SOLUTION
STRUCTURAL
DYNAMIC
OF NONLINEAR
SYSTEMS
BY A HYBRID FREQUENCY-TIME DOMAIN APPROACH
by
JAMES DANIEL KAWAMOTO
Bachelor of Science in Civil Engineering
University of California, Berkeley
(1979)
Master of Science in Civil Engineering
Massachusetts Institute of Technology
(1982)
SUBMITTED TO THE DEPARTMENT OF CIVIL ENGINEERING
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
the
at
MASSACHUSETTS
INSTITUTE
OF TECHNOLOGY
June 1983
@
Signature
Massachusetts
Institute
of Technology
1983
of Author
Department
of
Civil \ngineering
March 25, 1983
Certified by
Connor
Jerome J.
Thesis Supervisor
y/
Accepted by
Francois M.
Chairman, DepartmentaMASSANU
M.
Morel
C-ommittee on Graduate Students
TTS
TITUTE
JUL 12 1983
2
To My Family
for
their
Understanding and
Encouragement
3
SOLUTION OF NONLINEAR DYNAMIC STRUCTURAL SYSTEMS
BY A HYBRID FREQUENCY-TIME DOMAIN APPROACH
by
JAMES D.
KAWAMOTO
Submitted to the Department of Civil Engineering
on March 25, 1983, in partial fulfillment of the
Doctor of
of
degree
the
for
requirements
Philosophy.
ABSTRACT
application
and
This thesis presented the development
solution technique for
domain
frequency
iterative
an
of
referred
The scheme,
nonlinear dynamic structural systems.
hybrid frequency-time domain approach, combines
the
as
to
superposition
mode
a
with
formulation
pseudo-force
the
Nonlinearities are evaluated in the time domain,
analysis.
domain,
frequency
the
in
derived
is
solution
the
and
numerical
exact
theoretically
a
of
use
the
implying
integrator.
In conjunction with the development of the new solution
of numerical
analyses
accuracy
and
stability
technique,
of an
presentation
the
to
leading
surveyed,
integrators were
time
The
domain.
frequency
the
in
scheme
alternate analysis
were
analyses
stability
and
and frequency domain accuracy
Extensions
problem.
vibration
free
shown equivalent for the
frequency
the
in
to arbitrary loadings are easily handled
domain.
systems
excited
seismically
of
Various case studies
hybrid
the
for
demonstrated the developmental considerations
that
indicated
results
The
frequency-time domain scheme.
the
reproduce
accurately
can
scheme
solution
new
the
when
attractive
particularly
was
scheme
The
response.
numerical integration considerations severely restricted the
time step size for a direct time integration analysis.
Thesis
Title:
Supervisor:
Connor
Jerome J.
Professor of Civil Engineering
4
ACKNOWLEDGEMENTS
of my
doctoral
Witmer,
Professor
I extend my appreciation to the members
Professor
Connor,
Professor
committee,
and
Buyukozturk, Professor Kausel,
particular,
doctoral
I
Professor
thank
committee
chairman
Connor
and
my
for
In
Sunder.
Professor
acting
research
as my
supervisor
during my stay at MIT.
also
I would
supporting
my
like
research
to
and
thank
INTEVEP
studies
for
partially
during the last three
years.
Finally, I express my gratitude to Kong Ann Soon,
research colleague and
joys,
and memories.
friend,
for
sharing the
as
a
frustrations,
5
TABLE OF
CONTENTS
TITLE PAGE
1
DEDICATION
2
ABSTRACT
3
4
ACKNOWLEDGEMENTS
TABLE OF
5
CONTENTS
9
LIST OF FIGURES
LIST
OF TABLES
17
LIST
OF SYMBOLS
18
CHAPTER 1:
INTRODUCTION
24
CHAPTER 2:
NONLINEAR CONTINUUM MECHANICS
31
2.1
2.2
Review of Tensor Analysis
32
2.1.1
Scalars and Vectors
34
2.1.2
Second-Order Tensors
39
Kinematics
2.2.1
2.2.2
44
Displacement, Deformation Gradient
and Tensors
45
Strain Tensors
53
2.3
Stres.s Tensors
58
2.4
Energy Equations
63
2.5
Displacement-Based Finite Element Formulation
of the Governing Equations
68
Constitutive Relations
71
2.6
6
CHAPTER
3.1
3.2
GOVERNING EQUATIONS OF MOTION AND THEIR
NUMERICAL SOLUTION
3:
Equations
75
of Motion
76
Formulation
3.1.1
Exact
3.1.2
Pure Unconventional
3.1.3
Pseudo-Force Formulation
80
3.1.4
Incremental Response Formulation
83
Formulation
Time Integration Analyses
3.2.1
Direct
3.2.2
Nonlinear Modal Analysis in
Time Domain
CHAPTER 4:
79
86
Solution Methods
3.2.3
4.1
74
the
Nonlinear Modal Analysis in the
Frequency Domain
ANALYSIS OF NUMERICAL INTEGRATION METHODS
Stability and Accuracy Analysis
Schemes
Integration
86
103
110
112
of Numerical
113
4.2
Survey of Integration Schemes
122
4.3
Linear Systems Theory Approach for Accuracy
and Stability Analyses
130
Relation Between Time and Frequency Domain
Analyses
Stability
151
4.4
4.5
4.6
Extension of Accuracy and
to Nonlinear Problems
Stability Analyses
153
4.5.1
Analytical Approach
153
4.5.2
Numerical Experiments
159
Selecting a Time Increment
164
7
CHAPTER 5:
5.1
FOURIER TRANSFORMS AND THE HYBRID
FREQUENCY-TIME DOMAIN ANALYSIS SCHEME
5.1.2
Continuous Fourier
Transforms
Series
and
171
Fourier Transforms of Discrete Time
Series
175
5.1.3
Discrete Fourier
182
5.1.4
Numerical Evaluation of
Fourier Transform
5.1.5
Series and
Transforms
the Discrete
184
Frequency Domain Analysis and Nonlinear
Systems
5.2.1
Formulation of
5.2.2
Numerical Considerations for
an HFT Analysis
CHAPTER 6:
Applying
195
Solution Formulation
196
5.2.2.2
Zero Minimization Problem
198
5.2.2.3
Relaxation
205
5.2.2.4
Other
5.2.2.5
Stabilization by Artificial
Damping and Incremental Load
Application
209
Nonlinear Mode Updating in
Frequency Domain
215
Selecting a Time Increment
the
225
the HFT
227
235
STUDIES
Study,
207
Acceleration Schemes
Qualitative Evaluation of
Solution Scheme
SAMPLE
Feasibility
193
the HFT Approach
5.2.2.1
5.2.2.7
5.2.3
186
191
Hybrid Frequency-Time Domain Analysis
5.2.2.6
6.1
170
Frequency Domain Analysis
5.1.1
5.2
168
SDOF
System
236
8
6.2
SDOF
6.3
Ten DOF
6.4
MDOF
6.5
Cross-Braced Tubular Offshore Structures
Shear
Beam
281
Soil Amplification Study
Single-Bay Offshore
6.5.2
Two-Bay
to Taft
6.5.3
Structure
Offshore Structure
Earthquake
Subjected
Two-Bay Offshore Structure
to El Centro Earthquake
Subjected
6.6
Extremely Soft
6.7
Bilinear Elastic
6.8
Summary
of
SDOF
326
337
System
SUMMARY, CONCLUSIONS,
344
AND RECOMMENDATIONS
7.2
Conclusions:
Principal Results of
Present Study
the
352
Recommendations for Future Research
APPENDIX B:
357
359
REFERENCES
APPENDIX A:
349
349
Summary
Research
300
315
System
SDOF
300
312
Studies
7.1
7.3
270
Structure
6.5.1
CHAPTER 7:
258
Soil Amplification Problem
ALTERNATE ACCURACY ANALYSIS OF
STIFFLY-STABLE METHOD
THE PARK
ALTERNATE STABILITY ANALYSIS OF THE
CENTRAL DIFFERENCE METHOD USING THE
Z-TRANSFORM CONCEPT
APPENDIX C:
FOURIER TRANSFORM PAIR OF x
APPENDIX D:
ZERO MINIMIZATION TECHNIQUE
3
368
374
378
380
9
LIST OF FIGURES
Figure
Page
Notation for Reference and Deformed
Curvilinear Coordinate Systems Moving in
a Cartesian Coordinate System
46
Equilibrium Iteration Schemes Applied to
SDOF Systems
102
Spectral Radii of the Amplification Matrix A
as a Function of At/T (no physical damping)
124
4.2
Stability Region of Multistep Methods
125
4.3a
i as a
Algorithmic Damping Ratios
of
At/T (no physical damping)
126
2.1
3.1
4.1
4.3b
4.3c
4.3d
4.4
4.5
4.6
4.7
4.8
4.9
4.10
Function
Relative Period Errors T as a Function
of
At/T (no physical damping)
126
Algorithmic Damping Ratios i as a Function
of
At/T (1.5 per cent physical damping)
127
Relative Period Errors T as a Function
of
At/T (1.5 per cent physical damping)
127
Damping and Frequency Distortion Parameters
versus
wAt
128
Transfer Functions for Central Difference
Method (5 per cent physical damping)
136
Transfer Functions for Central Difference
Method (10 per cent physical damping)
137
Transfer Functions for Newmark Method
(5 per cent physical damping)
138
Transfer Functions for Newmark Method
(10 per cent physical damping)
139
Transfer Functions for Houbolt Method
(5 per cent physical damping)
140
Transfer Functions for Houbolt Method
(10 per cent physical damping)
141
10
Page
Figure
4.11
4.12
Transfer Functions for Park Method
(5 per cent physical damping)
142
Transfer Functions for Park Method
(10 per cent physical damping)
143
4.13a Bias as a Function of
(no physical damping)
4.13b Artificial Damping as
(no physical damping)
4.14a
oAt
146
a Function of
iAt
146
oAt
Bias as a Function of
(2 per cent physical damping)
4.14b Artificial Damping as a Function of
(2 per cent physical damping)
147
5At
147
4.15a Bias as a Function of
Z!At
(5 per cent physical damping)
4.15b Artificial Damping as a Function of
(5 per cent physical damping)
148
GAt
148
4.16a Bias as a Function of
5At
(10 per cent physical damping)
4.16b Artificial Damping as a Function of
(10 per cent physical damping)
4.17
149
wL&t
Conditional Stability Region of Four Methods
for Linearly Extrapolated Pseudo-Force
Procedure
Fourier Transforms
Time Signals
of Continuous and Discrete
5.2
Nonlinear Response
to a Harmonic Excitation
5.3
Effect of Appended
Domain Analysis
Zeroes in a Frequency
5.1
149
158
180
190
200
Damping Ratio vs. Number of Cycles Required
to Reduce Amplitude by 50 per cent
201
5.5
Zero Minimization Technique
204
5.6
Segmented History Analysis
226
5.4
11
Page
Figure
5.7
Flow Chart for Hybrid Frequency-Time Domain
Analysis Package
231
237
Brace Model
6.1
Tubular
6.2
Bucarest Acceleration History
6.3
Linear Elastic Response
6.4
Elastic-Plastic Response
5 ksi)
241
6.5
Elastic-Plastic Response Using a Time
Domain Analysis (Fy = 2 ksi)
242
Elastic-Plastic Response Using a Frequency
Domain Analysis, One Iteration (Fy = 2 ksi)
242
Elastic-Plastic Response Using a Frequency
Domain Analysis, Ten Iterations (Fy= 2 ksi)
243
Elastic-Plastic Response with Pseudo-Force
Correction to T = 2.0 s.
243
6.6
6.7
6.8
to a
(Fy
238
Sine Wave Loading
=
1 Iteration
240
245
6.9
Pseudo-Force History,
6.10
Pseudo-Force History, 2 Iterations
245
6.11
Pseudo-Force History, 3 Iterations
246
6.12
Pseudo-Force History for an Elastic-Perfectly
Plastic Material Model
247
Elastic-Plastic Response to the Bucarest
Earthquake
(Fy = 36 ksi)
249
Elastic-Plastic Response to the Bucarest
Earthquake, 6 Iterations, N = 12
252
Elastic-Plastic Response to the Bucarest
Earthquake, 5 Iterations, N = 12
252
Tubular Member Response to the Bucarest
Earthquake, Time Integration
253
Tubular Member Response to the Bucarest
Earthquake, 3 Iterations, N = 12
253
Tubular Member Response to the Bucarest
Earthquake, 12 Iterations, N = 13
254
6.13
6.14
6.15
6.16
6.17
6.18
12
Page
Figure
Tubular Member Response to the Bucarest
Earthquake, 15 Iterations, N = 13
254
Elastic-Plastic Response to the Bucarest
Earthquake, 10 Iterations, N = 10
259
6.21
Ramberg-Osgood Material Model
262
6.22
Taft Acceleration History
263
6.23
SDOF
6.24
Soil Amplification Results
Time Integration
Analysis
6.19
6.20
6.25
6.26
6.27
264
Soil Model
Using a Direct
268
Soil Amplification Results Using an HFT
Analysis (20 iterations, artificial viscous
damping ratio of 0.50)
268
Soil Amplification Results Using an HFT
Analysis (60 iterations)
269
Soil Amplification Results Using an HFT
Analysis (75 iterations)
269
Beam Study
271
6.28
Ten Degree of Freedom Shear
6.29
Response of DOF
and HFT Results
1, Direct Time Integration
Response of DOF
and HFT Results
10,
6.31
Deflected Shape
from Time 1.0
6.32
Linear Mode
6.33
First Nonlinear Mode Obtained
Load Distribution Method
6.30
6.34
6.35
6.36
273
Direct Time Integration
273
sec.
to
1.5 sec.
274
276
Shapes
from Static
276
First Nonlinear Mode Gram-Schmidt
Orthogonalized with Respect to Lowest Five
Linear Modes
276
Response of DOF 1 Using an HFT Analysis
(5 linear plus 1 nonlinear mode)
277
Response of DOF 10 Using an HFT Analysis
(5 linear plus 1 nonlinear mode)
277
13
Page
Figure
6.37
6.38
6.39
Converging Deflected Shapes for Iterations
5, 10, 15, and 20 (5 linear plus
1 nonlinear mode)
278
Deflected Shapes Obtained with 5 Linear
Modes Using a Direct Time Integration Analysis
279
Refined Multidegree
Amplification Study
of Freedom Soil
282
Linear Response to the Taft Earthquake
(surface displacement)
283
Nonlinear Response to the Taft Earthquake
Using a Direct Time Integration Analysis
(surface displacement)
283
Nonlinear Response after 5 Iterations
of 0.75)
(artificial viscous damping ratio
285
6.43
Nonlinear Response after 10 Iterations
285
6.44
Nonlinear Response after 20 Iterations
286
6.45
Nonlinear Response after 10 Iterations
(artifici al hyster e tic damping ratio of
time inc rement of 0.01 s)
6.40
6.41
6.42
0.75,
288
Iterations
288
Nonlinear Response
after 50 Iterations
290
6.48
Nonlinear Response
after 80
Iterations
290
6.49
Nonlinear Response
(10 linea r modes)
after
Iterations
Nonlinear Response
(5 linear modes)
after 10
Nonlinear Response
(1 linear mode)
after 10
6.46
Nonlinear Response after
6.47
6.50
6.51
10
10
291
291
Iterations
292
Nonlinear Response after
(1 nonlinear mode)
50
6.53a Nonlinear Response after
(1 linear mode)
50
6.52
Iterations
Iterations
294
Iterations
294
14
Page
Figure
6.53b Nonlinear Response after 80
(1 linear mode)
Iterations
295
6.54a Nonlinear Response after 40 Iterations
(time increment of 0.02s, 1 linear mode)
6.54b Nonlinear Response after 80
(1 linear mode)
296
Iterations
296
6.55
Stress-Strain Response of Ramberg-Osgood
Material Model
298
6.56
Single-Bay Offshore Structure
301
6.57
Linear Response to the Taft Earthquake
(lateral deck displacement)
303
6.58
Nonlinear Response to the
(direct time integration,
displacement)
Taft Earthquake
lateral deck
303
Nonlinear Response after 5 Iterations
modes, artificial hysteretic damping
(6 linear
of 0.5, time increment of 0.02 s)
ratio
305
6.60
Nonlinear Response after 15 Iterations
305
6.61
Nonlinear Response after
30 Iterations
306
6.62
Nonlinear Response after 30 Iterations
(6 linear modes, time increment of 0.05 s)
306
6.63
Nonlinear Response after 30 Iterations
(1 linear mode, time increment of 0.05 s)
307
Nonlinear Response after 25 Iterations
(3 linear modes, time increment of 0.02 s)
307
Nonlinear Response after 30 Iterations
(3 linear modes, time increment of 0.02
308
6.59
6.64
6.65
s)
6.66
Two-Bay Offshore Structure Subjected to
Taft Earthquake
313
6.67
Linear Response to Taft Earthquake
(lateral deck displacement)
316
6.68
Nonlinear Response to Taft Earthquake
(lateral deck displacement, time integration)
316
15
Page
Figure
6.69
Nonlinear Response after 3 Iterations
(1 linear mode, no artificial damping)
317
318
Centro Acceleration History
6.70
El
6.71
Two-Bay Offshore Structure Subjected to
El Centro Earthquake
320
Linear Response to El Centro Earthquake
(lateral deck displacement)
321
Nonlinear Response to El Centro Earthquake
(lateral deck displacement, time integration)
321
Nonlinear Response after 45 Iterations
(7 linear modes, artificial viscous damping
ratio of 0.50)
322
6.72
6.73
6.74
6.75
6.76
6.77
6.78
6.79
Nonlinear Response after
(1 linear mode)
10 Iterations
Nonlinear Response after
(3 linear modes)
10+12
324
Iterations
324
Nonlinear Response Using Direct
Time Integration
327
Nonlinear Response after 10 Iterations
(linear stiffness)
329
Nonlinear Response after 30
(linear stiffness)
329
Iterations
Nonlinear Response after 20 Iterations
(least-squares updated secant stiffness)
330
6.81
Pseudo-Force History after 4 Iterations
332
6.82
Pseudo-Force History after 5 Iterations
332
6.83
Pseudo-Force History after
6 Iterations
333
6.84
Pseudo-Force History after 50 Iterations
333
6.85
Nonlinear Response after 3 Iterations
(updated secant stiffness)
334
Nonlinear Response after 4 Iterations
(updated secant stiffness)
334
6.80
6.86
16
Page
Figure
6.87
6.88
6.89
Nonlinear Response after 5 Iterations
(updated secant stiffness)
Nonlinear Response after 49
(updated secant stiffness)
335
Iterations
335
Nonlinear Response after 50 Iterations
(updated secant stiffness)
336
338
System
6.90
Bilinear Elastic SDOF
6.91
Bilinear Elastic Linear Response
340
6.92
Bilinear Elastic Static Response
340
6.93
Bilinear Elastic Nonlinear Response
(time integration, time increment of 0.02
6.94
Bilinear Elastic Nonlinear Response
(HFT, 3 iterations, time increment of
1 s)
s)
341
341
D.1
Convolution with Continuous Functions
382
D.2
Zero Minimization Technique Viewed
in the Time Domain
384
17
LIST OF TABLES
Page
Tables
Resonant Frequencies and Amplitudes of
Numerical Integration Transfer Functions
144
Computation Times for Elastic-Perfectly
Plastic SDOF MOdel
251
6.2
Computation Times for Tubular
Brace SDOF Model
256
6.3
Computation Times for Elastic-Perfectly
Plastic SDOF Model Using the Zero Minimization
Technique
257
Residual Displacements When Using Relaxation
Schemes
260
6.5
Computation Times for
Beam Model
280
6.6
Computation Times for MDOF Soil Amplification
Study
299
4.1
6.1
6.4
6.7
6.8
6.9
6.10
10 DOF
Maximum Response Values
Offshore Structure
Computation Times
Structure
for
Shear
Single-Bay
309
for Single-Bay Offshore
311
Computation Times for Two-Bay Offshore
Structure Subjected to El Centro Earthquake
325
Summary of
345
Studies
18
LIST OF SYMBOLS
Description
Symbol
a
n
a
n
approximate acceleration response at
chapter 4
cosine Fourier series coefficient,
time t
n
chapter 5
amplification matrix
A
amplitude decay
AD
traction vector
surface where
A
is applied
0
b
sine Fourier series coefficient
B
body
B
left Cauchy-Green
n
force vector
deformation tensor
complex Fourier series
c
coefficient
n
C
damping matrix
C
right Cauchy-Green deformation tensor
C
inertia coefficient
C
artificial viscous damping matrix
di
difference between residual forces
successive iterations
d
approximate displacement
M
-v
response at
surface area
dA
differential
dm
differential mass
dV
differential
volume
of deformation tensor
D
rate
e
Almansi
strain tensor
Eijkl
fourth order elastic modulus tensor
E
secant Young's modulus
sec
of
time
t
19
fh
artificial hysteretic
fN
Nyquist frequency
f
cutoff
damping force
frequency
0
f
artificial viscous
fy
Ramberg-Osgood
fu
damping force
material model parameter
residual force vector
force vector
F
external
Fh
artificial hysteretic damping
FNL
pseudo-force vector
FOD
off-diagonal
Fv
artificial viscous
F
yield stress
force vector
force vector
damping force vector
y
F
deformation gradient
F~
spatial
deformation gradient
tensor
in the frequency domain
F(w)
forcing function
AF
effective load vector
-eff
tensor
g
gravitational acceleration constant
g.
base vectors in the reference configuration
of a convected body-fixed coordinate system
G
quasi-Newton updated
G
base vectors in the current configuration
of a convected body-fixed coordinate system
h
impulse response
H
transfer function
H
approximate numerical
function
H
exact
-e
stiffness matrix
function
integration transfer
transfer function
20
S,
1.,
~
i
i'
-l1
base vectors for a rectangular Cartesian
coordinate system
internal force vector
J
transformation matrix from a local
global coordinate system
K
stiffness matrix
eff
effective stiffness matrix
K
linear stiffness matrix
Kni
nonlinear stiffness matrix
-1
secant stiffness matrix
K
-sec
stiffness matrix
Kt
tangent
L
load operator
M
mass matrix
N
number
N
outward normal
NB
number of
-
q
q*
Aq
to a
of
points
in
load history
vector
points
in frequency spectrum
element nodal generalized displacement
node m
global nodal
of
generalized displacement
ith displacement correction
Q
response in the frequency domain
r
position vector in deformed configuration
R
position vector in undeformed configuration
R
rotation tensor
S
number of
S
2nd Piola-Kirchhoff
t
time
cycles
stress tensor
21
t
traction vector with respect to undeformed
configuration
t
nth discrete time
step
n
time increment
At
T
structural period
T
traction vector
T
approximate period
T.
ith structural period
u
displacement
vector
velocity vector
acceleration vector
u
Ramberg-Osgood material model parameter
u
y
stretch
tensor
U
right
v
approximate velocity response at
V
velocity vector
V
left
n
volume
reference
W
convergence residual
W
spin tensor
W
e
t
stretch tensor
V
0
time
external virtual work
W
internal virtual work
x
reference coordinates for a rectangular
Cartesian coordinate system
current coordinates for a rectangular
Cartesian coordinate system
y
generalized displacement
y
unconverged
generalized
response
n
22
yi
difference between residual
successive iterations
Y
generalized
z
state vector, chapter 4
z
z-transform, chapter
forces of
response in frequency domain
5
Green-Lagrange
strain tensor
e
elastic strain
tensor
P
plastic strain
tensor
r
generalized damping matrix normalized with
respect to the mass matrix
6
Dirac delta
6
logarithmic decay
au
infinitesimal virtual displacement
6W
e
function
vector
infinitesimal external virtual work
6W.
infinitesimal internal virtual work
A
diagonal matrix containing eigenvalues
viscous damping
h
ratio
artificial hysteretic
damping ratio
curvilinear coordinates for a convected
body-fixed coordinate system
artificial viscous
generalized
damping ratio
force vector
p
mass density in present configuration
p
mass density in deformed configuration
p
spectral
radius
Cauchy stress tensor
0
Jaumann rate of the Cauchy stress tensor
- 23
Kirchhoff
T
stress
tensor
0
Jaumann rate of
T
-nl1
stress
tensor
(mode shape)
eigenvector
-1
Kirchhoff
linear eigenvector
nonlinear eigenvector
displacement
interpolation function
circular frequency
four times highest
load freq'uency
C
structural
D
w
max
frequency
frequency
approximate
structural
approximate
damped structural
frequency
highest load frequency
highest
structural
frequency
max
circular frequency multiplied
increment
by time
fundamental or metric tensor
{
s}
m n
Christoffel
symbol
of
the second kind
24
CHAPTER 1
INTRODUCTION
capability
Considerable advances in the analytical
introduction of
necessary
highly
complex
the
status
of
to
decades with the
problems,
time
the
reduced
the tools
and provided
previously
This
interest.
academic
extension
the
encouraged
has
capability
of
into more severe environments and simultaneously
structures
resulted
calculations
to analyze
relegated
increased
last two
computer, have
the digital
tedious
to
allocated
the
during
achieved
engineers,
of
in
less
conservative
designs
the
approaching
ultimate member capacities.
The analysis of
techniques
elastic
easily
predicted and
threats
life.
to
these
is
structures by
adequate
containment
in
the
systems,
event
of
situations,
uncertain load conditions and extreme
ecosystem
of
aircraft,
linear
when the design loads are
the possibility
Many
traditional
failure
however,
consequences
failure.
buildings,
Nuclear
and
poses
no
involve
to
the
reactor
offshore
25
platforms
drilling
even a flock of birds.
an earthquake, tornado, hurricane, or
The
for
designed
material
of
accurately
1000 year earthquake?),
the
accurate
is
(do
we
(how
reproduce the structural
response?).
the
sizes as
load
Simple enlargements in member
expanded.
have
increases,
level
due
technological
limitations,
of
These
considerations
often
level,
small
always be
and
possible
the increased
exemplified by
design
the
development
strength level design,
the
as the ductility level
nature of the
probabilistic
level
corresponding
probability
of
of
first
The
levels.
the inherent
design,
load
a
for
expected once during the structure's life.
the
engines.
larger
requiring
to remain linear elastic
and provides a measure of
load
as
cannot
increases
engendered
two
to
referred
level, referred to
for
have
consisting of
requires the structure
level
as
structures
aircraft
weight
design codes
cost
to significant
the
allowing
although
continued use of a linear elastic analysis,
justified
analytical
the
Design philosophies have transformed as
capabilities
tools
our analytical
(do
system
'the
and
specify?),
we
structure
entire
quality
construction
the
the
the
be
structural members
model?),
actually construct
response
year or
100
the
structure
the
(should
level
design
of individual
behavior
the
the
to
related
system prompts questions
the
of
nature
probabilistic
The load may be
are but a few examples.
The
second
accounts
structure and loading
structural
safety.
A
to a situation with an extremely
occurrence
is
specified,
and
the
26
other words,
become inelastic, rather
increased structural
systems subjected to
including
both
incurring
the
and
for nonlinear
expanded,
material
are available
codes
response
impact and longer
and
kinematic
term loads
for general purpose
Additional
contributions to the time domain solution technique
of
numerical
alternate
integrators
such as
schemes
and
induce
continuously
equilibrium
in terms
iterators
and
the incremental and pseudo-force
formulations and the development
models
of
nonlinearities.
applications.
specialized
extremely
and
the
to estimate the nonlinear transient
computer
by
redesigning
problems has been developed,
structural
Numerous
load
this
cost.
mechanics
and refined
the
energy absorbing
the computational capability
As a result,
continuum
than
elastic
linear
remain
to
structure
hence
and
yield and buckle, in
to
members
structural
the
allowing
environment.
withstand
to
structure
the
properties of
the
the
exploit
to
prefers
often
designer
load without
level,
strength
the
of
higher than that
this
a load level much
stipulates
design
level
ductility
on
impact
of lives and minimal
any loss
The
designed to withstand
be
must
structure
of more realistic
revisions
in
this
material
numerical
solution library.
The
ability
particular
computational
to
structural
cost
for
reproduce
systems
actual
the
indeed
is
conducting
a
response
amazing.
nonlinear
of
The
analysis,
27
often
is
however,
prohibitive and limits
the analyst
the underlying
which
by
all
develop
engineers
behavior, are usually
out
problems
from numerical stability and accuracy constraints
stem
relation to the actual physical behavior
that have no
integration
when the time
Even
as
such
the
frequency
dependent stiffness and damping
any desired level
the
excited
systems.
has
being
in
the
can produce results with
response
scheme,
analysis
nonlinear
for
are
Applications
transient
solution
domain
solution
nonlinearities
that
of accuracy.
proposed
frequency-time
iterative
problems
nonlinear
toward
This
consequently,
frequency domain solution technique
structural
dynamic
the
the feasibility of an alternate
to develop and examine
iterative
prevent
terms
properly.
of this research study,
The purpose
handle
to
inability
limitations
system from being modelled
efficient,
fairly
is
approach
various
been
of the
but unnecessarily increase the solution cost.
system,
actual
of the
The apparent inefficiency of
nonlinear problems.
the, direct time integration approach for particular
may
a
cannot
today
Even super computers on the market
question.
handle
foundation
studies,
Parameter
structure.
original
the structural
for
"feel"
the
of
a
simplified
few extensive large scale analyses if not highly
renditions
to
of
called
oriented
seismically
the
approach,
consists
frequency
domain
evaluated in the
hybrid
of an
with
time domain combined
28
with
superposition
a mode
nonlinear
implies
modes.
A
use
of
the
and
integrator,
by
followed
current solution techniques and a
continuum
presented
establishes
2
with
from
fundamental
displacement-based
relations
A
kinematic
using
discusses
recent
finite
element
form
technique
on
discussion
condensed
measures
definitions.
of
is
The
the
the governing
the
constitutive
concludes the chapter.
Assumptions
defined.
each
form
various
proposals
Components
strain
the energy equations and develops
In Chapter 3 the different forms
motion are
nonlinear
of
tensor analysis
and
and then the various stress
derived
equations.
foundation
the
A brief review of
mechanics.
chapter proceeds on to
in
concludes
and
studies.
Chapter
are
review of
formulation
theoretical
the new solution scheme,
description of
case
the
first presents
a
kinematic
Both
size.
are considered.
and material nonlinearities
The thesis
the
superposition approach allows a
reduction in the problem
significant
of
numerical
exact
theoretically
mode
the
updating
domain solution in essence
frequency
a
and
approach
are
of
the
in deriving and
emphasized.
The
"exact" solution methods,
such
as
of the standard
are examined
nonlinear
direct time
in detail.
equations
mode
of
limitations
chapter
then
including
some
superposition.
integration solution
29
Numerical
schemes
integration
associated with numerical integrators,
problems
standard time domain form, and
in the
with case
domain
chapter begins with a review of the
The
scheme.
analysis
frequency-time
hybrid
the
5 presents
The chapter concludes
analyses.
studies of nonlinear
Chapter
in their
then an alternate formulation
discussed.
frequency domain is
exact
analysis.
presented
are
analyses
stability
and
Accuracy
domain
frequency
a
of
integrator
numerical
and hence to
theoretically
the
use
to
incentive
an
provide
time
Chapter 4 to identify
studied in
approaches are
integration
the basis of
forming
frequency domain analysis and a mathematical presentation of
series
Fourier
The
implementation.
the
detail
numerical
A
development.
6.
the
of
domain analysis to seismically
Chapter
-and examines
with
in
its
distinguishing features
hybrid
frequency-time
excited structural
Each
allowing
a
systems is
lists
pertinent
reasonable
comparison
study
numerical
parameters,
between the
direct time integration and
Results
describes
the chapter.
The actual application
in
chapter
associated
of its
discussion
numerical
scheme
domain
considerations
and suitability concludes
presented
the
second part of
frequency-time
hybrid
tcheir
and
transforms
and
proposed
are presented in response history form.
approach.
30
Chapter 7 summarizes the
conclusions
related to the new
research project
solution scheme.
and
provides
31
CHAPTER 2
NONLINEAR CONTINUUM MECHANICS
This
and
strain measures
and the
the
of
formulation
equations are presented in their most general
all kinematic nonlinearities
rotations
time
is
development
equations
governing
form, allowing
are
the
that
ensure
to
followed
mathematical
rigorous
A
behavior.
and
path and
and material nonlinearities consisting of
dependent
stress
finite strains
terms of
in
analyzing
The various
structural problems.
dynamic
nonlinear
for
the foundation
chapter establishes
consistent with the theoretical
governing
Tensor
basis.
analysis is applied throughout the presentation to elegantly
transform
complex
equations
while
abstract
into
definitions
the
maintaining
generality
practical
their
of
application.
The governing
to
admit
all
equations are
As
nonlinearities.
maintaining mathematical
times
derived in a rigorous
rigor,
appear wordy and pedantic,
the
a
form
consequence
of
may
at
presentation
but a full appreciation of
32
the
Nonlinear
principles.
for
definitions
this
classical
Bathe
follows
the outline of
recent report by Rodal
those
by
(25,44,67,73,74) and the
(72)
Witmer
and
as
such
mechanics
texts on continuum
the
inconsistency
This chapter
premise.
turn
mathematical
Flugge, Malvern, and Prager
Sedov,
more
any
in
actual observed
the
reproduce
to
and consequently
behavior,
defeats
is
analyses
nonlinear
a host of
conducting
of
purpose
The ultimate
selected definitions.
more
motion.
with
consistent
relations
constitutive
entailing
quantities,
strain
and
stress
of
rotations engenders
and
strains
finite
couple
equations
structural matrices to the governing
Accepting
a
"adding"
by
obtained
analyses
linear
extensions of
simple
analyses are not
underlying
their
of
qualities
subtle
the
understanding
and
text
by
(6).
2.1 REVIEW OF
The
reference
TENSOR ANALYSIS
laws
physical
since the
reference
frame
reference
configuration.
independent
independent
independent.
the
of
properties
media, however, are frame
the
governing
be
must
media
continuous
the
attained only by
generality of nonlinear analyses ca-n be
deformation
of
the
itself
media
The kinematics
frame
These
properties
governing
laws
geometrical descriptions are
or
of
and
satisfied
the
of
are
of deformable
dependent being defined
configuration
of
current
either
in
deformed
possessing
reference
reference
dependent
by tensors
and tensor
33
by
the physical
the invariance of
other
the use
Two
system
coordinate
a
in
generally
more
or
the
initial
spanning the
Cartesian
rectangular
a
in
defined
be
tensor space can
vectors
The basis
configuration.
(undeformed)
and
the
as
defined
also
continuum
in
configuration
(deformed)
configuration,
reference
only
laws is satisfied
necessary
are
reference
current
the
mechanics,
Euclidean
in four-dimensional space-time.
of tensors
frames of
The following
three-dimensional
to
the
frames accelerating relative to each
reference
In
space.
limited
is
presentation
tensors while
fourth order tensor.
a
is
modulus
elastic
second order
are
measures introduced later
strain
Stress and
tensors, vectors.
first order
and
scalars
are
tensors
order
Zero
dependent.
components are frame
whose
but
quantities,
invariant
are
which
equations
curvilinear
coordinate system.
The notation used here is
Scalars
(72).
Witmer
vectors, by underlined
by
while
tensors
are
letter with double
the
is
tensor
letters.
their corresponding
identified by
letters;
second
order
tensors,
Vector
components
indices.
in
of
by
their same
The indices are
the reference
upper case when defined in the current
selection
by
letter with a single
identified
also
defined
and
and
simple
denoted
are
letters;
underlined
doubly
similar to that of Rodal
are
index
simple
lower case when
configuration and
configuration.
upper or lower case letters for
The
a rectangular
34
Cartesian coordinate system are
("'")
sign
A
identified by
circumflex
a
and Vectors
of
scalar can be defined as a quantity consisting
that
component
single
in a
components
of the kernel letter.
on top
2.1.1 Scalars
Tensor
configuration.
the
of
independent
remains
basis
the
since
arbitrary
is
system
Cartesian
coordinate
a
under
invariant
is
a
transformation.
Vectors consist of more than one
can
component and
be
represented as
n
v = v1 b
-1
-
where the
b
contravariant
+
basis
a
form
(2.1)
vb.=vb.
=T
-- 1
-n'--a-1
k=1
components.
2.1
third equality in Eq.
summation
is
The RHS
(right hand
a
Notice
that in an n-dimensional space
infinite
set
of n
number
of
bases
gn
b such that
linearly independent vectors b.
be orthonormal or
the
index.
exists
an
each b consists of a
.
need
not
terms
of
its
b
as
The b.
even orthogonal.
The vector v can
covariant components v
v = v b
there
of
notation
repeated
any
over
implied
where
side)
indicial
is written in
the
are
v
the
and
r
for
also
be
written
with respect to
in
the basis
(2.2)
35
Given two
next.
presented
equalities of
and
Some general definitions
vectors u and v where
(2.3)
V = vib.
u =u b.
are
vectors
we have
(2.4)
= u Ab..
u-v = u'vib .b
-i-j
13
--
operation
where the
-1 3
ij
b -b1 =
-k-
the
1
k
(2.5)
base vectors b
the Kronecker delta
f1,
r=s
{0,
r#s
basis is
the dual
b
vector
vk
vk
defined such that
and
defined by
(2.7)
identical
Taking the dot product
-i
are
(2.6)
special case of an
For the
or
dot
61
k
is
6)
k
with
as
defined
31
(or reciprocal)
where
is
= b. b. a b..
b
Dual
2.4
u and v and
product of
scalar
id Eq.
orthonormal set of basis
vectors,
to the given basis.
of Eq.
2.1 with bi
using the equality in Eq.
and Eq.
2.2
2.6, we obtain the
components
=
v-b
"k
(2.8)
v
= v-b
36
tensor components g j,
The fundamental
ij
,
g
i.
g.
,
j
defined as
are
g.
b.-bj
-b
and
2.8,
2.2,
2.1,
contravariant and
the
between
b-
g
b.b.
Using Eqs.
be
g
following
2.9 the
(2.9)
relations
covariant components
of v can
established:
.
b
-i
-j
v.. = v
=
21
= v.b -b. =
-i
J-
g
gb..
ij
v.g'
J3 -
v..
