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
StructuralStructural dynamicsdynamics
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)
StructuralStructural dynamicsdynamics
[ ]UUUU ttttttt
+ += 21
2&&
[ ]UUU tttttt
+ += 21&
UCMUMKRUCM ttttttttttt
+
=
+ 2
112211
222
2.1 Central difference method
StructuralStructural dynamicsdynamics
2.1 Central difference method
StructuralStructural dynamicsdynamics
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)
StructuralStructural dynamicsdynamics
[ ]UUUUU tttttttttt
++ +=2
2 4521&&
[ ]UUUUU tttttttttt
++ +=2291811
61&
UCMUCM
UCMRUKCM
tttt
ttttt
tttt
tttt
++
++
+
+=
++
222
22
311
234
356112
2.2 Houbolt method
StructuralStructural dynamicsdynamics
2.3 Wilson method
It is assumed that
and integrating we obtain
In this case, equilibrium equation at time t+t must be used.
StructuralStructural dynamicsdynamics
[ ]UUUU &&&&&&&& ttttttt
+=++
[ ]UUUUU &&&&&&&& ttttttt
++=++
2
2
( )UUUUUU &&&&&&& tttttttt
+++=++
32
61
21
2.3 Wilson method
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
( )[ ] ttttttt ++= ++ UUUU &&&&&& 12
21 tt ttttttt
+
++= ++ UUUUU &&&&&
2.4 Newmark method
StructuralStructural dynamicsdynamics
2.4 Comparison between methods
StructuralStructural dynamicsdynamics
2.4 Comparison between methods
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
Transformation of equilibrium equation
with
)()( tt XPU =)(~)(~)(~)(~ tttt RXKXCXM =++ &&&
RPRKPPKCPPCMPPM TTTT ==== ~;~;~;~
3.1 Change of basis
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
Eigensolutions pairs
Eigenvectors M-orthonormalised
Eigenvalues
),(...,),,(),,( 222
12
21 nn
==
jiji
jTi ;0
;1 M
222 ...021 n
3.1 Change of basis
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
Dynamic load factor (response of a 1 d.o.f system)
Staticresponse
High frequencyloading
3.2 Effect of damping
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
Damping vs. frequency
Coefficients
4. Analysis of Direct Integration Methods
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
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
StructuralStructural dynamicsdynamics
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
4.2 Stability analysis
StructuralStructural dynamicsdynamics
Stability criterion Application.
4.3 Stability analysis
StructuralStructural dynamicsdynamics
Spectral radius vs.
4.3 Accuracy analysis
StructuralStructural dynamicsdynamics
Period elongation & amplitudedecay
4.4 Accuracy analysis
StructuralStructural dynamicsdynamics
Displacement response
Recent developments and application (in nonlinear domain)
2.5 Coupling of different integration operators
Top Related