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Finite element method for structural dynamic
and stability analyses
1
Prof C S Manohar Department of Civil Engineering
IISc, Bangalore 560 012 India
Module-12
Closure
Lecture-40 Closure and a few topics for further study
Topics which were proposed to be covered in this course
• Approximate methods and FEM
• Dynamics of truss and planar frame structures
• Damping models and analysis of equilibrium equations
• Dynamics of Grids and 3D frames
• A few computational aspects (solution of equilibrium equations,
eigenvalue problems, model reduction, and substructuring)
• Dynamic stiffness matrix and transfer matrix methods
• Dynamics of plane stress/strain, plate bending, shell and 3d elements
• Applications (earthquake engineering and vehicle structure
interactions)
• FEA of elastic stability problems
• Treatment of nonlinearity
• FE model updating
• FEM in hybrid simulations
2
Course modules
1. Approximate methods and FEM (4)
2. Finite element analysis of dynamics of planar
trusses and frames (3)
3. Analysis of equations of motion (3)
4. Analysis of grids and 3D frames (3)
5. Time integration of equation of motion (4)
6. Model reduction and substructuring schemes (2)
7. Analysis of 2 and 3 dimensional continua (6)
8. Plate bending and shell elements (4)
9. Structural stability analysis (9)
10.FE Model updating (2)
11.Nonlinear FE Models (3)
12.Closure
3
Acknowledgement 1. M Petyt, 1990, Introduction to finite element vibration analysis, CUP, Cambridge.
2. G J Simitses and D H Hodges, 2006, Fundamentals of structural stability, Elsevier,
Amsterdam.
3. R D Cook, D S Malkus, and M E Plesha, 1989, Concepts and applications of finite
element analysis, 3rd Edition, John Wiley, New York
4. S S Rao, 1999, The finite element method in engineering, 3rd Edition, Butterworth-
Heinemann, Boston.
5. W McGuire, R H Gallagher, and R D Ziemian, 2000, Matrix structural analysis, 2nd
Edition, John Wiley, New York.
6. T Belytschko, W K Liu, and B Moran, 2000, Nonlinear finite elements for continua and
structures, Wiley, Chichester.
7. J N Reddy, 2004, An introduction to nonlinear finite element analysis, Oxford
University Press, New York.
8. J N Reddy, 2013, An introduction to continuum mechanics, 2nd edition, Cambridge
University Press, New York.
9. K J Bathe, 1996, Finite element procedures, Prentice Hall of India, New Delhi.
10. J M T Thompson and G W Hunt, 1973, A general theory of elastic stability, John
Wiley, London
11. http://www.colorado.edu/engineering/CAS/courses.d/NFEM.d/Home.html [Professor
C A Felippa, University of Colorado at Boulder)
4
2 2
1 1
2
1
2 2
Hamilton's principle: Minimize the functiona
1
2
l
t t
t t
t
t
t dt T t V t dt
mu t ku t dt
L
u t
m
k
Example-1
6
0 0, 0 & , 0AE u t AE L u L t
0 0, 0 & , 0AE u t u L t
0, 0 & , 0u t AE L u L t
0, 0 & , 0u t u L t
Geometric, forced, or kinematic boundary condition: , 0 on the boundary
Free or natural boundary condition: , 0 on the boundary
u x t
AE x u x t
2
2; : ,0 & ,0 specified
u uAE x m x u x u x
x x t
IC-s
Boundary
conditions
7
, ,L AE x m x
M1k 2k
Exercise: set up the governing equation for the system shown below
,u x t
v
2 2
0
2 22
1 2
0
Hint
1 1,
2 2
1 1 1V , 0, ,
2 2 2
L
L
T t m x u x t dx Mv
t AE x u x t dx k u t v t k u L t v t
x
8
x
y
, ,L EI x m x
2
1
2
0
22
2
0
22
2
2
0 0
0 0
1Kinetic energy: ,
2
1Potential energy:
2
1 1Lagrangian ,
2 2
, ,
L
L
L L
tL L
t
T t m x v x t dx
vV t EI x dx
x
vT V m x v x t dx EI x dx
x
F v v dx F v v dxdt
L
L
Euler-Bernoulli beam
9
The 16 classical single span beams
0& 0EIv EIv
0& 0v v
0& 0EIv v
0& 0v EIv
0
0
, 0
, 0
L
L
EIv x t
EIv x t
0 EIv mv
10
2
2
2 2
1
2 2
Discrete MDOF system
has units of (rad/s) .