J2i
=
(2.10)
= v-b
V
= v b -b' = v g'
Sb.-b
- -
and raising
lowering
of
to the basis
applied
b.
b
bi = ga -1
-
-
Notice
b
.
Therefore,
j
i zi
holds
= 9ik
NikS
=
ikA
jl
k
(2.11)
=
and
for any arbitrary vectors b
jg
and b
=
g.,
we
have
(2.12)
9
11
.51
jl9 6 k
gikkj
process can also be
ji
k
j
This
indices.
Ji-
= g.
g
2.10 represent the process
= g. .b
In particular, if bi
J
of Eq.
vectors as follows:
2.11
that Eq.
=v.6.
J J
J.
lines
and third
The first
vg
(2.13)
37
In other words
[
[
where
and
=
gi]
(2.14)
1] ~
[g
] represents a matrix.
2.1,
2.2,
b v =
g
g
b
= V i. bJ-g..b bj
= v-g. bk'b
=
ii*1
=
1-v
(2.16)
= g'jb.b.
bb
g
=
(2.15)
can show that
Similarly we
1
where the
unit tensor 1 is
or metric
tensor.
defined as
The partial derivatives of
coordinates
spatial
a
derived
the fundamental
vector
next.
space with
is
{s
and contravariant
s
m n
obtained.
bases are
to
When a vector is
respect
The derivatives
tensor
respect
with
to
another vector expressed in terms of the
basis vectors
3- =
n
are
in Euclidean
differentiated
variable,
2.9,
then have
2.13 we
vv
From Eqs.
of the
a
scalar
original
covariant
therefore
s
(2.17)
am
n s
38
the
{
the
where
of
second kind given as
=
-s
mn
vectors
sc
= 0.5g s(mP +
n
_.
n
a
in
Notice that
Cartesian
m
p
{
are constant, and hence
s}
m n
)
(2.18)
system
coordinate
=
derivative of a covariant vector component
as
symbols
the Christoffel
are defined as
S}
m n
0.
the
basis
The cov.ariant
can be derived
v
follows:
J
s
=
g-
.
13(l
k
3
5vi
a3j
(2.19)
k
J)
k3
i
i
= - -v
Therefore,
v
=
avi
-- j
Similarly, the
vector
vk
k
.
.
covariant
component
v.
(2.20)
derivatives
can
be
of
derived,
a
contravariant
with
the
final
expression given as
v
. =
,J
(2. 21)
+ vk
J
k
39
Also
g,
bi
letting
i
to
Ej
and
we obtain
,
-\
We can also
m
m
- ni
ns
-*--g
= -g
3s
3
=
_(2.22)
show that
m
S
,---
=
3En
(2.23)
.1(.3
Second-Order Tensors
2.1.2
A
second-order
function,
tensor
is defined
as
a
which given one vector, assigns
linear vector function F has
the property
linear
vector
another vector
(a
F(au+bv) =
that
for arbitrary vectors u and v and scalars
aF(u)+bF(v)
b).
2.6 with respect
by differentiating Eq.
a and
In other words
u = T-v
(2.24)
the vectors u and v can be expanded in
Since
covariant,
their
components,
in four
u.
ui and vi ,
1
and
v.,
J
or
the components of
their
terms
of either
contravariant
T can be
expressed
forms
u.= T
vi
ui
ij V
u
= Tv.
j
u
= T.jv.
u
= T !vi
-J
(2.25)
40
that T'J
Notice
is not necessarily equal
to Ti:.
-.
J
1.
second-order
Any
dyadic.
A
is
the
dyad
dyadic is
be
higher
process
identical
Trsb b
of
- T. .b b
13--
similar
In mathematical notation
raising
and
to that shown for vectors.
-r-s
where a
order tensors by using
(2.26)
bs = T'sbrb
brbs = Tr.b
T = Trsb b =T rsr.--s
. sr-r-s
The
A
a
by
represented
two vectors u and v.
for
possible
and polyads.
polyadics
can
a linear combination of dyads
open product of
is-
expansion
T
tensor
lowering
From Eq.
= 0=
indices
is
2.26 we have
(2.27)
which implies
Trsb b - T..b irbjsb b
-r-s
ij
r-s
(2.28)
0
=
=
Therefore,
b
bjs)b r-s
(Trs
rb
13
(T - T..b
(2.29)
0
=0
Taking the dot product
with bt
Tr
and then with bs,
= birbjsT..
iJ
we
of both
sides
of Eq.
2.29 first
obtain
(2.30a)
41
similarly
and
T rs= b.i b.i T i
rs
irjs
(2. 30b)
Tr. = birb. T-3
.S
3js i.
T s = b. bisT'
r.
ir
The
2.1,
Eqs.
to
b
of
components T..
2.24,
2.7,
both sides
of
the tensor
and 2.25.
of Eq.
2.24,
T are derived
By applying the
by using
dot product
we obtain
-i~
b..u = u.
i
-i-
(T.b vk)
= b
-i=-k
= (b..T-b )vk
-i
Comparing Eq.
T
and
ik
(2.31)
=k
2.31 with 2.25 we conclude that
=b.-T-b
-i =-k
(2.32a)
similarly
Tik = b
kT-bk
Tk = b.T.bk
1-~~
T * = b 'T-b
=-k
.k
(2. 32b)
42
The
transpose S of a tensor
T is defined such
that
for
v
any vector
(2.33)
T-v = v-S
In other words,
T
T
(2.34)
St
T
In the particular case that
(2.35)
T-v = v-T
T
is a symmetric tensor, and T is
skew symmetric when
T-v = -v-T
(or operational)
The tensor
tensors
(2.36)
product
of
two
second
order
is defined by
=
T
ij
mn
San
4S n
= T iSmng
.k
J=
T S
(2.37)
.
n
=T.. Sjn i n
43
Notice
the tensor product of any second
that
"unit"
of second
scalar products
Two
the
referred to as
1 is often
tensor,
original
tensor with 1 produces the
or higher order
and consequently
tensor.
order
tensors are
defined
by
gk 1
j g.):(S
S-J
k-
T:S = (T
= T JS
(g.-gk
kk
kl -1
*g1(
-
-j
(2. 38a)
= T'jS
= T'jS
and
d1
6k
i
kl
=
ii
j
T
S
13
= T':S.j = T-.dSl:
1- -j
-J 1.
similarly
- -
Finally,
= T'S'
S.. = T
iJ
Ji
T--S = T
T rsg g ,
-r -s
Since T =
3T T
gs
--
rs
+ T
ks ak
-
1.
g
+ T
r
-1
tensors
order
second
of
derivatives
obtained.
= T'.Sj'
-J.J
(2.
38b)
can
be
we have
+T
rk
~
_k
+ Tr
(2.39)
ks
rs
~~~s+
+
prs =
p
T
Tk
r
rk
+ T p
s
r
k
rx s
44
Therefore,
Trs
s
Tks k r}Trk
Trs
Trs
a
3T
-ks
r.
.
3T r.S+
.s'
p9
r
Trk s
P1
rT.s
- k p
Tr.
.k s
p
Tk.
(2..40)
such that
(2.41)
Trs
rs =s'
rs'
= Trs'
-
2.2 KINEMATICS
Describing a continuum's motion and its
a reference frame
to
respect
Measures of motion include
medium
terms
the
and
convenient
is referred to as kinematics.
the velocity of particles
in
the
the continuum, expressed in
deformation of
to
follow particles
respect
is
approach
called
formulation in contrast
consider
the
motion
control volume.
fluid
flow
it
continua
solid
related to
their motion with
for
with
of strain.
For problems
This
location
is
often
in the continuum and describe
a
to
the
reference
Lagrangian
configuration.
(or
material)
to the Eulerian formulation where we
of
particles
through
The Eulerian approach is
problems.
If
more
a
stationary
appropriate
the reference configuration
45
used, while
used.
Lagrangian formulation is being
tensor components
the
denoted
by
from one frame
upper
configuration,
reference coordinates
i
-I
= i
.-i
coordinate
will
=
= i
-
A
iP.
yet,
case
letters.
or
system,
will
be
by
X
inertial
defined
.
convected
The
deforming
system
in
the
while
the
vectors
are
x.
by
base
(intrinsic)
in the current
-I
particle's location
given
in the
by
configuration.
deformed
Tensors
and
Therefore
undeformed
the position vectors
from the origin of the Cartesian system X
2.1.
a
reference
in the
2.2.1 Displacement, Deformation Gradients and
is
such
The curvilinear coordinates for
configuration and G
configurations
continuum
common with the
i with base vectors g
in Fig.
system,
body-fixed
system are given by
defined
rectangular
a
For
be
current
the
in
and
will
terms
all
-
also be used.
A
another.
of reference to
letters
coordinates,
current
and
case
coordinate
Cartesian
since
configuration
reference
lower
now become evident,
system dependent
are coordinate
easily transferred
In
tensors
with
medium
deformable
a
of
kinematics
the
describing
of
advantages
The
an updated
then
configuration,
considered
recently
most
the
of
that
the reference configuration is
if
is being
formulation
Lagrangian
total
a
configuration),
undeformed
(initial,
t=0
time
at
that
to
corresponds
,
r and R,
as
shown
46
Reference Configuration
1l
x i
r
Deformed
Configuration
R
Xi*
-1 t-1
3'13
Fig. 2.1
Notation for Reference and Deformed Curvilinear
Coordinate Systems Moving in a Cartesian Coordinate
System
47
R=X 1 i~
r = x1i
(2.42)
Also
.
dr = dx.i. = d
dR = dXi
As
ar
Since
=--4i.
dg.G
3R
G
1g -J
-
-L
3X.
3
(2.44)
i-
are vectors, all equations
and G
g
=
ax.
1
1
(2.43)
the basis vectors can be expressed as
a result,
g=
.
I
i
dE G
=
=dE
from section
2.1
are applicable.
The displacement vector u
is
defined next as
(2.45)
u = R - r
= U G=
UG
u g=
u = u
The velocity vector v is
V
-
[--at
-
we have
vectors
of basis
In terms
=
(R-r)
the time derivative
=
(2.46)
of u such that
(2.47)
R
.
and
V =V. -
P&1
=
i
V
i
.VIG
I
I
V GI
-
(2.48)
48
2.19,
Using Eqs.
the
2.47,
deformed basis vector is
0J
3
39 RI
G
-I -
derived as
of
follows:-
V ,G
'I--J
1
B(2.49)
35
=
the time derivative
3
-1
-Iig
and 2.48
V,
J'III
by differentiating
Furthermore,
.I
with
6
the contravariant
(2.50)
G
,-V5
The deformation gradient
dr
vector
each
=
IJ
J
G
=-V
of
II
-G
is given as
basis vectors
G
the time derivative
time,
to
respect
the equality G
r
at
a
F is
a tensor associating with
vector dR at R.
In mathematical
symbols
dR = F-dr
(2.51)
drF T
By definition the rectangular
given
Eq.
by
--.
3x.
2.51 as
The same result
can be derived by
F
are
expanding
follows:
dR = F-dr = (F
= F
Cartesian components of
dx i
i
i )(dxk4k)
(2.52)
49
then have
2.42 we
Using Eq.
dX i
=
I-i
(2.
F dx.i.
-IJ
3-1~
53)
implying
aX
F
Ij
=a
(2.54)
3x
Furthermore, by using the
Cartesian coordinate
F
+
equivalent
of
Eq.
2.45
in
a
system, we can show that
(2.55)
3x
Also
T
(2.
56)
implying
The
(2.57)
i
F =
components of
expanding Eq.
GJ
=
From Eqs.
GI -&j
F in a convected
are
derived
by
2.56
(F
.
system
-=
(2.58)
2.44 and 2.45 we have
=
(R - r)
=
(2.
59)
50
G
:-I
follows:
i =u i
U.*
j = it j
-
2.59 is transformed as
Eq.
2.34,
Using Eq.
(2.60)
,jli
Therefore,
+ u
Gq =&j+u oj=(g
)
(2.61)
j
2.58
Comparing Eqs.
F
i
i
iu
i. +
1 = (I3 +
3 = ,J
=g
and
,J
i
2.61 we conclude
+u
(2.62)
F '. = 6. + u
and
.
similarly
Fi
=g i
+u
i,
(2.
F
i.
+u.
jj
1,
1
=
We next examine
-1
63)
the spatial
deformation gradient
tensor
with the properties
-liT
-1
dr
dR-(f
-dR
F
)
(2. 64)
F
1
-
=
(E- 1)T
51
procedure as before
By a similar
it can be
shown that
-x.
(T?
2
iJ
ax
(2.
ij
65)
ax.
3
of convected coordinates
In terms
G
(F~1)
(F
=
iJ
gij
(2.66)
=
6
(F ):
=
gikG
(F
=
G
(F~l)i*
.J
j
KJ
-1 .J
and
)
U
- UI
= G
(F~1 )
J
(2.67)
=6
(F~ )0
I.
The
-
i
U
I,
right Cauchy-Green deformation tensor C is
defined
as
C=
= U-U = FT-F
(2.
68)
52
left Cauchy-Green deformation tensor B as
and the
by
defined
are
V
and
U
where
(2.
= V.V = F.FT
B = V
that
theorem to F such
decomposition
polar
the
applying
(2.70)
F = R-U = V-R
R
U
stretching
and
application
of
stretch
(similarly defined for V).
dR
F,
rotation
From Eqs.
2.68,
(2.71)
B = R-C-RT
Using
rigid
that
is evident
2.70 it
tensors.
F to dr can be viewed as a
of dr by U and then R applying a
obtain
2.69,
rotation tensor and
left
and
right
the
as
the
Therefore,
the
symmetric, positive definite tensors defined,
are
V
respectively,
to
tensor defined as
is an orthogonal
and
69)
the previously derived results
for the
components
of
we have
A
C
T
(FiK) F
=
(
k
+
= FKi FKj
uk
k
)(k
ax.
31
i
X
X
A
A
A
=
x
j
uk
(2.72)
+ -)
3x.
3A
3x,
.
4u
3x.
3x
53
In
convected coordinates we
C
(g
Fk. = G
TFk. = F
(F)
=
j + uk,i
)
can also show
k+u k
(2.73)
j+
k
=g . +13u.i,j + uj,i. + u ,j .u k,i
and
C
= g
+ uk 9j uk9
+ uj
+ ui
(2.74)
u,i +k,k9 uk 9k
91
Cjg+S
Strain Tensors
2.2.2
tensors
definitions
the
Using
a
For large deformation and
unique
measure of
therefore strain measures
are
numerically
various
should
In
the
easy
strain
to implement
rotation
longer exists,
should be selected such
and
that
and
they
and compatible with the
These
measures
strain
display no variation under rigid body motions.
special
definitions
strain no
relations.
constitutive
also
gradients
in section 2.2.1, various measures of
given
will now be established.
problems
deformation
of
case
should
of
reduce
common strain measures
are
strain
small
to
the
defined
problems,
same form.
in
terms
all
Many of the
of
squared
54
which
distance
is generally measured in
terms
first define the Green-Lagrange
We
=
From our
0.5(C
squared values.
as
strain tensor y
then have
76)
- 1) = 0.5(FT F - 1)
by noting
-
Euclidian
(2.7 5)
previous definitions we
(dS)2
of
-
Y = 0.5(U
Also
appealing since
intrinsically
is
length,
that
(ds)2 = dR-dR
dr-dr
-
(2. 77)
= dr.G-dr - dr-l-dr
= dr(G - 1)-dr
and comparing this
(dS)2
dr-Y-dr =
-=
as
-
2.75,
we deduce that
(ds)2
(2.78)
2
-
The components
result with Eq.
of the Green-Lagrange strain
tensor are given
follows:
k _-6..)
Y.. = 0.5(C j - 6..) = 0.5(
3x.i 3x.lJ~j 1J
1J13
13
3a.
3u
+
J
3 k Da k
10.5(7
la1x
1
k
(2.79)
55
-i-gij) =0.5(u. .+u. .+u
J,1
,
ijG
)=0.5(G
j = 0.5(C i-gj
ijIJ
ij. =0 5
)=0.5(u
.+u.
*+,
j.)0,J~
= 0.5(C. .- )=0.5(gikGG3jKJ _jiji
S= 0.5(C
-g
In contrast
in
reference
g
GLK-g
the
coordinates,
+uk .u k,
+u
)
(2.79)
)
)
)=0.5(u
the Green-Lagrange
derivatives
of
terms
)=0.5(g
to
i
ii
iljk
V
.u
,k
i+u
+u k juk,
given
tensor
strain
of displacements with respect
respect
with
displacements
is
tensor
Almansi
derivatives
of
coordinates.
Therefore, the Almansi
e is
tensor
defined
to
to
by
current
given as
e= 0.5(1 - B~)
(2.80)
= 0.5(j - ( 2 -1)
= 0.5(1 - (E~1)T-
it
Using a similar process as before,
e = (dS)
dR..dR =2
2
- (ds)
can be
shown that
2
(2.81)
and
3x
e9
=
= 0
e
= 0.5(G1I
e
0.5(6-
e
=
0.5(G
3xk
)
- gj
)
5
-xJ 0.5(U
=
GIKgk)
=
- GILG
gk)
0.5(u
=
iU
-
3Uj
-
3k
---3Y1 3x
+3
+ U
+ U
0.5(U
- U
-
+ U
JUK,I
(2.82)
UK $fK,
- U
UK
56
to
but we
length also exist,
logarithmic
of
terms
in
Other strain measures
linear
shall
the most commonly used Green-Lagrange
length
or
limit ourselves
strain
and Almansi
tensors.
The
equations.
constitutive
rate-type
strain
of
derivatives
time
The
are
for
required
rate of deformation
tensor D is defined as
D
where
and
An
=
U
=t (t)
subscript
covariant component
-
2DidE d
where D
y.
=
IJ
iJ
ds2)
-
I
deformation
time t.
tensors
in
given as follows:
=
(2.84)
(dS2)
Therefore from Eq.
-
e1
the
of
form is
(dS
=
the configuration at
t refers to
definition
alternate
previously
stretch tensors defined
the
V are
the U and
the
(2.83)
Vt (t)
=
2.79
and
Eq.
2.49
D
Ii
ij
=0
=
=
0.5(G
...
ii
g..)
-
ij
=
0.5G
Ii
.5(G :G)
-I -J
0.5(G 'G + 6 -G )
_IJ
- -J
-I
-V
L'G
G)
G(G+ V
LI- -J
K,J-
= 0.5(V9 + V
)
(2. 85a)
57
show that
can
Similarly we
)
+ V
= 0.5(V
D
9
9
= 0.5(V
D
D
+ V
,J
3V I
= 05(---+
IJ
X
3
(2.85b)
VI
-_3X
look at the material rate of the Almansi
We now
tensor in
)
J,
its covariant
terms of
GI
= (e
G G
contains
J
does not vanish
-I
J(2.86)
(
G
for
rigid body motions
since
.
G
rigid body components.
Green-Lagrange
the
can
strain,
be
such that
.
j
~=(iY).y
and
+ e
G
+ e
However, for
expanded
components
)
I
Therefore
strain
problems since it
j
(2.87)
a more appropriate measure for
is
Y
therefore
i
ig-&
vanishes
for rigid motions.
rate-type
We now derive
the relation between D and
i=
ig = ikjg
ik
= Dg
ik
L ik
= GKLD g
+2y,)DL 9ik
(gkl + 2y
(6
)D g
+ 2y i)D
(2.88)
58
Similar expressions can be found
for
the
other
component
types.
2.3
STRESS TENSORS
traction vector T is
The
defined as
the
force
surface
area
per unit
dP
T = d(2.89)
the
is
dP
where
force
across
exerted
the
differential
surface with area dA and outward normal N.
Beginning with this definition
T
=
can be
defined.
vector,
traction
The Cauchy
stress
is defined by
a
tensor
tensors
stress
various
of the
N'
(2.90)
Consequently,
a
is
defined
-IJ
IJ i
in the deformed space,
and
IJ
(2.91)
I.
G
I.
. J;I-
Using Eqs.
J
=I'
JI
GG
I--Ji
2.89-2.91, we then have
dP = N-adA = N G -a JKG G dA
(2.92)
IK
=Nya
GdA
59
the
in
dV
the
is
mass
mass
dm, the
deformed
the
area dA 0
surface
Using Eq.
.
present
the
density
of
to the
2.93 with 2.92,
and
with
n.,
the
differential
the
following
derived
IK
0
IK
GdA 0
G dA = p i a -K
dP = N a -K
The Kirchhoff
of
the
in
body
the
configuration;
components of the unit normal
covariant
in
density
mass
= dm/dV0
configuration;
reference
is
is
p0
configuration;
volume
(2.93)
= pN dA
p = dm/dV
where
undeformed
the
of
relation
configuration by using Nanson's
p n.dA
terms
in
can be expressed
dP
Similarly
stress
stress
the Cauchy
p0
tensor
(2.94)
2.4
r
is next defined
in
terms
tensor as
(2.95)
0
Therefore,
p
-
T=
P
N -T
0
(2.96)
and
I G dA 0K
dP = nTIK
-
(2o7
(2. 97)
60
A
t
vector
traction
respect
with
configuration dA0 can be defined such
to
undeformed
the
that
dP
0
=n
IK
i
T
(2.
98)
S
is
G
-K
Finally the 2nd
Piola-Kirchhoff
stress
tensor
defined by
(2.99)
t = n.(S-FT)
It
2nd
can be
shown that
Piola-Kirchhoff
components
sij=
the contravariant
stress
of the Kirchhoff
are equal
components
to the
S j of the
contravariant
stress
J
(2.100)
Therefore,
lj
GKJ
T
(2.101)
or
-
=
.
Sg1G
'19
= s1 (6
+
+ 2 Y)
(2.102)
61
the following relations
In general,
a
T =
S
S
F
The
1
(F-1)T = U-1 RT.-R-U-
-r-
- -
use of
T
- - - -
(2.104)
of
relations
must
dependent
of
rate
T
(2.106)
tensors allows an efficient
of motion.
governing equations
the
compatible
more
are
(2nd Piola-Kirchhoff
(Cauchy) give a
the
with
stress
tensor)
physical
"true"
stress.
In addition to
tensors
(2.105)
stress
different
other stress tensors
value of the
1
0
tensors
stress
constitutive
the
P
solution
numerical
rate
(2.103)
0 F-l. a.(F-1)T =--U-1.RT.a.R.U-1
=
=
F-S-FT = R.U.S.U.RT
=
P
P
0
while
the various
tensors can be derived:
stress
Some
between
be
the stress
defined
tensors, the
constitutive relations
the
traction vector
of
the
traction
that
such
stress
rates of
incremental
or
rate
can be employed.
The total
the rigid body
rate gives
less
vector
corresponding
to
pure
deformation
T-(-T-W)
where W is
W=
(2.107)
the spin tensor
(t)
(2.108)
62
Therefore,
= (N.a) + (N.a).W
T - (-T.W)
= N-a + N-a + N-(cT-W)
(2.109)
where
a is
=
N-(a + o-W - W-a)
=
Nc
rate of
the Jaumann (co-rotational)
the
Cauchy
stress
a
rigid
the
a-W - W-2
(2.110)
rigid body rotations
implying that
0,
=
_ +
that for
Notice
a
=
=
-T-W,
and therefore
the Jaumann rate properly accounts for
Furthermore,
body rotations.
can be
expanded
and
following obtained:
=
a G G J + D a + aD + W- a - C-W
(2.111)
and therefore
*
Notice
a
IJ
=r
that
--
G
D'+ D-*
a*25
a =
0 for rigid
rigid body motion.
a=
c .=yG G + D--
T
(2.112)
body rotations
since D
= 0
for
Similarly
rD
=--I--
(2.113)
63
stress
Kirchhoff
the
the Jaumann rate of
find
We can also
tensor
0
Po
=
p
0
P
p
=
a +
1)
=
-
-
(2.114)
p + 0
p 1. '
which can be reduced to
(2.115)
+ (tr D)T
(trD)a
-~ =+
2.4 ENERGY EQUATIONS
are
developed in this
body
Beginning with a
forces
of dynamic
governing equations
The
equilibrium
prescribed
displacement boundary
principle
of
virtual
virtual work, 6W
SW..
e
work
approach.
subjected
conditions, we
which states
, must equal
the
employ
conditions, are
SW.
*e
*i = SW
6u,
satisfying
internal
imposed on
the
virtual
essential
the body and we
the
the
that the external
Therefore, a set of compatible, infinitesimal
displacements
body
to
tractions and satisfying
surface
external
and
section using a variational
in
systems
structural
work,
virtual
boundary
obtain
(2.116)
64
The
forms
stress and
strain.
and
be
specified definitions
(2.117)
dt dVf
a
D
dt dV
D
dt dV =
a
D
dt dV
can
also
I
(2.
=
Vt
a
Equation 2.117
of
form
in component
t
in
expressed
the Cauchy stress we have
of
In terms
can
dt dV
j:
W=
W
on the
depending
different
w=
work
virtual
internal
vft
be
expressed
in
the
118)
reference
configuration
W.
cy:D -0dV dt
0
= =p
1fj
Using Eq.
expressed
,
=t
2.95,
in terms
internal
of the Kirchhoff
i
stress
work
tensor
can represent
fft-V9
dV dt
- -
can
T
be
as
(2.120)
internal
the
the 2nd Piola-Kirchhoff
terms of
virtual
:D dV dt
fV
Similarly, we
the
(2.119)
virtual
work
in
stress
0.5S:6dV dt
(2.121)
65
Notice
strain
power
2.119,
2.120,
unit
per
current volume V while
T
respect
to
the
and S and
C
are
respect
to
the
with
variables
conjugate
energetically
,
S and
and D,
are
with
variables
conjugate
from Eqs.
P
a
that
evident
is
internal
denote
therefore
and
volume,
and 2.121 it
energetically
":
products
scalar
the
that all
and
reference volume V .
The
of
components
W
of
terms
in
energetically conjugate variables can now be
W=
0
t
for
I Di'dV dt =
the conjugate variables
t
i dV dt
i
Vf
-Jfv
dd
0
f Vf
=ff~
for
the
D
siJ dVoat
0od0S
V
Y
and
;
W =t
f
V0
=
S
T
t V a 1 edV dt =
t
J
~O0
dV dt
=
dV
d
fvfYa
,we
.dVo
d YdV
S
-j
0
0ij~
dY
x
and
S
also have
ijdV dt =
V
f.J
and D and
variables
conjugate
=
t
JdV dt
TID
JV
d
= f'.S
0(2.123)
dV dt
SJ
ft
(2.122)
=S
0
=t
given as
0
0
other
D dV dt
IJDIJdVdt =jT
t
the
dy. dV
fT
JVJ.T
0
a
Y1j
de
dV(
dy.dV
i
0.5
C.
Since
66
=
A
=
4f
0
t
6W.
1
for
V
1
UJdVodt
VoS
conjugate variables
6Vi =
IJ
= JfejS' de1gdV0
Sy
in Eq.
of
i0
.
For example,
if the
dV0
terms
2.125 all
strain,
are defined with respect
(2nd Piola-Kirchhoff
),
reference volume V
2.125 applies to the total Lagrangian
6
P B- du dV
W e=
=
t-
+
o
p B 6u.dV
+
vector.
traction vector is
~
given as
u dA0
6u.dA 0
B is the body force vector
forces, using
formulation.
(2.126)
t
0
A
and t the external
the
corresponds to
applied.
stress,
<therefore
0
o
traction
and
virtual work is
The corresponding external
where
W
are used, we have
reference configuration
Green-Lagrange
Eq.
f
above
dV0
o
0(2.125)
that
the
S and
presented
the equations
Q:6:
=S'
to
dV0
(2.124)
expressions
various
the
de
T
Si D
1 dV0 dt
can now be derived using
Notice
A
t fV -T e 1 dV0dt =JV
=
Since
B
surface
surface where the
contains
all
body
D'Alembert's principle we have
+1
(2.127)
or
B
= -u
+ f
(2.128)
67
is
where u
contains all
other body forces
Therefore,
forces.
magnetic
and using
such as
the conjugate variables S and
V f:61 dV
0
p (f -
V
*).Su dV0 +
Taking the variation of Eqs.
=
the gravitational
in the undeformed
0.5(f '6F + 6E
,
f
and
term)
(inertia
vector
the acceleration
and
configuration
we have
(2.129)
t.Su dA
2.76 and 2.79 we obtain
(2.130)
F)
and
kj(.
.3[ 61k+ +uuk 1)sukJ
)uk+
Sy.. = 0-5
that
Notice
since
energy
the
derive
the
equilibrium equations
variational
particular using Gauss's
, + p
[Sjk( 6 i + uk
for
the components of the
t
for
)u k9 i]
principle
(2.131)
was
(prescribed
the boundary conditions.
In
2.129
theorem it can be
shown 'that
2.132)
= p0
equilibrium equation and
(2.133)
of the nonessential
surface
configuration.
to
the
Eq.
)
= n.Sjk i + u
,k
3k
the components
used
contains
equation,
and also
+uk
+ uk
tractions)
boundary
referred
to the
conditions
reference
68
2.5 DISPLACEMENT-BASED FINITE
GOVERNING EQUATIONS
The
the
using
computer.
the digital
finite
method consists of
by an assemblage
has
idealizing the
Each
of finite elements.
the use
behavior by
the
other words,
at
basis
The
finite
OF
THE
solved
in
(FEM)
of
the
continuum
element
a finite num ber of nodes, with each node describing the
element
are
are
method
element
finite
conjunction with
element
FORMULATION
motion
of
equations
governing
effectively
ELEMENT
coefficients of
the generaliz ed nodal
point
any
the
in
the geometric
substituting
of interpolation
the
responses,
finite
and hence the
coordinates of
for t he
functions.
In eq uation form we have
is
response
obtained
the point
response given in terms of
=2t)b
In
interpolation functions
element
expression
functions.
by
into the
interpolation
)J {q.(t)J
(2.134)
= Loi( J)
where the
and
of
the q
4(Ei)
's are
element m.
column
(L
vector).
any point
{qm(t)}I
are displacement
interpolation
functions
the element nodal generalized displacements
J
(d
symbolizes a
represents
in the element.
row
vector
and
{ },
a
the convected coordinates of
69
Using
2.134 we have
Eq.
6u(
t) =
L
j {6mt)}
(2. 135)
and
1 (E,t)
(Ed)
=)$
(2.136)
(t)
{
We can now express various kinematic
0.5(u.
2.79 is
in Eq.
Green-Lagrange strain
y.=
For
functions.
interpolation
the
quantities
+ u
+ u
in terms
of
the
example,
given as
u
(2.137)
or
y
(
,t) = 0.5 p
(
k
+4 j(Qk)j {q(t)}
jj {qm(t)}
+ 0.5Lqm(t)]{$i' }
{qm} + 0.5Lq
-D
where we used
{D
} D1
the differential gradient
LD j
= 0.5
LDliJ
=
$9
,J
(2.138)
{q
operators
+
(2.
139)
70
have
We also
6y .
=
LD .J{qm} + LSqmJ{Dl }D
continuum is
The entire
summation of
n
(6
=
(6W.
m
m=1
by
to the
the finite elements
n
Mi=1
2.129
by Eq.
em
(6W, )
2.125 and
Eq.
(2.141)
)
)m is the internal virtual work of element m given
(6W
where
finite
set of
described by the
the principle of virtual work applies
and
elements,
(2.140)
{q.}
the external
the total Lagrangian
(for
virtual work given
formulation).
2.134 to 2.140 into
Substituting Eqs.
obtain the
is
Eq.
2.141
we
finite element formulation of the energy equation
form) given as follows:
(total Lagrangian
n
T
Lsqm
f(V ) {D
(V)o
+
(v
where subscript
i
corresponds
dV0 + J(v){Dli} LDjJS
}S
'io
(A )
* i}t
dV{m}
dAo
(2.142)
p 0 {p }L+JdV 0 {{} ) = 0
o m
s indicates
to
that
the surface.
the interpolation
function
71
2.6
CONSTITUTIVE RELATIONS
strain tensors,
and
stress
in other words the
be as
The presentation will
relations.
since the development of
possible
and
highly problem dependent
relation
the
last section examines
This
between
constitutive
general as
brief and
constitutive relations
poses
the
is
areas
of
problem
of
strain tensor.
We
significant
research currently in progress.
of
The theory
evaluating
portion
plastic
the
of the
y . exists
that a plastic strain tensor
assume
e +
= y
y
yj. is
where
such that
p
(2.143)
+ Y
the
elastic
S
tensor
stress
the
considers
plasticity
strain tensor
using
obtained
conventional
from
the
elastic
linear
stress-strain relations
= E
Si
Notice
that
relating
(2.144)
Y
Eijkl is
the
Piola-Kirchhoff
Green-Lagrange
stress
strain
tensor.
modulus tensor are available for the
measures.
tensor
the fourth order elastic modulus
with
Corresponding
other
the
2nd
elastic
stress and strain
72
one-to-one
a
exists
there
that
assumes
plasticity
strain tensors.
relation between the stress and
(2.145)
= f(Si)
a
applies
only
=
ij
plasticity,
considers
incremental
,
f(S.
plastic flow as a
load
previous
provides the
given
stress
the magnitude
a
stress
space
surface,
outside,
history.
direction of
history.
of the
surface
define
the
in
(2.146)
)
ingredients:
basic
two
The flow rule
flow rule and the strain hardening rule.
describes the
and
1
13
Plasticity theory consists of
the
relation
and strain rates
stress
...
.
dS
strains,
and
stresses
incremental
a
assumes
and
dy .
and
the
theory
(flow)
strains,
total
considers
particular,
dy.
of
dye
,
the
between
proportional
using
than
strains dy
simple loading cases such as
incremental
The
loading.
rather
to
theory
this
assumed,
is
relation
one-to-one
Because
for
available
are
The deformation theory
the plastic strain tensor.
deriving
of
theories
different plasticity
Two
where
Stated
the
for
strain
rate
flow.
In
stress
plastic flow occurs.
general,
combinations
continuum remains elastic and
This surface
by
a
rule supplies
three-dimensional
the
flow rule
produced
The strain hardening
plastic
inside
the
differently,
and
stress
function of the
we
can
principal
within
the
for combinations
is
called
the
73
the yield
evolution of
the
We
rule
shall not examine
stress and
on
examine
available,
which
rules
particular material types.
relations may, in general,
relation between
the
numerous
It suffices to mention, that
the
are
is
that
be of the
under
considerable
the
constitutive
form
(2.147)
f(P,1)
In
other words,
stress
2.147 with Eq.
to
applies
the constitutive law may relate
and
rate
the
2.144,
small
rate
it
of deformation.
is evident
strain problems
that
stress
tensor
is more appropriate
a and D,
but
the
derive
constitutive
to
resulting
expressions being less
Kirchhoff
stresses
T.
Jaumann
the
Comparing Eq.
only
2.144
is constant.
the
relation between
in most cases Eq.
since thermodynamic
employed
Eq.
since Eijkl
Furthermore, we may use a constitutive
Cauchy
strain
to
note
Finally,
investigation.
the
applicable
relations
constitutive
of
flow
the
deriving
the details of
strain nor shall we
hardening
field
plastic flow occurs.
surface as
assumptions
various
given
the hardening rule describes
Consequently,
surface.
yield
principles
2.147
are often
relations, with the
complicated in terms of
the
74
CHAPTER 3
GOVERNING EQUATIONS OF MOTION AND THEIR NUMERICAL
equations with initial conditions.
value
initial
in closed
unavailable
are
problems
used
commonly
systems is
usually
solutions
and
temporal
by
for
are
Finite
elements
discretization while
spatial
the
finite difference methods
spatial
system.
discretization of the continuous
are
these
of
form, and hence approximate
numerically
derived
solution
The
continuous
for
differential
ordinary
order
second
by
mathematically
represented
are
problems
-dynamic
Structural
SOLUTION
employed
the
for
temporal
discretization.
This
chapter develops
the governing equations of motion
by expanding
the energy equations
deriving the
exact governing
rearranged
to
formulations:
tangent
all
be
pure
stiffness.
assumptions
emphasize
of
section
equations,
equations are
the
compatible with three different
unconventional;
the limitations
throughout
solution
pseudo-force;
A rigorous development
are stated
After
2.4.
the
is
pursued
and
and
presentation to
and applicability of
each form of
75
the
governing equations.
Following the derivation of
as
adapted to nonlinear problems are
techniques
various
approaches
reliable
solutions.
the
governing
examined.
The
of
the
obtaining
for
techniques
discusses
and
implicit
properties
accuracy
the
emphasizes
presentation
such
mode superposition
as
and alternate methods such
equations
to
directly
applied
integrators
numerical
of
or
explicit
using
integration
time
direct
solution techniques
popular numerical
current
motion,
the governing equations
3.1 EQUATIONS OF MOTION
Problems
the
or
state
steady
time
with
varies
problems can be
structural
through
the structure.
of
the
modes
driving and
exhibit
a
in
the
propagation
Consequently, the
short duration and is
all
characterized
on
elongation and bending of
response
Transient response
problems the
a
more
of
stress
response
by
Structural
global
structural
persists
include pile
type
scale
members
waves
excitation
an
Examples
structure.
blast type problems.
response
the
In wave propagation
type problems.
governed by
considers
into wave propagation and
further subdivided
is
study
problems where
not periodic.
and is
response
for a
This
transient response.
of transient response
class
categorized into
in structural dynamics can be
problems
such as the
and
usually
76
only the lowest
consequently
the
structural modes contribute
to
response.
compared
increment
response
approaches
and
freedom
of
degrees
of
number
Each
section.
different
the
among
Preference
viewed
is
structural
for
well
equally
used
in this
perspectives
different
equation
governing
exact
The
per
and
excitations,
seismic
and
wave,
by wind,
represented
loadings
limited frequency content
term,
long
from
arise
from three
form can be
problems.
depends
allowable
the
on the
time
number of numerical calculations
to the
time step.
