Dinamica Lineare FEM 2

Dr. Dr. Ing.Ing. Valentina Valentina SalomoniSalomoni

Dipartimento di Costruzioni e TrasportiDipartimento di Costruzioni e Trasporti

UniversitUniversit degli Studi di Padovadegli Studi di Padova

DINAMICA DELLE STRUTTUREDINAMICA DELLE STRUTTURELAUREA SPECIALISTICA LAUREA SPECIALISTICA -- INGEGNERIA CIVILEINGEGNERIA CIVILE

Soluzione delle equazioni di equilibrio nellSoluzione delle equazioni di equilibrio nellanalisi dinamicaanalisi dinamica

dei continui dei continui

Equilibrium equations

System of linear differential equations of second ordergoverning the linear dynamic response of a continuum or a structural system discretized by FEM.

In terms of forces we have

Solution methods

- Direct integration- Mode superposition

1. Introduction

StructuralStructural dynamicsdynamics

RKUUCUM =++ &&&

)()()()( tttt EDI RFFF =++

The previous system of equations is integrated using a step-by-step numerical procedure.

Direct means that prior to the numerical integration, no transformation of the equations into a different form isrequired.

Initial conditions (known)

Time stepping

2. Direct integration methods


nTt /=Ttttttt ,...,,...,,3,2,,0 +

UUU &&& 000 ,,

2.1 Central difference method

It is assumed that

Substituting in equilibrium equation (written at time t) weobtain (explicit integration method)


[ ]UUUU ttttttt

+ += 21

2&&

[ ]UUU tttttt

+ += 21&

UCMUMKRUCM ttttttttttt

+

=

+ 2

112211

222



The central difference method is CONDITIONALLY STABLE

Tipically it is assumed

with Tn = smallest period of the F.E.M. assembling with n d.o.f. Example

10nTt =

n

crTtt =

2.2 Houbolt method

It is assumed that

Substituting in equilibrium equation (written at time t+t) weobtain (implicit integration method)


[ ]UUUUU tttttttttt

++ +=2

2 4521&&

[ ]UUUUU tttttttttt

++ +=2291811

61&

UCMUCM

UCMRUKCM

tttt

ttttt

tttt

tttt

++

++

+

+=

++

222

22

311

234

356112

2.2 Houbolt method


2.3 Wilson method

It is assumed that

and integrating we obtain

In this case, equilibrium equation at time t+t must be used.


[ ]UUUU &&&&&&&& ttttttt

+=++

[ ]UUUUU &&&&&&&& ttttttt

++=++

2

2

( )UUUUUU &&&&&&& tttttttt

+++=++

32

61

21

2.3 Wilson method


2.4 Newmark method

It is assumed that

In this case, equilibrium equation at time t+t must be used. Particular cases:

= and = 1/6 linear acceleration method = 1 and = constant-average-acceleration method


( )[ ] ttttttt ++= ++ UUUU &&&&&& 12

21 tt ttttttt

+

++= ++ UUUUU &&&&&

2.4 Newmark method


2.4 Comparison between methods


2.5 Coupling of different integration operators

Numerical integration of fully coupled system of equationsfor describing deformable soil behaviour under appliedloads/thermal loads. (refer to applications)

3. Mode superposition


In direct integration, the number of required operations is

nmkswhere

= ( 2 depends on the type of matrix)n = matrix ordermk = half-bandwidths = number of time steps

Try to minimise the computational effort (half-bandwidth) by changing the structure of the dynamics fundamental equation.

3.1 Change of basis


Transformation of equilibrium equation

with

)()( tt XPU =)(~)(~)(~)(~ tttt RXKXCXM =++ &&&

RPRKPPKCPPCMPPM TTTT ==== ~;~;~;~

3.1 Change of basis


Choice of P matrix solution of undamped eigenvalueproblem (free vibrations of the system)

Solution

By substitution we obtain the generalised eigenproblem

)(sin 0tt = U0KUUM =+&&

MK 2=

3.1 Change of basis


Eigensolutions pairs

Eigenvectors M-orthonormalised

Eigenvalues

),(...,),,(),,( 222

12

21 nn

==

jiji

jTi ;0

;1 M

222 ...021 n

3.1 Change of basis


Posing

we obtain

With orthonormal eigenvectors

Hence if FEM equations are decoupled.

=

2

22

21

...

n

[ ]n ...,, 21=

2T K =

2MK =

IMT =P =

3.1 Change of basis


The system becomes, with

and initial conditions are

The system may become a sequence of decoupled equations (if eigenvectors are also C orthonormal).

)()( tt XU =

UMX 00 T=

)()()()( tttt TT RXXCX 2 =++ &&&

UMX && 00 T=

3.2 Effect of damping


Dynamic load factor (response of a 1 d.o.f system)

Staticresponse

High frequencyloading

3.2 Effect of damping


Number of modes to be included in the analysis depend on: Considered structure Distribution and frequency content of LOAD

Earthquake loading 10 lowest modes(with either n > 1000)

Blast or shock loading p modes(with p > 2/3 n)

Vibration excitation analysis all frequencesbetween

l and u

3.3 Damping vs. frequency


Damping vs. frequency

Coefficients

4. Analysis of Direct Integration Methods


Stability and accuracy Approximation and load

operators

Starting from

posing

We have

MK 2=)()( tt XU =

)()()()( 2 tttt TRXXX =++ &&&

4. Analysis of Direct Integration Methods


Establish the recoursive relationship

where

A = integration approximation operator

L = load operator

Solution at any time station is given by applying recoursivelythe previous equation.

( )rtttt ++ += LXAX

4.2 Stability analysis


Spectral radius concept and stability criterion Previous equation without loading is used. Then find the

spectral decomposition of A. It is

With P the matrix of eigenvectors of A and J the Jordan formof A (eigenvalues i of A on its diagonal).

The spectral radius of A is

and Jn is bounded for n , if and only if (A) 1 (Stability criterion).

1nn PPJA =

,...2,1,max)( == ii A



Stability criterion Application.



Spectral radius vs.

4.3 Accuracy analysis


Period elongation & amplitudedecay

4.4 Accuracy analysis


Displacement response

Recent developments and application (in nonlinear domain)

2.5 Coupling of different integration operators

Dinamica Lineare FEM 2

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