If .
t
t
t
n nn nt
n n
N
n n
u KuR u
u Mu
R u
Ku R u
M
R u
u y R u O
2
2
1
0
2
0
2
0
2
0
Axially vibrating bar
Euler Bernoulli beam
L
L
L
L
AE x x dx
R x
m x x dx
EI x x dx
R x
R x
m x x dx
Rayleigh’s quotient
Rayleigh’s principle
11
How to lower the value of ?R x Rayleigh - Ritz Method
2
0
2
0
1
, 1, 2, , : a set of known linearly independent functions
whic allh satisfy the boundary conditions.
, 1,2, , : a set of unknown co
L
L
N
n n
n
n
n
EI x x dx
R x
m x x dx
x a x
x n N
a n N
nstants which need
to be determined
Strategy: Select , 1,2, , such that is minimized.na n N R x
12
What MWR achieves?
A PDE governing the behavior of a continuous system
has been replaced by an equivalent set of ODE-s (IVP-s)
with a view to obtain an approximate solution.
0 0
Field equation: , , , ,
ICS: ,0 , ,0
BCS: Appropriate geometric and natural BCS
EI x v x t m x v x t c x v x t f x t
v x v x v x v x
1
,N
n n
n
v x t a t x
; 0 , 0Ma Ca Ka P t a a
Drive the residue , to be small
in some sense (MWR).
e x t
13
0
0
0
1
0
,Least squares , 0 for 1, 2, ,
Collocation , 0 for 1, 2, ,
Galerkin , 0 for 1, 2, ,
, 0 Subdomain collocation
for 1, 2, ,
Petrov-Galerkin
L
n
L
n
L
n
L
n n
n
e x te x t dx n N
a
x x e x t dx n N
x e x t dx n N
U x x U x x e x t dx
n N
x
0
, 0 for 1, 2, ,
L
e x t dx n N
14
0 Weight Residue
Method of weighted residues
, 0 for 1,2, ,
; 0 , 0
L
nw x e x t dx n N
Ma Ca Ka P t a a
Assumed mode method and Lagrange's equation
Strong (operational) form, Weighted residual form, and
weak (variational) form of governing equations
15
,u x t
Mesh
Element
, ,
BCs+ICs
OR
Min A=
Lu x t F x t
Ldt
Mesh
16
1
1
, is approximated
in terms of
, ; 1, 2, ,
;
Within an element
, is approximated
as , ,
An ap
FINITE
proximate
ELEMENT
METHOD
n
e
i
r
i i j
i
Ne e e
i i
i
u x t
u x t i N
i j
u x t
u x t N x u x t
umerical method
to obtain solutions to PDE-s or
variational problems
: Elements
: Nodes
1,2, ,
i
ix
i N
17
Element level EOM
in local coordinate system
Element level EOM; ; ;
in global coordinate system
Global EOM after
s s s s s s s
s s s s s s s
t t t
s s s s s s s s s s s
t
s s s s
M u C u K u F t
M U C U K U F t
U T u M T M T K T K T
C T C T
Summary
1 1
1 1
0
assembly
of structural matrices and
before imposing boundary
condtions;
Equations for unknown
reactions
p pt t
s s s ss ss s
p pt
ss s sss s
I I
MU CU KU F t
M A M A K A K A
C A C A F t A F t
M U C
0
0 0
0 0
Equations for unknown
displace0 ; 0
ments
I I I I
II I II I II I I
MU CU KU F
U K U F t
M U C U K U F t
t
U U U U
18
A
B
A
B
Neglect axial deformation
16 17
1 4
7 19
19 22
22 10
5 17
11 17
2 8
8 14
13 14 15
cos sin
0
u uu uu uu uu uu uu uu uu uu u u
12
6
5
43
2
1
11
109
8
7
24
18
17
16
15
14
13
23
2221
20
19
Hinge
19
Stress
x
Discontinuity introduced by FEM
Displacement based FEM introduces
discontinuities in spatial variation of
quantities which are truly continuous.
20
LTI
LTI
LTI
LTI
)(tf
t
dfthtx0
)()()(
)()()( FHX )(F
)(t )(th
)exp( ti )exp()( tiH
( ) ( )
( ) ( )
( ) ( )
f t F
x t X
h t H
Input-output relations for linear time invariant systems
Linear Damping models
21
Viscous Structural
Classical Non-Classical Classical Non-Classical
Classification into viscous and structural depends upon behavior
of energy dissipated under harmonic steady state as a function of
frequency.