3.1.1
Exact
Formulation
Beginning with
Eq.
by
given
equation
the finite element
2.142
and
form
using
of
the
the
energy
following
substitutions
f(om
{D
h= fV)
{D
I
S=
we
J
(3.1)
dV9
mj
M
} S
}L D -J
f idV
{
m
(3.2)
dV9
(3.3)
SidV
{
+
,}st
dA
(3.4)
obtain
n
L6
m=
m
+
~
-- f)
= 0
(3.5)
77
is
generalized
displacements
displacements
the
to
by
{q*}
transferring the
local
using
the
displ-acements
global
nodal
global
of
terms
in
then rewritten
Equation 3.5
relation:
following
(3.6)
where J
local
the transformation matrix from the
is
global coordinate
the
to
Therefore,
reference frame.
n
(3.7)
L
m=l
+
* + h*j*
-
f*)
=
0
where it can be shown that
(3.8)
M* = JT J
(3.
T
h
=
JT hJ
(3.10)
Tg
(3.11)
=
Next
we
group
together
contributions that are
-p
and
9)
the
linear
functions of the
and
nonlinear
response and let
(3.12)
+
similarly
* = JT
T
+ JTh
(3.13)
78
Therefore, we
obtain
n
L
(m
6
(*j*
0
(3.14)
f*) = 0.
(3.15)
f
+ i
and
n
L
+
i*
We now define
n
M =
m*
(3.16)
1*
(3.17)
f*
(3.18)
1
J=
n
1 =
1
3=
n
F =
3=1
Equation 3.15 can then be
[
Since
Eq.
q* J
L 61J
3.19
rewritten as
(3.19)
j* + I - F) = 0
contains independent
and
arbitrary
components,
becomes
* -I + F
Equation
governing
made
(3.20)
3.20
equations
represents
of
concerning the nature
motion.
of
the
No
exact
form
of
the
assumptions have been
loading or the
response and
in
79
(72)
and
a
represents
formulations.
Formulation
Unconventional
The pure unconventional
Eq.
refers to solving
implicit numerical
M
the mass matrix
(left hand
RHS
that
compatible with specific solution schemes.
3.1.2 Pure
or
three
following
The
the exact equations
sections present alternate forms of
are
are
strains
infinitesimal
Other formulations which assume
"conventional"
numerical form.
efficient
and
compact
problems,
strain
"small"
or
infinitesimal
as to
called
to finite strain
of the equation of motion applies
this form
as well
"unconventional" formulation;
the
3.20 as
refer to Eq.
Witmer
and
Rodal
particular, the constitutive relations.
side)
terms of
3.20.
I.
When
RHS of
Eq.
entire
equation.
economical.
mass matrix
form,
the
reciprocals
and q*
are known
3.20
A solution by an
quite
using
As
F are known.
and
I,
explicit
The
inverse of M is
its diagonal
easily
the
term on
are defined
of motion
in
the LHS
explicit integrators,
other words,
a result, all
is the
terms on the
in the
therefore,
integrator,
in
time
at a
only unknown
only matrix inversion
(undamped problem).
of
only unknown
The
step,
the previous time
when M,
Observe that
the only structural matrix on
the numerical equations
however,
explicit
either
with
integration operators.
of Eq.
vector
the
is
is
directly
3.20
equation
form of the governing
is
involves the
When given in a lumped mass
obtained by evaluating
terms.
Even
if
M
is
in
the
a
80
In
general,
high
to
ensure
stability
problems
of
explicit
also accuracy, as exemplified
to
duration is
a
"long"
If the
short.
therefore
procedures
and operators
other
K
is
the
Ms*
by adding Kq*
Kj*
=
of
solution
schemes.
-I
pseudo-force formulation
to both sides of
initial linear elastic
+
types
over
may become
burden
such as implicit
The governing equation for the
obtained
evaluated
Formulation
3.1.3 Pseudo-Force
is
where the load
problems
response must be
requiring
excessive,
time
propagation problems,
the computing
time,
period of
by wave
response
structural
small
small
the
stability but
only to ensure
necessary not
are
limit
thus
integrators
3.20 to problems where extremely
solution of Eq.
severe
The
stability.
required
a
time
small
extremely
the
increment
and
results, but at
accurate
due to
the pure
Extensions
systems.
small
reliable results for
computational cost
increments
is
produces
and
economical
is
systems also produce
large
to
of explicit operators with
the use
formulation
unconventional
accurate and
M
time).
of
independent
that
(assuming
analysis
the
of
beginning
the
only once at
executed
inversion is
the
form,
consistent mass
+ F + Kg*
Eq.
3.20, where
stiffness matrix
(3.21)
81
where
n
n
LIJ
*=
K =
and
in Eq.
is given
Eijkl
FNL = K*
can
3.21
Equation
2.144.
by grouping all
further
simplified
o-
kl
i3
0 m
m=1
m=1
22)
LD jdV 0 J
D.}Eijkl
(V )
response terms
be
together
-I
(3.23)
Therefore,
+ K
q*
=
Notice that
and
problems
is
assumes
that
is
therefore
referred
Eq.
2.144 is
restricted
conventional
This
the
form,
Another
form.
finite
to
applies
3.24
form,
than the unconventional
steps
may
and
3.24 no assumptions have been
Eq.
in
well
as
to as
as
infinitesimal
the modified
made
a result, Eq.
As
relations.
constitutive
the
concerning
(3.24)
F + FNL
strain
unconventional
conventional pseudo-force form,
valid
evaluating
when
to infinitesimal
I, and
strain problems.
in addition to being more
restricted
form, requires many more computation
a larger storage.
Furthermore,
often be numerically ill-conditioned,
the
calculations
requiring extended
precision.
By adding Kq*
convergence
convenient
to
problems
form
is
both sides
are
of
reduced
produced,
Eq.
3.20,
stability and
considerably and a more
appropriate for approximate
82
-schemes that predict FNL
extrapolative
solve for new q* until
that
integrators
implicit
only once since
factorized
and
hence this
is unknown when using
with
an
However,
by
thus
Eq.
the solution accuracy
repeating the analysis with
reducing
3.24
can
amount
since the
a force extrapolation
implicit operator;
increments,
time,
they remain independent of
is questionable
error
only
are
solution approach is fairly economical.
reliability, however,
verified
3.24.
LHS
the
on
matrices
and
Explicit
converges.
are equally compatible with Eq.
structural
the
that
Notice
FNL
or iterative schemes
also
be
solved
of
approach
can
be
smaller time
technique's
the
Its
efficiency.
iteratively
to
convergence within each time step by using various available
iterative
(72,78).
techniques
with
transient
implicit
operators
83
3.1.4 Incremental Response Formulation
The incremental
is
derived
by
formulation for
subtracting
governing equations
the
the equations
exact
at time t-At from those
form
at
of motion
for
time
the
t
to
obtain
_MAI*= -AI
+AF
(3.
25)
(3.
26)
where
Al* =1* -11tat
-t-
-t
=
AF
AI
is
first
AI
t
-
(3.27)
At
(3.28)
F
then approximated by a Taylor
series
expansion to
the
order
=
to yield the
equations
(3.29)
(31/33*)Ag*
tangent
stiffness
form
of
the
incremental
of motion given as
(3.30)
84
K
and
tangent
the
where
stiffness Kt
(3.31)
.I/3D*
=
AI by a finite difference expansion.
the derivative of
and
(v{
D
{q,} S
}L D
dV
(3. 32)
3.31 we have
Then from Eq.
ai/3g
=
j(V)m{ D.}
Also
dV +
} S
(V)m{ D
3.3,
3.2,
From Eqs.
level
3.12 we have at the local
k
K t.
next evaluate
We shall
approximating
to
due
the residual force
represents
f
defined by
is
bS1
/ aykI
+ fV0)M{
D
} D1
+ fV){
D
}L D1 j{q}
i
akl/ 3
(3.33)
dv0
idV 0
[3Si/Ykl] L'ykl/_Sj dV0
2.14Q we have
from Eq.
aykl/
]S
L
J
Dkl
=L
q
+ (1/2)
j{D
k [cD
1
(3.34)
+ (1/2)
3
.0[Sm
} [Ykl
(V
=
+ (1/2)
(V
) {D
Dck
1
we have
level
the local
at
Finally
[q J{DC
LDkl
/
}
3S/
Dykl
dV
LqitD ck
LDc
1 jdV
(3.35)
+ (1/2)
{D
}
S
o M
{D
+
o
m
}LD 1j S'dV
/ 3Yk
q
{Dc, }LDekJ dvo
85
+
+ (1/2)
} LD'
(V{)
D . JD1 j{ q }
+(1/2)
On a global
}
{(v
D
) {D
{D(V
1
o m
{q
S
i/
Dy
Dklj dV
/
(
[q {D
LDJ{q} [js. /Dyk
1
Ddv
]
qJ{Dc1 } LDckJdvO
scale
(3.36)
J k
t.
and
n
(3.37)
t=
M=1
3.30, notice
Recalling Eq.
is
the
equation of motion
to the residual
solved exactly due
not
that
term given
force
as
f
=
Various
tag*
equilibrium iteration schemes
conjunction
in
used
minimize f
-u1
to
the zeroth
such
0
0
o,
q
,
and
iteration.
that Eq.
3.38
the
with
3.30.
in Eq.
for A4
solved
(3.38)
-AF
Next a correction
is satisfied.
repeated
until
may
be
times
fu satisfies
that,
numerical
integrators
to
3.30
and
Aq
computational cost
first
The new Aq 1
the
depending on
q 0 +Aq
=
analysis,
considerably.
is
is
criterion.
We
the solution
therefore
1
process
above
some convergence
evaluated
is
scheme,
reevaluated and refactorized a significant
throughout the
is
Aqo where the superscript o refers
to evaluate a new AF
in general
be
In particular, Eq.
then used
note
therefore,
must,
Kt
number of
aggravating
the
86
3.2
SOLUTION METHODS
solutions
to the governing equations
3.24,
3.20,
of
characteristics
briefly
discussed, and
(equilibrium
for
numerical
is
presentation
parameters
the
increment
techniques,
governing
size
considered.
also
limited to solution techniques
theoretically of providing the
and
"exact"
by
response
the solution process
produce solutions deviating
limitations
the
structural modelling and constitutive relations.
entire
governing
the
The adjective direct
geometric
structural
with generalized matrices
the
of
is customarily arrived at by
techniques.
from
in
Integration Analyses
An accurate solution
motion
time
These
actual observed behavior only because of
Direct Time
varying
(e.g.,
the
3.2.1
capable
tolerance).
convergence
consequently,
3.38
Eq.
Alternate techniques
are
solutions
are
operators
solving
for
iteration) are examined.
stability
and
integration
techniques
deriving time history
The
of motion given by Eqs.
accuracy
The
3.30.
and
numerical
for obtaining
section discusses methods
This
direct
refers
matrices,
equations
time integration
to manipulating the
opposed
to working
(mode superposition analysis),
time integration pertains to temporal
governing equations to derive the
of
and
integration of the
response.
87
Time
of motion at time
equations
the solution is
which
explicit methods
of
form
stiffness
the
at
time
for
of
In terms
time t.
for
for
we consider
MIg*(t-at) + K tAg*(t-at) = &F(t-At) + f (t-At)
and
the
equations,
governing
the
the
implicit methods
already known) while
equation of motion at
consider the
tangent
(i.e.,
t-At
or
on
operate
methods
Explicit
operators.
implicit
involve either explicit
techniques
integration
(3.39)
implicit methods,
Mjak*(t)
Explicit
+ Kta*(t) = &F(t) + f (t)
integration
predictor
difference
Wilson-e
variation
Houbolt
(10,87),
stiffly-stable
All
(59),
numerical
of
the
at
the
at
current
result, dynamic equilibrium
time steps
include
tn'
in
Newmark
the
time
(33,34).
general
assume
is satisfied
(50),
time increment,
previous
step
type
(35),
a-method
during the
the
central
Runge-Kutta
backward-difference
operators
response
the
and
and the Hilber
then given the response
response
methods
the
include
operators
Implicit operators
operators.
(3.40)
is
time
step,
calculated.
only
at
Park
some
and
the
As
a
discrete
88
the stability and accuracy characteristics.
integrator
are
Instability
is defined
the
truncation error,
and
multistep methods
(60).
explicit methods are conditionally stable
unconditionally
are
methods
implicit
stable
for
linear
on how the equations of motion are
solved.
In
At/T
At
where
structural
that vary as
the
increment
time
and
formulated
a function
and
T,
of
a natural
period.
Accounting
of
the
is
schemes
implicit and explicit
of accuracy, both
schemes have accuracy parameters
may
problems
implicit
depending
terms
(unstable
while
the
in
stability
conditional
induce
to
step)
to nonlinear
The extension
elastic problems.
now
time
critical
than a
step greater
time
for a
of numerical
suffices
and it
for
roots
spurious
A detailed examination
is given in Chapter 4,
integration
say that
frequency dispersion,
of
effect
and
the solution,
of
as a divergence
refers to numerical damping,
accuracy
numerical
a
selecting
in
The two main considerations
for
integration
operators,
governing
differential
algebraic
equations.
response at
For example,
time
accuracy
the stability and
equations
we
now
are
limitations
consider
transformed
how the,,
into
All integration operators express the
t in terms of
previous or
the central difference method
future responses.
assumes
U(t) = (1/At 2)(u(t+At) - 2u(t) + u(t-At))
(3.41)
A(t) = (1/2At)(u(t+At)
(3.42)
- u(t-At))
89
This
integration operator
motion,
most often used
M(l/At )(u(t) -
2
equations
of
to give
3.20, at time t-At
Eq.
conjunction
in
the
of
form
unconventional
pure
the
with
is
u(t-At) + u(t-2At))
+ F(t-At)
= -I(t-At)
(3.43)
and
(1/At 2)Mu(t)= -I(t-at) + F(t-At) + (1/At 2)M(2u(t-At)-u(t-2At))
(3.44)
that
Notice
the only unknown is
to
applied
of motion, Eq.
equations
n (t)
tangent
the
= 6_(t-At)
where
Eqs.
a
and
6
3.45 and
_MA(t)
1(t-At)at
are
3.30.
+
3.46 into Eq.
+ a!(t) At2
AF(t)
technique
the incremental
(3.45)
+ 6u(t )]At
(0.5-a)Ui(t-At)
+ a-(t) At2
parameters.
3.30 at time
+ AA(t-At)At
=
integration
The Newmark method assumes
integration
+ Kt An(t-At)
on the LHS.
stiffness form of
(l-6)u(t-At)
+
u(t) = u(t-At) +
Newmark
the
consider
now
We
u(t)
+
+ f (t)
t we
[(0.5-a)6A(t-At)
(3.46)
Substituting
obtain
(3.47)
90
3.47
After many manipulations, Eq.
the incremental
((l/a
displacement at
t )M + K )aS(t)
=AF(t) +
+ (1/aat)Acj(t-At)
form
a
only unknown,
the RHS and the
known quantities on
all
with
is rewritten in
on the LHS
time t,
M((/at2),q
+ ((1/2a)-l'<(t-At))
t(3.48)
+ f
or
are given as
AF
K
=
-eff
AF ff=
(l/Ct)2 M+ K
2nd term in
(3.50)
t
f
-u
static analysis
decomposition
Aq(t).
by employing
of
Notice that
K
-ef f
3.49 is
now
can
3.49
Equation
.
3.48
brackets on RHS of Eq.
The approximate equality in Eq.
of
and effective load
effective stiffness matrix K
where the
vector
(3.49)
AFff + AF(t)
Aq(t)
K
be
in Eq.
to the omission
solved easily as
techniques
and then back
due
equations
Eq.
3.49
3.44, we have K
represents
the
of motion for all numerical
a
Gauss
substitution to obtain
_-eff
result,
as
such
in a
algebraic
=
At 2 M.
A
As
a
form of the
integration operators.
91
of
pseudo-force
the
in
since
t, implying
response at time
the
on the RHS,
is present
response
the
function of
while in the
also
is
f
3.30,
Eq.
expansion
series
in
therefore,
and
methods,
with a
time
previous
3.30
Eq.
it
is a
represents
the
by truncating
the
Taylor
must
be
used,
3.24
Equati on
conjunction
FNL(t) from FNL at
(f.u
t
time
at
form,
stiffness
tangent
an unkno wn variable since
I).
of
function
is a
F
that an unknown variable
obtaining K
force due to
residual
3.24,
form, Eq.
the
considerations
additional
introduce
motion
of
equations
of
forms
stiffness
pseudo-force and tangent
The
technique that
steps
by
estimates
extrapolation
must be combined with a iterative
technique to minimize f
-u
The force
vector FNL is
estimated by
a
Taylor
series
expansion
F
NL
(t)
=
t}
NL
F Ct-&t) + AtF
using,
Then
for
approximation, we
F
(t-At) + 0(lh
difference
obtain
(t-At) = (1/at) (FNL (t -At)
Substituting Eq.
)(3.51)
backward
a
example,
2
3.52 in Eq.
FNL(t) = 2FNL(t-at) - FNL(t-
- F
(t-
3.51,
2At)
2
At))
(3. 52)
we have
(3.53)
92
3.53
and substituting Eq.
time
at
3.30
into Eq.
we
t,
arrive at
+ Kg(t) = F(t) + 2F NL(t-At)
1(t)
by
series
Taylor
truncated
a
replaced
3.54 is approximate since jNL was
that Eq.
Notice
(3.54)
FNL (t-2Lt)
-
a finite difference
using
equation.
f
is
variable
for a correction displacement
solve
-u
response Aqi
Then the updated
i
-1
=
to
until
-u
a function of Aq
referred
,
of
reevaluated
is
as
to
next
Eq.
using
The procedure
).
equilibrium
Eqs.
solving
proceeds
to
the next
is reevaluated after each
Newton
given as
3.55,
iteration,
3.56,
and
3.38
some convergence criterion is satisfied, whereupon the
solution
Ki
-t
f
consists
therefore
is
(3.56)
that AF is
(notice
minimize
(3.55)
~i
The residual force vector
3.38
Aq satisfying
(static analysis)
= f (t)i1
A(t)
K,
-t_
unknown
the error in this case, we
minimize
To
.
the
formulation,
stiffness
tangent
For the
iteration
reevaluation and
expensive,
and
scheme.
time step.
refactorization of
3.55,
if
are
using
a
we
iteration,
It
In Eq.
is
apparent
K
is
that
the
computationally
-t
may
render
the tangent stiffness approach
93
-t
in more
The following pages examine
reevaluated.
the
detail
need not be
K
the
However,
unconventional approaches.
constantly
pseudo-force and pure
the
to
compared
inefficient
highly
techniques
iteration
equilibrium
different
available.
f
minimizing
Atq
of
estimates
Eq.
3.38
in
Eq.
3.55.
numerically
solving
approximation
to
investigate
shall
We
Aq.
and
3.55
find
to
to
due
are
Eq.
in
main
the
general
efficiency
equation
the
accurate
obtaining
In
(K)
Jacobian
the
reestablishing
and
techniques'
the
to
obstacles
in
efficiently
for
developed
Numerous schemes have been
better
a
Newton,
the
modified Newton, and quasi-Newton methods.
In Newton's method, K
in
-t
highly accurate results and
approximation
solution and
This
after each iteration.
refactorized
if
I'
iteration consists of
technique produces
reforming
the tangent
3.55.
or
Evaluating
operations
the
factorizing
while
numerical
approximation requires O(n
for
large
systems
can
intractable problem, thus
effective techniques.
3
stiffness matrix
then deriving the
stiffness matrix to solve Eq.
the new stiffness
the
each
However,
(21).
corresponding to the new configuration and
inverse
an
once
the neighborhood of the
in
continuous
is
and
updated
converges quadratically
found
been
has
is
3.55
Eq.
matrix
decomposition to
) operations.
easily
requires
produce
necessitating
O(n
find
2
)
the
Such
calculations
an
economically
the
use
of
more
94
As
that
each
the
the
stiffness matrix is not reevaluated after
tangent
rather at preselected
iteration but
begins
solution
This
diverge.
to
when
the
technique,
the cost
of
implementing
the
3.55
is
Eq.
stated
Simply
decomposition.
numerical
and
matrix
stiffness
the
or
intervals
modified Newton's method, significantly reduces
reforming
such
modified
been
has
a result, Newton's method
rewritten as
(3.
KAq (t) = fut)
time step or
is a previous
where T<t
when
cycle
iterative
570)
equilibrium was satisfied.
each
after
once
properties of
during each time
step),
the
modified
obviously attained by reevaluating
often,
but
technique's
consequently
this
must
nonlinearity present
efficient
process
have
(update
Newton's
method
a
simultaneously
good
"feel"
in the problem to
updating interval.
=
efficiency
is
the stiffness matrix more
effectiveness.
economic
t-At
convergence
Greater
linearly.
converges
the modified Newton method are
the case where T
In
quadratic.
than
less
step,
cycle or even during each time
iterative
the convergence
reevaluated
not
stiffness matrix is
the tangent
Since
reduces
The
for the
the
analyst
degree of
judiciously choose an
95
In
of
the limit, T=0
This
the
to as
referred
approach is
and
the Aitken
modified
the
Newton
the
imposes
acceleration
Aitken
the
where
technique,
with
combined
are
acceleration (2)
such as
schemes
acceleration
systems,
stiffening
for
even diverge, especially
consequently
"initial stress" method.
solution may converge very
For highly nonlinear problems the
slowly or
at time 0.
matrix is performed only once
stiffness
the
decomposition
the evaluation and
and
following correction:
A~
and
a
+q(t
t
8)
A~C)(3.
is the acceleration coefficient matrix given as
a
A
(t) /(t
Due to
(3.59)
(t)
the inefficiency of Newton's method
the
the modified Newton method,
of
quality
and
a
poor
convergence
third
scheme, the quasi-Newton method, has been investigated
and recently applied to structural problems
speaking,
Historically
first
by
introduced
Fletcher
and
Powell
in
generalized the technique
quasi-Newton
the
Davidon
1963
in
proposed
its
mechanics problems.
1959
and applied
application
it
to
Mathies
to
method
was
and popularized by
In
(21).
of nonlinear systems of equations.
1979
(7,21,22,29,45).
1965
Broyden
the optimization
and
nonlinear
Strang
in
continuum
96
stiffness matrix and derive
each
-1
.
-t
number
secant approximation, yielding a
G
q (t)-q
_1
= f1 - f
the
operations
of
of
follows:
(3.60)
The Jacobian G'
(t).
at
first
the
1
Kt
to
approximation
an
is
G'
where
_1
(t))d
it
evaluation of
The quasi-Newton equation is given as
O(n 2).
tangent
factorize
cost after
to the
only
due
therefore
is
iteration
The computational
to K
approximation
its inverse or
the
the quasi-Newton method calculates a secant
step,
time
quasi-Newton
the
Rather than reevaluate
simple.
is fairly
method
underlying
framework
The conceptual
is
and
equals
d
in essence obtained
from
secant
using a multi-dimensional generalization of the
method given as
i-2
zz ~'
z z Z
Various
- z(z i
~ Z(z
z(z i-2)-
-
rank one updates were
(3. 61)
)
initially presented, with
the following form:
i
G
i-1 +
+
=G
ST
where
i-1 - G
d
)rT
-
-
r
.
r is an arbitrary vector such
f i-f 1(29).
-u -u
(3.62)
-d
i1
that rTd=O
and
y
=
97
rank one
In Broyden's
i-i
T
i
-Ci-1 1 di-1 d-1T
)d'
G' d
-
(i-1
(
i-1 +
+
G
-B =-
iT
d
r=d and
update,
(3.63)
di-1i
Similarly Davidon has
suggested an update with r=y -G
ld
such that
i1-
Notice
+
=G
G
that G
-B
I-1i1 i-I) (~
d
-
i-1iI-i T
- G
)
d
(3.
GI-1 i-1 T i-1
i-i
) d
d
-G
(jy
is
unsymmetric
G
and
is
-D
not
64)
necessarily
positive definite.
have been proposed which
Rank two updates
to being
symmetric, are
positive
definiteness,
of
also positive definite.
i-T
"M
+
i
(I
G
1-
-d
-BFGS
=G
-
i-i
)
(3. 65)
-
T
d
and the Broyden-Fletcher-Goldfarb-Shanno
i
T
i-1~
lT 1 1
i-lTd
di-1
two updates
Ii
v+
v:J
G
insuring
given as
(DFP)
T
i
i -d
G F
popular rank
The most
are the Davidon-Fletcher-Powell
i-1
By
addition
the algorithm has a greater guarantee
stability.
numerical
in
+
x
i-1 i-1
x
i-lT i-
T
1
(BFGS) given as
i-1 i-1 I-1 T i
C
y
y
G
S~
iT
i-1 i-i
(3.
66)
98
In
actual practice, the
updates
to the
matrix.
Using the property
(A + arv)~
where
x=A~1r,
inverse
flexibility matrix rather
produces
than the
stiffness
(3.
xzT
-
= A~1
scheme
can
a=(l+avTA~lr1, we
z=ATv,
derive
67)
the
rank one update as
=G
i-
-1
(d
+ :
1-1
and similarly
(T
-BFGS
-1
i-l
- G
i-1
T
y
i-1
i-i
i-1
1-1T
i-1
i-1
(3.
i-1
.
1-1
for implementing
68)
In particular
T
)G
-
T
)
for rank two updates.
The procedure
is as
quasi-Newton
i- id
T
T ..
i-i
d i-1 i-1
d
d
Td
i-1 i-1
the quasi-Newton
T
(3. 69)
method
follows:
1.
Evaluate di
= -(G*)~f
2.
Set qi+l(t)
= qi(t)
3.
Compute fi+l(t),
-u
I
-u
+ d'
and
therefore
y
f i+1
-u
(t)
f (t)
-u
4.
Derive new Gi+1
5.
Check convergence
6.
Repeat
steps
criteria
1-5 until
convergence is attained
-
99
line
a
Rather than
search schemes have been proposed.
i+1
i+
d
performed in the
scalar multiplier
.T|
d1
i
various
setting
(3.70)
line search is
optimal
rate,
convergence
the
further accelerate
To
i
direction d
an
to obtain
S such that
= 0
(3.71)
Then
±
1+1
i +
+
reduce
line search should
Although the
of
(3.72)
d
determining
the process of
iterations,
(7)
Bathe and Cimento
that
state
the
satisfactory without the line search
expensive.
is
convergence
rate
is
scheme when
3.73)
papers discuss the
the
occurs,
either
convergence
In
be
can
it
or
linearly
properties
general,
satisfied by the Jacobian
solution,
convergence
i+1
are
(22).
method
quasi-Newton
conditions
of
#
number
n=0.5.
Numerous
the
required
d(f
i
with
the
if
of
various
in the neighborhood
shown that local
superlinearly.
convergence
Superlinear
occurs when
-
C*
(t)
-
3*
(3.74)
100
is
converges
to zero.
for
the
The reader is
to
referred
{a.}
sequence
the
the references
further details.
Finally,
the
and
solution
exact
where q
convergence
After each iteration,
extremely slow.
the
set
tolerance, the update
matrix
iterations
until
tolerance,
and
then
the
quasi-Newton
to
all
iteration
the
of significant
certain
a
than
can
scheme
n
for
used
be
less
is
residual
the
can
method
Newton
the
nonlinearities,
(otherwise
cases
such
In
singular).
becomes
than a
found larger
implemented
not
is
number
the condition
matrix is evaluated and if
new
of
may become
scheme
quasi-Newton
a
using
cases,
particular
in
it should be noted that
be
reimplemented.
regard
With
acceleration for mildly
dynamic
response
nonlinear
technique
BFGS
will
analyses
since
the
should
not
inertia
less
smootherners,
inducing
BFGS may still
be economical.
Notice
be
combined
with the Aitken
combined
However,
problems.
be
exhibit
terms
used.
In
iterations,
to
pseudo-force method.
the
force
By doing
extrapolation
so,
force extrapolation are offset by the
in the
act
as
and therefore the
that -the equilibrium iteration methods
with
general,
sudden changes
tend
if
cost
less
the
then
exist,
nonlinearities
significant
effective
Newton
modified
the
recommends
(7)
Bathe
schemes,
can
also
scheme of the
approximations
in
the
iteration techniques.
101
the
tangent stiffness,
how
often
can be interpreted as
secant flexibility,
produces
that
to a previous
by adding terms
systems
SDOF
for
of
convergence criteria may
the
energy
or
force,
displacement,
of
terms
definition
some
require
stated,
Briefly
in
the
an updating scheme
iteration techniques
defined
reevaluated.
3.1.
are shown in Fig.
convergence.
The
is
These three approaches
stiffness.
All
uses
stiffness
secant
exact
secant
be
stiffness
actually
method
two being
the difference between the
tangent
the
quasi-Newton
the
employ
Both the Newton and modified Newton methods
system.
but
considering a SDOF
readily by
grasped
is
operates
method
iteration
equilibrium
each
how
-of
feel
physical
A
residuals such that convergence occurs when
1 (t)
IW(t) - w
IlW ~_ (t)
or we
<
2
E
(3.75)
2
can use
-(0)
2
- 11
(t)12
<
(3.76)
11i-1 (t) 12
where W
the
is
tolerance,
convergence
Euclidean
,
modest
converging
the
subscript
it
can
be
Eq.
3.76.
residual,
the
and
may
quantity,
and
3.75
Eq.
than
displacement
fairly
Since
norm.
Qxyj-jyJJ<jJx-y
criterion
defined
is
even
a
W
When
convergence
give
a
the
2 refers
to the
shown
that
easily
usually
is
E
more
stringent
represents
the
requirements
are
sense
of
false
102
F
Fot
F-------------- ----------- -
tangent stiffness
u
a.
Newton
F
F0
initial
b.
tangent stiffne iss
Modified Newton
u
F
F0
.....
secant stiffness
u
c.
Fig. 3.1
Quasi-Newton
Equilibrium Iteration Schemes Applied
to SDOF Systems
103
are more
criteria
Force
convergence.
stringent since
the derivatives
stresses are derived from strains, which are
of
displacements.
for
stiffening
while
are used for softening
forms
integrator exhibits
formulation
convergence
and stability
However,
worst
does
not
unconventional
and
for
form
pseudo-force
cases.
pseudo-force
and
matrix,
the stiffness
with
the
pure
small
systems
The
tangent
systems.
formulation is extremely uneconomical when used in
stiffness
conjunction
the
with
Newton
indicate that the
Recent
method.
tangent stiffness method
with a quasi-Newton iterator becomes highly
problems with significant nonlinearities
3.2.2 Nonlinear Modal Analysis in the
The
analyses,
has
good
converge in all
efficient,
large
tangent
the
extremely
being more efficient for
form
pseudo-force
however,
fairly
are
approaches
but good
stability
the pure unconventional
because
methods never involve any updating of
both
solution
of
properties, while the
but
is also stable,
the
demonstrates
stiffness
method
types
implicit integrators,
of
terms
In
accuracy.
(7).
the pure unconventional form combined with an
formulations,
explicit
three
energy
and
displacement
structures
on the
a general comment
As
force criteria are often used
Energy and
structures
the
excessive
for
combined
attractive
for
(78).
Time Domain
performing
some
nonlinear
particularly when parameter studies are conducted,
prompted various
techniques
cost
studies,
to
the
researchers
solution
to
of
adapt
modal
nonlinear
analysis
problems
104
good results economically while others
relatively
produced
results at a cost
have yielded marginal
adapting
with
associated
issues
section
This
methods.
integration
have
approaches
Some
(3,8,30,48,49,51,52,53,75,80).
time
to
comparable
identifies
the major
modal
analysis
linear
techniques to nonlinear continuum mechanics problems.
a
Given
freedom.
but behavior
complex,
to model
matrices
is relatively
number
degrees
of
of
whose geometry is fairly
simple,
requires
the structural geometry,
direct
a typical
structural
large
though
even
the
be derived
by a linear combination of a small
of basis vectors --
more commonly referred to as mode
can
response
number
system
large
technique
integration
time
small
relatively
a
analyzed with
structures can be
mainly because complicated
attractiveness
its
gains
technique
superposition
mode
linear
The
derived
and
shapes
the
from
following
linear eigenvalue
problem:
(3.77)
SMAO
and
K
where
A,
matrix of mode
stiffness
shapes or
the diagonal matrix containing the
From a mathematical
vector
structural
the
are
the
(,
matrices;
and
M
q
eigenvectors
defined
is
.
point
as
a
of
view
linear
and
mass
eigenvectors;
eigenvalues.
the
displacement
combination
of
the
In other words
(3.78)
105
in
jargon
engineering
vector space
the response
concept of representing
This
applied
to
produces a
smaller
dynamic
problem.
substitute
q by its
of
equations
v + r;
A
Eq.
3.79
can
be
is
L=
A=
T
D
such that
to M
F
is
solved
uncoupling
TM
when
and
=
+
has been
I.
Notice
of n
Even
uncoupling
n(<m
since
for
when advantage
does
the
not
not
occur,
matrix
Fawzy
sizes
which
numerical and
(single
is
3.77
Consequently,
various
SDOF
the
equations
uncoupled
using
normalized
from Eq.
when
diagonal
als o
solution procedures
or
,
T
independently
systems.
freedom)
F
satisfied (KM~1 C is symmetric).
represents a set
analytical
results
to obtain a new set
(3.79)
is diagonal.
criterion
motion
generalized coordinate representation Oy
+ Av
with respect
that
of
given as
T
Cc
_=
where
reduced
specifically,
More
pre-multiply all matrices by OT
and then
by a
equations
the
when
space
vector
n
to a reduced vector space
[m
n
the
from
transferred
is
q
q,
to represent
eigenvectors
only
using
By
n<m.
where
eigenvectors,
n
of
reduced
a
by
represented
m
In
generalized coordinates.
m
the response q can be
most cases
exist
there
then
m,
size
therefore
and
eigenvectors
If
the generalized coordinates.
as
the matrices K and M are of
set
referred to
the linear combination coefficients,
y are
where
degree
take
economy
of
of the
still
and computer
106
requirements
storage
y is q= jD.
solving for
much
represented adequately
D, and
eigenvectors
is
stresses
must
shapes
approach may no longer be attractive
initial conditions
but
be expected
adequately reproduce
changes occur
the
that
the actual
in
the
initial
updates
method for
selecting shapes should be
significant
the
problems.
matrices
mass
in
may
response
economy results
is
Ritz vectors.
nonlinear
response if
structural
matrices.
shapes or an alternate
implemented.
adequately
from the
As a
shapes may
the initial mode
Otherwise
if
performed
is
to nonlinear
selected
the
still
event,
analysis
shapes,
it would
to
of
longer represent all deformed
result
no significant
number
from the
considered
be
may
no
or
the eigenvectors derived
change with time,
general
and
damping,
stiffness,
Since the
the
(85).
is applied
superposition
mode
when
the
structure
modal
type analysis
essence a Rayleigh-Ritz
In
when
a large
the
and
employed,
be
must
structural
or
the
over
contains very high frequency components,
mode
selected
by the
accurately,
considerably
varies
excitation
the
in fatigue analyses where
determined
be
small,
local
the
when
desired, such as
response
be
the analysis must be performed again
Especially
n.
with a larger
n should
is too
n
if
However,
m.
than
smaller
response q cannot be
n
economical,
that the solution is
insure
To
after
final solution
The
are reduced.
In
represented,
any
a
reduction in the matrix
107
the possibility of using larger time steps.
sizes and
The major
accurate
obtaining
and
analysis
appropriate basis vectors.
should
time periods
The first
the Rayleigh-Ritz
of
convergence
(52).
economically
and
easily
substantial
a
span
and
vectors
basis
the
of the
response
requirement
ensures
remain good representations
and
generated,
selecting the
results is
general
the solution space, be
portion of
over long
In
independent
linearly
be
a nonlinear modal
consideration in performing
solution while the
others
enable economically derived and accurate answers.
Four methods currently described in
be
method employs
become
following
only the linear
The simplest
paragraphs.
eigenvalues
Although the
(8,48,75).
analysis
adequate
the
in
presented
the literature will
throughout
linear basis
the
vectors are
tend
for mildly nonlinear problems, variations
to
unacceptable for problems with significant geometric
nonlinearities.
material
the number
response
stiffening
structures
increases,
Problems with
behavior
are
localized
highly
also poorly reproduced.
of eigenvalues adequately reproducing
are
also sufficient for
in
such
In general
the
linear
the nonlinear analysis of
structures, but may be inadequate
since
nonlinear
cases
requiring more eigenvalues
the
(8).
for
softening
structural
period
108
with
significant
economies, but
judgement on
requires considerable
response
structural
expected
the
knowing
without
vectors
basis
nonlinear
the
of
selection
the
is
provides
method
This
updating.
no
performed
these
analysis
the
vectors,
basis
initial
the
as
its
some of
Using
(49,52).
configurations
nonlinear
deformed
derived
with basis vectors
the structure in
eigenvalue problem of
from the
vectors
combined
vectors
basis
linear
employs
method
first
the
of
similar modification
A
the
the part of
analyst.
slight modification of the
third method is a
The
vectors
than
a
respect
with
the current
to
stiffness
matrix must
be
throughout
problem solved
to
expensive
is
it
and
regenerated
the residual
similar
to
since
rather
the
f
Although
first
the
than
the
entire
eigenvalue
and
the
the
initial
the analysis.
becomes excessive
third method,
linear
by updating
the analysis
(51).
Although
this method is more economical
the complete eigenvalue problem
the
to
normalized
since
perform
then carrying out
whenever
greater
then
and
results
The last method involves deriving
eigenvectors
becomes
set of basis vectors.
accurate
method yields more
method,
residual f
generated
-u are
residual f
initial
the basis vectors necessary
tolerance,
certain
reduce the
this
Whenever the force
(3).
to the
linear updates
basis vector approach and uses
linear
super-variational
is never
approach
solved,
but
generates
109
(k-1) T
2(k)
where
K(q
k is
),
eigenvalues using
(3.80)
normalized
basis vector.
defined as
follows:
-o
2(k)
Wj
Finally
then
M)$
1
eigenvectors
updated
the
is
the
(k-1)
2(k)
(K( Q)
2(k)
(k)
a.
factor
4.,
and
step;
this
correction
A
(k-1)T
ai
at
number;
mode
the
i,
number;
matrix
stiffness
the
the following algorithm:
(k-1)
iteration
the
first
approach
super-variational
the
Briefly stated
the
can
response
the
result
a
be represented adequately.
theoretically always
updates
As
(84).
vectors
previous
the
from
vectors
basis
current
the
to
approximations
are
derived
the
using
following expression:
n
[i
(-1)
(k) = C(k)
where n is
a (k)
number
the
of
basis
vectors
and
c (k)
i
using
studies
Various
factor.
normalizing
(3.82)
super-variational approach indicate a quadratic
is
a
the
convergence
(51).
