIGH FREQUENCIES Filtering of high frequencies may be desirable Exact sol.:

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HHT Hilber-Hughes-Taylor α method HHT filters high frequencies without damping low frequencies

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NUMERICAL ( ALGORITHMIC) DAMPING CAN EXIST HHT(-0.3) IT MAY BE WANTED OR NOT and varies with parameters ( γ,β)

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Algorithmic data for Newmark …or HHT(under CONTROL/AN..

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Mass can be CONSISTENT (as obtained by FEM) or LUMPED (concentrated at (some) nodes) Only lumped masses are available in ZSOIL Lumped masses tend to lead to underestimate frequencies

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Lumped masses tend to lead to underestimate frequencies: ILLUSTRATION

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RAYLEIGH DAMPING a) Recall: Ma(t)+Cv(t)+Kd(t)=F(t) C=αM+βK is RAYLEIGH DAMPING α,β:constants This form of damping is not representative of physical reality, in general. Its success is due to the fact that it maintains mode decoupling in modal analysis

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RAYLEIGH DAMPING b): PARENTHESIS ON MODAL ANALYSIS

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RAYLEIGH DAMPING d) COMPARING THE MODAL EQUATION WITH THE 1DOF VISCOUSLY DAMPED OSCILLATOR YIELDS:

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RAYLEIGH DAMPING e)

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RAYLEIGH DAMPING f) this can be plotted

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2 (ω,ξ) pairs are used to define α 0, β 0 in ZSOIL

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NONLINEAR DYNAMICS

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E y CONSTITUTIVE MODEL: ELASTIC-PERFECTLY PLASTIC 1- dimensional this problem is non-linear

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FROM LOCAL TO GLOBAL NONLINEAR RESPONSE

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SOLUTION OF LINEARIZED PROBLEM, static case hence the following algorithm: Linearize at, w. Taylor exp. Nonlinear problem to solve F d i: iteration n: step

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THE PROBLEM IS NONLINEAR & THEREFORE NEEDS ITERATIONS i: iteration n: step tends to 0 d FnFn F n+1

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NEWTON- RAPHSON & al. ITERATIVES SCHEMES i: iteration n: step 1.Full NR, update K T at each step & iteration, till d FnFn F n+1 d FnFn 2.Constant stiffness,use K T o till 3.Modified NR, update K T opportunistically, each step e.g.,till KToKTo 4. BFGS, optimalsecant scheme

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TOLERANCESITERATIVE ALGORITHMS

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STATICS (nonlinear case) N(d)=F DYNAMICS (nonlinear case) Ma(t)+Cv(t)+N(d(t))=F(t) >>>> MATRIX FORMS (e.g.)

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DIRECT TIME INTEGRATION (nonlinear case) Using Newmarks algorithm (or Hilbers): At each time step, solve:

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STATICS (nonlinear case) N(d)=F DYNAMICS (nonlinear case) Ma(t)+Cv(t)+N(d(t))=F(t) or Ma(t)+N(d,v)=F(t) >>>> at any t n+1, we have an equivalent static problem N*(d n+1) =F* n+1 a n+1 =………… v n+1 =………… MATRIX FORMS Like for linear case

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SEISMIC INPUT a >>Fin+Fdamp+Fel = Fext equilibrium >>>

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SEISMIC INPUT b yields

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