Classification into classical and non-classical depends
upon
orthogonality (or lack of orthogonality) of damping matrix
with respect to undamped normal modal matrix.
22
ResponseQuantity
Receptance
Displacement Admittance Dynamic stiffness
Dynamic compliance
Dynamic flexibility
Velocity Mobility Mechanical impedance
Acceleration Accelerance Apparent mass
R FR F R
Nomenclature for FRF
23
12
FRF calculations (valid for both viscous and structural damping models
(a) Viscously damped system
exp
b Structurally damped system
MU CU KU F i t
M i C K
MU K i
S
Direct calcul
umma
t n
ry
a io
12
2 21
2
exp
c Viscously damped system with classical damping
2
; ; ; Diag
Nrn sn
rs
n n n n
t t
i
D U F i t
M K iD
i
K M M I K
Calculation based on mode superposition
24
2 21
2
d Structurally damped system with classical damping
; ; ; Diag
e Viscously damped system with nonclas
njk rk
jr
k k k
t t
i
iD
K M M I K
Calculation based on mode superposition (continued)
* *
*1
* *
*
*
1 2
2 21
2
sical damping
; ; ; Diag
Diag
f Structurally damped system with nonclassical damping
;
nrk jk jk rk
jr
k k k
t t
n
njk rk
jr
k k
t
i i
B A A I A
s
K iD s M
2 2 2
1 2; Diag t
nM I K iD s s s
25
Dynamic stiffness matrix for an Euler-Bernoulli beam
Focus : Steady state behavior
1 1,u t p t
2 2,u t p t 4 4,u t p t
3 3,u t p t
exp1,2,3,4
exp
k k
k k
u t i tk
p t P i t
1 2 1 2, , , , , ,EI m l c c h h
26
State vector and transfer matrix
2 exp i t 1 exp i t
2 expP i t 1 expP i t
, ,AE m l
,u x t
x
2 2
State vector=
Transfer matrix
R L
P
TP P
T
27
P t
1 1,u t P t
3 3,u t P t
4 4,u t P t
5 5,u t P t 2 2,u t P t
6 6,u t P t, , , ,EI AE m l c
2D beam (frame)
element
4 4,u t P t
3 3,u t P t
6 6,u t P t
5 5,u t P t
1 1,u t P t 2 2,u t P t
2D beam (grid)
element
28
2 2,u t P t
2-noded element with 3 dofs per node
1 1 4 2
3 1 2 2 6 3 5 4
,
,
x t u t x u t x
v x t u t x u t x u t x u t x
3 3,u t P t
5 5,u t P t
6 6,u t P t
1 1,u t P t 4 4,u t P t
29
2 2
2 2
3 2 2
2 2
2 2
2 2
0 0 0 0
0 4 6 0 2 6
0 6 12 0 6 12
0 0 0 0
0 2 6 0 4 6
0 6 12 0 6 12
140 700 0 0 0
0 4 22 0 3 13
0 22 156 0 13 54
70 1404200 0 0 0
0 3 13 0 4 22
0 13
m m
m m
JGl JGl
EI EI
l l l l
l lEIK
l JGl JGl
EI EI
l l l l
l l
I I
m m
l l l l
l lmlM
I I
m m
l l l l
54 0 22 156l l
30
3D beam element
1
11
10
9
8
7
6
5
43
2
12, , , , , , , ,x y zE G I I I A J l
z
y
x
Translation in mForce in N
Rotation in radForce in Nm
, , , , , , ,u x t v x t w x t x t
31
2 2
0 0
Axial deformation Twisting
2 22 2
2 2
0 0
Bending@z Bending@y
2
1 1
2 2
1 1
2 2
with =
L L
L L
z y
uU AE dx GJ dx
x x
v wEI dx EI dx
x x
J y zx z
2
2 2 2 2
0 0 0 0
Axial deformation Twisting Bending@z Bending@y
2 2
1 1 1 1 + +
2 2 2 2
with
A
L L L L
m
m
A
dA
T mu dx I dx mv dt mw dt
I y z dA
0 0
Consider a -dof system
, ,
0 ; 0
This equation constitutes a set of semi-discretized system of
coupled
Numerical integration of equations of equillibrium
Remark
sec
s:
ond
N
MU CU KU R U t U t t F t
U U U U
order ode-s. That is, these equations have been
obtained after discretizing the spatial variables.