In
the
general
economical
in
may not
be
updates
must
linear
feasible
be
mode
technique
superposition
analyses when n<<m, but the
for
nonlinear
implemented.
Even
problems
if
no
is
solution
when
updates
modal
are
110
required,
the response must
reevaluate
the nonlinear terms
further
in the pseudo-force
the
Although
technique
in the time domain may be more economical
time integration solution, a numerical
still
used,
and
addresses
a nonlinear modal
domain,
subsequently
frequency-time domain
integrator given
approaches
the
exact
can be
In
of
form
to
Moreover,
it
as
frequency
the
hybrid
exact
the
in
o
transfer
H
(H
the
frequency
the frequency
increment
the limit
and
numerical
function
the analytical
as
form of H can be
equals 0 for values above a
other words,
the
the frequency domain has
theoretically
a
extended
zero).
to
the
technique.
(HFT)
in terms
modified such that
frequency.
is
section
This
results.
referred
using
spectrum range is
to
than the
integrator
analysis method in
Obtaining the solution in
of
analysis
therefore accuracy and instability problems
present and may invalidate the
advantage
Domain
modal
nonlinear
a
of
use
direct
tends
approach
reducing the solution efficiency.
3.2.3 Nonlinear Modal Analysis in the Frequency
are
to
stiffness matrix, thus
reform the
to
necessary,
when
and,
coordinates
to the natural
coordinates
generalized
the
from
transferred
be constantly
an infinite
imposed at the higher frequencies.
specified
artificial damping
111
In general
employs
scheme
the proposed
-
(3.83)
~ denotes a
function H
is therefore
where
w
consists
generalized
examines
transfer
(3.84)
is the circular frequency.
of
obtaining
transferring FNL
back and
The
matrix.
2-.
_)
(-W 11 + iW~C + K
domain, transferring Y(w)
FNL,
motion in
+NL
where the
=
domain
form
generalized
H(W)
of
equations
pseudo-force
the
frequency-time
hybrid
forth until
the
characteristics
the mechanics of
the
to the
time
to the frequency
the
solution
the HFT method
frequency
to
evaluate
domain, and
iterating
domain
converges.
efficiency
and
procedure
in the
response Y(w)
computational
of
The solution
5
accuracy
and
describes
implementing an HFT analysis.
Chapter
in
detail
112
CHAPTER 4
ANALYSIS OF NUMERICAL INTEGRATION METHODS
These
differential
equations
Structural
dynamic
numerically
by
a
and
elements
difference
temporal
methods.
discretization
discretization
More recent
to
produce
precise
analyzed with standard
approach in the
of
the
finite
employing
finite
(31).
combining
a
series
This chapter examines
their
techniques and
solutions.
ability
Stability and accuracy are
time domain schemes,
frequency domain is
selecting an appropriate
using
(26,28,56,95) and spatial
approach
numerical integration
solved
include temporal
approaches
by finite elements
solution with transform methods
are
consequently,
discretization
discretization by a spectral
popular
for the most basic problems.
problems,
spatial
a
problem whose
pose an initial value
solution exists only
analytical
a
ordinary
nonlinear
order
second
continuum.
of
behavior
nonlinear
dynamic
the
mathematically
represent
2
Ch.
in
derived
equations
energy
The
and
presented.
time increment
is
an alternate
The practice
reviewed,
chapter concludes with a summary of case studies
and
and the
113
of
considerations
to nonlinear analyses.
4.1
stability
and
accuracy
extending
implications
STABILITY AND ACCURACY ANALYSIS OF NUMERICAL
INTEGRATION
SCHEMES
Stability of
integrators
numerical
these
the
parameters
to natural period
convenient
and accuracy separately, it
purpose
of
requisites
consequently
are' a
these
and
themselves
have no significance.
considered
a
prerequisite
results are unattainable for
converse
emphasize
increment
statement is
untrue.
to
examine
stability
is
converges.
solution,
be
the time
integrators
solution
convergence
for
investigating
stressed that the
is
analyzing numerical
numerical
the
functions of
as
between
ratio.
is often
Although it
if
is achieved by
conflicting requirements
accuracy
A compromise
increments.
time
large
dictate
considerations
increments, but economic
of
use
the
The second feature
Accurate solutions are attained by employing
accuracy.
time
the
system, whose errors
lead to instability.
and assumptions may
small
involves approximating
equations by an algebraic
differential
is
inherently
the
Applying
progresses in time.
it
remains bounded as
solution
that
implies
integrator
a numerical
stable
two
fundamental
to
determine
The
necessary
and
accurate
characteristics
Furthermore,
by
stability can
for accuracy since accurate
an unstable solution, while the
The
following discussion will
the interrelation between stability and accuracy.
114
simply
integrator
numerical
the
of
Convergence
numerical
solution approaches
increment
tends
the exact
rate
to zero. Consistency implies that
We now examine stability by
time
finite
the
(the
order of
the
considering
be
must
terms
error.
truncation
time increment
to the same
approximated
the
the
as
all
differently,
Stated
the
infinitesimal
defined
is
that the error decreases
that
differential
exact
becomes
the time increment
accuracy).
means
solution as
difference approximation approaches the
equation as
consistent.
is
equation
governing
the
of
approximation
difference
finite
the
when
convergence
for
condition
sufficient
and
necessary
a
is
stability
that
theorem states
This
(71).
satisfied
theorem is
equivalence
Lax
the
systems converges when
solution of linear
The
governing
equation of motion for a linear problem
M'
+
C; + Kq -
conditions _1(0)
with initial
1
Fawzy criterion (KM
produces
rewritten
in
structural
Eq.
4.1
problem
can
be
(normal-mode) form
(4.2)
defined as in Eqs.
matrices
stability can be
the eigenvalue
F
=
the
satisfies
If C
and
modes,
normal
generalized
term is
and 4(0).
symmetric),
C is
classical
14y + Cy + Xy
where each
(4-1)
F
are diagonal.
by
examined
As a result
considering
3.79 and
of Eq.
separately
all
4.2,
the
each mode
independent equations of
+ k y
m.y. + 171c.y.
i
ili
3.78 and
=
F.
1
(4.3)
115
The
i and
subscript
(~)
tilda
are omitted
in the
subsequent
development.
analytically
solution
vibration
The free
Eq.
to
4.3
is
given
as
7
+ y(o)(t
(0)
y(t) = e-gt
s
(4.4)
sinMDt + y(0)cosDDj
WD
where
=
Ik/m
natural frequency
=
c/2wm
damping ratio
2 055
l
D
damped frequency
function F(t)
For an arbitrary forcing
given by
(4.5)
the exact
solution is
Duhamel's integral
t
y(t) = (1/m D)fF(T)e-W(t-T) sinwD (t-T)dT
(4.6)
+ damped free vibration (Eq. 4.4)
or
y(t) = C (t) sinwD
+ C2(t)c
wheret
+ damped free vibration
(4,7)
e
C
1
(t)
=(1/mW
D fo t
F(T)
-
V
e.
cosw TdTr
D
(4.8)
e
F()
F
C (W =-(1/MW D o0
2
i~
sinj- -r
116
solution
rewritten in
At using state vectors z.
obtained with a numerical
The discrete
E=0
between successive
time steps,
Considering
integrator.
variation
therefore becomes
4.7
Eq.
(4.9)
= Az(t) + Lf(t)
z(t+At)
form
the discrete
and assuming a linear load
the undamped case
stability
terms of discrete
then be compared with
solution can
of the exact
is
4.7
Eq.
characteristics,
time increments
evaluation of accuracy and
the
To facillate
with state vectdrs
z(t)
[ y(t),
-
Aty(t)
]
(4. 10)
-
f(t)
[ F(t),
as
A is defined
In
operator.
F(t+At)
the amplification matrix
the general
e
=
_D
and
L,
the underdamped
case of
the
load
problem,
is given as follows:
the amplification matrix
A
]T
os 2 D
in
-FsinQ D
D
+
-U
e
. sinD(
cos D
(4.11)
c
27
1
Q.
-0.5
0
where
Tl =
The
-WAt
corresponding expression for
involved.
For the undamped
sin
L
=
the load
however,
problem,
quite
we have
if /QI
(1/k)
-co s
operator is
- 1
1
(1/k)A
(4.12)
117
free vibration problem the
In the
is
we have
zero and
(4.13)
= Az(t)
z(t+At)
being denoted by d
time nAt
response at
,
and
a
n
for the
The
(4.14)
--n
--n
conditions
d
=
q(O)
v
=
q(O)
(4.15)
-cv o
(F
o
o =
kd
-
)/m
0
also be written in terms
can
4.14
conditions
n
+ Lf
= Az
Equation
v
4.9 becomes
recurrence relation given by Eq.
a
,
acceleration, respectively.
displacement, velocity, and
with initial
n
approximate
the
solved numerically with
Equation 4.2 is
z
-n+l
load
the
state vector for
initial
of the
to give
n+1
n+1
=A
z
-n+1
Notice
on
z
-0
+ e+
n+l-i
A
.416
that the actual
the
(4.16)
Lf.
form of A
integration operator.
selected
In
particular, the
and
load
of Eq.
4.14
difference between the exact amplification matrix
operator
of
Eqs.
provides a means
and
of analyzing
characteristics of
be
4.11
4.12
the
and
the chosen operator.
construed as an error,
and
those
stability
depends
4.14
and L in Eq.
and
accuracy
This difference
the propagation of
the
can
error
118
as
to
n tends
form
solution behavior for the homogeneous
4.16
of Eq.
(4.17)
- -0
In other words,
finite difference approximate response
the
non-forced system with nonzero
of the
examined
to assess
predicted
response.
the stability and
Assuming the eigenvectors
conditions
error features of
the
obtain
(4.18)
where c contains
the eigenvectors of A
matrix consisting of the eigenvalues
a
n
the numerical
=
and A
is
of A.
X
a
(4.19)
that the
state vectors
integrator are of order
three
corresponding
such that
(dn, Atvn, At 2 an)T
Therefore, A is
(4. 20)
a 3x3 matrix, and
derived from its characteristic
det (A -
diagonal
Therefore,
-1
Notice in general
z
is
apply
we
of A are distinct,
to A and
similarity transformation
initial
A =
to
the
examine
- Anz
z
-n
a
to
suffices
it
In a stability analysis
stability.
a measure of
infinity furnishes
XI) = -3
+ 2A
2
-
the
eigenvalues of
A
are
equation
A 2 X + A3
0
(4.21)
119
where
I =
identity matrix
Ai-
half
A2=
sum of
the trace of A
of A
the principal minors
A 3 = determinant of A
4.17 and 4.19 we have
Using Eqs.
+ c2 n
cn
d=
c.
the
where
(4.22)
+ c3 n
constants
are
initial
the
from
derived
conditions.
In the
An
by
being bounded as n
for An
,
0
tends to
S
such that
Notice
to be bounded.
the numerical
is
integration scheme
integrator
is
less than
or
that since
p
equal
to
depends
on
1
O=WAt.
given in terms of
A is
the
Therefore,
infinity.
must be
=max{X.}
spectral radius p
one
defined
following discussion, stability will be
stable
integrator is
Oc
exists
the
stable for O<Q<Qc,
and
stable,
conditionally
for
finite
If a
G,
all
the
the
if
is
scheme
unconditionally stable.
stability definition
The above
by
plotting
plane.
XI=l,
let
4.21.
the complex eigenvalues
For a given
corresponding
can be depicted visually
i
and
time increment At,
if
Physical
we can
the complex
evaluate
X
the
they all fall within the unit circle
the scheme is stable.
X=(1+z)/(1-z)
onto
X
An alternate
and substitute
stability is
this
procedure
is
to
expression into Eq.
then defined as
the
left
half
120
plane with the
complex 0
of the
Notice
systems.
to undamped
stable
outside of
4.21,
while
axis corresponding
imaginary
that
their stability boundary, described
schemes
explicit
are
are
schemes
implicit
by Eq.
their
inside
stable
stability boundary.
accuracy
examine
next
We
integrators.
numerical
characteristics
Assuming that the
approximation satisfies consistency and
0<2<2
with
QB >0
an
consequence, it
As a
X12 = A + iB = e
-
stability occurs
X1
roots
IX 3 1<1X1
21<1
for
convergence.
mathematically
conjugate
such that
spurious root X 3
finite difference
the Lax theorem ensures
can be shown
complex
two
has
4.21
,
the
of
and
that
X2
Eq.
and a
and
(4.23)
D
where
ID
-
=
Atan (B/A)
-ln (A
-D(
0 - =Q0(1i-
-
2
2
S=
+ B )/22D
E
2 )0.5
-0.
(4.24)
WD
o
D/At
D
=
/At
i
=
D
n
Notice that the tilda ()denotes
Equation 4.22
d
=e-
approximate quantities.
can then be rewritten as
cosw tn + c'sinw tn) + c3 X~
(snc
(33)
.25
121
this
in
included
is
method
directly the numerical solution
=
G
when
equations
i
E
D=w D.
and
4.25)
(Eq.
(the
Newmark
can
compare
we
category),
The numerical
4.4.
of Eq.
solution
schemes
integration
root (A 3 =0)
two
For
solution is
two
ratio
where
Other equivalent accuracy parameters
and
/d
AD=l-d n
where nD21T/
the
include
dissipation
numerical
logarithmic
decrement
I
=
d
where wD
to rewrite Eq.
ce (a+ib)FDnAt +
4.22 as
two root
for
(23,59,60,62)
(4.26a)
c2e(a-ib)FODnAt
natural
the damped
is
)
=ti(d n/d n
w At.
would be
schemes
for
decay
amplitude
An equivalent approach to analyze accuracy
In
the
are characterized by the algorithmic damping
and T-2rI7/.
T=2Tr/w
b
only
the relative period error (dispersion) (T-T)/T
and
and
exact
between
Discrepancies
exact
with the
frequency of Eq.
are the numerical damping and
phase
shift
4.3 and
a
parameters.
particular,
a
=
(4.26b)
b =
From Eq.
when
are
D
4.26b it is
evident
a=-(1-E2 )-o-5 and b=1
when a>O
b
D
or b is
functions
imaginary.
of
D*
that the
and
also
exact solution
occurs
that instability
occurs
Notice once again
that
a
and
122
4.2
INTEGRATION SCHEMES
SURVEY OF
define
methods
(for
methods
t).
shown in previous
As
time
small
statement
stability
and
Although the
schemes
are
general
problems.
implicit
schemes
discussion
limits
it
methods,
and
applicability
Recent developments
applied
coupled
with
fluid-structure
methods.
The
explicit
Newmark, Wilson-O,
linear problems
applied to
systems
to
regions
of
others
Semi-implicit methods combining
to
more
the
(11)
with some having
limited
to
specific
include the implicit-explicit
with regions
high
problems
itself
should be noted that new
under constant development
methods
the
the
of
require
section.
integration
popular
time
while
instability,
and implicit
(3 point)
are examined in this
step, but
characteristics
Park, and Hilber
Houbolt,
time
displacement at
time
per
for
holds
accuracy
difference
are called corrector
sections, explicit methods involve
eliminate
to
steps
converse
central
of the
expense
computational
minimal
other
acceleration at
the velocity and
defined as functions
be
may
quantities, and thus
example,
implicit
of
terms
in
response
and
response,
predictor methods, while
current
the
response
current
t
referred to as
are
hence
the previous
response in terms of
the
express
explicit methods
The
explicit.
and
imp-licit
categories:
basic
two
into
classified
are
schemes
Integration
of high rigidity
flexibility,
such
as
in
(13,14,15,36,37,41,61,63).
the stability
and
accuracy
123
basic
the same
subject to
nevertheless, are
new integration schemes,
These
(58,64,83).
consideration
under
also
are
methods
explicit
factorizing property of
non-matrix
the
and
methods
implicit
of
characteristics
stability
and
4.1,
the
accuracy questions.
as a
derived
the
complex
for
equivalent
w,
to our
(T
4.25
Eq.
Eq.
A.16)
can
plots
and
derived
be
shows
an
period
method
displays
computational
high-order Taylor
Krieg (40)
better
cost
states
Fig.
the
than
frequency.
series,
and
4.3
by
elongation defined
into the
favorite
Similar accuracy
ratios.
rationale behind
central
The
among
explicit
that the central difference
stability
the
4.2 is
the algorithmic
integration scheme.
tends to be
stability
physical damping
for other
Park (60) demonstrates
schemes.
T,
4.26.
Eq.
for
appropriate
difference method
2 in
that
our
same as
Figures 4.1-4.4 offer insight
selecting
the
results such as Fig.
yielding
the
T,
4.4
Fig.
4.2
the exact undamped natural
is the
and
damping ratio,
the
Notice
onto
spectral radius
depicts
Fig.
various schemes.
Accuracy is analyzed next,
for
4.1
Figure
can be
results plotted
the
and
At/T, and
as a function of
regions
U,
function of
U-plane.
amplification matrix A
the
of
Ps
radius
spectral
section
in
outlined
procedure
the
Using
and
less
requires
predictor-corrector schemes,
4th-order Runge-Kutta
that the central difference method
scheme.
has
the
124
p
1.
ABBREVIATIONS:
S
R
R: orooosed p-method
(p-0. 8)
P
N
A: Hilber's. a-method
(a--0.1111)
A
N: Newmark's algorithm
0.9
(y-0.6111, 3-0.3086)
w: Wilson's algorithm
(0-1.4)
P: Park's algorithm
H: Houbolt's algorithm
0.8
0.1
0.01
Fig.
4. 1
W
W
H
1
At/ T
10
100
Spectral radii of the amplification matrix A as a function of AI/T (no physical damping)
(11)
125
GEAR
-
3
rd ORDER
GEAR
2
nd ORDER
4.0
2.0
STABLE
-2.0
Re (.&t - n)
0
-2.0
PARK
-4.0
WILSON (6 =1.5)
HOUBOLT
TRAPEZOIDAL R!JLE : ENTIRE LEFT - HAND PLANE
INCLUDING IMAGINARY AXIS
CENTRAL DIFFERENCE : - 2 j < S -,&t < 2 j
Fig.
4.2
Stability Region of Multistep Methods
( 60 )
126
W
N
H
P
0.3"{
ABBREVIATIONS:
R:
oroDosed p-mrethod
(P-0. 8)
0.02--
A: Hilber's- a-method
(a -- 0.111
N: Newmark's algorithm
(7-0.6111. A3-0.3086)
W: Wilson's algorithm
(6-1.4)
0.01
A
P : Park's algorithm
H: Houbolt's algorithm
R
0
Fig.
4.
3a
0.05
0.15
Q10
A/T
Algorithmic damping ratio~ as function of
H
At/T
(no physical damping)
(11)
P W
0.15 T
ABBREVIATIONS:
0.10--
A
R
: oroposed p-method
(p -0.8)
N
A: Hilber's- a-method
(a--0.1111)
N: Newmark's algorithm
(7 -0.6111, /3-0. 3086)
W: Wilson's algorithm
(0-1.4)
0.05-
P: Park's algorithm
H: Houbolt's algorithm
0
Fig.
4.
3b
0.05
0.10
Relative period errors r as a function of
0.15 A/T
At/T
(no physical damping)
(11)
127
NH
W
0.03ABBREVIATIONS:
R:
orooosed
p-method
(p-0.8)
A: HiIber's. a-method
Q.02..
(a--0.1111)
N: Newmark's algorithm
(y-0.6111, A-0.3086)
W: Wilson's algorithm
(0-1.4)
0.01-
p : Park's algorithm
H: Houbolt's algorithm
0
Fig.
4. 3c
005
Algorithmic damping ratiom
0.10
0.15 AIrT
as function of Ar/T (1-5 per cent physical damping)
H
(11)
W
0.15 -T
ABBREVIATIONS:
A
R
R :rocosed
(p-0.8)
p-method
A : Hilber's- a-method
N
(a--0.1111)
N: Newmark's algorithm
0
()y-.6111, /3-0.3086)
W: Wilson's algorithm
(0-1.4)
P : Park's algorithm
0.05--
H: Houbolt's algorithm
Fig.
4. 3d
0
Relative period errors
r
0.05
010
as a function of
At/T
0.15 A/T
(1-5 per cent physical damping)
(11)
128
0aa
If
a.
Damping Coefficients of Multistep Methods
.0
I.-
1.2
I,-
z
Lu
0
LU
0
b.
Fig. 4.4
0.4
-
0.8
12
Frequency Distortion of Multistep Methods
Damping and Frequency Distortion Parameters versus wAt
(60)
129
stable.
are conditionally
schemes
method to
range from -21
undamped
free
In particular, Fig.
4.2
(imaginary axis) we have
problem
vibration
<
cr-
2/w
=
The
and
(33)
it appears
4.3
Fig.
From
scheme of Hilber
of Bazzi and
that
demonstrate good algorithmic damping and
these
Moreover,
control
the numerical dissipation by
parameter ( a
or p ).
rule),
trapezoidal
The Newmark
however,
imposes no error in the
in Fig.
subject to
that
the recent
Anderheggen (11)
low relative period
a nonphysical external
method
(a=0. 2
of problem under
that
stable
physical damping, as
it
is
emphasized that the
an appropriate numerical integrator also depends
expected
, 6 =0.5,
4.4a.
As a consequence,
is
5
is also unconditionally
specified
to
the capability
two schemes have
errors.
modelling
scheme
is
implicit scheme
selection of a "best"
interpretation.
shown
difference
central
the
that
period compression.
a
exhibits
(4.27)
T/7
4.4
Notice from Fig.
and
the
for
result,
a
As
21.
to
central difference
the
implying for the critical time step
wAt<2,
At
order
of
region
stability
the
denotes
explicit second
all
that
furthermore
and
schemes,
order
second
explicit
all
of
time step
critical
largest
consideration.
accurate
up
to
If
the
spatial
of
choice
on the
type
structural
the highest modes and it
is
the loading will excite practically all modes,
130
consist of a
structural problems, however,
represents
that adequately
geometric
the
Most
numerical
model
structural
shape
but has no physical
lowest structural modes,
of the
and many
is preferred.
integrator
numerical
"exact"
an
then
significance for the highest modes. Although the loading may
excite
only the
numerical
to
approximate
an
of
the use
leading
integrator may induce numerical resonance
This
response.
class
solved preferably by numerical
the
high frequency
displaying
consequently,
integrators
that damp out
in
errors
no
at
stable
remain
and
response
high
problems,
of
is
frequencies,
insignificant
physically
the
of
amplification
frequency
modes,
lowest
these
the high frequency
contribution.
AND
ACCURACY
FOR
SYSTEMS THEORY APPROACH
4.3 LINEAR
STABILITY ANALYSES
stability analysis scheme presented
The accuracy and
is
section 4.2
usually
are
quite cumbersome,
often
however,
if
not
Accuracy,
impossible.
varies with the load history since
free
load histories
to arbitrary
Extensions
vibration problem.
the homogeneous
to
applied
in
load
the
vector
is also expanded by the finite difference approximation, and
hence
the
effect of
the load vector should
in an accuracy analysis.
to a general
experiments,
load
load vector
but
histories
problems
cannot
Extending
is
possible by
this procedure
since
be
the
the
derived
accuracy
in closed
included
analysis
conducting numerical
is limited
analytical
also be
to a few simple
solution
form.
for
most
Furthermore,
131
ratio should
the algorithmic damping
by
consists
procedure
basic
the numerical
corresponding to
the ability
to
approach
The
theory.
exact
frequency
the
transfer
as
integration scheme.
theory
systems
approach
the algorithmic damping as a
to examine
excitation
the
of
function
an alternate
frequency response function
linear
the
of
Advantages
include
referred
approximate
function) with the
excitation
of comparing the
(often
function
response
and
linear systems
using
accuracy
examining
presents
section
This
frequencies.
a quantity
structural
of the
that varies as a function
be viewed as
frequency
and how
difference expansion of the load vector affects
the finite
the solution
accuracy.
procedure
The
numerical
the
Consider
will
be
now
approximation
differential equation given by Eq.
m j
n
+ cqn +
n
All numerical
other
kq
=
F
described
of
in
the
governing
4.3.
(4.28)
n
integrators expand
response quantities,
detail.
and
the response
in
terms
of
in general we have
k
n
=
k
4n=Ea
_~q
(4.29)
132
where a
Eq.
6
are
constants.
4.28 we obtain
k
k
k
and
where
the y
the
frequency
transform,
+ kq
6 q
atq. + c
M
are
is
equation
corresponding
the
using
in
(4.'30)
Equation 4.30 is
at
particular
in
4.29
Eq.
ZY F,
constants.
domain
and
=
Substituting
transferred
Fourier
discrete-time
any
obtained
frequency
by
the
to
0
the
following
substitutions:
Q(Q)
> e
> e~
-2
Q(2)
e
Q-2Q
(4.31)
F
' 0
e F(G)
->
n
> e
F n-1
, F(Q)
-1i2 QF (Q)
F -2
yielding
k
m
i=;
k
.eiG(n-i)Q(_)
+ke0Q(2)
Finally,
+ c
k
1
6. e-iGn-iQ(
y
i=0 i
integrator is
Eq.
4.28
H
32)
function corresponding
to
given by
(4.33)
H(Q) = Q(P.)/F(Q)
The analytical
(4.
i(n-i)F()
the approximate transfer
the numerical
)
i=01
form for the
exact transfer function He (W)
in
is given by
(Q)
=
(-w 2 m +
iWc + k)~1
(4.34)
133
can
approximate with respect to the
damping
algorithmic
frequency of
the
transfer function.
An
obtained
by
comparing the
4.33 and 4.34.
frequency domain accuracy analysis was applied
This
central
Houbolt
exact
is
ratio
imaginary parts of Eqs.
the
by
resonant
the
of
shift
the
considering
In this
is derived
the period elongation
development,
particular
dissipation
4.33 and 4.34.
by comparing Eqs.
derived
be
elongation and numerical
the period
Measures of
(H),
difference (CD),
and Park
Newmark
the Park method
calculations for
(trapezoidal rule, N),
integration schemes.
(P)
detailed
follows:
C4. 35)
(cosG - 1) + 12a 2 sin2 + a 3 )
iCD)
=
(2a
HN(g)
=
(1 + cos) /((4aI + a 3 )cosQ + i4a
HHG2 )
= (2a1 +
1
The
are provided in Appendix A.
transfer functions are as
The approximate
2
sinQ - 4a
+ a3)
-iQ
-i20
H(0)
=
(25a /9 + 10a2/
3
+ (-a 1
2a 2 /3)e
-i30 -1
)
+ a 3 + (-50a /6 - 5(42
+(115al/12 + 2a 2 )e-i2Q +
+ lla 1 /6e~i4s
-
(4-36)
(4.37)
lla 2 /3 + a 3 + (-5a-6a2)e
+ (4a1+3a 3 )e
to
(.38)
(-50a 1 /9 - a2/3)e-i3Q
- a1 e-i5/3 + a1e-i60/36)~
where
Q
= Wat
W
= natural frequency
a,
= 1/47T2
nAtina
2
a
3
= At2 a
ni
=
viscous damping ratio
(4.
39)
134
The
H (2)
+ i2a 2 Q +a 3 ) 1
2
=
(4.40)
nondimensionalized
4.40 have been
4.35 to
that Eqs.
Notice
function is given as
transfer
exact
by multiplying the usual transfer function
H(w) = (-w2m + iwc + k)
(4.41)
in the
period
compression
to
difference between
(
the
specified
has
Similar
Also,
damping because the
imaginary
term
the
imaginary
and
ai
is not
The corresponding
and
Park
schemes,
viscous damping is
directly
induces
are
terms
transfer function
exact has
not
2a 2 Q-
Notice
damping
real term, and
for F-0,
in
not
is
conversely
associated with any
the central difference
no numerical dissipation when
impose
schemes
viscous damping
viscous
with any
Since a 2 =0
terms.
Newmark
a3
and
the
the
than
less
to the Newmark method.
algebraically associated
that
while
to al
difference
central
the
damping
specified
the
that
is due
corresponding
approximate
2a 2 sinO
statements apply
particular
terms
algorithmic
an
exhibits
scheme
the
the
that
evident
it is
central difference method
-Q2).
cf.
2(cos Q-1)
been
has
2 = wAt.
4.35 with 4.40,
Comparing Eq.
W
frequency
load
the
.
Also
-m
At 2
nondimensionalized to
by
specified.
transfer
Eqs.
4.37
functions
and 4.38,
associated with
period
for
Houbolt
indicate that
real terms
elongation
the
and
the
and therefore
the
terms
135
Newmark;
third,
figure of
each
the corresponding
maxima and
in
Numerically
plotted.
when
to
the exact
is
to
demonstrate
artificial
Houbolt
Newmark,
use,
Its
it
an
the
of
is the worst
Houbolt,
the
inverted
the
transfer function.
resonant
of
the
of
those
and
provides
restricted
is
because
The Newmark method
frequency
implicit
distortion
In
Park methods deamplify
tends
and
while
the
general,
the
methods,
considered.
the
frequency
frequency (least
however,
least
the
damping
for
values
frequency are provided
resonant
in tegrator.
explicit
exactly
captured
that the central difference method
estimate
the maxima of
the
line
distortion).
of
approximate transfer function and
the
best
representations
"v"
to
Notice
for
hat
corresponds
solid
similar plots
obtained
plots
all
In
4.1.
Table
a
for
be
not
could
Figures
phase plots
a result,
As
4.40.
and Eq.
function
transfer
each
g=0.10.
give
d
viscous
specified
a
discrete
are
These plots
4.35-4.38
Eqs.
The first
second,
the
and
c
Figs.
and
At/T =0.20.
Park.
figure give the amplitude and
and b of each
=0.05,
and
=0.05
second,
to
corresponds
set
difference;
fourth,
and
Houbolt;
ratio
damping
central
the
being
set
first
two with
four sets of
consist of
eight figures
The
to 4.12.
4.5
are given in Figs.
transfer functions
Plots of the
At/T
Newmark schemes.
central difference and
to the
numerical
induce
directly
3
imaginary components, contrary
the
dissipation by affecting
the
a
and
al
to
corresponding
the response
b.
a,
3500
3000
2500
:
2000
-1
-2
fo 1500
1000
-2
500
-3
0
0.0
0.5
1.0
1.5
Frequency
2.0
2.5
3.0
(rad)
3.5
At/T
1.5
n
=
0.05
2.0
Frequency (rad)
H
c.
350
300
d.
0.0"'M
-
-0.5
250
-1.0
200
T -1.5
0
n
~0
150
L
to
-2.0
100
-2.5
50 -
-3.0
0
0.0
I
0 .5
-3.5
1.0
1.5
Frequency
Fig. 4.5
2.0
frad)
2.5
3.0
3.5
At/T
-
0.0
= 0.20
0.5
1.0
1.5
Frequency
2.0
2.5
3.0
3 .5
[rad)
Transfer Functions for Central Difference Method (5 per cent physical damping)
2000
b.
0.0
1800
-0.5
1600
-1.0
1400
rN 1200
T0
-1.5
1000
.0
|i -2.0
800
600
-2.5
400'
-3.0 -
200
0
0.0
0-5
-
-3.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0
0.5
1.0
At/Tn = 0.05
Frequency (rad)
1.5
2.0
2.5
3.0
3.5
Frequency (rad)
H
C.
180
-.1
d.
0.0,
160
-0.5
140
-1.0
.120
:E
0
-1.5
100
Ci
L
to
80
-2.0
60
-2.5
40
-3.0
20
0
0.0
0. 5
-3.5
1.0
1.5
2.5
3.0
3.5
L
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
(rad)
At/T = 0.20
Frequency (rad)
Transfer Functions for Cenpral Difference Method (10 per cent physical damping)
Frequency
Fig. 4.6
2.0
b.
a.
I
I
I
0.0
1
transfer function
-0.5
Exact transfer function
-1.0
N
3 -1.5
0
.0
0
- -2.0
-2.5
-3.0
0 '
0.0
I
0.5
1.0
1.5
Frequency
2.0
2.5
3.0
Crad)
3.5
At/T
0.0
=
0.5
1.0
1.5
2.0
2.5
3.0
3.5
2.5
3.0
3.5
Frequency (ead)
0.05
c.
d.
250
0.01'
-0.5
200
-1.0150
el%
:E
0
L
0
100
1.5-
-2.0
-2.5
50
-3.0
0 '
0.0
'I
0.5
1.0
1.5
4.7
2.5
-3.5
3.0
3.5
-
0.0
0.5
1.0
1.5
2.0
Crad)
At/T = 0. 20
Frequency (rad)
Transfer Functions for Newmark Method (5 per cent physical damping)
Frequency
Fig.
2.0
a.
2000
b.
0.0'
1800
-0.5
1600
-1.0
1400
1200
0
.0
P0
-1.5
1000
- -2.0
00
600
0
-2.5
-3.5
0.0
0.5
2.0
1.5
1.0
2.5
3.0
3.5
At/T
Frequency (ead)
0.0
n
=
0.5
1.0
0.05
2.0
1.5
Frequency
2.5
3.0
3.5
rad)
HLA
110
d.
C.
140
0.0'
120
-0.5
100
-1.0
i
80
-1.5
.0
60
aY
to
L.
to
-2.0
N/
40
0
0.0
-
-3.0
0.5
1.0
1.5
2.0
Frequency (rad)
Fig. 4.8
2.5
3.0
3.5
At/Tn = 0.20
0.0
0.5
1.0
1.5
2.0
2.5
Frequency (rad)
Transfer Functions for Newmark Method (10 per cent physical damping)
3.0
3.5
b.
a.
3500
3000
2500
Y 2000
L
1500
1000
500,
0
0.0
0.5
1.5
.0
2.0
2.5
3.0
3.5
At/Tn
Frequency (rad)
1.5
0.05
2.0
2.5
3.0
3.5
Frequency (rad)
0
c.
d.
250
0.0
'u
-0.5
200
-1.0
Tj
150
-1.5
IL
0
-2.0
100
-2.5
50
-3.0
3.5
0 '1'
0.0
0.5
1.0
1.5
2.0
frequency (ead)
Fig. 4.9
2.5
3.0
3.5
At/Tn = 0.20
0.0
0.5
1.0
1.5
2.0
2.5
Frequency (rad)
Transfer Functions for Houbolt Method (5 per cent physical damping)
3.0
3.5
b.
a,.
2000
1800
1600
-1.0
1400
1200
L2
0
(0
-1.5
1000
-2.0
800
600
-2.5 -3.0 -3.5
0.0
0.5
1.5
1.0
Frequency
2.0
2.5
3.0
AtIT
[rad)
0.0
3.5
1.0
= 0.05
1.5
2.0
2.5
3.0
3.5
2.5
3.0
3.5
Frequency trad)
0.0'
140
f-%
lb
0.5
120
-0.5
100
-1.0
-1.5
80
r%
M0
-2.0
60
-2.5 -3.0 -
20
-3.5
0
0.0
0.5
1.0
1.5
2.0
Frequency (rad)
Fig. 4.10
2.5
3.0
3.5
At/T
0.0
= 0.20
0.5
1.0
1.5
2.0
Frequency [rad)
Transfer Functions for Hou~olt Method (10 per cent physical damping)
b.
a.,
4000
-1.0
T -1.5
e-%
n-
L
M
-2.5 -3.0 0.0
0.5
-3.5
1.5
1.0
2.0
Frequency
2.5
3.0
3.5
At/T
frad)
0.0
0.5
1.0
= 0.05
1.5
2.0
2.5
3.0
3.5
Frequency [rad)
H
d.
C.
250
-0.5
200
-1.0
-1.5
'I,
n
0
L
iv
100
-2.0
-2.5
0 '
0.0
'L
0.5
-3.5
1.0
1.5
4.11
2.5
3.0
3.5
(rad)
At/T
0.0
0.5
l.0
l.5
2.0
= 0.20
n
Frequency frad)
Transfer Functions for Park Method (5 per cent physical damping)
Frequency
Fig.
2.0
2.5
3.0
3.5
b.
a.
2000
1800
-0.5
1600
nct ion
1400
r function
-1.0
1200
T0
i -1.5
1000
-2.0
800
600
-2.5
-3.0
200
-3.5
0
0.0
1.0
0.5
1.5
2.0
2.5
3.0
At/T
Frequency (rad)
0.5
0.0
3.5
1.0
= 0.05
1.5
2.0
2.5
3.0
3.5
Frequency (rad)
H-
d.
0.0 1"
-0.5 -1.0
I-
T0
L
M
-1.5-2.0
-2.5
-3.0
-3.5
5
1.5
1.0
2.0
Frequency (rad)
Fig.
4.12
2.5
3.0
0.0
3.5
At/T
1.0
1.5
2.0
2.5
Frequency (rad)
= 0.20
Transfer Functions for Park Method
0.5
(io
per cent physical damping)
3.0
3.5
At/T
Maximum Frequency
n
p
I
I
IH(Q) |
0
Exact
CD
N
H
P
0.05
0.00
0.3142
0.3155
0.3116
0.3023
0.3090
0.20
0.00
1.257
1.359
1.122
0.9144
1.013
0.05
0.05
0.3138
0.3155
0.3108
0.3007
0.3082
0.20
0.05
1.255
1.359
1.120
0.8873
1.002
0.05
0.10
0.3126
0.3155
0.3085
0.2978
0.20
0.10
1.250
1.359
1.113
0. 8608
CD N
H
P
-
r
r
I
.
Exact
CD
N
r
I
H
P
16090
88720
00
61
221
3313
3987
00
00
4004
4050
4005
250
321
250
54
0.3059
2008
2025
2010
1857
2006
0. 9878
126
161
126
48
83
central difference
Newmark
Houbolt
Park
Table 4.1
P
Resonant Frequencies and Amplitudes of Numerical
Integration Transfer Functions
120
4-
145
for
amplify for those
more accurate integrator than
that
analyses.
Extensions
the Park method,
it
as
should
a
be
linear
to
limited
are
results
these
remembered
a reverse
the Newmark method
Although these plots portray
trend.
Figs.
to
4.7 a,c
difference method depicts
central
The
a,c.
