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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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Far-field BC needed 2-phase medium Ground motion

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**For time being in Z_Soil: limited structural dynamics**

a, or d t

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with some extensions analysis by geomod

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STATICS RECALL

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**STATIC EQUILIBRIUM STATEMENT, 1-PHASE**

Boundary value problem displacement imposed on u Equilibrium 12 +(12 /x2)dx2 12 f1 traction imposed on 11 11+(11/x1)dx1 x2 x1 dx1 direction 1: (11/x1)dx1dx2+(12 /x2) dx1dx2+ f1dx1dx2=0 L(u)= ij/xj + fi=0 (Differential equation of equilibrium)

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**FORMAL DIFFERENTIAL PROBLEM STATEMENT**

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

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**Kd=F 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**

Boundary value problem displacement imposed on u Equilibrium traction imposed on 12 +(12 /x2)dx2 12 f1 11 11+(11/x1)dx1 x2 x1 dx1 direction 1: (11/x1)dx1dx2+(12 /x2) dx1dx2+ f1dx1dx2=0 L(u)= ij/xj + fi=0

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**FORMAL DIFFERENTIAL PROBLEM STATEMENT**

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

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**Kd=F Ma(t)+[Cv(t)]+Kd(t) =F(t) COMPARING MATRIX FORMS 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) 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 Newmark’s algorithm : At each time step, solve:

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**DIRECT TIME INTEGRATION (linear case)…b)**

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**Ma(t)+Cv(t)+Kd(t)=F(t) >>>>**

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

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**NEWMARK IS A 1-STEP ALGORITHM**

All information to compute solution at time tn+1, is in solution at time tn , restart is easy

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**NUMERICAL ( ALGORITHMIC) DAMPING CAN EXIST**

and varies with parameters (γ ,β) ● ● Newmark(0.6,0.3025) ● ● HHT ● ● ● ● Newmark(0.5,0.25) ● ● ● ● ● ● ● IT MAY BE WANTED OR NOT

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**DISCRETIZATION APPROXIMATES HIGH FREQUENCIES**

Exact sol.: Filtering of high frequencies may be desirable

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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**

and varies with parameters (γ ,β) ● ● ● ● HHT(-0.3) ● ● ● ● ● ● ● ● ● ● ● IT MAY BE WANTED OR NOT

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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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**C=αM+βK is RAYLEIGH DAMPING**

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 α0,β0 in ZSOIL**

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NONLINEAR DYNAMICS

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**CONSTITUTIVE MODEL: ELASTIC-PERFECTLY PLASTIC**

1- dimensional E y 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**

Nonlinear problem to solve d Linearize at , w. Taylor exp. hence the following algorithm: i: iteration n: step

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**THE PROBLEM IS NONLINEAR & THEREFORE**

NEEDS ITERATIONS tends to 0 Fn+1 Fn i: iteration n: step d

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d Fn Fn+1 NEWTON- RAPHSON & al. ITERATIVES SCHEMES d Fn Fn+1 KTo 2.Constant stiffness,use KTo till i: iteration n: step 3.Modified NR, update KT opportunistically, each step e.g.,till 1.Full NR, update KT at each step & iteration, till 4. BFGS, “optimal”secant scheme

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**TOLERANCES ITERATIVE ALGORITHMS**

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**Ma(t)+Cv(t)+N(d(t))=F(t)**

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

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**DIRECT TIME INTEGRATION (nonlinear case)**

Using Newmark’s algorithm (or Hilber’s): At each time step, solve:

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**Ma(t)+Cv(t)+N(d(t))=F(t) or Ma(t)+N(d,v)=F(t) **

MATRIX FORMS 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 tn+1, we have an equivalent static problem N*(dn+1)=F*n+1 an+1=………… vn+1=………… Like for linear case

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

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SEISMIC INPUT b yields

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