This set of equations constitutes a set of initial value problems.
We consider solution of the ab
0 1 2 1
ove equation at a set of
discrete time instants with .
The basic idea is to replace the derivatives appearing
in the above equations by finite difference approximations
an
n n n nt t t t t t t
d then solve the resulting algebraic equations. 32
Explicit method with first order accuracy
Implicit method with first order accur
Forward Euler
Backward
acy
Explicit method with second order accurac
Euler
Cen a
y
tr
Discussion on following methods
l difference
Newmark's family of methods
HHT- method and genera
Implicit method with second order accuracy
Explicit-Implicit method
lization
HHT- with operator sp
s
litt ing
33
34
At least second order accuracy
Unconditional stability when applied to LTI systems
Controllable algorithmic damping in higher modes
Investiga
Desirable features of numerical integration schemes
No overshoot
te spectral radius as frequency
Spectral radius 1 as driving frequency 0
Excessive oscillations during the first few steps
Self-starting
No more than one set of implicit
equations to be solved at each step
35
0 0
1
2
; 0 & 0
m
m
s
t t t t
m m m
r m r m r m r
ss sm
ss ss
MX CX KX F t X X X X
X tX t X t
X t
M X t C X t K X t F t
M X C X K X F t
I
K K
I
K M
Recall : model reduction
Static condensation
Dynamic condensation 12
1
sm sm
m t t
m m m
s
K M
SEREP
Coupling techniques
• Spatial coupling method
• Modal coupling method
– Fixed interface (component mode synthesis)
– Free interface
36
37
x
y
z
transverse load
x
y
z
longitudinal load
Membrane action
Bending action
Membrane action can be analysed using plane stress elements
Bending action: topic for the study
38
Linear triangular plane stress element
12
3
1 1,x y
2 2,x y
3 3,x y
1u
2u
3u2v
1v
3v
, , ,x u x y t
, , ,y v x y t
Field variables
, ,
, ,
u x y t
v x y t
1 2 3
4 5 6
, ,
, ,
u x y t t t x t y
v x y t t t x t y
Thickness=h
39
, , ,x u x y t
, , ,y v x y t
,x y
1 11 ,x y
4 44 ,x y
3 33 ,x y
2 22 ,x y
1, 1 1, 1
1,1 1,1
,
Linear quadrilateral element
t
e
A
t
e
A
K hB DBdA
M h N NdA
: transform the element domain to square
to facilitate evaluation of the integrals.
Idea
4 3
21
A
B
40
8
54 3
21
7
6
Rectangular hexahedron
Pentahedron
Isoparametric hexahedron
Tetrahedron
41
Isoparametric hexahedron element
1
2
1
3
23
4
4
5
5
6
7
68
78
x
z
y
8 noded element with 3dofs/node
42
Axisymmetric problems
3D axisymmetric solid
Not necessarily prismatic
Not necessarily thin or thick
Geometry
Surface tractions , , = ,
Body force: , , 0,
, , = , , , , = ,r r z z
f r z f r z
F r z
F r z F r z F r z F r z
Loads
, , 0
, , ,
, , ,
v r z
u r z u r z
w r z w r z
Displacements
Linear, homogeneous
elastic, isotropic
Material
Rotational
Symmetry
about an axis
,r u
,z w,v
43
Four noded linear quadrilateral thick plate bending element
4
3
2
1
3 3,x y 3 4,x y
2 2,x y
1 1,x y
x
y
1,1
1, 1 1, 1
0,0
1,1
4
1
4
1
, ,
, ,
i i
i
i i
i
x x N
y y N
31 1
2 12 2
t t s
A A
hV D dA kh D dA
2a
c
, , yy v
e
,z w
, , xx u
Examples: Bridge deck, building floors, ship hulls,
Beam centroidal axis is placed eccentrically to the middle surface of the plate.