4.12
Figs.
shown in
as
less
and
frequency
resonant
the
than
frequencies greater
nonlinear analyses are presented
to
in section 4.5.
As mentioned
period
the
of
plots
previously, it
T
algorithmic damping
in
0.05,
0.02,
exhibits
method
when
and
nonzero
a
elongation
(or
versus At/T
.
Eq.
A.16)
is
damping
viscous
and
of 0.00,
difference
E=0.
artificial damping when
no
derive
plots are given
Such
the central
that
Notice
0.10.
bias,
to
specified damping ratios
for
4.13-4.16
Figs.
is also possible
However,
the central
specified,
difference method produces a negative artificial damping.
summary
In
analysis
accuracy
period
examining
scheme
provides
time
the
resonant
as
to
how
throughout
effect
of
the
the
response
entire frequency
expanding
approximation
is
is
also
the
deamplified
spectrum.
load
included.
by
of
dissipation.
domain method where the
approach
this
frequency,
means
elegant
period elongation
algorithmic damping ratio and
to
an
numerical
and
elongation
linear systems theory
the
that
the conventional
to
Contrary
note
we
a
correspond
gives
or
insight
amplified
Furthermore, the
finite
difference
Finally, notice that
the
146
0D
301
I
25
x
Central Difference
v
Newmark
*
Park
15
10
I
I
I
I
iI
-
r
A
Houbolt
5
A
0
x
x
x
-5
-10
I
I
I
I
I
i1
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.1+ 0.16 0.18 0.20
At /Tn
Fig.
4.13a
Bias as a Function of 1At
(no physical damping)
16
-
-
L
14
0
12
10
0CI
8
6
A
0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.1+ 0.16 0.18 0.20
At /Tn
Fig. 4.13b
Artificial Damping as a Function of
(no physical damping)
oAt
147
30
25 15
0
x
Central Difference
v
Newmark
*
Park
A
10
Houbolt
5
A
0
x
-5
A
-A
x
-
-10
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.1+ 0.16 0.18 0.20
At/Tn
Fig.
4.15a
Bias as a Function of At
(2 per cent physical damping)
18
o0
0.
16
I+4
12
10
8
-
I
--
2
0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.1+ 0.16 0.18 0.20
at /Tn
Fig. 4.15b
Artificial Damping as a Function of !At
(2 per cent physical damping)
148
30
25
20
15
0
10
5
-5L
'
-10'
'
'
'
''
'
'
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.1+ 0.16 0.18 0.20
At /Tn
Fig. 4.14a
Bias as a Function of MAt
(5 per cent physical damping)
18
18
14
CO
12
CL
10
C
8
6
2'
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.1+ 0.16 0.18 0.20
At/Tn
Fig. 4.14b
Artificial Damping as a Function of
(5 per cent physical damping)
At
149
0
35
30
25
20
15
x
Central Difference
v
Newmark
Park
A
10
5
Houbolt
A
f
A
A
A
x x
0 -5
-to
0. 00 0.02 0.04 0.06 0.09 0.10 0.12 0.1+ 0.1 0.19 0.20
At
Fig. 4.16a
/Tn
Bias as a Function of EAt
(10 per cent physical damping)
20
18
0
16
A.2
I+
CL
A
12
A
10
I
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.1+ 0.16 0.18 0.20
6
t
At /Tn
Fig. 4.16b
Artificial Damping as a Function of EoAt
(10 per cent physical damping)
150
these
Consequently,
themselves.
impulsive free vibration problem
is
where h(t)
h(t)
by the
by
multiplied
or
deamplifies
Stability
the inaccuracies
can
where
the
z is a
region
x(n)
of
is
the
given by
Stability
which
region
of
has
no
An illustrative
difference method
is
then defined
z-transform
by
or those
converges.
z-transform
the
of
the
In
convergence cannot contain any
a stable right-sided sequence
therefore
The
transform (57).
Fourier
(4.42)
convergence
particular,
circle.
the
of
the
using
by
investigated
be
complex variable.
z for
n>0)
transfer
z-n
sequences of
poles, and
which when
function,
in the approximate
of a sequence x(n)
z-transform X(z)
n=- o
also
place
in
z-transform
=
representation
each frequency.
function at
X(z)
with
transfer function, amplifies
approximate
the
domain
by
multiplying
comparing He(w)F(w)
then
weighting
a
represents
load
the
The
loadings
general
frequency domain,
to the
the frequency
essence,
In
H(w)
of
(the Fourier pair
more
to
transfer function, and
H(w)F(w).
of
load
the
transferring
to the
accuracy plots apply
the impulse response function).
extended
be
can
analysis
is
by
functions
transfer
the
to
correspond
given here
plots
poles
located
(defined only
outside
stability analysis
is given in Appendix B.
of
the
the
for
unit
central
151
4.4 RELATION BETWEEN TIME
ANALYSES
DOMAIN
FREQUENCY
AND
Time and frequency domain analyses are
section 4.3 should
therefore be
time domain analysis given in
section 4.1,
domain analysis presented in
to the
equivalent
assuming that we are
in
the
free vibration problem
conventional
stability and accuracy
considering
The
cases.
both
analyses were restricted to
hence the
forms
equivalent
The alternate frequency
problem.
same
the
analyzing
for
STABILITY
free vibration problem,
the
limited to
following discussion is
and
this particular
problem.
The stability analysis
or equal
than
less
homogeneous
the
to
ensured when the
Stability is
equation.
the
corresponding
4.22,
Eq.
to one.
governing
spectral radius
Let us now transfer Eq.
X z -n
z-n + c2
dz-n = C
n=0
Eq.
is given by
The response Q(z)
Q(z)=
is
4.22 to
by
frequency domain using the z-transform, defined
4.42.
with
begins
time domain
in the
n=0
n=0
CO
n
+ c3
-n
n=0
c
z
1 z-X
(443)
z
z
+c
2 z-X2
z{c 1 (z-X 2 ) (z-X
3
) + c 2 (z-X
(z-X
For
the
free vibration
regarded as an impulse,
F(z)
=
1
z
3 z-.3
1
assuming
) (z-X
3
z
) + c 3 (z-X
> 1
1
) (z-X
2
1 ) (z-X2 ) (z-X
3
problem,
the
excitation
F
can be
implying that
(4-44)
152
The
transfer function H(z)
=
H(z)
is the
is
a right-sided
convergence of H(z)
is
the
definition in the
poles
of
zero may occur,
possible for
the
(load
the
vector)
mathematically
expanded load
vector to
the
accuracy
system stability.
analysis.
Equation
4.4
vibration
response,
and
response.
integrated
itself gives
and
of Q(z)=
both
where
it is
for
A similar argument can be presented
4.4
therefore the
and
situation
finite difference
of H(z)
Notice
from the poles
therefore
and
than or
z-plane.
(transfer function) and numerator
denominator
affect the
a
particular,
In
differ
may
Q(z)=H(z)F(z)
H(z)-l.
equal
be an impulse,
that F need not
in general
the
in
or on the unit circle
lie within
poles
that all
frequency domain is
the
stability
equivalent
the
Therefore,
one.
to
equal
ensure
in
less
is
of
located at
are
stability
radius
spectral
that the
implies
time domain
poles
to
circle
poles of H(z)
unit
the
of
implying that all
As mentioned previously,
X3.
and
X2,
the
4.43,
From Eq.
stability.
the
sequence,
exterior
on the unit
inside or
located
H(z) must be
Xi,
the z-plane,
in
response
impulse
h(n)
Since
centered
circle
unit
the
z-transform of
h(n).
of
region
(4.45)
Q(z)/F(z)
H(z)
function
defined as
is
The
Eq.
4.25,
impulse
the
to
corresponds
the
impulse response
can
be considered
functions.
Rather
free
numerically
response function h by
the free vibration response.
4.25
exact
the exact
Therefore,
Eqs.
and approximate
than comparing
the
two
153
in
to
domain
time
the
of the artificial
obtain estimates
equations
damping
and frequency distortion, we
transfer both
the
frequency domain and compare
the properties
to
corresponds
therefore
the
4.5
limited
was
4.3
section
presentation
in
to the Fourier transform H(w),
and
circle
in
the
that
Notice
H.
transfer functions
their
of
behavior on the unit
to the
z-plane.
ANALYSES
STABILITY
AND
ACCURACY
OF
EXTENSION
TO
NONLINEAR PROBLEMS
and
properties,
of section
analysis
problems
characterized
matrices.
This section examines an analytical
surveys the
nonlinear problems and also
structural
dependent
time
by
nonlinear
for
modified
be
must
4.2
system
and accuracy
stability
the
consequently
the
on
depends
A
matrix
amplification
The
approach
to
results of numerical
experiments.
4.5.1 Analytical
The
stability of nonlinear problems is
interaction of
formulation
of
Unconditionally
unstable
for
the
local
the numerical
the
stable
analysis
governing
the
Park
motion.
of
equations
integration
to
governed by
integrator with the approximate
for nonlinear analyses.
stability
the
Approach
schemes
may
(59,60,62)
become
extends
nonlinear systems by accounting
change in stiffness with
instability (instability
time
and
occurring in a
then
small
examining
sequence
154
of
problems
those
is
approximation
modulus
tangent
where a pseudo-force or
two cases:
evaluated exactly and
are
nonlinearities
the
where
approach considers
His
steps).
time
involved.
the nonlinearities
The exact evaluation of
the
stability
Fig.
4.2,
z
discussed
For a linear undamped system the equation describing
first.
2
is
of the
region
ref.
(see
given as
is
rule,
trapezoidal
in
shown
60 for details)
2
-2
W
At
+
(4.46)
= 0
4
polynomial
but
included,
are
nonlinearities
When
the
in
is derived
4.46,
same fashion as for Eq.
for the numerically expanded governing equation where we
with
varies
w
that
assume
4z2 + (At 2) (
where
is
W
nn
4.47
n )z nn+ (Lt 2/2)((
the natural
that
coefficient
-
if W
of
n
z
< E
n-1
is
n-l
Eq.
4.46
time t
at
negative,
is
stability.
(60)
+wn-1=
0
n-
(4.47)
from
Notice
.
(a softening system),
and
obtained.
On the
Also,
if
other hand,
the trapezoidal
the
the
consequently
local instability.
S> (a hardening system),
local
n
the
particular,
In
frequency at time t
trapezoidal rule produces
equals
time.
trapezoidal rule becomes
stability equation for the
Eq.
characteristic
the
En
if
rule retains
155
is
Houbolt method
the
for
equation
corresponding
The
(60)
12z
and
the
3
+ 4z
2
2
-2
+w At (z + 1)
Park method
3
=0
(4.48)
(60),
+ 1)
256z6 + 576z5 + 516z4 + 216z3 + 9W 2t
n 2(z
Notice
are
tangent
m'odulus
Park
Houbolt and
the
corresponding
from
the
nonlinear
equations
now
W
only because
stiffness,
structural
time dependent
A
varies with time.
positive.
linear stability
equations
the
since
remain
always
Wn
Furthermore,
only affects the accuracy.
therefore,
instability of
apparent that the
from the admission
in
used
pseudo-force and
the
tangent
is
it
4.49,
and
4.48
trapezoidal rule arises
of past-step derivatives
considers next
Park (60)
integrators
Eqs.
4.47 with
Comparing Eq.
wn-l'
numerical
the instability of
with
conjunction
stiffness methods,
approximate
the
that
based
are
finite difference extrapolations and differentiations
the external
difference
approximations
linearize
a result introduce
derivatives.
combining
force
internal
and
of motion, and as
step
(4.49)
involved
are
approximations
containing
coefficients
on
0
unconditionally stable when no pseudo-force or
methods
differ
that the
4.49
and
4.48
from Eqs.
=
It
the Houbolt and
is
vectors.
the
Park methods
finite
These
governing equations
errors
expected,
of
in terms
of past
consequently,
that
no
past
containing
156
derivatives
This
The
examining
by
confirmed
the
for the pseudo-force approach.
derived
equations
stability
past
unstable.
produces a system that may be
suspicion is indeed
introduce
that
derivatives with solution approximations
stability equations in terms
of
X are given as
follows
(60):
Rule:
Trapezoidal
X2 + -At
4X4 - 8
+ (461-
+ 2(1 +
62
+ 1)
]
+ 2(6i 7 62 )-21
2
(4.50)
(
Houbolt Method:
(2X3 - 5X2 + 4X - 1) +
2
X)
w2 (X3 + 26 1 X
(4.51)
= 0
Park Method:
152 +16X -1)2 + 36
(13-
2
At 2 (6
+2
5
_
4)
=
(4. 52)
where
6=
2
(1/w
@f(q n-1-i)
)
(4.53)
f(q)
=
-j
2
Notice
that Eqs.
terms
of
the
conditionally
qn + FNL
4.50-4.52
.,
and
contain past
therefore
stable behavior.
all
step derivatives
methods
exhibit
in
a
157
Figure 4.17
the
S.
III
correspond
respectively,
case.
(area
for two different wAt values.
plane
In
to
while the other
particular,
inside the diagrams)
rapidly
than that
indicating
that
nonlinearities)
softening
for
in
that
govern more severely
contracts
more
is
increased,
(for example,
geometric
quadrant III
effects (for example,
stability region
the
quadrant I
stiffening effects
systems,
to the combined
regions refer
observe
on
Regions I and
softening
and
hardening
plotted
regions
depicts the stability
as
the
wAt
time increment
than
plastic material behavior).
158
TRAPEZO1DAL
WILSON
HOUBOLT
I'
3.0
'I
'I
I
PARK
2.0
I
/
/
/
Region
I
1.0
~
Region I
-2.0
( a)
.At
2.45
Region IV
Region III
Region II
Region I
-2.0
'31
(b)nAt a 4.5
Region III
Fig.
4.17
1.0
Region IV
Conditional Stability Region of Four Methods for
Linearly Extrapolated Pseudo-Force Solution Procedure.
(6 0)
159
4.5.2
Experiments
Numerical
numerical
and
of
interact with
integrators
identifies
iterator,
equilibrium
experiments.
admit
summary
a
The
assesses
numerical
conducting
presents
section
This
consequently,
by
readily
more
instability
research,
of
state
current
however,
derivation.
pseudo-force
and
a source
the material model,
such as
instability
of
sources
formulation
derivatives as
problems,
Nonlinear
to
into how the
insight
the solution
the inclusion of past
error propagation.
other
valuable
provides
problems
nonlinear
approach
the analytical stability
The extension of
of
these
numerical experiments.
Stricklin in 1971
to
subjected
a
step
nonlinearities were included,
derived
6
=0.5),
the
by
and
pseudo-force
4th
Houbolt
schemes
linear load extrapolation.
scheme
was
difficulties.
was more
also
The
and
of
applied
the
results
equations of motion with
A quadratic
implemented,
indicated
but
that
load
acceptable
errors.
extrapolation
produced
numerical
the Houbolt method
stable and accurate than the Newmark
Runge-Kutta method
(a =0.25,
conventional
a
to
kinematic
solution was
the numerical
Runge-Kutta, Newmark
order
formulation
Only
loading.
pressure
revolution
of
(81) analyzed a shell
required extremely small
scheme and the
time
steps
for
160
problems
examined structural
(93)
1972
Wu and Witmer in
with kinematic and material (elastic-plastic) nonlinearities
central difference, Newmark (trapezoidal),
the 3-point
using
methods.
Houbolt
and
formulation was employed for
produced accurate
the
method
Also in 1972 Weeks (86)
The
central
an
and Houbolt
Only
the
method
Newmark methods
enough
stiffness
stable.
remained
remained stable
formulation,
considerable
large time
damping.
the
but
at
solution
the
Weeks
method with load extrapolation
and
while
the
Houbolt
and
Both
the
time
increments
when using
concluded
small
method
that
time
were
Newmark
the
increments
Houbolt
a
excited by
nonlinearities
formulation
pseudo-force
reproduce
to
kinematic
to
subjected
history and a cantilevered rod
load.
In
system
SDOF
a
to
method became unstable for
Houbolt
tangent
difference, Newmark (trapezoidal),
impulsive
included.
pseudo-force
iteration.
load
rectangular
the
equilibrium
applied
methods were
linear problems.
studied both
Newton
with
formulation
stiffness
for
increment
time
for
than
central difference
load extrapolation and the
linear
with
formulation
the
scheme again
increment
time
critical
to that
nonlinear problems compared
extrapolated
were
the Houbolt
Furthermore,
smaller
a
exhibited
for
formulations were used
larger
at a
results
method.
Newmark
implicit schemes while both
results,
Based on their
linearly.
the
pseudo-forces
The
scheme.
the explicit
conventional
unconventional
and
conventional
pseudo-force
A
small
the tangent
exhibited
the Newmark
increments
161
computational
results at minimal
the most accurate
produced
cost.
McNamara analyzed in 1974
(trapezoidal),
operators
combined with the
modified
a
using
reformulated after
an
the same beam with
of kinematic nonlinearities while the
In
both
cases
the Houbolt method was the
the
step
load
case
for all
results
time steps
time
increment 1/30th of
was the most
Park developed in 1975
the
combining
scheme
Gear
compared
was
conventional
for the
pseudo-force
the
external
impulse
clamped at
the edges
apex.
Kinematic
load problem.
stiffly stable
scheme
method
Houbolt
with
supported
using
linear
by
The
a
load
cylinder under a
and also a shallow spherical cap
and subjected to
and
method
Newmark
impulsive
formulation
extrapolation applied to a simply
uniform
The
and three-step methods.
two-step
with
a
(60)
1/5th
central difference, at a
the largest.
unstable scheme
Newmark and
increment
a time
largest considered and the
the
For
stable
produced
considered while the
Wilson-e methods became unstable at
of
model).
best scheme.
method
Houbolt
the
second case also
(elastic-plastic
nonlinearities
included material
case consisted
The first
impulsive load were considered.
only
both
A beam clamped at
step).
step load and
to a point
subjected
ends
each time
(stiffness
scheme
iteration
Newton
formulation
stiffness
tangent
(9=1.5)
Wilson-O
and
Houbolt,
Newmark
difference,
the central
(46)
a
material
step
load
at
its
(elastic-plastic)
162
only kinematic
with
stiffness
each time step).
at
The
to a cubically hardening
then
and
the most
stable results
gave
also
for
instabilities
At/T>1/4
At/T>1/6 in the softening
Gere,
Adeli,
trapezoidal
rule with
Runge-Kutta
and
formulation of
edges
free
bilinear
1978
in
(1)
central
conducted
difference,
4th order
Houbolt,
tangent stiffness
and
material
the
other
model
two
was
Newton
problem consisting of
its plane at
a
in addition
cycle iteration, and the
plane stress
plate uniformly loaded in
and
case
the equation of motion and a modified
A
global
display
case.
were combined with the
method.
iteration
two
elongation.
hardening
the
method
exhibited a
the implicit Newmark (trapezoidal),
schemes
Park
to
began
The explicit
schemes.
the implicit
and
in
but
period
greater
Weaver
and
The Houbolt
study of popular explicit schemes
comprehensive
to
a
Park
cases and provided
in both problems,
method
Newmark
the
Finally,
and
damping
significant
The
spring problem.
response.
accurate displacement
softening spring
to a
results in both
stable
produced
scheme
(trapezoidal),
Newmark
Houbolt, and Park methods were applied
stiffness
(reformulate
iteration
Newton
modified
formulation
increment.
tangent
the
employed
(59)
studies
Additional
accurate
an
time
larger
a
for
response
displacement
while
second case.
the
linearities were admitted in
indicated the Park method produced
results
The
problem
the first
nonlinearities were considered for
a
the mid-line with two
clamped
employed
was examined.
and
A
kinematic
163
The
included.
nonlinearities were
while
4th
the
rule with
time
best
The Park method was deemed
numerical damping.
freedom was
increased.
Recently in
the
1981 Steigmann
unconventional
pseudo-force
was
The base case
The
an
impulsive
load at a
rate
Analyses
approach
of
with
the
employing
the
were also conducted.
central
the
Houbolt,
central region of the
dependent
more
a study using
combined
to a doubly clamped beam
elastic-strain hardening, and
strain
concluded
(trapezoidal),
nonlinearities were included,
-
(78)
using
derived
Newmark
methods were applied
number
as
degrees
of
the
schemes.
iteration
quasi-Newton
method
the Park
formulation
stiffness
tangent
method.
when
inclusion of
The
problems without kinematic nonlinearities.
particularly
next,
competence for
Park methods, both schemes were of comparable
effective,
the
central difference and
Comparing the
nonlinearities, however, rendered
being
the Newmark
implicit schemes, with
of the
displayed
scheme
Runge-Kutta
the
and
the Houbolt, last.
and
large
two-cycle iteration exhibited instability at
increments,
excessive
trapezoidal
central difference and
Both the
effectiveness.
similar
demonstrated
and
next
were
with
rule
trapezoidal
order Runge-Kutta and
iteration
two-cycle
explicit methods,
of the
the best
be
to
method
of
central
the
computational efficiency and accuracy indicated
difference
in terms
results
final
difference
and
subjected
Park
to
span. Kinematic
and elastic-perfectly plastic,
elastic
material
-
models
strain
hardening
were
employed.
164
of
the study
schemes,
the
DFP method was most
behavior
independent
-
stiffness
central
difference
methods
were
while
Houbolt
the
transient
strain
stiffness
-
BFGS
the
the
best
results
general,
when
the
Park and Houbolt
tended
for
approach
to
accurately
peak
the
the tangent
applied
to
the Houbolt method combined
best
was
next best.
was
combination
gave
iterator
(50-100 times
reproduced
iteration
quasi-Newton
dependent material,
strain rate
with
In
response.
independent
rate
response
better
method
Newmark
the
method
displacement
transient
the
estimate
the
Park
The
preferred.
scheme)
strain
increment)
time
tangent
the
the time increment
increments
time
for large
material, while
for
effective
most
was
method
difference
central
for the
required
Park
and
For small and
approach.
times
rate
effective for
stable when using
(up to 30
step sizes
strain
Houbolt
The
equilibrium iteration
moderate time
was most
models.
unconditionally
were
schemes
quasi-Newton
the
appropriate for
BFGS
the
and
dependent material
rate
that for
indicated
Results
and
the
Furthermore, the
Newmark/BFGS
method
Park
using a pseudo-force
linear-
extrapolation non-iterative procedure.
4.6
SELECTING A TIME INCREMENT
The procedure for selecting
reviewed
in
this
section.
an appropriate time step is
Different
criteria are applied
depending on whether a wave propagation or
problem
is
under consideration.
structural
Any problem must
type
consider
165
the
pseudo-force
motion,
formulation of
such as the
considerations
additional
of
equations
the
of
excitation.
the
entails
analyses
nonlinear
to
extension
of
the frequency content
structure, and
The
the mathematical model
of the integrator,
characteristics
accuracy
the
integrator,
numerical
of
type
the
procedure, equilibrium
extrapolation
iteration technique, and material modelling.
practically all
structural
and
consequently the
structural
that
it accurately
In wave propagation problems
excited,
are
frequencies
model must be
selected such
through
propagation of stress waves
size
step
structural
frequency
explicit integration
satisfies
be
also
their
the
used
stability
excludes
smallest
the
highest
mathematical model
employed
accuracy
since
of
criterion,
the small
requirements inevitably
requirement.
therefore
the
the governing
are
time
Implicit schemes may
propagation problems,
is
stability
no
but
longer
their higher computational
cost
them from consideration.
The critical
time
problem using the central
At
analagously
as
of unconditional
advantage, and
based on
in the
solving wave
for
property
this
schemes
increment chosen for
time
an
embedded
Using
structure.
the
or
The
the structure.
is
problem
size,
element
geometric
a
such
for
the
captures
cr = 2/w max
increment
for
a
difference method
linear
undamped
is
(4.54)
166
W
where
In nonlinear
frequency.
of
Witmer
recommends
(92)
the following
critical time
structures with large
of
)
(2/w
max
0.8 x
(4.55)
explicit schemes require
Other
smaller time steps
than
that
by
only
central difference method.
the
Structural dynamic problems are characterized
the
usually
cases
The structural model
This approach to modelling
the structure, however, may include areas
stiffness
small
or
excite these
to
Explicit
high.
for
extremely
time increment
though
only
areas,
localized
corresponding
Although
mass.
extremely
the
stiff
of localized
the
structural
structural
is governed
the
lowest
Furthermore, the highest
by the
modes
modes
frequencies
or light members
dynamic
highest
high
loading may not
integrators, consequently, are
large
in
geometric
appropriate
an
provides
the structure.
of
reproduction
excited.
being
modes
lowest
such
of the
frequency
and material nonlinearities:
deflections
=
must be
4.54
in Eq.
max
nonlinear analyses
for
At r
cr
structural
and w
rule-of-thumb
convenient
increment
the
instantaneous stability.
system for
as a
structural
instantaneous
largest
the
as
linear-system
analyses, however,
change with time,
frequencies
regarded
highest
the
is
max
are very
rarely
problems since
frequency,
used
the
even
contribute to the
response.
no
physical
usually
have
167
resemblance
to
structure.
the actual
procedure for selecting a time increment
following
and evaluate its highest
2.
Define w -
3.
Assuming
(
4w
c
w
frequency),
4.
frequency w max
W/w<0. 2 5
for
5.
frequencies up
Select a time increment
at
=
c
structural
obtained
using
C
such
Discretize the structure
captures all
the
w,
and
response can be
the dynamic
static
response is
the
frequency
loading
based on w
At
a
domain
max
that
the
is
(6):
frequency
the
load history to
Transform the
1.
the
recommends
Bathe
formulation.
stiffness
tangent
combined with a pseudo-force or
schemes
implicit integration
employ
often
therefore,
Structural dynamic problems,
that
adequately
it
to wc
satisfying
C
where
in Fig.
From section 4.5 we note
that c,
solution
formulation
scheme.
Furthermore,
behavior
may
modal
1/20
has a minimum value of
accuracy plots given
the
the
c1
employed.
complex
restrict
analyses the
w
and
the
the
path
time
and
upon
4.3.
is also
a function
of
equilibrium
iteration
dependent
material
step size.
would correspond
based
is
to the
In nonlinear
highest
mode
168
CHAPTER
5
FOURIER TRANSFORMS AND THE HYBRID
FREQUENCY-TIME DOMAIN ANALYSIS
Nonlinear
by
an
impressive
equilibrium iterators,
numerical
in
simplifications
by
structural discretization, and
computational
may
be
and
divergence
the
accuracy
the
material
theoretical
number
preclude
the
of analyses.
As a
and
requiring extensive parameter
of
convergence
is
limited
modelling,
The
basis.
cost for producing accurate results, however,
excessive
considerable
integrators,
whose
detectors, providing solutions
only
increments,
accelerators,
convergence
criteria,
time
selecting
for
algorithm
generated
approach has
This
equilibrium iteration schemes.
with
combined
techniques
integration
solved
often
structural dynamic problems are
time
direct
SCHEME
modelling uncertainties
studies to
execution
of
a
result, applications
evaluate the
usually resort
effect
to other solution
169
techniques
load
efficient,
conducted
analyses must be
of
(HFT)
Unlike the
problems.
timewise
accurate
hybrid
domain approach for solving
nonlinear
numerical
not
integrator,
by
the
analyst
their energy
appropriate
option of examining
the
for
begins
domain
solution
then
values
damping
chapter
and
and
spectrum form,
and
stiffness
frequency
the
in
soil-structure
with
a
scheme
concludes
with
time
steps
domain
offers
response quantities
also
can
frequency
the
problems.
standard
in
dependent
selected,
be
interaction
review of
a mode
considerations.
numerical integration
Furthermore, the solution
or
"exact"
theoretically
selection of
allowing the
damping
combines
scheme
a
with
analysis
constrained
integration
in accuracy only
is limited
The
considerations.
superposition
false
without
solution
frequency distortion and, hence,
by modelling
time
direct
case of the
frequency-time domain method provides an
the hybrid
scheme,
alternate
an
presents
chapter
frequency-time
initially to ascertain the extent
results.
the error in the
This
integration
time
direct
detailed
hence
and
in the
actual
the
on
and numerous assumptions
material modelling
behavior,
simplifications
involve
they
behavior,
structural
the
of
estimate
good
indeed may provide a
and
remarkably
are
schemes
these
Although
approaches.
static
spectra or equivalent
the response
such as
as
is
The
frequency
its numerical
implementation
a description of
the HFT method.
and
170
5.1
FREQUENCY DOMAIN ANALYSIS
A
scheme
solution
strictly
linear
to
governing equation of
frequency
the
in
time-invariant
motion for a
domain
systems.
SDOF system is
The
solution in the
frequency
linear
given as
(5.1)
m'<(t) + c4(t) + kq(t) = f(t)
whose
applies
domain
is
easily
grasped
conceptually by letting
f (t)
= f(ow)
e
(5..2)
i t
a transfer function H(w) such that
and defining
q(t) = H(w)f(t) = H(w)f(o) e
Substituting Eq.
5.3 into
Eq.
(5.3)
5.1 we obtain
kH()f()eio
- 2 mH()F(w) eiWt + icH(w)f() eiWt +
=
()
(5.4)
iWt
implying
(-W2m + ioc + k)H(w) = 1
(5.
5)
There fore
H(w) = (-2m + iwc + k) ~6
(5.6)
171
for MDOF
In general
systems we have
H() H~)= (-W2M + iWC +K
The procedure,
to
history
force
Fourier transform;
product
the
by
history
force
back to
-1
(5.7)
the
transferring
therefore, consists of
domain by evaluating
its
multiplying the Fourier transform of
the
frequency
the
and transferring
the transfer function;
the time domain using an inverse Fourier
transform to obtain the response history.
following
The
analytical approach producing the
is
time
providing
presented
the
and
first,
Fourier Series
continuous
mathematical
An
scheme.
continuously
response
in
a numerical approach
then
response at discrete times
5.1.1 Continuous
In the
solution
domain
frequency
the
for
basis
the
examine
subsections
is described.
and Transforms
time approach the
f
forcing function
is expressed in terms of its Fourier series
2m t
2mTt
(amcos
f(t) = a /2 +
m=1
o
where
t is
function;
is
the current
and
amm and
from
-T/2
orthogonality relations
(5.8)
T
T
the
period of
time;
T,
bm are
constants.
evaluated by multiplying Eq.
integrating
)
+ bmsin
5.8 by
The
cos
to T/2 with respect
of
the cosine and
the
forcing
coefficient am
2mrt
then
and
T
to t.
sine
Using the
functions
we
172
obtain
T/2
a.
and
2nr t
(2/T)
f(t) cos
1-T/2 ~
-
dt,
n = 0,1,2, ---
(5. 9)
dt,
n = 1,2,3, ---
(5. 10)
T
similarly
bi = (2/T)
JT/2
2nrt
f(t)
-T/2
sin
5.8 is
Eq.
Convergence of
piecewise
continuous
period T.
The Fourier
points
T
guaranteed
convergence are
to
and
the
that
sufficient, but
Equation 5.8 can also be
at
its
(f(t +)+f(t'))/2
at
its
above
to
conditions
for
not necessary.
rewritten in terms
exponentials by using the following
Cos 2rmt
T
0.5(ei2rmt/T+ e-i2rmt/T)
sin 2inmt
T
0 .5 (ei2nmt/T
=
f(t)
converges
series
Note
discontinuities.
are
f'
for -T/2<t<T/2 and f is periodic with
continuity
of
and
f
when
of
complex
identities
e i2lmt/T)
(5.11)
Hence,
fw e =
c ei2nt/T
(5.12)
n= - oo
wher e
cn = 0.5(an - ib n),
n = 0, +1, +2, ...
(5.13)
f(t)e -i27rnt /T
= (1/T)
T/2
dt
173
approach is
time
discrete
F(w)
transform
The Fourier
=ff(t)e-i2Wt/T
a
of
to infinity in Eq.
derived by extending T
apply
to
f(t)
is
function
5.12,
resulting
dt
(5.14)
(5.15)
=(l/21r)f F(w) ei2lt/T dw
f(t)
in
inverse Fourier transform by
the
and
that
of this
functions.
aperiodic
F(w)
transforms
the
when
The remainder
examined.
Fourier
considers
discussion
later
to
referred
be
will
Equation 5.12
Equations 5.14 and 5.15
Fourier
transform
imply
that
f(t)
and
The existence of F(w)
pair.
are
F(w)
a
is assured
when
fIf(t)
Idt
< -o
Once again, Eq.
(5.16)
5.16
is
a sufficient,
but not
a
necessary
condition.
The Fourier transform of
if
we
use
the
theory
of
applications the Dirac delta
00,
exists
impulse functions also
distributions
function
(17).
For such
6 is defined by
t = t0
0, otherwise
0
(5.
6(t-t 0 ) dt = 1
17)
174
Extending these concepts
solution
dynamic system is
'a
of
structural
to
integral
time domain by Duhamel's
t
given analytically
in the
(18,38)
(5.18)
dT
h(t-t)
D)
q(t) =.J f
the
problems,
where
h(t-t)
=
(1/m~)e-
referred
is
5.18
Equation
tTsinD(t-t)
(5.19)
to
mathematically
the
as
convolution integral and can be rewritten as
q(t) = f(t)*h(t)
(5.
5.14
Using Eq.
the response in the
20)
frequency domain is
given by
Q(W)
q(t)
=
Substituting Eq.
Q(W)
=J
5.18 into Eq.
001f(T)h(t-T)
0f(T )
J-0
f(T)
(Co
= Co
H(w)
5.21 we obtain
dT)e-i2Trt/T dt
h(t-T)e-i2t/T dt dT
h(t-T)e-i27(t-T)/T
f(T)H(w) e
=F(w)
(5.21)
e-i2nt/T dt
-- 27r-r/T
dT
dt e- i27T/T dT
(5.22)
175
Therefore,
and
pairs
corresponds
the
that
Notice
pairs.
simple
a
to
transform
Fourier
transform
multiplication
the mathematical
history
discrete
Fourier
presentation on
discrete
For
apply.
of
solution
at
given
Fourier
provide
transform,
agrees
5.1-5.7
a
basis
we
must
Fourier
The
force
a
involves
the previous
does
transforms
and
series
to
resort
series
not
the
and
for understanding the discrete
transform is not
discrete-time
Fourier
the continuous
Fourier transform.
that
emphasized
is
it
but
Series
and hence
times,
series
time
5.1
Eq.
discrete-time Fourier transform.
transforms
frequency
formulation.
5.1.2 Fourier Transforms of Discrete Time
The actual
the
in
domain
time
in the
convolution
the presentation for Eqs.
domain, and hence,
with
are Fourier
are
H(w)
and
h(t)
moreover
w)F(w)
H(
and
h(t)*f(t)
the
an approximation to
Although it
is
possible
to begin with the continuous Fourier transform when deriving
Fourier
discrete-time
the
foundation
independently
furthermore
exact
the
of
of
the
results for
discrete-time
the
the
transform,
continuous
discrete-time
theoretical
Fourier transform exists
Fourier
Fourier
transform,
and
transform produces
discrete time series.
discrete-time
Fourier
transform by defining the discrete time series as a
sequence
We begin our discussion of
x where x(n) is the nth term in
of
the linear
the
the sequence.
system represented by Eq.
5.1
The
response
is given as
176
(5.23)
q(n) = T(f(n))
where T
linear transformation such
is a
T [afl(n) + bf 2 (n
= aT f 1 (n)]
that
+ bT f 2 (n)]
= aql(n) + bq 2 (n)
(5.24)
for arbitrary constants a and b.
6(n),
is
6 (n)
If hk(n)
defined
unit-sample
The
sequence,
by
1,
n = 0
0,
otherwise
=
(5.25)
is defined
as the
system
response
to
6(n-k),
we
have
q(n)
=
=
k=-m
Since we
have
5.26 can be
q(n) =
f(k)6(n-k)
T[
f(k)T 6 (n-k)]
= k
=-
f(k)hk(n)
assumed a linear time
invariant
(5.
26)
Eq.
system,
rewritten as
f(k)h(n-k)
(5.27)
k=-"0
where Eq.
5.27 is
q(n) = f(n)*h(n)
the convolution sum or
(5.
28)
177
A
defined by
stable system is
z
k=-
_h(k)
00
discrete-time Fourier transform of a
The
is
x(n)
sequence
defined by
the
(5.30)
x(n)e-in
= E
n=-- 0
X(e'i
and
(5.29)
< 0
inverse Fourier transform by
7r
X(e U2)
x(n) = (1/2n)f
eio
5.30 is
The convergence of Eq.
< 00
Equation 5.27 is represented
5.30
Q(e
guaranteed when
(5.32)
z: Ix(n)
n= - 0
Eq.
(5.31)
2
in the frequency
using
domain
to obtain
) = H(e
)F(e
(5.33)
)
where
h(n)e
H(e P2) =
and H(e')
unit-sample
is
the frequency response
response is h(n).
(5.34)
of
the
system when
the
178
has
continuous
presented independently of the
been
Fourier
transform is applied to structural
excited
by
time.
in
continuously
systems that are
and
history
load
continuous
a
Therefore,
the
discrete response and the continuous
Fourier
discrete-time
the
In actual practice, however,
transform.
transform
discrete-time Fourier
The development of the
respond
relation between the
will
response
now
be
examined.
Fourier
=
We
At.
increments
and
5.14 and
5.15 we
its
have
(5-. 35)
(5.36)
x (t)e-itdt
a
x(n) be the sequence
let
Xa (i W)
history
Xa (iw)eiwtdw
(1/27)
=
X (iw)
a
Eq.
Then by Eqs.
transform.
x (t)
Also
the continuous
x a(t) be
Let
from
derived
now compare Xa (iw
x a(t)
) with X(e
at
time
).
From
5.31 we have
7r
x(n) = (1/27r)
X(e1)e Mn d!
(5.37)
-7r
and
from Eq.
5.35
00
x(n) = x (not) = (1/2T)
X (iw)e iwnlt dw
Equation 5.38 is expanded further as
(5.38)
follows:
CO
x(n)
=
(2r+l)r/At
iwnAt
dw
X (iw) e
f
(1/2T) X
(2r-1)?r/Ata
r=-o
iwnAt i27rn
27r
TT/At
00
d
At
Xa
f
(1/2T) I
r=-o -iT/At
(5.39)
179
Tr/At -
f
= (1/2r)
(I
X (iw
+ i
))eint
dw
-w/At r=-o
= (1/2E)
3
(1/At)
5.37
X(e
)
X (ion + i
dt
and 5.39 we conclude that
(1/At)
X (
+ iE
a At
r=-Oo
=
) e
r=-_-0
-7T
From Eqs.
