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Transcript of NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures...
![Page 1: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/1.jpg)
NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL :
with application to geomechanics & structures
Th.Zimmermann
copyright zace services ltd
![Page 2: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/2.jpg)
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 imposedon u
11 11+(11/x1)dx1
12 +(12 /x2)dx2
12
f1
direction 1:
(11/x1)dx1dx2+(12 /x2) dx1dx2+ f1dx1dx2=0
L(u)= ij/xj + fi=0
x1
x2
dx1
Equilibrium
Boundary value problem
(Differential equation of equilibrium)
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FORMAL DIFFERENTIAL PROBLEM STATEMENT1-phase,linear or nonlinear)
Txonfijij 0,Txonuu ukk
Txontn tijjij ,
epD
Incremental elasto-plastic constitutive equation:
(equilibrium)
(displ.boundary cond.)
(traction bound. cond.)
tu
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 forcesd=vector of nodal displacements)
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DYNAMICS
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DYNAMIC EQUILIBRIUM STATEMENT, 1-PHASE
11 11+(11/x1)dx1
12 +(12 /x2)dx2
12
f1
direction 1:
(11/x1)dx1dx2+(12 /x2) dx1dx2+ f1dx1dx2=0
L(u)= ij/xj + fi=0
x1
x2
dx1
1u
1 1 2u dx dx
iu
Boundary value problem
Equilibriumdisplacement imposedon u traction imposed
on
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FORMAL DIFFERENTIAL PROBLEM STATEMENT
, 0ij ji if o xTnu
Deformation(1-phase):
k k uu u on xT
,ij j j i tn t xon T
epDIncremental elasto-plastic constitutive equation:
(equilibrium)
(displ.boundary cond.)
(traction bound. cond.)
( ) ( ) ( )u tt t t
0 0( 0, ) ( ); ( 0, ) ( )u t x u x u t x u x (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)
v = d; a = d
COMPARING MATRIX FORMS
optional
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SOLUTION TECHNIQUES
-MODAL ANALYSIS-FREQUENCY DOMAIN ANALYSIS
both essentially restricted to linear problems
-DIRECT TIME INTEGRATIONappropriate for a fully nonlinear analysis
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DIRECT TIME INTEGRATION (linear case)…a)
Using Newmark’s algorithm :At each time step, solve:
2
1.
2. ( / 2)[(1 2 ) 2 ]
3. [(1 )
:
: ,
: 0.
]
5, 0.25( lg .)
algorithmic
with
wh and areere
typically trape
paramet
zoidal
e
t t
s
a o
r
t
n 1 n 1 n 1 n 1
n 1 n n n n 1
n 1 n n n 1
Ma Cv Κd F
d d v a a
v v a a
1 ( 1)nt n t
![Page 15: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/15.jpg)
DIRECT TIME INTEGRATION (linear case)…b)
2
[ , (........) ]
[ , (.
,
2. ( / 2)[(1 2 ) 2 ]
3. [(1 ) ]
..
1
.....) ]
.
* *
functionof
functionof
at t
t t
t
n 1
n 1 n n n n 1
n 1 n n n 1
n 1 n 1 n 1 n 1
n 1 n 1 n
n 1 n 1 n
d d v a a
a d
v v a a
Ma C
d
v Κ
v
d F
Κ d Fn 1 n 1
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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=…………
v = d; a = d
MATRIX FORMS
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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
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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Err>10%
fi Contin. Discrete N=…
Err>10%
fi Contin. Discrete N=…
;2
ii i
Ef i
L
; 2sin2 1 2N i
i N i
k if
m N
k
Continuous case Discrete case
L = 10 L = 10m
particular case: 1; / ; 1, ,1
LE k EA l E A l m Al
N
;2
ii i
Ef i
L
; 2sin2 1 2N i
i N i
k if
m N
k
Continuous case Discrete case
L = 10 L = 10m
particular case: 1; / ; 1, ,1
LE k EA l E A l m Al
N
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
![Page 22: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/22.jpg)
NUMERICAL ( ALGORITHMIC) DAMPING CAN EXIST
● ● ● ● ●● ●
●
●
●
●
HHT(-0.3)
IT MAY BE WANTED OR NOT
and varies with parameters (γ ,β)
●
●
●
●
![Page 23: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/23.jpg)
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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Err>10%
fi Contin. Discrete N=…
Err>10%
fi Contin. Discrete N=…
;2
ii i
Ef i
L
; 2sin2 1 2N i
i N i
k if
m N
k
Continuous case Discrete case
L = 10 L = 10m
particular case: 1; / ; 1, ,1
LE k EA l E A l m Al
N
;2
ii i
Ef i
L
; 2sin2 1 2N i
i N i
k if
m N
k
Continuous case Discrete case
L = 10 L = 10m
particular case: 1; / ; 1, ,1
LE k EA l E A l m Al
N
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 itmaintains mode decoupling in modal analysis
![Page 27: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/27.jpg)
2
2
0 1...