Membrane and bending action gets
Plate stiffened by beam elements
coupled
A Steady state of
rest
periodic motion
random motion
An externally imposed
disturbance
perturbation
The state is stable if
response to perturbation
dies and original state
is restored
Original state
is not restored
Motion grows
without limit
The state is
unstable
Motion neither
grows nor decays:
The state is
unstable or stable?
q x
P P
PP
A
B
C
D
q x
q x
q x
EIy M x
EIy Py M x
ivEIy q x
ivEIy Py q x
3 2
A
B cos sin
C
D cos sin
y ax b PI
y a x b x PI
y ax bx cx d PI
y a x b x cx d PI
P
EI
47
0 03
0 02
0 0
Summary
3 tan=
2 1 cos0
cos
tan
2
u uu
u
uu
u u
l uM M M u
u
P P
Q
B
Q l cR
l
A
QcR
l
BA
cl c
0 0.5 1 1.5 210
-1
100
101
102
u
Sta
bili
ty f
un
ctio
ns
chi
epsilon
xi
48
y x
0y xP P
P P
e e
P P
q x
These three problems are mathematically equivalent.
The
• transverse load
• eccentrically applied axial load
• initial imperfections
are manifestations of departures from an ideal situation.
How about the study of the ideal situation itself?
49
What should be such that a adjoining equillibrium position becomes
possible?
Assume: an adjoining equillibrium position is indeed possible.
P
x
y x
P P
2
20; 0 0; 0
This is an eigenvalue problem.
For what values of does this equation admit a nontrivial solution?
d yEI Py y y L
dx
P
50
, ; ,
Equillibrium points
*, * 0
*, * 0
* ; *
exp
x f x y y g x y
f x y
g x y
x t x t y t y t
f f f f
t t tx y x yst
t t tg g g
x y
Stability of equillibrium points
If 0, lim the fixed point *, * is unstable
If 0, lim 0 the fixed point *, * is stable
t
t
sg
x y
s a ib
ta x y
t
ta x y
t
51
Consider a system with generalized cordinates.
Focus attention on statically loaded structures.
A stationary value of the total potential energy with res
n
Energy methods for stability analysis
Axiom -1
pect to the
generalized coordinates is necessary and sufficient condition for the
equillibrium state of the system.
A complete relative minimum of the total potential energy with respect
to the g
Axiom - 2
eneralized coordinates is necessary and sufficient for the stability
of an equillibrium state of the system.
J M T Thompson and G W Hunt, 1973, A general theory of elastic stability, John Wiley, London
52
Buckling due to bifurcation of equillibrium
x
y x
P P
sinL LL
P
k
cosL cosL
-1.5 -1 -0.5 0 0.5 1 1.5-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
theta
P/4
kl
1P
3P
2P
st1 buckling load
rd3 buckling load
nd2 buckling load
Limit load buckling
53
2 2 2
0 0
0 0 0
0 0 0
1
2
0 01
0 0 with =2
0 0
xx zx
V
e e e
xx xy xz
t
xy yy yz
V
u v w u u v v w wU dV
x x x y z y z y z
u
u
v N u N u G u
w
s
U s dV s
s
0 0 0
0 01
0 02
0 0
0 01
with 0 0 Geometric stiffness matrix2
0 0
xz yz zz
tt t
e e e e
V
t
V
s
U u G s G u dV u K u
s
s
K G s G dV
s
54
Imperfection sensitive structures
P PP
1k
2k
3k
A AA
B BB
l
mP
mP
55
Parametrically excited systems
P t
,f x t
P t
gx t
gy t
gm g y t d
d
y
h
x
y
A
,a v
vk
sm
vc
um
u t
, , ,EI m c l
,y x t
PP
P
P
Line of action of remains unaltered as beam deforms. Static analysis canbe used to find cr
P
P
Line of action of remains tangential to the deformed beam axis. Static analysis does not leadto correct value of .cr
P
P
“Follower”
forces
57
0 0
How to characterize resonances in systems governed by equations of the form
0; 0 ; 0
when the parametric excitations are periodic.
How to arrive at FE models for P
M t X C t X K t X X X X X
Problem 1
Problem 2
DE-s with time varying coefficients?
Are there any situations in statically loaded systems, wherein one needs
to use dynamic analysis to infer stability conditions?