(5.39)
at
)
(5.40)
or
00
X(eiWAt)
= (1/At) I
r=-o
xa(iW + i
has nonzero
Suppose Xa (i w)
that if
2n
/At
is greater
Xa(iW + i 2r)
all r # 0.
for
)
will be
time
the
is
large
that
reproduces
Nyquist
(5.42)
incorrect.
x(n)
and
w),
and
w
history.
is
given
Fourier
,
low
from
rederived
sampling period,
is
The minimum sampling rate
the actual history is
to
referred
as
the
frequency given as
(5.43)
fN = 1/2At = w /2 r
where
w
overlapping phenomenom when
This
aliasing.
called
high
of
2
than
less
is
overlaps
spurious
of X a(i
At
if 27T /
increment, or equivalently the
too
5.41 we conclude
5.40 or
than. 2wo,
Otherwise,
frequency components
X(e
and
< w
< w
0
=
include
will
X(e ')
-w
for
values
Then from Eq.
zero elsewhere.
is
(5.41)
corresponds
to
frequency
the highest
A schematic representation of
in Fig.
5.1.
transform;
Figure
Fig.
5.la
5.1b,
the
in
the
time
the aliasing effect
shows
the
continuous
discrete-time Fourier
180
(a)
1/T
V
-2
nT
2V
4w
SW
rIT
(b)
X(e o)
1/T
A,'>
A,/>
4w
2v
-2w
aOT
nOT
(C)
(a) Fourier transform of a continuous-time signal.
(b) Fourier transform of the discrete-time signal obtained by
periodic sampling. The sampling period is large, so the periodic
repetitions of the continuous-time transform overlap. (c) Same as
(b), but with the sampling period small enough so that the periodic
repetitions of the continuous-time transform do not overlap.
Fig. 5.1
Fourier Transforms of Continuous and
Discrete Time Signals (57)
181
and
frequency;
Notice
greater than the Nyquist frequency.
tends
frequency
reproduced
values xa (nAt)
(t)
=
x
k-coa
that the Nyquist
recovered from
the signal
signals
can
signals
the
discrete
its
using the following interpolation formula:
(kAt)
(5.44)
sin((r/At)(t-k))
(T/At ) (t -kAt )
Fourier
results for discrete-time series.
continuous-time
be
is
summary, discrete-time
In
exact
rate is
bandlimited
For
exactly.
a
a
sampling
the bandwidth of
to infinity as
continuous-time signal x (t)
x
Nyquist
to infinity, and hence only bandlimited
extends
be
the
when
5.lc,
Fig.
the
than
less
rate is
transform when the sampling
signals
provide
transforms
The application to
rate
the sampling
requires that
At
less than one-half the Nyquist frequency
At <
(5.45)
N
where
in
corresponds
fN
the
continuous
time
involve fN
approaching
frequency
f.
0
exists for
f>f
must
to
the
highest
Actual applications may
signal.
infinity,
and
be selected such
and At equals 1/2f
present
frequency
.
therefore
a
cutoff
that negligible energy
182
Series and Transforms
Discrete Fourier
5.1.3
Fourier
transform
requires
the time and
frequency
domains.
the
that
Consider the Fourier
x(n)=x(n+kN)
that
such
integer.
There
e
exponentials
N-1
x(n) = (1/N) 1 X(k)
k=O
Multiplying both
summing
from n=0
transform
are
extended
(DFT)
transform
series.
of a periodic sequence
where
N
i(2ff/N)nk
as
in Eq.
exponentials
Fourier
series
the
is
N
only-
exist
now
the
time and frequency
operates with discrete
opposed to
5.12.
of
to N-1, we
x(n)
period and k an
distinct
complex
the infinite number of
Therefore,
(5.46)
ei(27/N) nk
sides
of
Therefore,
discrete Fourier
to derive the
subsequently
a
sequences both in
discrete
Fourier
discrete-time
and
series
implementation
A numerical
frequency output.
continuous
and
input
time
discrete
a
on
based
is
previously
presented
transform
Fourier
discrete-time
The
Eq.
5.46
by
e
i(27/N)nr and
can show that
N-1
X(k) = T x(n) e-i( 2 f/N) nk
n=0
Notice
(5.47)
that in addition to x(n) being periodic,
periodic with period N.
By defining W
X(k) is also
-iC2N /Ne
Eqs.
5.46
and 5.47 can be rewritten as
N-1
x(n) = (1/N)T X(k)W-kn
N
k=O
N-1
x(n)W kn
X(k) =
N
n=O
Z
(5.48)
(5.49)
183
are
5.48 and 5.49
Equations
series
Fourier
discrete
the
the periodic sequence x(n).
(DFS) of
extension
The
the
of
sequences
finite-duration
Fourier
discrete
produces
Fourier
discrete
the
to
series
(DFT) given by
transform
N-1
(1/N) jX (k) W-Nkn
k=0
0
N-1
11x(n) WkN
n=0
0
x(n) =
X(k)
=
Fourier
discrete
the
examine
now
We
otherwise
0 <k <N-1
(5.51)
otherwise
N-point
convolution
linear
Let
transform.
two
convolution of
(5.50)
0 <n <N-1
the
using
x 3 (n)
be
the
linear
x
(n)
and
x
sequences,
2
(n).
Therefore
N-1
x 3 (n) =
(5.52)
x 1 (m) x 2 (n-m)
M=O
Notice
that x3(n) has
X 1(k)
and
X 2(k)
Substituting Eqs.
5.52,
length 2N-1,
and
therefore
are computed on the basis
5.50 and
5.51 for
the
DFT's
of 2N-1 points.
2N-1
points
1,N
0 <n <2N-1
into
Eq.
we obtain
N-2
1/(2N-l)
x 3 (n)
=
O2
=k
(X (k) X2(k))
W
k= 0
(5.53)
otherwise
184
In
the
application, however,
progresses
points
is
response at
The computation of the
the
inverse
evaluation of Eqs.
numerical
increases,
in
its
The
actual
of
points
as
response
it
load history of N
zeroes such that the
discrete Fourier
DFT,
examined
is
therefore,
N
or 4N
The number
additions.
increases
and
A direct
next.
complex additions
and N(4N-2) real
operations,
transform
requires approximately
rapidly as N
feasible
and the direct evaluation of the DFT is
Numerous approaches have been devised
computational
of
than
rather
small values of N.
only for
Many
the
the Discrete Fourier Transform
5.50 and 5.51
real multiplications
of
actual
multiplications and N(N-1)
complex
x3
negligible.
t=MAt is
5.1.4 Numerical Evaluation of
similarly
and
number
the
attenuates
extended M points by adding
time
f,
domain.
finite
Therefore, the
in time.
impulse
directly
given
is
involves a
damping
system
because
the
because
frequency
the
in
form
analytical
X1(w)
domain,
frequency
is
domain and then transferring to
the time
in
x1
evaluating
to
theoretically be evaluated
points
of
number
infinite
an
history
load
the
the convolution must
q,
response
for
x2
function h,
response
x
where
problems,
dynamic
structural
convolution
linear
the
of
application
the
effort
the
for the DFT (16,20,54,55,69,77,90,91).
these algorithms gain
advantage
for reducing
their
efficiency
by
taking
of the periodicity and symmetry properties of
the
185
W
N
- Wk(n+N) _
N
particular
In
trigonometric functions.
(5. 54)
(k+N)n
N
and
k(N-n)
N
where
(5.55)
-kn)*
N
5.55
Eq.
in
*
supercase
complex
the
to
refers
conjugate.
the computational
increasing
N/2
until
Since
x(n).
into
smaller
two-point sequences remain,
such that
length
of
sequence,
N
must
approximately
appended
to
number of points
extended
history
the DFT of a two-point
The total number of
two.
for
the
the
Notice
evaluation of the DFT.
is
FFT
total
end
of the
up to a power
provides
a
of
record
two.
finer
zeroes
In such cases
of 2.
to bring the
The
DFT
resolution
the
that
sequence x(n) may not have a
equal to a power
terms
to yield the full DFT of
additions
and
The
DFT.
obviously a significant decrease from
duration
finite
number of
are
Nlog 2 N,
for a direct
actual
a power of
be
N
two-point
a
stage computes
multiplications
complex
that
then recombined
final
the
(20).
in 1965
the FFT consists of successively
the final computation only involves
two-point DFT's are
fast Fourier
efficiency is the
x
sequence
the
decomposing
sequences
behind
concept
basic
for
available
Cooley and Tukey
transform (FFT) presented by
The
currently
algorithm
popular
The most
total
of
since
the
the
186
increased
have
components
frequency
of
number
while
the
and
can
remains the same.
sampling rate
available
are
FFT
the
of
forms
Various
be classified into decimation-in-time algorithms,
generally
the sequence x(n)
into smaller
subsequences,
that
decompose X(k)
that
rearrange
and
decimation-in-frequency algorithms,
into
smaller subsequences.
not
a
More
(16,69).
prime number
Winograd
fast
efficient
than
the FFT
efficiency
in
actual
the form of
FFT employed, the
the
with
conjunction
applications
more
demonstrates
less
(47,76).
computer
Regardless
of
has
the
FFT
of
in
rendered the
alternative
viable
a
the
is
application
digital
domain
time
traditional
but
N
8 or a
include
which
(90,91)
theoretically,
approach
domain
frequency
approaches
recent
transform
Fourier
on
are based
example N may be a power of
For
2.
of
power
Some algorithms
to
the
solving linear dynamic
schemes for
systems.
5.1.5 Frequency Domain Analysis
Nonlinear systems
frequency
at
excited
and
frequency
W
frequency w
that
w19 w 2
>
w
addresses
linear
A
w
3
also
contain
eventually die
, and so
the
unexpected
system with
frequency w
forth that
analytical
harmonically
respond
will
system, however, may
nonlinear
completely
often display
behavior.
of
patterns
and Nonlinear Systems
initial
transients
damped
out in
respond at other
persist in time.
solution
of
systems.
at
with
A
frequencies
This section
Duffing's
problem
187
(65,79)
distinguishing
time
Essential
and
identified,
domain
frequency
and
domain
the
to
background
nonlinear problems.
of
features
between
differences
introductory
an
provides
and
underlying motive for
the
solutions
are
exploiting
the frequency domain solution, and in
particular
scheme, is presented.
the HFT
Duffing's problem consists of the
system defined by
the
following equation:
3
m
+ cc + aq +
the
damping;
the
Sq
a
are
3
nonlinear material
type
stiffening
The
component.
introduced
system
by
6
greater
is
than
can be viewed as a geometric
type nonlinearity.
Let
solution
us
in
less than
first consider
the
time
the
addition
the
is
an elastic
zero and elastic
In another sense,
if
the
can be regarded as a
zero.
softening
mass;
and f,
stiffness;
problem, where the material
if
the
to
corresponds
linear
the
,
Nonlinearities
excitation.
of
(5.56)
structural problem, m
where in a
c,
ft
=f(t)
linear problem,
Sq
the
6
=
3
0.
term
A
domain can be obtained analytically
using Duhamel's integral
q(t)
=
t
0 h(r)f(t-T)
= h(t)*f (t)
(5. 57)
dT
(convolution)
188
have
impulse
the
is
where h
a few
for
integrated
state forced
causal
a
assumed
response
response
q(t) =
we
5.57
must
be
to obtain the steady
integral is often cumbersome,
Fourier
the
J
to substitute
into
Eq.
q and f
(use
the
transforms of
for periodic forcing
(1/2Tr)
even
therefore an alternate solution scheme is
and
series
Fourier
cycles
A possible alternative is
(#=0)
that
response.
numerically,
desired.
Notice
Equation
system.
Evaluating Duhamel's
5.56
function.
functions)
Q(W) ei t dw
(5.58)
J
f (t) = (1/27)
F(w) eiot dw
to obtain
M
C -w2 Q(w)eiot do + c fJ*iQ(o)eiWt
dw
(.5. 59)
+ a
and
Q(W) e
dWt=
f/
(~
iot
F(w) eiot dw
after simplifying
p0
2 +
(-
m + ic
+ a)Q(w)e
-
p
do = f
iwt
F(M) e
dw
(5.
60)
or
Q(M) = (-W2m + iWc + a)~1 F(w)
= H()
F(W)
(5.61)
189
where H
the transfer function.
is
and
implies
convolution in
the time
solution
the
in
solution
since
course,
simplicity
the Fourier transforms must
be evaluated,
but
the FFT.
straightforward since
on
longer
no
depends
the impulse response function
approximations
Analytical
amplitude.
have
using iteration and perturbation techniques,
that
indicate
the steady state
(subharmonic)
in
illustrated
ultraharmonic
function given by Eq.
an
w, but may also have
that
implies
a
5.61 where an input
frequency
w
,
and
subharmonic
This
is no
simple
frequency
transfer
only
w
longer valid.
In
we may have
particular,
Q(W1 )
output
schemes
(ultraharmonic) components,
5.2.
behavior
with any degree
response does not necessarily
and higher
Fig.
derived
been
these solution
of the frequency component
only
generates
general,
In
of accuracy attainable.
lower
time
system frequency which now varies with the response
the
consist
is
integral
Duhamel's
by
A
a O.
now consider the nonlinear problem,
solution
domain
provides an
of
the numerical computation is minimized by using
Let us
of
exists,
A tradeoff
exploit this approach.
incentive to
a simple algebraic
by
frequency domain
the
same
the
obtain
can
conceptual
The
in
solution
a
domain
frequency
multiplication of H and F.
evaluating
we
domain,
the
evaluating
than
rather
that
domain
frequency
a
to
corresponds
5.61
Equation
=
H(uywo ) F(w )
(5.62)
190
////""
-2up
t
a.
excitation
.q
t
harmonic response
b.
q
t
subharmonic response
c.
q
t
d.
Fig.
5.2
ultraharmonic response
Nonlinear Response to a Harmonic Excitation
191
Following
5.56
Eq.
same except
the same procedure as before,
to
the
for
aq
3
q
do 2 d
3)
5.63,
integral given by Eq.
multiplication
involves
once again
confronted
(convolution)
posed
Eq.
5.61.
with
the
same
by
time
the
is now
an
the evaluation of
contrasted
by
denoted
(5.63)
3
frequency domain
the
5.56 in
the
that
is given by
process and, moreover,
iterative
the
3
2 ~43) Q4
2 0
2
of Eq.
The solution
terms remain the
We can show (see Appendix C)
.
Fourier transform pair of
( 1Q-
All
frequency domain.
transfer
let us
to
the
simple
As a result, we are
problem
integration
domain
solution of
the
linear system.
is
The proposed solution
evaluation
$q
3
by
component
frequency
to
the
solving
and
in the
remaining
in
system
the
In other words, we introduce the hybrid
domain.
in
integral
the
reverting to a time domain evaluation of the
frequency-time domain scheme where
evaluated
eliminate
the time domain, and the
frequency domain with
nonlinearities
all
are
solution is executed
only simple multiplications.
5.2 HYBRID FREQUENCY-TIME DOMAIN ANALYSIS
The
in
the
standard frequency
previous section is
domain solution scheme presented
limited to linear
time
invariant
extensions to nonlinear analyses have been
systems.
Various
proposed
recently,
with
specialized
applications
to the
192
derive
applications
represents
solution
the
whe're
iterative
in
of
the
drag
The proposed solution schemes consist
of an
frequency
domain.
avoid
schemes
these
in
the
transients,
initial
the
eliminating
of
problem
obtained
inherent harmonic
By exploiting the
transform,
the Fourier
basis of
load vectors on the RHS
solution is
the
equations of motion and
evaluated
are
nonlinearities
process whereby the
time domain and expressed by
the
the
hydrodynamic
the
to
nonlinearities are due
damping effect.
behavior,
state nonlinear
steady
the
converged
the
loading, and hence
state excitation, the wave
to a steady
response
structural
the
These
(5,24,70,82).
structures
offshore
of
response
characteristic of time domain solution schemes.
Most nonlinear frequency domain
for
developed
dependent
are
response problems
structural dynamic
small
excessively
the
behavior
numerical
or
soil-structure
frequency
integrator,
actual
stiffness
the
frequency
are
rather
tremendously
than
domain.
large
the
in
material
Moreover,
characterized
and damping
terms
Other
size
conducting any significant parameter studies.
an
the accuracy
response.
problems
interaction
require
by
determined
structural
dependent
exactly only in
problems
time increment
problems
solution schemes.
however,
problems,
are
with response
response
system matrices while transient
usually addressed by time integration
Various
of
steady state
schemes
solution
and
are
by
solved
structural
preclude
193
the
An alternate solution scheme,
been
consequently,
to
analysis
schemes,
nonlinearities
steady state
the
applicability and
Eq.
numerical
its
of
review
domain
scheme
solution
pseudo-force approach given by
3.24 and rewritten here as
where M
vector
the
(5.64)
Kq = F + F
and
matrices;
K
F,
represent
the
linear
the
external
load
containing all nonlinear
global
nodal
in its generalized
mass
vector;
terms;
displacement.
mode superposition approach, Eq.
3.83
associated
a
with
time
unconventional
the
describes
the HFT Approach
frequency
hybrid
Mj+
as
examines
concludes
first
section
limitations.
5.2.1 Formulation of
employs
This
scheme,
and
considerations,
The
and is applicable to both transient and
type problems.
solution
HFT
initial
embraces
The scheme
system.
of
development
from the
material nonlinearities
the nonlinear
system to
linear
the effect
the
for
In
domain
frequency
proposed
the HFT method also admits
and
kinematic
transient
systems.
structural
transients, and hence accounts
initial
all
scale
previously
the
contrast
of
large
of
response
dynamic
to
has,
section
the
evaluate
developed
frequency-time
this
in
presented
method,
(HFT)
domain
hybrid
and q
In
5.64 is
stiffness
and
NL
F
,
is
the
force
redefined
conjunction with a
rewritten
form including damping
as
Eq.
194
where the
the
=
+
+
generalized
is
The corresponding
response.
displacement
y
and
denotes generalized matrices
(~)
tilda
(5.65)
+ FNL
transfer matrix R is
(-w 2 M +
_()=
(5.66)
K)1
iWC +
Assuming the eigenproblem has
are available, the basic procedure for
analysis is
1.
as
HFT
follows:
Compute
FFT,
the
using
domain
frequency
the
obtain
therefore
representation of
2.
the
conducting
Evaluate Fourier transform of F(t)
and
the
5.65 and transfer matrix H(w)
in Eq.
matrices
generalized
and
completed
been
the force history, F(w).
the response in the
frequency
domain
Y(w)
by simply multiplying H(w) and F(w).
3.
Transfer Y(w)
to
the time domain by
evaluating
its
inverse Fourier transform
4.
Derive
generalized
the
geometric
y(t),
response
nonlinearities.
Store
response
the
and
from
the
determine
all
q(t)
nonlinearities
as a
force
time history FNL(t).
5.
Transfer F NL(t)
to the frequency domain
using
the
FFT
6.
The
forcing function
is
now F(W)+FNL(W).
Multiply
195
forcing
the
response Y(o).
frequency domain
Repeat
7.
new
the
obtain
to
H(w)
by
function
criterion
a convergence
3-6 until
steps
is
satisfied
The HFT
solution
existing
direct time
minimal effort.
scheme to
integration computer programs
requires
The only program dependent attribute
evaluate the nonlinearities
phase
the
for
load vector
efficient
results,
procedure
addressed
ingenious
during the equilibrium iteration
incremental
application of
however, entails
in
outlined
in the next
formulation and as
the HFT method
steps
stretches
FFT
the
of
acquiring
practical
the
basic
of the HFT solution scheme involves
the mind, nor does
the
solution.
solution
of
no
its application
numerical
recondite
the nonlinearities
and
Despite
in the
resorts
to
its simple
has never been
applied
structural dynamic
problems
formulation, the HFT solution scheme
to
to
for Applying an HFT Analysis
The HFT method evaluates
for
accurate
subsection.
same manner as a time integration approach
the
An
These modifications are
1-7.
require extremely complex and powerful
methods.
part of the
producing
numerous additions
Considerations
development
the
schemes already
for explicit and pseudo-force formulations.
5.2.2 Numerical
The
is
aspect
Even this
step 4.
time integration
implemented since
easily
and
simple
adapting the HFT
evaluation of the nonlinearities,
is
conceptually
is
Furthermore,
executed.
easily
scheme
196
(based on a
review of
engineers
are
unacquainted
domain
frequency
theoretical
scheme is
for
with
the
developed
been
has
to
and additions
the basic procedure
to
achieving a practical solution scheme can be classified
categories
These changes are
Steps
1-7 of
different
two
of efficiency and
examined in the
(accuracy).
following subsections.
the solution process can be
The
forms.
the dual displacement
response
to
y = Y
7 to derive
the
2, and
obtains
linear response produces
the
y 2,
k 1=
the
linear
which
when
nonlinear response
(5.67)
+ y2
1+
in
then iterates from steps
a correction response
governing equation for the
+
approached
first approach, referred to here
formulation,
yi from steps 1 and
3 through
added
stability
Solution Formulation
5.2.2.1
The
familiar
structural problems with accurate
dynamic
Modifications
under the
as
the
considerations.
nonlinear
results.
with
unfeasible, based on cursory
The HFT method presented here
solve
uncomfortable
others
and
concept
the
believe
concept
or
many
possibly because
the literature)
first cycle is
(5.68)
197
successive iterative cycles
the
and during
This
formulation
the
analyst
an estimate of
the
linear
the
the nonlinear
response
yi
Notice
response.
errors
round-off
thus
for F,
that
are
the Fourier
points
less
force history FNL may require
frequency domain than that
the
independently of
the evaluation of
furthermore,
and,
transform of the
the
numerical
response,
linear
69)
provides
consequently
and
the nonlinear correction is evaluated
minimized
in
calculates
cycle,
first
during
since
(5.
~
F NL
+
-+
reducing the
computational cost.
The
dual displacement
computational
slowly
converges
significantly
Furthermore,
only in the
is
linear response and since F NL
succeeding
cycles
the
approach its
true value.
appreciably
from
the
response
response y.
actual
response
departs
first
based
analysis, the
Y 1 +Y
on
y
if
words,
differs
response,
reasonable accuracy
y1
may
during
2,
may never
y2
correction
then
_2
to produce
stem from an
the
mode
appended zeroes
being
included
number of
load
cycle to obtain the
nonlinear
being
diverge
actual
the
since
Inaccuracies in
insufficient number of modes
superposition
other
In
cannot be calculated with any
the correct
solution
linear response, and may even
from the
employed
is
the
However,
nonlinear
the
when
and become unstable.
history
features.
accuracy
favorable
possesses
formulation
in
198
of
significant
Nyquist frequency
the
on
operates
evaluates
-
initial
succeeding
that
later
the
displacement formulation may converge faster because a
can
response
nonlinear
the
Notice also
employing artificial damping.
by
produced
one
that
sense
during
be shown
In addition, it will
"better" initial guess to
the
formulation
the
in
corrected
are
approximations
iterations.
total
forgiving
solves Eq.
total force history,
to the
response
this approach is more
,
and hence
displacement
total
-NL
F+F
formulation,
displacement
total
the
the
solution formulation,
displacement,
total
Since
5.65 directly.
always
the
as
to here
referred
dual
the
of
difficulties
the second
formulation,
displacement
a
component.
various
the
Considering
producing
increment
time
excessive
an
selection
the
or
histories,
the
resolve
adequately
to
deficient
be
that
storage requirements are reduced considerably since only
quantity y is
response
5.2.2.2
rather than both y1 and
Zero Minimization Problem
the
evaluating
appended to
the
the
represents
tr ansform
Fourier
of
end
excitation
radix-2 FFT,
terms,
a power
increase
the
or nonlinear
sequence
of
two,
zeroes
extra
x(n),
where
force history.
x(n)
such
allowing the
resolution
that
demonstrated
requires
sequence
t he
additional zeroes extend the
N
5.1.4
a nd
5.1.3
Subsections
contains
stored,
by
x(n)
The
that
it
use of
the
decreasing
the
199
the
end of
The
is appended,
of zeroes
significant amplitude
Fourier
inverse
the
linear
(,
natural
Let
u
0
and
0
-
is
amplitude
decreases
A plot
of
shown
w,
and damped natural
frequency
The free
conditions.
for
ratio
damping
wD'
vibration
possible to determine
in
one
cycle
derived with such an
The
the
(5.70)
sinw t + u cosw t)
5.70, it
ratio
is
Consider the
'A + u 05
Using Eq.
5.4.
approach.
a SDOF system with viscous
u(t) = et
Fig.
will
then given as
response is
ratio.
response
appended zeroes can be derived
the initial
be
of
5.3.
frequency
u
have a
use
the
process
This
is
insufficient
conditions
transient
to
time
decay time
reproduce the
to
using an analytical
free vibration of
an
If
and hence
t=NAt,
final
number of
systems
finite
of
free vibration will
response.
schematically in Fig.
The actual
the
transform
initial
the
alter
at time
that the
history implies
This
zeroes.
additional
the
amount
finite
a
to a negligible amplitude.
provided by
initial
to the
the load history creating a damped
at the end of
free vibration that requires
number
is due
item in a physical sense
last
at the
effects
the beginning.
response history from entering
conditions
decay
transient
prevent
and
increment,
frequency
analyst,
final
to
much
the
for a specified damping
approach
therefore,
amplitude
how
is
shown
in
selects an allowable
the
initial
response
200
force
T
a.
t
excitation
response
T 0T
b.
2
T1
actual response
response
T
c.
t
response with zeroes appended up to
time T
response
t
d.
Fig.
5.3
response with cutoff at time T2
Effect of Appended Zeroes in a Frequency Domain Analysis
201
and
produce such an amplitude attenuation,
equals
points
of
number
(zeroes)
of cycles
S to
calculates
the
the number
5.4
from Fig.
obtains
amplitude,
cycles,
which
1
20
in those
contained
S/At.
6
5
S
0E2
Ze
Fig.
5. 4
is
It
become
Furthermore,
therefore,
efficiency of
In
of
ntio
zeroes
extra
analyses the
in nonlinear
zeroes
of
0.5
1.DamPing
may
may
system
may
to increase, and
periods
necessitating the use of even more
number
large
.5
systems with high fundamental
structural
the
causing
for
1
00
number
the
evident that
prohibitively large
periods.
soften,
Damping ratio vs. number of cyclrequired to reduce amplitude by 50
(18)
peL
4
This
zeroes.
significantly
reduce
the
the frequency domain solution.
to
response
the
extra
zeroes
minimization technique was developed
appended zeroes in the
of
points N up
the
actual load
to
problem,
such that
ideal case brought the
the next
history.
smallest power of
More
zeroes
can
zero
a
the number of
total
number
2 greater
obviously
than
be
202
that the
insufficient,
resolution is
if the
appended
it
but
is noted
the
proposed zero minimization technique eliminates
of zeroes
the number
choosing
of
task
analyst's
based on
free vibration considerations.
The procedure for
follows:
scheme is as
history
excitation
of
number
Let N be the
1.
minimization
zero
the
implementing
representing
NB
the total number
and
F(t)
the
points
of
points used in the frequency domain (N<NB).
NB is
2.
is the smallest
Using
3.
transform
Fourier
exact for
transform is
This
transfer
its
by
function
representation
domain
frequency
the
NB
increment and NB.
Multiply F(w)
obtain
the
evaluate
such an NB,
the given time
In other words,
of two greater than N.
power
of the load history.
F(w)
4.
that N<NB<2N.
chosen such
H(c)
to
of
the
response Y(w)
5.
obtain
6.
y(t)
is
an incorrect
beginning
of
Obtain the correct
the
linear
produces a significant
modifies
the
response by realizing
that
the
be
zero
(or
at t=NBAt
that
the response.
conditions
whatever the analyst
of
NB
the
because
response
with
executed
free vibration component
initial
to
the response history y(t)
convolution
7.
Y(o)
Evaluate the inverse Fourier transform of
Y
and
y
specified).
free vibration component by
should
Therefore,
purge y(t)
subtracting out
this
203
5.70.
component using Eq.
5.5.
given in Fig.
is
technique
easily reduced by a factor
of two
only difference between the
is theoretically sound since the
y
by
minimization
numerically, however,
if
final free vibration,
vibration
consists
harmonic
be
still
may
it
is
t=NAt
probability
of some
the
combining
such that they produce
is
technique
is negligible,
applies
to
some of
these
In
A
conditions
most
cases,
vibration components
initial
conditions
at
and the procedure outlined in Fig.
Notice
acceptable.
zero
of
number
the initial
free
free
the
conditions.
initial
vibration.
however,
t-O
flushing
damped free
the
that
possible
therefore, would use
to evaluate the free
of
finite
a
of
even
Furthermore,
Since
present.
numerically
components,
better approach,
5.5
from y(t)
some components
components when combined yield zero
time
is derived
adjusted to zero by
initial conditions are
vibration
at
economic
finite difference approximation.
the
out the
on
Based
available, but y
is
y
considerations
zero
the
implemented
correction
This
approximate.
is
on y
initial conditions
explanation of
for a mathematical
technique).
rederived
is
y
is the
y
response
incorrect
imposing the known actual
(see Appendix D
using a
the
and
the free vibration component.
of
presence
from
y
response
exact
procedure
The
to four.
can
points NB
total
the number of
Using this technique,
be
minimization
zero
the
of
representation
schematic
A
both
that
the
linear and
zero
minimization
nonlinear analyses,
204
Q[,~
A~AAA/MAANAPAAA
VI Wv
vI vv 4'
a.
I
2N
-
f
excitation with minimum number of appended zeroes
U
t
b.
incorrect response obtained by a frequency domain
solution with minimum zeroes
c.
u(t)
=
-e
-uf
t
G" si
f+
f
t+u
sin
+ u Cos
D
u
-1
\",Ooel
d.
Fig.
5.5
correct response
Zero Minimization Technique
11
in
t
205
with no
iterative process involved
linear analysis.
for the
5.2.2.3 Relaxation
In
addition
the
accelerating
is defined as
procedure
+
yi = ay
where
number;
a
applying
(5.71)
(1-a)yi-1
coefficient;
a<l,
over-relaxation;
relaxation.
yi
than
~
y
called
a=1,
no
larger weighting factor
reverse
the
a<l
when
and
before
is,
and
under-relaxation;
In general, when a>l a
is applied to
procedure
the
a>l
For
relaxation.
the response
y,
response
the
y,
and
relaxation;
cycle
iteration
the
refers to
is the acceleration
after applying
of
follows:
right superscript
the
means
relaxation
The
process.
convergence
a
as
investigated
were
procedures
relaxation
technique,
minimization
zero
the
to
statement holds.
Relaxation was applied
pseudo-force
histories.
over-relaxation
the
to both the displacement
simultaneous
over-relaxation
pseudo-force
histories
at
the
significant
began
of
the
to
instability.
During the
moreover
latter cycles,
similar
of
the
the
convergence process
diverge and eventually
A
and
response
accelerated the
an even greater rate.
response
and
cycles,
early
the
in
the
accelerated
an
demonstrated
response
convergence
applications
both
In
and
however,
displayed a
application
of
206
rate with
under-relaxation demonstrated a slower convergence
no
instability.
time
vector
at
t<tn,
the
t =nAt
n
In
progresses
forward
significantly
appreciable
scheme,
from
,
where
n
consequently,
amplifies
the
When no
cycle.
hence the
are not
to which
time up
pseudo-force
differ
may
and
over-relaxation
incorrect pseudo-force
from the
the problem by creating
for
to the pseudo-force history
relaxation is used, the
create these
pseudo-force
differences
the pseudo-force
a response drastically different
response, in turn exacerbating
next
an
of
The use
.
n
history
amplified
to
correction
the actual
t
last
the
is
pseudo-force history for an
exact
the
the
is
tn
after
interval
time after t
process may still
the
time
result,
resembles
an even worse approximation
the
As a
time
response at
solution
the
words,
time
history, producing
true
history
is correct,
short
for only a
response
in time.
t
history after time
response
from the
other
converge.
the
is evaluated
the
of
end
a
Since the pseudo-force
scheme.
solution
domain
frequency
with
combined
formulation
pseudo-force
the
on
based
HFT approach being
from the
stems
accelerate the convergence
to
scheme
over-relaxation
the
of
inability
The
significant
after
in
time
differences
t
succeeding
solution eventually converges.
iterative
,
but
cycles,
in
these
and
207
Based on
the
results
that
any
scheme
appears
accelerate
5.2.2.4
or
simultaneously
the
structural
to
presented by Hall
transform size
greater
L
2 M-L,
Fourier
particular,
two real arrays of
in
stored
transform of
of
algorithm
versatility in choosing
thus
a
advantage
transform
the
terms
different
particularly
are
improved
2 or 3),
takes
approach
second
evaluating
and
=
8-point
reducing the
time.
computation
The
(N =
An
dynamics.
offers
(32)
that
first
The
history length.
each
for
an N/2-point
a
since
arrays
real
and
4-point
schemes
FFT variations include
suited
history.
chosen
optimally
be
force
is probably impractic al
transforms, but
will
array by
example,
for
exploits,
real
N-point
an
and segmenting
transform,
method
two N-point
the Fourier transform of
FFT's,
radix-2
than
other
using
solution
the
increasing
for
approaches
include
evaluating
the
solution scheme.
Other Acceleration Schemes
efficiency
Recent
the HFT
rate of
cannot
cycles
latter
in
and
history
force
actual
differences
the convergence
Alternate
FFT
'
these
amplifying
it
a pseudo-force history
producing
significantly from the
differing
study,
relaxation
the
of
complex
the complex array
is
the real
FFT
In
be
rearranged
of length N,
the Fourier
can
evaluated,
and
the complex array recombined to obtain
Fourier transforms of
the
of a complex array.
length N
array
of
arrays.
The FFT,
then
the
the actual
therefore,
208
is
FFT also evalutes
response history y(t),
transferring
when
and
exclude any
array may
scheme proposed by Hall, however,
and
computer storage
the
evaluating
and
history
corrects
history
latter portions
parts,
the
convergence
would
be
This
half
the final
cycles
It appears,
would
for
half
will
and
do
therefore,
example,
evaluated
be
in the same
occur
would
approach
The
to
initial
first
pseudo-force
the
they begin
only in
considered next,
because the
segment
in
number of
that
into two
and
first
cycles as
if
second
half
convergence may be more
rapid
history were being evaluated.
the entire
each
to
time progressing manner.
history were segmented,
load
the
response
force
the
the pseudo-force history are evaluated
to the exact history.
converge
if
of
but
cycles,
all
segmenting
of
the
in a
itself
requiring
arrays,
is based on the observation that
separately
to
computation time.
scheme
The final efficiency
The
is particularly adapted
problems with real and complex symmetric
half
in the complex
efficiency.
in
gains
overall
arrays
real
of the
terms
of
recombination
subsequent
the
frequency domain.
the load history to the
rearrangement
only
applies
and hence the approach
the algebraic
In addition,
to obtain the
array,
complex
a
Y(w),
spectrum
response
the
of
transform
Fourier
inverse
the
problems the
dynamic
structural
to
application
particular
the
in
but
in some cases,
solution efficiency
the
increases
approach
This
arrays.
real
two
for
once
executed
response
is
be examined again
The
already
later
established.
in relation to
209
nonlinear mode updating schemes.
Application
Load
described in this
process)
actually
may
particular,
the actual
infinite
an
possess
of a 4th,
the use
approximation
hence
and
zero,
improperly and
the
yield
not
the
unconverged
neighborhood
of
the
the correct velocity at time
This
the
of
beginning
history,
destabilize.
the discrete
The
increment
time
with
significant
numerical
second
Fourier
large
the
causing
eventually
transform
is
converged
entire
exists
in
the
vector
response
solution
to
approximation of using
occurs when an
selected
magnitude
the end of
pseudo-force
previously
the
the response
"leakage" results
vector toward
pseudo-force
works
technique
end of
free vibration at the
response history creating an incorrect
at
finite difference
minimization
zero
that
such an extreme degree
contaminates the entire response.
the
In
tenth order
5th, or even
will
frequency content.
displacement in
zero may be oscillating to
time
that
series
transform of
extracting the Fourier
and
minimization
zero
the
for
(used
expansion
difference
finite
a
by
zero
time
at
velocity
evaluating -the
such
involves numerical approximations
application, however,
in
Its
theoretically.
work
should
5.2.1
subsection
basic procedure presented
The
section.
are
process
solution
the HFT
for stabilizing
Schemes
as
Incremental
and
Stabilization by Artificial Damping
5.2.2.5
such
at
that a
the
excessively
component
corresponding
210
frequency,
cause an
improper transform of
the pseudo-force history, and
solution.
to an unconverging
eventually contribute
Furthermore, the use
finite
of a digital computer with
length and, hence, the presence of truncation error and
word
and lower limits
upper
sense
be numerical in the
solution process, but
the
to
Due
scheme,
solution
these numbers
stabilization
divergence
prevent
stabilization scheme
large numbers during
would
employs
of the equation of motion
both sides
f+(C+
C
C+
artificial
that
both
first
The
viscous
(force)
damping.
is added
viscous damping
(5.72)
force equal
is
to C v-y.
Notice
Eq.
5.72.
are unknowns on the RHS of
Cv is not given
explicitly, since
on a mode superposition approach, but
in
such that
v
terms of artificial
4--T C v
is equal
to
follows:
- ~NL
-v
+ F
+ Ky= F + F
~NL and F
In general,
expressed
as
but
process,
is the artificial viscous damping matrix and F
the
is based
developed that
properly.
artificial
HFT
the
of
nature
the convergence
mathematically, a damping matrix
Stated
the
remain finite.
schemes were
used
when
iterative
analytical
numerical
inherent
the
prevent
may
This divergence may only
the
that
necessarily accelerate
do not
sizes
number
would also induce
process
solution
where
on
process from converging.
solution
V
This aliasing may
producing aliasing.
Nyquist
to 2E v w
viscous
the HFT
scheme
rather it
damping
is
ratios
211
In addition, artificial hysteretic damping may be used.
