Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Similar presentations


Presentation on theme: "NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd."— Presentation transcript:

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

2 Ground motion Far-field BC needed 2-phase medium

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

4 analysis by geomod with some extensions

5 STATICS RECALL

6 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)

7 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

8 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)

9 DYNAMICS

10 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

11 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

12 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

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

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

15 DIRECT TIME INTEGRATION (linear case)…b)

16 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

17

18 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

19 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)

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

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

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

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

24 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

25 Lumped masses tend to lead to underestimate frequencies: ILLUSTRATION

26 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

27 RAYLEIGH DAMPING b): PARENTHESIS ON MODAL ANALYSIS

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

29 RAYLEIGH DAMPING e)

30 RAYLEIGH DAMPING f) this can be plotted

31 2 (ω,ξ) pairs are used to define α 0, β 0 in ZSOIL

32 NONLINEAR DYNAMICS

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

34 FROM LOCAL TO GLOBAL NONLINEAR RESPONSE

35 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

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

37 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

38 TOLERANCESITERATIVE ALGORITHMS

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

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

41 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

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

43 SEISMIC INPUT b yields

44 Time-history


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

Similar presentations


Ads by Google