Problem 3
58
,a v
vk
sm
vc
um
u t
nl
1
1
1
n
n i
i
x l
1
n
n i
i
x l
0
1
n
i
i
x l
1
1
1
0 0
1
0 1
position of the wheel in local coordinate system
n
i
i
n
i
i
x l
x l
1
1
1 0
1 1 0
1
Element level equation of
Let &
0
mo
0
tion
n
n
n n
n
n
s
t
n n v
n n v
t
n n
V
t Md F
u t V
M
tM M t
u tm
tC C t I t t t c
u tI t t t c
K K t I t t t
0
1 1 0
1 0
00
Further steps: assembly, imposition of BCs
v
n n v
tnn n u s
tk
u tI t t t k
FI t t t m m g
60
Mathematical
models
Experimental
models
Can we combine them?
What are the issues?
Studies on existing structures
Finite element
model updating
61
To be determined To be determined from Unknownexperimentally FE model I updating
parameters
Experimental
determination
of
S S
S
Tasks
Analytical determination of elements of the matrix
Sensitivity analysis
Solution of the( often overdetermined) set of equations
Iterations
Pseudo - inverse
Singular value decomposition
Forward proble
Tikhonov Regula
m of design sen
rization
sitivty
Sources of nonlinearity
• Nonlinear strain-displacement relations
(geometric nonlinearity)
• Nonlinear constitutive laws (nonlinear
stress-strain relations)
• Nonlinearity associated with boundary
conditions
• Nonlinear energy dissipation mechanisms
62
63
Stress
StrainMaterial nonlinear; small displacementsand small strains
Material linear/nonlinear; large rotationand small strains
Material linear/nonlinear; large rotationand large strains
Nonlinear boundaryconditions Stress
Strain
65
Nonlinearly elastic systems and systems with hereditary
nonlinearities
, , , ;0 ;
(0) & 0 specified
, , Nonlinear function of instantaneous values of &
mx cx kx g x t x t t h x x t f t
x x
g x t x t t x t
.
, ;0 Nonlinear function of response time histories up to time .
x t
h x x t t
-5 -4 -3 -2 -1 0 1 2 3 4 5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5Restoring Force Vs. Displacement
Displacement,mm
Re
sto
rin
g F
orc
e, kN
66
Base & referenceconfigurations coincide(Taken to remain fixed)
Currentconfiguration
Total Lagrangian
approach
67
Baseconfiguration
Reference configurationupdated at each incrementwhile solving the equilibrium equation
Currentconfiguration
Updated Lagrangian
approach
68
Baseconfiguration Co-rotated configuration
(base configuration undergoesrigid motion; cg-s overlap)
Currentconfiguration
Co-rotational
formulation
• Base configuration is used as reference to measure rotations.
• Co-rotated configuration is used as reference to measure
current stress and strains
Strain measures
• Infinitesimal strains: for body under rigid
body rotations, the strains would not be
zero.
• New measures needed:
– Rigid body motions imply zero strains
– For small strains, the infinitesimal strain
definitions are to be restored.
69
Stress measures
• Cauchy stress tensor: defined with respect to deformed
geometry. This would not be known in advance.
• Two alternatives:
– Stress as a measure which conjugates with a measure of strain
to produce internal energy
– As a quantity which produces a traction vector in conjunction
with a normal vector defined with respect to a surface element
70
71
2 2 2
2
2 2 2 2 2 2
2
Sum of virtual external work done on a body and the virtual
work stored in the body should be zero.
Consider configuration C
W : V V S
e d V f u d V t u d S
Principle of virtual displacements
2 2 2
2 2 2 2 2 2
2
0
W 0ij ij i i i i
V V S
e d V f u d V t u d S
72
2 0
2 0
2 0
0
0
2 2 2 2 0
2 0 0
2 2 2 0
0
2 2 2 0
0
2 2
0 0
All quantities reckoned with respect to undeformed configuration (C ).
ij ij ij ij
V V
i i i i
V V
i i i i
S S
ij ij
V
e d V S E d V
f u d V f u d V
t u d S t u d S
S E
Total Lagrangian approach
0 0
0 2 0 2 0
0 0 0i i i i
V S
d V f u d V t u d S
73
2 1
2 1
2 1
1
2 2 2 2 1
2 1 1
2 2 2 1
1
2 2 2 1
1
2
1
All quantities reckoned with respect to the latest known configuration (C ).
ij ij ij ij
V V
i i i i
V V
i i i i
S S
ij
e d V S d V
f u d V f u d V
t u d S t u d S
Updated Lagrangian approach
1
1 1
2
1 1
2
1 1
2 2 1 2
1 1 1
2 2 1 2 1
1 1 1
updated Green-Lagrange strain tensor
body force refered to in C .
surface traction refered to in C .