The
of
concept
theoretically in the
examined
is
becomes k+kh where k
the generalized
is
k
is
the hysteretic term given by
Eh
is
the
hh
is
the
force
damping
hysteretic
displacement
Equation 5.72
response
in
Y is
where
the generalized
+ f
domain.
one mode becomes
v
c
response;
The
ratio.
frequency
the
NL
V1- and
12EhkY(w) where Y(w)
in the frequency domain for
Y + (c + 2Ew)Y + k(1+ i2E )Y = f + f
5.72
(L Kj) and
k where i=
damping
equals
fh(w)
stiffness
12
hysteretic
artificial
Eq.
The stiffness now
domain.
frequency
defined
only
frequency domain, and hence
the
in
is
damping
hysteretic
+ f
h
(5.73)
T
equals
;
and
on
the
other terms are as defined previously.
The
choice of using Ev or
damping
actual
Artificial
for
Eh or
dominating
the
hysteretic damping tends
dynamic
structural
both
structural
to be
the
material
conceptually since
more appealing
time
domain.
in other words,
equivalent
functions
by
it
such problems
dissipation
energy
type damping may be
is well
defined
in
the
Notice that the
zero minimization technique,
the use of Eq.
5.70 is still
application
demonstrated
A viscous
behavior.
response.
appropriate
more
problems since
type
usually exhibit a frequency independent
in
depends
for both
deriving
valid.
This
types of damping is
easily
the
for the undamped problem.
corresponding
We have
transfer
212
viscous damping and
for artificial
for artificial
(5.75)
~
+ i2
W
[2+(1
Hh (
(5.74)
+ W)
H (w) = (-W2 + iw2
hysteretic damping.
equals Hh(w) for o=t.
Hv(w)
Therefore,
turn, Yv (W)
In
F(w)=l.
This situation corresponds to
problem
(the
impulse
response
Fourier
transform
and
function),
h'
vibration
pair of H(w) is h(t),
hence
the
still
5.70
Eq.
if
Yh (W)
equals
free
the
Ev
when
applies.
amplitudes
on
the
damping
the
of
LHS
force
sound.
actual
the
response,
histories
pseudo-force
the
approach
is
succeeding cycles may be
and
response
result
the
a
numerical
HFT
end of
scheme
the
the loading,
to
the
the
divergence.
problem by providing
the
considerably the initial
pseudo-force
solution converges.
evaluates
the
from
history.
is a pseudo-force history containing small
amplitudes until
the
consequent
the
this
attenuating
of
means
in
if
subjected to
significantly
differing
resulting
history,
a
the
previously,
mentioned
As
Artificial damping alleviates
analyst
that
implying
cycles produce poor approximations
initial iterative
actual
governing equation is balanced by
on the RHS,
theoretically
vibration
The damping matrix
the iterative process.
during
simply
method
by preventing excessive
solution
the
stabilizes
stabilization
a
damping as
Artificial
The
incorrect
Furthermore, since
the pseudo-force history only to
the artificial damping approach
aids
213
for example,.artificial
provided by,
cycle
will
initial
the
damping
is
hysteretic
damping.
As
the
progresses,
cycle
iterative
the
the free
only
where
response
linear
produce the damped
out
Notice that when
the
present,
is
damping
hysteretic
loading.
the
the end of
vibration at
damping
by
technique
minimization
zero
the
hysteretic
actual
NL
damping
is
by
the
NL
that
last
h
v
evaluated
approximated
evaluating
As
of applying
scheme,
the
c (w)=i2E
v
v
components
component
process
are
use of
applied
evaluated at
large
in the response may be
instability
before
artificial viscous
carefully.
A
w,
by
but an
finite
from y(t),
again.
frequency
during
if
significant
incorrect
w
the
pseudo-force
a large high
frequency
generated, in turn creating
a larger high frequency viscous damping
producing
iwY(w),
present
are
result,
a
As
Y(w)
from
depends on the
wwY(w)
frequency spectrum.
iterative
and must be
the FFT is incurred).
previously
never
is
y
therefore, approximates y(t)
the problems mentioned
Furthermore,
obtained
be
shown by Eq.
process
inverse Fourier transform of
additional cost
in
can
from y (y(t)
'the
difference
solution
the
in
damping
artificial
However,
depends on the velocity y.
f
actually
and
the
of
involves numerical approximations.
concept
5.73,
zero.
implementation
The actual
hysteretic
all
contains
F
iteration,
contributions and F is
damping
such
the pseudo-force vector F
in
manifested
force, and
possibly
the solution converges.
damping should,
consequently,
The
be
214
is
directly
of
the
frequency
associated with artificial viscous
problems
two
The
.
Its
damping are avoided.
in the
no
since
employed,
force cannot be
free vibration components
incorrect
cycle,
succeeding
the
transferring
hysteretic
The
domain.
time
that
be
response
corrected
frequency domain and then multiplying
force vector
other
applications
two
involves
f(t)+f NL(t)
both y(t)
and
for
the
contains
the
for
the
evaluated
y(t)
by 12E h.
and f(t)+f NL(t)
array
are
implemented
storing,
f(t)+fNL(t) as its
by
only
the
to
back
This process
the FFT, one to transfer the
frequency
domain
the
and
of using
hysteretic damping are avoided by realizing
one FFT need be
complex
to the
with
removed in
The apparent inefficiences
to transfer y(t).
artificial
of
Y(o)
force
damping
can
therefore,
are
by
domain
conjunction
however,
technique,
minimization
zero
the
In
force
damping
frequency
the
the response Y(w) by 12Eh*
multiplying
technique were not
the hysteretic
in
easily
pose
disadvantage would
minimization
in such a case
obtained
be
would
zero
the
if
problems
This
time domain.
not
is
however,
implementation,
straightforward since the hysteretic damping
evaluated
better
damping
to Y(w) and also independent
proportional
W
the
since
behavior
numerical
a
demonstrates
damping
hysteretic
Artificial
that
real histories, and hence only
if
advantage
taken
of
real part
The actual
procedure
transferring two real arrays simultaneously is given
references
(17).
a
as its
for example, y(t)
imaginary part.
is
in
215
studies
Case
included were
artificial damping was
when
in Ch.
discussed
conducted and are
load
diminished
response during the
incorrect
the
where
considerably
such that
scaled
is
damping
concept of artificial
the same
based on
is
technique
structure
stabilization
This
incrementally.
applied
and
load
incremental
loading the
than
Rather
approach.
the
is
immediately with the actual load history, the
down
solution
the
6.
Another stabilization method
application
behavior of
the
demonstrating
initial
cycles is
resulting pseudo-force
the
history does not generate an excessive incorrect response in
the
course of
the
solution
identical
the
if
of
the frequency domain
f(w) + f
where
2
k
= -W
from
regarded
solution is
is
stiffness approach
perspective of a complex
k c(w)Y(w)
and indeed
static problems,
nonlinear
resembles
procedure
The
succeeding cycles.
(W) + f
(W) + fh (Wo)
-2
+ io2w(E + Ev) + w (1 + i2E
(.5.76)
)
(5.77)
5.2.2.6 Nonlinear Mode Updating in the Frequency Domain
The numerical considerations
concern the solution of the
subsection discusses
specifically,
schemes
consists
in the
the
given
up
to
point
this
independent SDOF problems.
This
or
more
the problem in its
development
frequency domain.
of
In
of evaluating modal updates
entirety,
nonlinear mode updating
general,
to the
the
problem
linear structure
216
as
it becomes
to
nonlinear and applying these
shapes
the
obtain
to
matrices
structural
geometric
the
mode
new
generalized matrices.
Beginning with
second
the
linear mode shapes, where n is
of freedom, are used
degrees
The
cycles.
mode
shapes orthogonal
The
obtained.
to
n
iterative
few
is stopped, and m nonlinear
the original
eigenvector matrix
T
linear
first
the
in
that
than the total number of
less
process
solution
suppose
problem,
i
n
are
shapes
mode
is given as
T
n shapes
(5.78)
m shapes
nonlinear
These mode shapes are
of
equation
motion
then applied
to
The generalized
motion.
only partially diagonal.
derived
were
from
the
to the original governing
obtain the generalized equation of
structural matrices,
however,
Although the linear mode shapes
eigenproblem
using
the
are
J1
linear
stiffness matrix
-2
-
(
=
(5.79)
implying that
T
-
1=0
(5.80)
217
or
2
T
lK
the
eigenproblems using nonlinear
be
longer
5.66,
cannot
the
of
the reciprocals
computing
by
simply
evaluated
are no
given by Eq.
transfer matrix H,
matrix.
diagonal
matrices
structural
generalized
the
diagonal,
(5.82)
is not necessarily a
Tul Kn
the
Since
stiffness matrices
=
-2
and hence
from
derived
were
Inl
shapes
mode
nonlinear
(K
(5.81)
= diagonal matrix
=
diagonal terms in H~1.
problem is
approach,
inversion
spectrum
to obtain H directly by
(10,000
requirements since
stored in
The
alternate
terms in
the
addition to
is
the diagonal
impracticality of
iterative
the
for
scheme
used.
process,
This
an
in the
has
terms
approach
storage
must
be
suggests
an
H
managing the off-diagonal
Rather than
approximate
approach
of
time
10,000
terms.
the first
transfer matrix.
solution
matrix
off-diagonal
the
frequencies
additional
entails
moreover,
and,
steps)
matrix
excitation
if the
times
the
since
all
for
executed
be
must
This
[H-1 ]1.
evaluating
costly
extremely
is
however,
H- 1
nondiagonal
One possibility for circumventing the
transfers
employing
diagonal
all
H
in
transfer
off-diagonal
218
to
H~
from
obtained
solution process is
the
since
problems
of motion for
The governing equations
Myf + (C + C
_+ K=F
superscript
the
where
+ F
off-diagonal
force vector
new
system is now
(5.83)
matrix
generalized
a
to
and
terms
F0D
is
the
(5.84)
+
the
where
an
iterative.
+ F
diagonal
its
only
+ F
already
the
~0D
v
refers
s
containing
~OD =
~NL
~
uO
u
the
of
any
pose
not
does
but
scheme,
solution
iterative
is easily
engenders
obviously
the
used
terms are
reciprocals
the
procedure
This
of
RHS
hence H
and
,
computing
by
terms.
diagonal
1
H
approximate
the
evaluate
remaining diagonal
The
of motion.
equations
the
to
matrices
generalized
the
in
terms
superscript
'
containing only off-diagonal
denotes
terms.
a
generalized
matrix
Therefore
C=C + C
= U +
(5.85)
C C
G+
-V
V
A
shapes.
final
approach is to
use
the
In other words, new eigenvectors
a nonlinear stiffness matrix K
are
only
employed
in the
and
nonlinear
are evaluated
mode
from
only these eigenvectors
succeeding iterations.
The equation of
219
is
motion
difference
the
evaluating
nonlinear
approach
This last
member forces.
response
is
represents
the stiffness
history.
The
fairly
uniform
problem
of
updating
updating schemes
approaches because
in
proceed
of
solution at
hence
all
commences.
cycle.
a
that
Knl
the
shapes.
mode
time
and
be
expensive
to
implemented whenever
domain
solution
Furthermore,
steps is obtained
the
mode
3.2.2,
in section
frequency
Updating is possible
the first
in contrast to the
substantially.
shapes must be
handling
Nonlinear
change
mode
entire
an
for
to
all time
implies
stiffness
after
matrix
an
K
the
very
The
that
simultaneously,
selected before the
only
the
Applications of
available
approaches
in the frequency domain,
nonlinearities begin
nature
the
6.
nonlinear mode updating schemes discussed
cannot
actual
when
such
transfer matrices, we now proceed
non-diagonal
the
eigenproblem, however, may be more
various
With
suitable
is
schemes are presented in Chapter
these
I,
q and
eigenvectors must be derived.
of
set
1
during a significant portion of
computationally than the other
new
K
between
by
derived
now
is
FNL
vector
pseudo-force
the
that
Notice
from Knl.
derived
were
mode shapes i
the
because
diagonal
are
matrices
structural
generalized
all
where
(5.86)
-NL
~
the
and
solution
iterative
is
never
220
and only
evaluated,
These
the pseudo-force history
significantly
considerations
the mode
updating
Of
second
restrict
presented in
shapes
mode
involving
eigenproblem of an imposed deformed shape
The
promising.
offers
HFT would
1.
HFT
the additional
due
scheme
the means for
to its
the
section 3.2.2,
selected
appears
from the
most
the
iterative process
advantage of providing
the
insight to
This updating approach adapted
response.
nonlinear
extracted.
shapes.
the updating schemes
approach
is
to the
be as follows:
linear
mode
load distribution
from
few cycles with
Iterate the first
the
shapes
2.
Stop the Analysis
3.
Derive an equivalent
static
the pseudo-force histories
4.
Apply
structure
this
and
static
load
obtain a
thus,
distribution
to
the
tangent stiffness matrix
K
-t
5.
Evaluate the first
few eigenpairs
corresponding
to
K
-t
6.
Gram-Schmidt
orthogonalize
eigenvectors with respect
to the previous eigenvectors
7.
Restart the analysis using
8.
Repeat steps
acceptable
2-7 until
convergence
nonlinear
these
the
the new basis
solution
vectors
demonstrates
221
on
insight
the part
same
the
of
is
easily
all
for
differing
selected.
members,
and
a
set
load distribution, or even
requires
A
considerable
second updating
history
perspective.
In particular,
stress-strain states,
are
histories
evaluation of
stiffness
employs one set of
thus
or
An
less
luck
and
preparation
time
was
involving
rational and
systematic
history
from the
current
stiffness
the
permitting
tangent and secant
eigenvectors
the
exploits
scheme
indirectly,
HFT
luck.
more importantly,
the exact tangent and secant
the exact global
matrices.
structure
deformed
since the pseudo-force
available
that
distributions,
forces, obtained
from the member
derived
of
more
a
from
pseudo-force
is
of a static
This
developed.
subsequently
the selection
minimal
demanding
and
insight
however,
histories
judgement and,
approach
load
static
pseudo-force
nonlinear
the
recreates
adequately
pseudo-force
Most analyses,
complicated
extremely
involve
equivalent
the
time for all members,
distribution
For simple
analyst.
the
involves considerable
however,
those exhibiting one peak occurring at
as
such
histories,
3,
Step
straightforward.
fairly
appears
described
as
procedure
The
solution
scheme,
during each
structural
however,
iterative cycle,
requiring a method for selecting a structural
stiffness
response
history.
stiffnesses characterizing most
of the
set of
Only one
stiffness matrix and
its
used if
an
transfer
chosen.
If
exact
diagonal
the approach
eigenvectors
matrix
of transferring
can be
approach
is
off-diagonal terms
222
to
used,
again only one stiffness matrix can be used in the
but
of H.
evaluation
In
either case,
is given as
1.
of
stiffnesses may be
to different
corresponding
eigenvectors
a host
selected,
the governing equation is
of
the RHS
the second updating
valid and
scheme is
follows:
from the
Obtain Esec for each member
the
of
generation
the
during
evaluated
pairs
stress-strain
pseudo-force histories
2.
and
3.
During
EE
2
sec
where
the summation
store
a
degree
of
least
Reconstruct
the
the results
secant
of
5.
Evaluate m eigenvectors
D
6.
Use
e
diagonal
directly
matrix
orthogonalize
cn
transfer
matrix
Knl
step 3
of Knl
the
next
function)
with respect
and
eigenvectors
in
for each
from step 2
stiffness
corresponding to the E secs
evaluated,
Esec
squares approximation of
freedom using
Z'E sec
is over time
After the entire pseudo-force history is
obtain
4.
the process of ddriving E sec,
proceed
to the
with
iteration
(exact
Gram-Schmidt
or
previous
the
set
of
analysis
(off-diagonal term transfer approach)
This updating scheme can be executed
the
analysis,
analyst
and hence
during
efficiency
can be
the
without
requires no participation
solution
process.
increased by implementing
The
the
stopping
from the
updating
process
only
223
after every
of
intervals
the history, eigenvectors corresponding
eigenvectors are
eigenvector
the
dominant
are
matrix
stiffness
secant
then
fitted
for specified
be evaluated
can
matrices
stiffness
secant
squares
least
Furthermore,
i cycles.
to each
and
derived,
the
This approach produces a total
recombined.
consisting of eigenvectors representing
matrix
behavior of different portions
of
response
the
history.
The
of
portion
contributed
by
the
an additional
cost
as
shapes.
recently
for both
analysis
HFT
the
eigensolver is
essential.
technique is
(9).
structural systems
appears
an
periods and mode
An efficient
mode shapes are updated.
updating
used
incurred in
is
The subspace interation
large
the lowest
as
such
properties
structural
therefore
Although
frequency domain solutions to determine
time integration and
as the
systems may be
large
problem.
eigenvalue
substantial
analysis is usually conducted initially
eigenvalue
shapes,
for
cost
solution
the
a
the mode updating scheme,
Regardless of
updating
modal
both
6.
is discussed in Chapter
schemes
of
implementation
actual
schemes,
attractive
the
initial
With respect
subspace
the
since
to the HFT mode
approach
iteration
also
previous mode shapes can be
the
shapes
often employed for
in
determining
the
updated
A more efficient eigensolver has been investigated
by
Wilson,
et.
al.
(88,89).
The
scheme
is
224
suited
particularly
to
vector,
such as
evaluating
the exact eigenvectors
Ritz
their shape is
natural
then
involves
Ritz
vectors
exact
eigenvectors.
Furthermore,
were
basis
smaller
opposed
the response
to
if
their
the loading, even though
load frequencies.
generating
that
one-tenth the effort
analyses
shapes
those
as
response,
indicate
al.,
et.
The
analysis.
to
may be near dominating
Studies by Wilson,
because a
vectors, which are
not participate in
orthogonal to
frequency
superposition
the
in
may
which
eigenvectors
new
correspond
only
vectors
participating
actually
the
system,
the
superposition
mode
used in the succeeding
generated
than
of
Ritz
orthogonal
temporal
a
Rather
in seismic excitation problems.
generates
scheme
by
vector multiplied
decomposed into a spatial
can be
the excitation
problems where
the
to generate the-
computation times
for mode
less when using Ritz vectors,
could
accurate
more
a
produce
response.
the second mode updating
scheme, we
segmenting method
introduced
in section
was
motivated
In conjunction with
now
reexamine
5.2.2.4.
the
segmenting
The
pseudo-force
history
incentive here
is
by
different
function
produces
a
response.
good
and
D
-nl
approach
link
is
representation
An overlap-add or
appropriately
such
the
the
its
overlap-save
response of the
The
exact diagonal
maintained
of
the
history
response
an
that
from
time.
in
itself
correcting
to model portions of
K
-nl
transfer
scheme
and
the
of
portion
(17)
method
different
Knl
the
would
segments.
225
time
to t
t
schematically in Fig.
shown
is
The procedure
.
for
and Knl
to t
t
for time
In particular, K 1 may be used
5.6.
Selecting a Time Increment
5.2.2.7
the
of
part
final
the
As
section, we discuss a procedure
load history by choosing the
corresponding to
Tn
period
for
increment, therefore, must
response
material
selection procedure is outlined as
Obtain Fourier transform of
2.
Let
3.
Choose
4.
followed.
Compare
.1/2fn
that
time
The
load spectrum
/wmax
with
Ati
A
load history
Wmax be highest frequency in
1
behavior.
follows:
1.
At
large
too
be chosen such
adequately
is
the
in
used
increment may be
time
material
path-dependent
with
problems
smaller time
the
mode
Time increment At1 , however, may be
appropriate.
negligible
then compared with the
highest
larger
that at
such
At
load component has
the
a
AtCTn /2,
If
analysis.
increment
increment is
time
This
magnitude.
time
the
frequency fN,
the Nyquist
for selecting an appropriate,
is obtained from the external
guess
A first
time increment.
considerations
numerical
the
of
period
highest
the
structural mode T n
5.
If
At
is
significantly smaller than T n/2,
time
increment
load
history
components set
may
should
be
appropriate,
have
its
to zero if a larger
and the
higher
At is
a
larger
external
frequency
chosen.
Also,
226
F
K
K2
K3
a.
excitation history
b.
response using K1 for interval 1
ul4
t
c.
response using K2 for interval 2
t
u3
d.
u
response using K 3 for interval 3
t
t
e.
Fig.
total response
5.6
Segmented History Analysis
227
is
theoretically consistent with steps
a
linear
higher
frequency
W
.
components
in the
exercised
new
practical
advantages
analytical
over accepted
evaluating
dynamic
its
technique
5 should be
to
must
provide
problems.
the
to
analyst
solving
considerable
offer
The
solution techniques.
applicability
structural
for
adapted
the HFT solution scheme
of
section
this
4 and
the
the HFT Solution Scheme
problems
engineering
.characteristics
in
load
carefully.
5.2.3 Qualitative Evaluation of
Any
steps
hence
and
history,
pseudo-force
from
originate
will
analysis
HFT
external
These ultraharmonic
(65,79).
generated
be
can
ultraharmonic
of the
components corresponding to multiples
frequency
highest
frequency component at
however,
analysis,
nonlinear
a
in
history).
F(W) has
if
Y(w)=H(w)F(w),
since
w 0, Y(w) also has a highest
In
3 since
(load
input
than the
frequency components
words,
approach
This
2, and
1,
never
was
a response will never be generated with
analysis
other
In
time
time increment.
selecting a
when
considered
consecutive
history
pseudo-force
the
that
and
may be necessary
time increment
steps, a smaller
Notice
changes for
considerable
demonstrates
dependent
path
is
behavior
material
the
if
salient
are accentuated
a
basis
for
the solution of nonlinear
228
frequency domain solution, the
a
with
the
exploits
the
offered by a
economics
problems,
dynamic
structural
but
response
loadings,
and wind forces.
be
can
behavior
response
to
that
in the sense
partially restricted
frequency
The
evaluated,
is
frequency
spectrum.
elongation
and
approach
is
solution
propagation
steady state
the global
are
nonlinear
localized effects
but
(85).
to
capture
Numerical
direct
the
accuracy
artificial damping,
the
appropriate
time
integration
and
if
the
in
the
problems
of
peaks
consequently, are
time increment
than the
approach.
transfer
integrator
chosen
and
often 5-10 times larger
a
employs
properly
is
sufficient
resolution
a
wave
spectrums
wave
as an exact numerical
frequency
Nyquist
eliminated,
domain
behaves
that
function
of
intense
to
of modes, producing an inefficient solution
period
the
fatigue degradation may require a substantial number
such as
the
include
to wave
Extensions
response
the
as
such
problems
are
therefore,
Applications
excluded.
are
problems
type
modes
all
excitations,
seismic
to
transient
practical
the
precludes
applications,
Potential
integrator.
easily accommodates
solution of wave propagation problems where
excited.
space and
vector
reduced
approach
superposition
mode
HFT solution approach
exact numerical
accuracy of a theoretically
The
superposition technique
the nonlinear mode
combining
By
time
of an HFT
increment
229
Furthermore,
(or
velocity
and
hence
dominating
a
from
and
of generalized displacement
measure
of
the response and readily portrays
the
a
quantitative
linear to a nonlinear
response.
solution is more easily examined by
spectra,
pseudo-force
acceleration) and
offers
domain
frequency
the
in
solution
response in terms
the
presents
the
modes
the
transition
accuracy of the
The
response
the
comparing
spectra of consecutive cycles.
the
solution
after each
parameters
examples include changing the time
the
frequency
of
number
frequency domain),
the
examining
if
the
spectra,
response
component at
of points in
By
shapes.
can ascertain
the analyst
frequency has
the Nyquist
the
increment, modifying
reselecting the mode
and
Particular
iteration.
(total number
components
of
updating
an
solution approach also allows
The HFT
an appreciable
the
magnitude
and evaluate the contribution of each mode to
response,
and subsequently choose a larger time increment
eliminate the non-participating modes.
verifying the
the
added, and the number
response
resulting
previous
storing
of
be
decreased,
zeroes
compared
the
can be
more modes
extended,
with
iteration
last
and
response
then
the
at
the
iteration.
This attribute of
however,
can
a final check for
during the
solution accuracy,
increment
time
As
or
originates
the
updating
from
the
the
solution
frequency
parameters,
domain
approach
entire response history, and hence implies
that
230
scheme demands
storage
requirements are easily
inside the iterative
embedded
over
the response history operates in the
contained
in
Two complex arrays
TRANSF and
transfer
functions
stores
TRANSF
the
FFT.
the
points in
of
if
the frequency domain
the transfer function is
number
are
CA
all
for
of
required.
modes and
application
of
and CA,
and NB
the number
(TRANSF can be
eliminated
of modes
NMODE is the number
loop
second
size NMODE*(NB/2+1)
Therefore, TRANSF has
size NB where
and
the generalized
the
frequency components and CA is used in
domain
the
over
is
RESPON
subroutine
loop
The
history.
force
The first
time
degrees of freedom and evaluates
generalized
response.
total
the
evaluates
loops
two main
loop 100.
are
the
for
5.7
in Fig.
Notice that
total displacement formulation.
its
examining
by
deduced
presented
program flow chart,
computer
The HFT
storage space.
a substantial
the HFT
it
time
each
reevaluated
is
used).
Since
degrees
of
mode
and
implemented
total
history.
because
alternate
the
the
transferring
an
procedure
force history and then
repeating
since the
force
the
force vector must be
sequence
this
entails
Furthermore,
generalized
succeeds
freedom and the entire
requires
approach
response loop
generalized
the
loop,
loop
the
the process
generalized
entire geometric force, and hence
for
stored for all
The
iterative
frequency domain
entire
history.
evaluating
response
the next mode
force is
evaluation
derived
for
the
one
cannot be
from
the entire geometric
the
force
231
no
RESTAR
QNON
QLIN
0
0
@
232
@
call RESPON
evaluate generalized response y
233
o
EXPAND
Fig.
5.7
Flow Chart for Hybrid Frequency-Time
Domain Analysis Package
234
each mode.
forces of
The result
generalized force history of all
where NP is
NMODE*NP
size
generalized
the
history must be acquired before evaluating
storing the
is an array GPF
GPF
has
in the
load
modes.
included
points
the number of
history.
by
storage
size of
problem
with
NB=8192,
the
is
TRANSF,
CA,
mainly
modes
required storage
not stored but
or 408,192
storage requirements,
compared
to
space is
those
of
typical
(NP),
implying
the
space
storage
problem.
for our particular
large
are relatively
consequently,
integration
time
direct
typical
TRANSF
If
613,042.
recalculated each time,
becomes NB+NMODE*NP,
The
steps
and 8000 time
a
For
NB+NMODE*(NB/2+1)+NMODE*NP.
50
to a minimum
amounting
GPF,
and
are determined
therefore,
The HFT storage requirements,
programs.
With
judiciously
select
nonlinear problems.
the
possess
implementing
can be
an
The HFT
the solution process.
can
scheme
for
does
not
however,
established direct time
valuable
insight
when
Considerable studies on
of problems are necessary before
employed with confidence.
analyst
solution
approach,
that can provide
the
mind,
appropriate
extensive background of
integration studies
a variety
in
these considerations
the
HFT
scheme
235
CHAPTER 6
SAMPLE STUDIES
Chapter 2 of
4,
the
and 5 discussed
solution
The
from
well
has
formulations
the actual numerical
techniques.
numerically
solution
with
then,
although
consisting
of
unpredictable
such as
involves
a
a
may
the actual
of
stemming
computer.
well
be
two
Even
formulated
nonlinear solution
and error
learning period
and
unforeseen
from
a
theoretical
established
trial
multitude
"bugs"
obtaining
of
of the digital
theoretically and numerically,
often
implementation of
process
problem
the
Chapters
possible only during the last
become
the introduction
decades
process
theoretical
for nonlinear continuum mechanics, while
foundation
3,
the
presented
thesis
this
numerical
often
limitations
finite length numbers and discrete modelling or
even
existence
and
restrictions
theoretical
uniqueness
of
the solution.
such
as
the
236
frequency-time domain solution scheme
in this chapter
are illustrated
system
SDOF
by
first
demonstrated
stability
and
Efficiency
studies.
several development
chronological
The
ideas.
by
acceleration
and
damping,
display
studies
studies are not
presented in
rather in a sequence emphasizing
but
order,
the
of
formulation
sample
Most
updating.
mode
nonlinear
then
examples.
system
artificial
by
stabilization
relaxation,
are
and
studies,
minimization,
zero
scheme-
solution
the
on
is placed
sample
of
characteristics
in MDOF
additional refinements are examined
Particular emphasis
described in Chapter 5
collection
by a
hybrid
the
for
considerations
development
The
the evolution of the HFT scheme as
it
is
more
to
applied
complex problems.
6.1
This
development
ability
to reproduce
tubular
6.1,
SDOF
structure
SDOF
viscous
damping.
brace
had
scheme
a
to
ascertain its
response of
SDOF systems.
period
of 1.72 seconds and 5%
Both an elastic-perfectly
model
material
were employed.
does not
the nonlinear
represent any
and
plastic
(39,43,66,94),
Although the
the
during
conducted
scheme
HFT
the
of
was used to demonstrate
HFT
studies
section discusses
initial
The
SDOF SYSTEM
STUDY,
FEASIBILITY
a
shown in Fig.
in
tubular brace model
realistic structure,
a
this model
the convergence capabilities
of
the
for a complex material model containing buckling
and a degrading stiffness.
Two
acceleration histories
were
237
employed
--
amplitude
10 and a
component
March
scaled to 0.25
acceleration
1977 Bucarest
g,
shown
P
E
in
wave with
sine
cycle
single
the
from
5 second history extracted
the
of
second
two
a
SN
earthquake with peak
Fig.
6.2.
CB
A
Fig.
The
plastic
first
6
SDOF
the
=0.50, using a
with
time
study consisted of
direct
The
with
Tubular Brace Model
model
material
excitation.
obtained
6.1
and
time
the
single
elastic-perfectly
integration
of 0.1
combined
An
identical
specified in
the HFT analysis.
minimization or artificial damping was employed.
were required in the frequency domain
effects
(20
to
a=0.25 and
seconds and
the Newton equilibrium iteration scheme.
increment was
represent
the
were
results
Newmark integration method,
time increment
wave
sine
cycle
zero
No
512
points
to eliminate transient
load and an additional
492
100
80-
60 40 -
-20 -40
-60 -80 -100
0
TIME
Fig.
15
10
5
6.2
20
(SEC)
- Bucarest Acceleration History
25
239
appended
displacement formulation was
used
for the linear case are given in Fig.
6.3.
A dual
zeroes).
in all HFT analyses.
The results
6.4
Figure
shows
time domain and hybrid
the
domain results for a yield stress
a
the responses are almost
To
hybrid
After
identify critical problem areas associated with
the
study
was
domain
frequency-time
6.5
substantial
yield
ksi
t=1.5
ten
iterations,
seconds.
carefully,
we
results in the
the
or
notice
additional
end of
that
the
with
solution diverges.
even
These
evaluating
6 seconds,
loading terminated.
obtained for the
the pseudo-force
the loading was
conducted.
response obtained in the
ten iterations.
by
interval of
response up to two seconds using direct
the
5
6.6 and 6.7 are perused
history for a time
the poor results
study
A
case.
compared to the
frequency domain were obtained
4 seconds after the
0.56 inches.
particularly after
inaccurate,
is
2
was
The frequency domain response,
In fact, when Figs.
pseudo-force
Given
the
of yielding occurred,
yield stress case.
after
stress
this
for
results
the
presents
amount
the
analysis,
to a yield displacement of
corresponding
Figure
five iterations
identical.
extended to an extreme case where the
ksi,
to
corresponding
of 5 ksi,
of 1.39 inches.
yield displacement
frequency-time
2 ksi
case,
an
calculated only to
Figure
6.8
shows
the
time integration and
frequency domain
after one
and
Notice that the HFT analysis converges when
240
2.5
1
1
i
i
i
1
2
3
4
5
2.0 1.5 1.0 -
z
LU
u
0.5 -
<
..J
0.0
(n
H
-0.5
C
-1.0
-1.5
--2.0
0
TIME CSEC)
Fig.
6.3
oinear-Elastic Iesponse to a sine wave Loadinq
6
241
2.5
1
2.0
1.5
z
w
1.0
W0.5
U/
00
..
-0.5-
/
Time Integration,
At
= 0.01
Frequency Domain, 1 iteration
Domain, 5 iterations
-a-Frequency
-1.0
1
0
2
3
TIME (SEC)
Fig.
6. 4
Elastic-Plastic Response (Fy = 5 ksi)
4
5
6
242
2.0
1.5
1.0
0.5
z
LU
a.
-J
0
0.0
-0.5
-1.0
-1.5
-2.0
1
0
2
3
4
5
6
TIME CSECJ
Fig.
6.5
20
Elastic-Plastic Response Using a Time Domain Analysis
(Fy = 2 ksi)
-
15 -2
10 5 z
U
0
-5
Hn
-10
-15-20-25 -
0
1
2
3
4
5
6
TIME CSEC)
Fig.
6.6
Elastic-Plastic Response Using a Frequency Domain Analysis, One
Iteration (Fy = 2 ksi)
243
100
50
0
-50LI
-100CL
-150-200-250
-
6
5
4
3
2
1
0
TIME (SEC)
Fig.
6.7'
Elastic-Plastic Response Using a Frequency Domain Analysis,
Iterat ions (Fy = 2 ksi)
10
1.0
0.5
0.0
z
U
-0.5
-n
-1.0
-1.5
-2.0
L
0.0
0.2
0.4
0.6
0.9
1.0
1.2
1.4
1.6
1.8
2.0
TIME (SEC)
Fig.
6.8
Elastic-Plastic Response with Pseudo-force Correction to T = 2.0s.
244
the
(after time 2 seconds).
free vibration part is neglected
frequency domain converges only when
is neglected can be
and
6.11.
.
first
the
during
portions correspond to plastic yielding regions.
sloped
from
portion
flat
linea r
initial
explanation
6.10
Figures
how
the
and
6.11
show the
that
to
identical
pseudo-force histories
linear
the
exciting
period
cause
the
displacement
hybrid
,
In other words,
structure.
which are imposing corrections
to obtain the nonlinear
structure
at periods
the
vibration
to
approach
corrections
that
are
frequency-time domain analysis
excited portion of
response,
stage,
upon
are
and consequently
yielding
resonance,
extremely
is
the
fundamental
its
close to
free
structure
almost
vibration
of
periods
response
during
the
the
of
for
three figures depict periodic
All
with
histories
pseudo-force
an
is obtained).
pseudo-force history
of
pseudo-force histories
6.12 for
to Fig.
The
to the
corresponds
seconds
response (refer
elastic
i terations
successive
0.5
to
0
to
and the
yielding
plastic
from
reversals
stress
elastic
correspond
6.9
The plateaus in Fig.
6.3.
shown in Fig.
the
response
elastic
the linear
resembles
6.10,
closely
how
Notice
iteration.
history
pseudo-force
6.9,
pseudo-force history
the
depicts
6.9
part
the free vibration
explained by observing Figs.
Figure
the
in
response
the
where
anomaly
apparent
This
large.
therefore
the response and cannot
The
limited
reproduce
to
the
the
free vibration response for an elastic-perfectly plastic
system
whenever
the
free
vibration
stage
consists
of
245
250
200
150
C
100
50
0-
0
-50 -100 -150 -200
1
0
2
3
4
5
6
4
5
6
TIME
Fig.
6.9
Pseudo Force History,
1 iteration
800
600
w
400
±
200
CD)
w 0
0-
-200 -400 -600 -800
0
2
3
TIME
Fig.
6. 10
Pseudo Force History, 2 iterations
246
3E3
2
C-,
1
CD
0
a-.
-1
-2
-3E3
0
1
2
3
TIME
Fig.
6.11
Pseudo Force History, 3 iterations
4
5
6
247
P
f
u
c
a.
-
Actual Response
- --
Linear Response
Force-Displacement Diagram
Ps = Pseudo-force - linear load - nonlinear load
P
s
5
--- ---
time
b.
Fig.
6.12
Pseudo-force History
- Pseudo-force History for an Elastic-Perfectly
Material Model
Plastic
248
deformations.
inelastic
next
The
at
time
increment
6
seconds
of
0.01
zeroes was
all
number
A
seconds.
Figure 6.13 shows
time
direct
iterations
inches
integration
seconds
CPU time while
of
iteration
equilibrium
for
response
the
After eleven
in
the
frequency
domain,
domain
observed
required 90
analysis
using
step required only 5
time
each
the
is 4.24x10~4
visually
analysis
HFT
time
the
be
can
The
at
(Y2)
displacement
figures.
two
the
loading and response.
analyses.
such that no differences
between
sufficient
a
that
that
HFT
nonlinear
the
7700
and
using 2 1 3 points
of
residual
approximately
displacement
the
beginning
excitation to ensure
the
of harmonics represented
of the
time
eliminated and
effects were
)
a
seconds, with a
string of
of the
added to the end
transient
11
ending at
and
g
0.25
(peak acceleration of
Bucarest earthquake
seconds
5
to
subjected
ksi
36
of
stress
yield
with
structure
same
the
study considered
seconds.
It
domain
loading
the
stage and
within the
even during
terminates
elastic
the
regime,
expended
consequently the study was
scheme.
Further
if
CPU
hybrid
the
that
provides
technique
loading
However,
evident
is quite
results
accurate
the
frequency-time
free vibration
the
during
the
stage after
free vibrations oscillate
which
usually
is
time
is
much
the
larger,
extended to optimize the
analyses determined
case.
and
solution
the minimum number of
249
15
Time Integration
-
10
-
.- ...-- Frequency Domain,
10 iterations, N = 12
5
0
o
_10-J-15
-
-2
0
1
2
3
4
5
6
TIME (SEC)
Fig . 6. 13
Elastic-Plastic Resonse to the Bucarest Earthquake (Fy = 36 ksi)
250
number
the
loading and
required in the
zeroes
trailing
of
iterations at which the responses were sufficiently similar.
The
the
The
Notice
that
the
to
that
identical
displacement
the
inches
after
compares well
toward
response
the exact
with
the
end
of
the history
of
forward progressing corrective behavior
18"
which
large error
This
response.
the
large,
is approximately
seconds
2.9
time
at
response
time step is extremely
last
corresponding to the
342
of
residual
a
Although the maximum response
iterations.