( )
i i ii i
i i
y c y y i n
c Rayleigh
+
modal transformation ,
modal matrix, generates a decoupled
system of modal equations;
for each mode we have:
with the
and the
u = Φy
Φ
C uM u+ + K u = 0
RAYLEIGH DAMPING b):PARENTHESIS ON MODAL ANALYSIS
![Page 28: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/28.jpg)
RAYLEIGH DAMPING d)
COMPARING THE MODAL EQUATION
2 2
2
2
( ) 0 1...
2 0 1...
2 ( )
i i ii i
i i i
y y y i n
y y y i n
i
+
+
WITH THE 1DOF VISCOUSLY DAMPED OSCILLATOR
YIELDS:
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RAYLEIGH DAMPING e)
20 1
0
1
2
2 ( ) ,
1/0.5
1/
2 ( , )
define α and β and the viscousdamping at any frequency
( ) / 2
i i i
i i i
j j j
k k k
from i one gets
therefore pairs
k
+
+
![Page 30: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/30.jpg)
RAYLEIGH DAMPING f) this can be plotted
![Page 31: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/31.jpg)
2 (ω,ξ) pairs are used to define α0,β0 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
epE
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FROM LOCAL TO GLOBAL NONLINEAR RESPONSE
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SOLUTION OF LINEARIZED PROBLEM, static case
N(d)=F
....)/()()( 1
1
1 ddNdNdN i
n
i
n
/ in+1( N d)Δd = F - N(d )
i+1 in+1 n+1d = d + Δd
Tn+1K
hence the following algorithm:
Linearize at , w. Taylor exp.i
nd 1
Nonlinear problem to solveF
d
i
nd 1
i: iterationn: step
![Page 36: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/36.jpg)
THE PROBLEM IS NONLINEAR & THEREFORE NEEDS ITERATIONS
... .TOL in+1 n+1
i( N/ d)Δd = F - N(d = ΔF)
i+1 in+1 n+1d = d + Δd
nd
i: iterationn: step
tends to 0
d
on+1K
1FFn
Fn+1
1
1nd
Fon+1K
![Page 37: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/37.jpg)
NEWTON- RAPHSON & al.ITERATIVES SCHEMES
i: iterationn: step
1.Full NR, update KT at each step & iteration, till .TOLF i
ndd
1FFn
Fn+1
1
1nd
To
nK 11F
2Fnd
d
Fn
Fn+1iF
2.Constant stiffness,use KTo
till .TOLF i
3.Modified NR, update KT opportunistically, each step e.g.,till
KTo
.TOLF i
4. BFGS, “optimal”secant scheme
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TOLERANCES ITERATIVE 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.)
![Page 40: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/40.jpg)
DIRECT TIME INTEGRATION (nonlinear case)
Using Newmark’s algorithm (or Hilber’s):At each time step, solve:
2
algor
2. ( / 2)[(1 2 )
:
:
2 ]
3. [(1 ) ]
,
: 0
ithmic
.5, 0.25
t t
with
where
typical
and are paramete
t
ly
rs
n+1 n+1 n+1
n 1 n
n
n n n 1
n 1 n n 1
+
n
11. Ma +Cv + = F
d d v a a
v v a a
N(d )1 ( 1)nt n t
![Page 41: NONLINEAR DYNAMIC FINITE ELEMENT ANALYSIS IN ZSOIL : with application to geomechanics & structures Th.Zimmermann copyright zace services ltd.](https://reader034.fdocuments.in/reader034/viewer/2022051014/551775ca5503460e6e8b5100/html5/thumbnails/41.jpg)
STATICS(nonlinear case)
N(d)=F
DYNAMICS(nonlinear case)
Ma(t)+Cv(t)+N(d(t))=F(t)orMa(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=…………
MATRIX FORMS
Like for linear case
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SEISMIC INPUT a
uG
FextFin+Fel+Fdamp
uG
uG
uR
uT
uG
FextFin+Fel+Fdamp
uG
FextFin+Fel+Fdamp
uG
uG
uR
uT
uG
uG
uR
uT
uG
FextFin+Fel+Fdamp
uG
uG
uR
uT
uG
FextFin+Fel+Fdamp
uG
FextFin+Fel+Fdamp
uG
uG
uR
uT
uG
uG
uR
uT
>>Fin+Fdamp+Fel = Fext
equilibrium>>>
T R R extM u +Cu + Ku = F
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SEISMIC INPUT b
,with R T G R T G R T Gu u - u ,u u - u u = u - u
yields
....
( )
with input as BC or
with input as inertia forces under seismic
T T T ext G G
R R R ext G
M u +Cu + Ku = F +Cu + Ku
M u +Cu + Ku = F - Mu
T R R extM u +Cu + Ku = F
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Time-history