0
i
i
ij ij
V
i i i i
V S
f
t
S d V R
R f u d V t u d S
Follow up
• Material nonlinearity
• Stability analysis: inclusion of nonlinearity
at different levels
• Hybrid testing
• Bayesian filtering
• Uncertainty modelling and fem
• Thermal loads: fire
• Anisotropy
74
Application of FE models in
structural testing • Pseudo-dynamic tests
– How to handle complicated inelastic
behavior?
• Real time sub-structure testing
– How to test interacting primary and secondary
system?
75
76
Hybrid simulations in earthquake qualification testing
Hybrid simulations
Shake table test Effective force test
(Reaction wall and strong floor
based)
Experimental partTest structure
Numerical part
Geometry Less payload
Time Slow down; less stringent requirements on hardware
Geometry and time
Scaling
Traditional
methods
77
Numerical solution
gMx Cx R x M x t
ComputedDisplacements x t
Earthquake excitations
MeasuredReactions
R x t
Pseudo Dynamic
Testing
-5 -4 -3 -2 -1 0 1 2 3 4 5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5Restoring Force Vs. Displacement
Displacement,mm
Re
sto
rin
g F
orc
e, kN
78
Experimentally explicit & numerically implicit scheme
Step size adjustment via a variational equation
Bending-torsion coupled piecewise nonlinear system.
mod
Inertial force + Dissipation force + Stiffness force = Applied forceExperimental modelNumerical model Numerical el
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2Restoring Force Vs. Displacement
Displacement,mm
Re
sto
rin
g F
orc
e, kN
0.5 1 1.5 2 2.5 3 3.5 4 4.5
-1.5
-1
-0.5
0
0.5
1
1.5
Displacement along X-axis Vs. Time
Time,s
Am
plit
ud
e, m
m
EFT
PSD
79
50kN, ±75 mm, Fatigue rated (2) 100kN, ±75 mm, Fatigue rated (1) 300kN, ±75 mm, Fatigue rated (1) Frequency up to ~50 Hz
2370MS, 32-Bit,
TMS320C5502, USB2,
16 DAQ, 5 kHz, 8 DAC
Reaction wall: L-Shape, 6x5x5x1 m Strong Floor: 200 PCC, 900 RCC
Hybrid Simulation Lab
96 Channels data acquisition system
Understood(numerical)
Not understood(experimental)
gx t
1x t
2x t 1x t
1m
1k
1c
2x t
2m
f x
2c
gx t
2x t
2m
f x
2c
1x t
1m
1k
1c
gx t
1x t
Measure reaction transferred to the support cf t
cf t
E
N
Real time interaction between N and E
Time delays
Noisy measurements
No need for scaling in space or time
Greater demands on hardware and FEA
What is new and challenging?
• FE modeling enters control algorithm of
the shake table/actuators • Accuracy, stability, speed,…
• Compulsions of real time operations:
modeling of time delays; need to have
“fast” solvers.
81
Frameworks for modeling uncertainties
• Probability theory
• Interval analysis
• Convex sets
• Fuzzy set
• Hybrid models
82
Challenge
How to combine these tools with structural analysis methods?
Note
In understanding structural failures one needs to model
structural nonlinearities as well as uncertaitnies.
83
185 190 195 200 205 210 215220
240
260
280
300
320
340
x1
x2
185 190 195 200 205 210 215220
240
260
280
300
320
340
x1
x2
185 190 195 200 205 210 215220
240
260
280
300
320
340
x1
x2
185 190 195 200 205 210 215220
240
260
280
300
320
340
x1
x2
No new
data are
likely to be
available
84
Interval models: ; 1,2, ,i i ix x x i n
1x 1x
2x
2x
1x
2x
85
Convex models:
: positive definite matrix; 0.
tx x a
a
1x
2x
The region need not be ellipsoidal.
It needs to be convex.
86
X
, ,0 1A AA x x x
Fuzzy models
A x
x x
1
x
The membership function needs to be convex with max value=1.
:
min :
max :
AF x X x
x f f F
x f f F
87
Fuzzy interval
At membership value ,
0 1
x x x
Fuzzy convex set
At membership level ,
0 1
tx x a
Structural
systems
Uncertain actions
Uncertain
System parameters
Uncertain outputs
Propagation of uncertainties must be consistent with the laws of
mechanics