5
Figure 6.15
analysis.
for
displacement response
depicts
for
except
6.13,
the end of the
toward
discrepancies
practically
is
response
Fig.
in
given
6.14.
is shown in Fig.
corresponding displacement plot
A
of 26 seconds).
(CPU time
minimal cost
results at
best
the
produces
after six iterations
of 157
residual displacement
required.
212 points are
a minimum of
results,
tabulated
seen from
can be
As
given in Table 6.1.
results are
in
characterizes the
the HFT
solution
HFT
solution
scheme.
As
scheme
a final
this
study of
the
section,
the tubular
was applied to a SDOF system composed of
earthquake
in Fig.
6.16.
response
diverges.
iterations
The
history.
after
Using 212
three
With 213
are
Response values
Bucarest
same
the
to
subjected
brace material model and
time integration results are shown
points
in
iterations,
points the
shown in Figs.
the
shown
response
in
and computation times
are
Fig.
6.17,
and
12
after
6.18 and 6.19,
the
scheme
HFT
15
respectively.
listed
in
Table
251
Iterations
No. of
Points
Solution
Scheme
________
I
I
Time
Integration
(nonlinear)
HFT
time
*
T
17-64
5
.0004
17-64
90
163
17.64
52
8192
11
8192
6
4096
11
1
17-64
42
4096
8
10
17-72
33
4096
7
20
17.74
29
4096
6
157
17.53
26
4096
5
342
54.97*
23
2048
11
4132
34.73
22
increment
=
0.01
seconds
occurred during last time
Table 6.1
CPU
Maximum
Displacement Time
(sec)
(inches)
Residual
Computation Times
SDOF Model
for
all analyses
step
for Elastic-Perfectly Plastic
252
15
10
5
|-LJ
LJ
0
C-
-J)
-5
10
15
0
1
2
3
-
5
6
TIME (SEC)
6.14
Fig.
Elastic-Plastic Response to the Bucarest Earthquake, 6 iterations,
N = 12.
60
50 40 z
LUJ
30 -
LU
...J
CL.
(/)
20 10 0
-10
-20
-
0
1
2
3
+
5
6
TIME (SEC)
Fig.
6.15
Elastic-Plastic Response to the Bucarest Earthquake, 5 iterations,
N - 12.
253
12
10
8
6
4
C,
2
0
U,)
-2
-4
-6
-8
-10
0
1
2
3
4
5
6
TIME (SEC)
Fig.
6.16 Tubular Member Response to the Bucarest Earthquake,
Time Integration
300
200
77-
w
CD
100
0
-100
-200
0
1
2
3
4
5
6
TIME (SEC)
Fig. 6.17 Tubular Member Response to the Bucarest Earthquake,
3 Iterations, N = 12
254
35
30
25
20
LU
0-4
15
10
L-
5
0
-5
-10
0
1
2
3
4
5
6
TIME (SEC)
Fig.
6.18
Tubular Member Response to the Bucarest Earthquake
12 Iterations, N = 13
12
10
8
6
:z
C-)
4
2
0
-2
-6
-8
-10
0 .
1
2
3
4
5
6
TIME (SEC)
Fig.
6.19
Tubular Member Response to the Bucarest Earthquake,
15 Iterations, N = 13
255
6.2.
reproducing
load
complex
for
SDOF
histories
and
represented
possible means of
considers
The
two cases where the
Bucarest
earthquake readily
solution in
and
illustrate
cost
only
500
the 5 second load history, a
and
and
6.2
the
for
elastic-perfectly
respectively.
As
inefficiency
stemming
from
the
minimization
technique
factor
a
doubles
consequence
for
each
of
this
a zero
zeroes,
appended
developed,
was
6.1
Tables
time
computation
the
two.
described
as
in
5.
Chapter
Applying
the
elastic-perfectly
Notice
system
SDOF
plastic
in general
the
that
the
-
500
for
the
to
to
subjected
results displayed
reduced by a factor of 4
has been
scheme
minimization
zero
earthquake yielded
Bucarest
points
the computation
Although
zeroes.
of
6.3.
the
plastic
that
additional
of
consisted
3596
tubular brace models,
indicate
complicated
by
the frequency domain required an additional
zeroes
7692
to
the efficiency.
excitation
necessary to model
are
points
improving
appended
the
contributed by
subjected
succeeding discussion
the
and
inefficient,
extremely
of
The solution scheme as presented, however,
material models.
is
systems
response
the
capable
is
scheme
solution
domain
frequency-time
hybrid
the
that
demonstrate
studies
These
in
the
the
Table
number of required points
(optimum
case
load history and 524
has
zeroes).
1024
The
256
No. of
Points
Solution
Scheme
Residual
Iterations
Maximum
CPU
Displacement Time
(inches)
(sec)
+
Time
Integration
(nonlinear)
HFT
time
*
11.2
8192
- 15
16
8192
12
206
4096
30
increment
=
0.01
occurred during last
Table 6.2
Computation Times
11.2
33.0*
1. 8x105
seconds
2.3x10
for all
6
115
95
105
analyses
time step
for Tubular
Brace SDOF Model
257
Solution
Scheme
No.
of
Points
Iterations |Residual
Maximum
CPU
Displacement Time
(inches)
(sec)
Time
Integration
(nonlinear)
20.75
5
Time
Integration
(linear)
17.64
6
512
10
5795
234
11
1024
6
202
18.53
10
1024
8
80
18.86
13
1024
10
100
17.40
15
2048
6
41
17.62
16
2048
7
15
17.50
17
HFT
time
Table
6.3
increment = 0.01
seconds
for all
analyses
Computation Times for Elastic-Perfectly Plastic
SDOF Model Using the Zero Minimization Technique
258
response after
Fig.
10 iterations using
1024 points
simple
was also
relaxation scheme
in
of
the
to any
form
over-relaxation or under-relaxation, usually producing
rate, if not
slower convergence
Tabulated results
Bucarest earthquake.
of 1.0,
coefficients
This
to
SDOF
are
direct
superposition
and that
approach
the use
of
analysis was not
6.2 SDOF
not be applied
can
a possibly
HFT
results
are
less
than
that
even when using the
noted
is
the
smaller time
that
to
mode
a
SDOF systems
increment
in the
a
soil
exploited.
SOIL AMPLIFICATION PROBLEM
This
study
amplification
(19).
analyses,
it
scheme,
minimization
zero
of
Accurate
systems.
integration
time
the
for acceleration
feasibility
the
derivable and although the efficiency may be
of
to
and 0.99.
1.01,
section demonstrated
applied
scheme
1.10,
the
for
subjected
system
SDOF
plastic
elastic-perfectly
a
shows
Table 6.4
characteristics
convergence
typical
divergence.
a
study
the
the system was highly sensitive
that
indicated
of
Details
Results
procedure were given in Chapter 5.
as
investigated
convergence.
the
accelerating
of
means
HFT
shown
6.20.
A
of
is
applied
problem
the
scheme
HFT
analyzed
by Constantopoulos
The problem exhibited a fairly
significant
nonlinearities
and,
to
in
complex response
moreover,
1973
with
had no viscous
259
15
At = 0.01 s
10
5
LtJ
0
L
0~-5
-10
-15
-20
0
1
2
3
1
5
6
TIME (SEC)
Fig.
6.20
Elastic-Plastic Response to the Bucarest Earthquake,
10 Iterations, N = 10
260
Accel eration
Coeff icient
Al pha
Iteration
2
3
4
5
6
497
758
777
627
460
202
I .00
1.10
1.01
0.99
Table 6.4
Residual
7
115
8
9
79
138
10
100
1
399
2
1360
3
2868
4
5421
5
8101
6
7
8
13261
23395
37698
9
70330
10
142800
1
640
2
3
4
5
6
7
,8
1039
985
952
986
1223
1466
1528
9
2022
10
2256
1
2
3
4
5
6
382
706
993
821
852
742
7
500
8
9
621
457
10
290
Residual Displacements When Using
Relaxation Schemes
261
the
scheme used in section 6.1.
basic HFT
the base of
applied at
f =1.264,
a=0.05,
r=2)
relationship governing
u-un
f -f.
u -_ u
f
characterized
the Ramberg-Osgood model
is
r
( 6 .1 )
cf
cf
cu
~
a
+
The
soil behavior.
the
of
=0.001,
(u
model
material
Ramberg-Osgood
A
s.
0.25
system with a period
SDOF
a
by
the
this example
In
a soil deposit.
represented
was
soil
excitation
to a seismic
response
surface
the
determining
of
consists
amplification problem in general
soil
The
for
problems
posed convergence
aspects
These two
damping.
yywhere
the u's correspond to
the
producing
The excitation
earthquake,
in
Fig.
the
was
the
of
component
load-deflection
first
record
Taft
shown
curve
No viscous
damping was specified.
parameters are provided in Fig.
The extension
of
the
problems with no damping is
frequency
g,
decay with
frequencies
do not
frequencies
avoids
these
develop,
time.
6.23.
domain
analysis
and
Assuming that
the
singularities,
it
is
to
theoretically inconsistent since
correspond
that
or
shown
Relevant
an undamped problem exhibits a periodic response which
not
N69W
Kern county
1952
the
of
6.21.
the
of
with peak acceleration scaled to 0.01
6.22.
structural
in Fig.
seconds
ten
to forces,
and f's
displacements
to
all
the
discretized
a
possible
resonance
to
does
significant loading
natural
frequency
effect
structural
spectrum
will
not
evaluate numerically the
262
P/P
y
(!
a)
y
x/x
(e
Fig.
6.21
Ramberg-Osgood Material Model
/
y
E )
y
263
60
C
0
20
40
-60
-80
0
4
2
6
8
Tine (sec)
Fig.
6.22
Taft Acceleration History
10
264
m =2
T = 0.25 sec.
k = 1264
=0.01 g
Tmax
Taft Earthquake
U
'Fig. 6,23
SDOF Soil Model
265
contribute
loads
problem is
particularly acute
spectrum
energy
seismic analyses where the
for
of
range
significant
a
contain
may
to
begin
This resonance
frequencies.
the natural
at
histories
pseudo-force
the
if
intractable
become
will
problem
stable
previously
a
and
shift,
a
frequencies will
the natural
however,
analysis,
nonlinear
In
domain approach.
standard frequency
response with the
frequencies.
As
result,, preliminary runs using
a
iterations
initial
process
appears
in the
consequence,
form of
artificial viscous
in
converging
the
damping
artificial
the solution
of
As
vectors.
load
in
hysteretic
The
latter cycles
pseudo-force
soil
the
same approach as
damping.
a
damping was employed in
the
as a buffer
to
to act
soil amplification studies
subsequent
extremes
the
during
the
using
section 6.1 progress without any
by
provided
problem contains hysteretic damping
damping
Although this
produced a diverging solution.
in section 6.1
model,
solution scheme
the
approximates
essence,
In
response.
actual
the
the
hysteretic
damping effect upon the response.
Furthermore, preliminary
linear
response
from the
that
that
the
response.
The formulation of
the
equations of motion was consequently modified such
rather
earthquake
indicated
(no viscous damping) differed significantly
actual nonlinear
governing
studies
than
and
evaluating
then
the
iterating
linear
with
response
the
to
the
resulting
266
to
pseudo-forces
when
all
obtain a correction displacement term which
added to the
iterations
actual
linear response gave the
(total displacement
was evaluated directly
response
the
and
loading,
earthquake
the
included
actual response,
formulation).
0.005
and
s
0.01
s
were
used.
The results
for
At=0.005. s,
differed slightly from those
that
concluded
the soil model
At=0.01
Consequently, the Newmark method with a=0.
damping)
artificial
of
the
that since
no
numerical
the
response.
5
(no
6=0.50
and
used in conjunction with a Newton
was
In
equilibrium iteration scheme.
tolerance
2
was
it
and
Notice
on
imposed
be
should
of
At=0.01 s
for
damping,
hysteretic
contains
damping
artificial
adequate.
was
s
increments
time
analyses
time integration
In the
a
addition
10~8 and iteration limit
of 15 was
displacement
specified in
equilibrium iteration.
For
was
the HFT analysis
selected.
The
response
the
of
peaks
captured for At=0.02 s and no further
obtained
corresponds
for
At<0.01s
Hz
to 50
for
since
the
could not
information
Nyquist
the
seconds
0.01
of
increment
a time
would
be
frequency
earthquake
given
be
history.
Based
on preliminary studies, an artificial viscous damping
ratio
of
eventually
artificial
0.50
was
diverge
since
selected
during
damping ratio less
the
the
iterative
than 0.25.
response
would
process
for an
267
the
47
of the methods,
efficiencies
the
75
and
60
between
converges
were required
for 75 iterations
should be mentioned
previous SDOF
displacement
Both schemes
formulation
estimate
excessive
during
again proved
analysis.
also
but
causing
nonlinear,
Even with
of 5%,
were
damping
and
by
as
a
in the
10th
the
the
of
the
inaccuracies
iterative process.
less efficient
than a direct
study.
by producing
nonlinear
in
total
the
demonstrated in this
stabilized the solution process
initial
preventing
artificial
of
use
The
history
included not
cycle.
iterative
better
It
analysis.
the solution would diverge
study,
s CPU time
specified,
damping ratio
the
required for
172
for the HFT analysis.
viscous
ordinary
specified
response
Comparing
using the HFT
highly
was
difficulties
convergence
The
that artificial damping was
problem
the
6.27 show
s CPU time were
simply because no viscous damping was
because
and
iterations.
integration analyses while
time
direct
6.26,
and 75 iterations.
60,
response after 20,
6.25,
Figs.
and
analysis,
integration
time
a
using
results
the
depicts
6.24
Figure
the
response
and
pseudo-force
The HFT scheme
time
a
once
integration
268
20E-3
l5 At =0.01 s
t
10
-05
--5
C -10
-15 -
-20E-3
0
1
2
3
+
5
TIME (SEC)
Fig.
7
8
8
9
10
D(F 1
Soil Amplification Results Using a Direct Time
Integration Analysis
6.24
~2L-'J
U
to
5
Cl)
0
0
-to
-15
-20E-3
'
0
1
2
3
t
5
TIlE (SEC)
Fig.
6.25
6
7
8
9
10
DOF I
Soil Amplification Results Using an HFT Analysis
(20 iterations, artificial viscous damping ratio
of 0.50)
269
15
S10
5
-5
-20- 3
.20E-3
0
+
2. 3
1
5
TIME (SEC)
6
7
8
a
10
DOF I
Soil Amplification Results Using an HFT Analysis
(60 iterations)
Fig. 6.26
15 -At
=
o.01 s
zt0
5
-0 -
-15 .2E-3
0
1
2
34
5
TIME (SEC)
Fig.
6.27
6
7
8
9
10
DOF I
Soil Amplification Results Using an HFT Analysis
(75 iterations)
270
6.3
DOF SHEAR BEAM STRUCTURE
TEN
response of
The
described
structure
was
models
3.65
was
examined
in this
seconds
and
seconds.
0.28
period,
10th
the
period
fundamental
The
study.
To induce
loading.
the
A
wave
sine
cycle
a single
duration two seconds was used as
of
a
elastic-perfectly plastic material
by
excitation represented by
seismic
freedom shear beam type
a ten degree of
localized nonlinear behavior and participation
of
higher
of
100
ksi was
specified
stress
yield
except at
for each member,
specifying a damping
Relevant
frequencies.
of
ratio
was employed by
6.28.
Fig.
the
direct
of
motion
increment
In
integration
time
solved
were
of 0.02
an
in
seconds was
of
0.1
Notice that
used.
case.
points.
seconds
However, for this
histories.
of
the
governing
equations
A
form.
incremental
time
was
analysis
domain
specified.
a
time
64 points were
in the optimum case we would use only
frequency domain were
shape
The
specified..
the hybrid frequency-time
increment
the
equilibrium iteration scheme were used for
Newton
the
and
6=0.50
and
The Newmark integration method with a=0.25
the
given in
parameters are
structural
4th
and
1st
the
for
5%
the
where
four
level
Rayleigh damping
1 ksi.
was
stress
yield
a
structure,
the
in
modes
particular problem, 32 points
insufficient
displacement
to
in
adequately resolve
response
No artificial damping was
32
and
pseudo-force
employed, and
the dual
271
U
m,l
.ul 0
1'
k = 133,
m = 1
E = 0.05
@
W, = 1.72 and
= 3.65 sec.
.
T1
0
= 0.28 sec.
1
m k
g
U
g
10
'2
t
Elastic-perfectly plastic material model
at level 4
F=
y
100,
Fig. 6.28
elsewhere
Ten Degree of Freedom Shear Beam Study
W = 11.52 rad/sec
272
formulation was used.
displacement
frequency
The problem of nonlinear mode updating in the
first
were
matrices
in
addressed
with
implemented
of
equations
the
of
first
the
was
updating
Mode
motion.
section
in
outlined
scheme
hence
and
transferring them to
off-diagonal terms were managed by
the RHS
Linear and
study.
this
simultaneously,
nonlinear mode shapes were used
all
structural
generalized
new
the
in
terms
off-diagonal
the
handling
of
problem
consequent
the
and
domain
5.2.2.6.
An HFT analysis with all
particular
problem.
As
can
histories
of dof's
1 and
10,
be
shown in Figs.
response
10th
of
degree
freedom
formulation).
its
nonlinear response resembles
6.31
converged
6.31 it is
in
corresponding
deflected
apparent that the nonlinearities are
element 4
(from the base),
distribution
resulting
existing
first proposed mode updating
readily adapted
to this problem.
the
because
its
shapes
s.
in a
only at
scheme,
linear
that
response.
linear
t=1.0 s to t=1.5
solution from time
pseudo-force
The
the
depicts
the
Notice
quickly
converges
response
The
iteration correspond to
(dual displacement
6.30,
6.29 and
the solution converges within 20 iterations.
histories after the first
response
the
from
seen
this
of
behavior
convergence
the
determine
to
first
ten linear modes was conducted
Figure
of the
Fig.
From
concentrated
fairly
simple
nodes 3 and 4.
consequently,
is
273
0.5
0.4
0.3
ILU
LU
C,
-J
0~
0.2
(/)
3.1i
LU
0.0
H-
-0.1
-j
LU
-0.2
-0.30.0
0.2
0.4
0.6
0.8
1.0
TIME (SEC)
Fig. 6.29
1.4
1.6
1.8
2.0
1.8
2.0
DOF 1
Response of DOF 1, Direct Time Integration
and HFT Results
0
5LUJ
1.2
Ii
-2
LU
F-
LUJ
-3
-4
H-_
-5
-6
0.0
0.2
0.4
0.6
0.8
1.0
TIME (SEC)
Fig.
6.30
1.2
1.1
1.6
DOF 10
Response of DOF 10, Direct Time Integration
and HFT Results
274
R
11
101
12
-2
-3
-51
0
20
40
60
80
ELEVATION
Fig.
6.31
Deflected Shape from Time 1.0 sec.
to 1.5 sec.
100
275
updating
first
5
analysis
derived,
the
to
applied
the
the pseudo-force histories
were
original structure, and
then the
the
tangent
matrix,
evaluated.
The ten
6.32
and
respect to
the first 5 linear mode shapes
restarted using
to obtain the
6 modes).
The corresponding response histories
in Figs.
6.35
6.38
first
a
is
that
apparent
(a
expended CPU
of
the
efficiency
times are
listed
first
total of
are provided
10
and
s to t=1.5 s
the
is
Furthermore, Fig.
obtained
with
the
integration
is
the nonlinear response.
As
analysis,
the
time
using
to properly represent
the
that this mode updating
response
the
modes
linear
five
measure
time t=1.0
effective for this problem.
is
demonstrates
inadequate
It
6.37.
given in Fig.
procedure
shape from
deflected
converging
and
1
dof's
for
6.36
was
analysis
nonlinear Gram-Schmidt orthogonalized mode shape
and
first
shapes and
5 linear mode
the first
The
6.34.
The
orthogonalized
Gram-Schmidt
nonlinear mode shape shown in Fig.
Fig.
6.33.
in Fig.
shape,
then
shape was
nonlinear mode
with
nonlinear
first
in
given
are
linear mode shapes
was
shape
mode
nonlinear
first
the
From
obtained.
tangent stiffness matrix was
stiffness
was
distribution
load
static
equivalent
An
examined.
and
stopped
was
cycles,
these
After
shapes.
mode
linear
the
with
cycles
first five
the
iterating
scheme by
mode
the
study implemented
portion of this
The second
of
the
in Table
HFT
6.5.
276
T, = 3.65 s
T2 = 1.23 s
T3 = 0.75 s
T
= 0.55 s
T5 = 0.44 s
2
K
T6 = 0.37 s
T
= 0.33 s
Fig,
6.32
T8 = 0.30 s
T
= 0.29 s
T10
0.28 s
Linear Modes Shapes
First Nonlinear Mode Gram-Schmidt
Fig.
First Nonlinear Mode Obtained Fig.
6.33 from Static Load Distribution
6.34 Orthogonalized with Respect to
Method
Lowest Five Linear Modes
277
0.5
0.4 -
0.3
0.2
2:
0.1
LU
0.0
-0.1
-0.2 -0.3 -0.4
0.0
0.5
1.0
1.5
2.0
TIME
Fig.
6.35
Response of DOF 1 Using an HFT Analysis
(5 linear plus 1 nonlinear mode)
0
-l
-2
CD)
-3
LU-
-4
-5
-6 0.0
0.5
1.0
1.5
TIME
Fig.
6.36
Response of DOF 10 Using an HFT Analysis
(5 linear plus 1 nonlinear mode)
2.0
h.i
0
U)
0.0
-0.5-1.0
-1.5
-2.0
I-
-2.5
-3.0
-3.5
-4.0
hi
F-
hi
I
0
-2
-3
LL
-4.5
-5.0
-5
0
20
40
60
80
100
0
40
60
80
100
b. 10 iterations
a. 5 iterations
1
0
hi
20
0-
co
0
U)
-2
-3
LU
C:)
hi
F-
hi
-6
-6
0
20
c.
Fig.
6.37
-2
-3
40
60
15 iterations
80
100
0
20
40
60
d. 20 iterations
Converging Deflected Shapes for Iterations 5, 10, 15, and 20
(5 linear plus 1 nonlinear mode)
80
100
279
0.0
-0.5
-1.0
L.
-1.5
-2.0
L--
-2.5
-3.0
-3.5
-4.0
0
20
40
60
80
i0d
ELEVATION
Fig.
6.38
Deflected Shapes Obtained with 5 Linear Modes Using
a Direct Time Integration Analysis
280
Solution
Scheme
Time
No. of
Modes
CPU Time
(sec)
0.02
all
11.9
0.02
all
15.1
0.1
10L
23.4
Time
Integration
Linear
nonlinear
HFT
0.1
5L+4NL
18.3*
0.1
5L+2NL
16.4*
0.1
5L+INL
15.2*
All HFT CPU times
L
NL *
Table
6.5
are for 20
iterations
linear mode
nonlinear mode
nonlinear solution
time
Computation Times for
10 DOF
Shear Beam Model
281
study, however, was
The
6.4 MDOF
and
soil model consisting of a
a stiffness profile varying with the
square root
freedom lumped
structural
fundamental
The
s.
No
the Taft earthquake scaled
viscous
The direct
time
Newmark
the
integration
analyses
integration method
studies
of 0.00001 and iteration
indicated
adequate.
at
the
soil
The
that a
time
used.
were
(a-0. 2
a
limit of
step
was
size
Relevant
6.39.
Fig.
Newton equilibrium iteration scheme with
tolerance
T,
period
damping was specified,
to 0.05 g was
properties are provided in
structural
in
structural
9 degree of
and T 9 , 0.031
0.357 s
considered
was
mass
the depth.
using
efficiency
an
to a fairly simple response,
limited
with a refined
but
closely coupled
and
context
time integration analysis.
soil amplification problem
study,
model
the
SOIL AMPLIFICATION STUDY
Another
this
of
stability problems.
producing no
of
of a direct
that
to
comparable
the
this
in
and
accuracy
sufficient
demonstrated
with
RHS
the
to
scheme
HFT
The
motion.
of
conjunction
terms
the off-diagonal
transference of
equation
in
used
scheme
updating
mode
first
the
feasibility of
the
section demonstrated
This
5
conducted
, 6-0.50) and
force
residual
15.
of
Preliminary
0.005
s
was
linear and nonlinear displacement responses
surface are given in Figs.
6.40 and 6.41.
282
Gg, y
k = 75.2
.
m
G8'
= 0.0125
m = 0.0250
G7, Y
G6'
y
G5'9 Y
G4,
y
G3'
Y
G2
Y
G,
Y
k increases
with square
root of depth
i.
40
m = 0.0250
k = 312.4
Taft Earthquake
g
(0.05 g)
p
no viscous damping
Ramber g-Osgood
Material Model
6
T, = 0.36 s,
Fig.
6.39,
T
= 0.03 s
Refined Multidegree of Freedom Soil Amplification Study
283
0.2
Cn
-012
-0.6
0
1
2
3
+
5
TIME (SEC)
Fig.
6.40
7
8
8
9
10
00F 9
Linear Response to the Taft Earthquake
(surface displacement)
0.3
C
4
0.1
0
V
0.0
-0.
1
-0.2
-0.3
-0 .40
2
twne
Fig.
6.41
6
4
8
[seo)
Nonlinear Response to the Taft Earthquake Using
a Direct Time Integration Analysis
(surface displacement)
10
284
this
viscous
the artificial
behavior provided by
the second mode updating scheme
apply
artificial
viscous damping ratio of 0.50
obtaining
the first
damping
ratio of 0.50.
20
the response
response
is
shape
frequency
low
and
the solution
to the extent
maintained,
10
and
5
with
high frequency components
significant
iterations,
begin to modify
in
Notice
and 6.44.
a
iterations
introduces a slow drift
component
further
after
but
iterations,
20
and
10,
after
converge
to
appears
response
the
6.43,
for a
diverged
5,
10
of
history
after
6.42,
are given in Figs.
iterations
response, a damping
entire
results
The
for
adequate
was
eventually
solution
the
because
of the
for the
of 0.75 was necessary
seconds
that
five seconds
Although an
0.01 s.
of
increment
time
a
with
HFT
in the
formulation was used
total displacement
analysis
ratio
secant
squares
(least
to
and
damping
approach).
stiffness
The
a
As
the stabilizing
offered an opportunity to examine
it
result,
response with
response.
linear
the
from
deviations
significant
6.3,
section
complex nonlinear
a
exhibited
problem
in
the MDOF problem discussed
to
Compared
the
but
that
the
proper
amplitudes
grow
that
the
without bound.
diagnostic
Further
artificial
high
viscous damping
frequency
contributed
the
response
low
studies
indicated
force contributed the significant
while
the
frequency drift.
pseudo-force
vector
Any static drift
in
285
W1.JS
04
C,,
-02
-0.3
1
0
2
12
3
5
34
5
TIME (SEC)
6
7
1
8 7 8
80
DF 9
Nonlinear Response after 5 Iterations
(artificial viscous damping ratio of 0.75)
Fig. 6.42
-
0.3 -
t
=
0.01 s
0.2
0 1
2
3
4
TIME (SEC)
Fig.
6.43
5
6
7
8
DOF 9
Nonlinear Response after 10 Iterations
9
10
286
0.3
.
t =0.01 s
0.1
w
S-0.2
-
0.3
-0.5
-0.7
0
'
1
2
3
4
TIE (SEC)
Fig.
6.44
5
6
7
8
9
DOF 9
Nonlinear Response after 20 Iterations
10
287
the
FNL
since
other
the
(which is derived
pseudo-force
terms
consisting
initially
then, since
estimates
a viscous
producing
response,
in the
artificial
As
a result
damping
the
2E vm is used,
(w
large)
it dominates
(low
pseudo-force
incorrect
the
inaccuracies
the succeeding cycle.
damping force for
of
to
response,
the higher modes
which further aggravates
drift)
frequency
correct
each iteration until
an
force
frequency
zero
no
the
type artificial
to increase with
the
that when the HFT solution
of
the artificial damping force for
continues
had
domain)
postulated
gives poor
of
of the response equal
from the velocity
is
It
components.
vector
damping
artificial
the
and
frequency
in the
iw Y(w)
loading
excitation
earthquake
the
the
solution can be attributed to
properties
convergence
unstable
introduced
by the artificial viscous damping, an artificial
hysteretic
damping
considered
was
definition the hysteretic
for
by
this
the
problem since
very
its
type damping appears more suitable
in essence all
damping
is
contributed
soil model.
Before
studies
executing the
were
increment.
10
By
next.
conducted
entire HFT
to
iterations
with
At=0.01
corresponding results with At=0.05
artificial
increment
hysteretic
damping
of 0.05 s appeared
and
s.
In
Fig.
both
6.46,
the
cases
the
ratio equalled 0.75.
acceptable
after
obtained
the results
s,
time
an appropriate
determine
Figure 6.45 depicts
preliminary
analysis,
and
was
A time
used
in
288
0.25
... 0.20
w 0.15
C
S0. 0
0.0
o-0.05
;-0.10
-0.15
-0.20
-0.25
0
2
3
+
4
5
tine (sec)
6.45
Fig.
6
7
8
9
10
9
10
DOF 9
Nonlinear Response after 10 Iterations
(artificial hysteretic damping ratio of 0.75,
time increment of 0.01 s)
0 .20
0.15
+C
<D
5 0.10
0
f 0.05
a
QJ)
-0.00
*-0.0
0
1
2
3
4
t ine' (ec)
Fig.
6.46
5
6
7
DOF 8
Nonlinear Response after 10 Iterations
(time increment of 0.05 s)
8
289
analyses.
succeeding
frequency
56
appended
and
earthquake
10 second
(200,for the
domain
256 points were used in the
zeroes).
in
to
portion of the
latter
and 80
50
for 80
converged
iterations are
convergence
This
seconds.
final iterations
the
correct
initial
the
that
response, implying
time progressing
and corrects in a
first
converges
portion
the
that
indicates
5
time
to
fairly similar up
behavior
after
results
the
that
Notice
6.48
solution parameters.
specified
shape for the
final
its
shown
solution history had
time the
by which
iterations,
Fig.
and
iterations
50
for
6.47
Fig.
are
linear modes
results using all nine
The RFT
form.
the direct
schemes,
and
the HFT
time integration analysis
analysis,
updating
study implemented
determine
reproduce
response after
and
adequately
amplitudes
in
Chapter
Initial
5.
Fig.
6.51,
reproduces
are
Figure
using all
the corresponding
9
6.49
shows
linear
response using 5
1
linear mode.
One
the
response,
although
slightly smaller.
to
s
linear modes necessary
response.
the
ten iterations
Figure 6.50 shows
modes,
of
number
minimum
the
adequately
described
scheme,
second
the
conducted with a time increment of 0.05
were
studies
required 650 s
540 s.
The next portion of this
mode
solution
two
comparison of the efficiency of the
As a
to
the
modes.
linear
linear mode
the
peak
290
0.3
C
0
0.1
03
0
-a
0.0
V
-0.1
0
-0
0
1
2
+
3
5
t ine (3ec)
Fig.
6.47
6
7
8
9
10
8
9
10
DOF 9
Nonlinear Response after 50 Iterations
0.3
c
C
(D
Q
0.4
0
0.0
*
-0.1
-0.2
-0.3
0
1
2
3
+
tine (sec)
Fig.
6.48
5
6
7
DF 9
Nonlinear Response after 80 Iterations
291
0.20
-'
C-
0.15
S0.10
0
0.05
S0.00
'0.05
0-0.10
c-0.15
-0.20
0
1
2
3
+
5
ttne (3ec)
6.49
Fig.
6
7
8
9
8
9
10
DF 8
Nonlinear Response after 10 Iterations
(10 linear modes) ~
0.20
c 0.15
5
0
0.10
-0.05
0.00
0-0.0
-0.25
-0.20
0
1
2
3
+
5
t ine (3ec)
Fig.
6.50
6
7
DOF 9
Nonlinear Response after 10 Iterations
(5 linear modes)
10
292
0.20
0.15
-
At = 0.05 s
Dr0610
0
-
f0.05
0.
- 0.00
*-0.15 -0.20
0
1
2
t
I
I
I
I
3
+
5
6
7
tine (see)
Fig.
6.51
DOF 9
Nonlinear Response after 10 Iterations
(1 linear mode)
8
9
10
293
Realizing that
this
particular
applied
the
fundamental mode was
problem,
the
scheme
was
ten cycles with one
first
linear mode.
1.
Iterate the
2.
Stop
the
stress-strain diagram of each member to obtain nine
new
secant
Apply a
the analysis.
least
Ksec
(s).
4.
Restart
the analysis with the
eigenvector
(one nonlinear mode).
for another 40
Iterate
The response after 50
linear mode and next 40
of
increment
comparison
of
increment
iterations
6.53b.
0.05
shown
s,
that
the
nonlinear mode (50
iterations
for
additional
yielding
6.53a
10
Fig.
response
iterations
accuracy gained
of 0.02
s.
with
and
response
80
converged
using a
a
time
after
50
iterations, Fig.
6.48 it
is
faster with only one
for one nonlinear mode and
Finally, Fig.
by using
For
conducted
iterations and
the
one
6.52.
was
6.52 and 6.53 with Fig.
one linear mode).
increment
in
analysis
all
during
Fig.
in
Comparing Figs.
apparent
time
mode
nonlinear
fundamental
(first
shown
purposes an additional
linear
one
with
nonlinear
nonlinear mode)
one
is
s
secant
cycles.
iterations
with
0.05
the
evaluate
and
eigenvector
time
to
fit
structural
corresponding
the
Establish
5.
squares
stiffnesses, k sec'
stiffness
a
updating
in
following form:
in the
3.
the
mode
acceptable
one
80
6.54 shows
linear mode
and
294
0.3
0.2
c
(D
c
(D
Q
0.1
CL
0.0.
0
0D
C-,
-0.3
-0.4L'
0
1
2
3
+
5
t Ine (sec)
Fig.
6.52
6
7
8
9
10
8
9
10
DF 9
Nonlinear Response after 50 Iterations
(1 nonlinear mode)
0.3
C
0
0
0.2
0.1
a
0.0
-0.1
0
1
2
3
+
t ine (ec)
Fig.
5
8
7
DOF 8
6.53a Nonlinear Response after 50 Iterations
(1 linear mode)
295
0.3
c
0.2
0
(-)
0.0
1
1
1
At =0.05 s
* -0.1.-0.2
0
1
2
3
+
ine (3ec)
Fig. 6.53b
5
6
7
DOF 9
Nonlinear Response after 80 Iterations
(1 linear mode)
8
9
10
296
0.3
4-
C
0.2
0
0.1
C
0.0
-0.1
-0.2
-0.3
-0.4-
0
1
2
3
4-
8
tine (e0)
Fig.
8
9
10
9
10
DF 9
Nonlinear Response after 40 Iterations
(time increment of 0.02s, 1 linear mode)
6.54a
0.3
7
-
4-
C
0
0.2 0.1
-
C
-0.1
-0.2
-0.3
-0.4
0
1
2
3
TiMe
Fig. 6.54b
+
(e6)
5
8
7
8
DCF 9
Nonlinear Response after 80 Iterations
Ctime increment of 0.02 s, I linear mode)
297
The
compared
to
that
of K 1
with 0.357
force-displacement
reductions
originates
various
analyses
are
in computational cost
compared to the
efficiency
This
analyses.
the direct time
of
are
scheme displays
ten times
from the use of a time increment
that
than
the
for
integration
time
direct
k sec'S
Notice that the HFT
in Table 6.6.
significant
the
6.55.
The computation times
provided
that isec
representative
A
which
from
diagram,
is shown in Fig.
derived,
indicating
structure.
softer, nonlinear
portrays a
s,
0.466 s
period
The fundamental mode of K sec has
response.
and
the nonlinear
estimates to
initial
closer
providing
hence
K1 ,
than
the overall structural behavior
representing
better
from K
opposed to one linear mode, stems
as
mode,
convergence when using one nonlinear
faster rate of
integration analysis
larger
combined
with a nonlinear mode superposition scheme.
section
This
concept
damping
problem and the
effect on the
scheme
updating
accelerated
Compared
squares secant
artificial viscous
negligible
mode
least
hysteretic
otherwise
intractable
an
stabilize
stiffness mode updating
adapted
damping proved better
artificial
than the
to
artificial
increase the analysis efficiency.
to
scheme
the
introduced
to
the
the
damping, and,
computational
was
this
problem
furthermore, had a
cost.
The
second
easily implemented and actually
convergence
first
to
The hysteretic
mode
of
the
updating
iterative
process.
scheme, this method
298
1.0
0.8 00.8
.0.2
-
- 0 .2 -
-0.2 -0.8
-8
E-3
-6
'
'
'
'
4
-2
0
2
+
6
strain or displacement
Fig.
6.55
Stress-Strain Response of Ramberg-Osgood Material Model
a
E-3
299
Solution
Scheme
CPU Time
(sec)
No. and Type
of Modes Used
0.005
Direct
0.05
9
linear
80
544
0.05
1 linear
80
133
0.05
*
50
100
80
315
Time
Integration'
HFT
1 linear
0.02
*
Iterative
Cycles
Time
Increment
first ten
1 linear,
1 nonlinear, next 40
Table
6.6
650
iterations
iterations
Computation Times for MDOF Soil
Amplification Study
300
from the
demanded less participation
self
in current use.
iterators
HFT
The
was
scheme
offshore steel
subjected
to seismic excitations.
previous
studies
in
to
may
where
the
loading
rather
than
the external
developed HFT
structures
from
differed
was
excitation
freedom had
be
the
external
no
study provided an opportunity
of
degrees
dominated
by
the
pseudo-force
this
Furthermore,
force.
point
for
freedom
study
the fully
applying
scheme.
first
determining
portion
the
of
to an offshore
initial analyses were
structure,
s.
shown in Fig.
A
stiff
this
study
convergence
general
scheme when applied
0.41
jacket
Single-Bay Offshore Structure
The
hence
to
of freedom were not
the
accuracy of
culminating
a
represented
6.5.1
degrees
rotational degrees of
the convergence
to examine
study
ground motion, implying that
a consequence, the
As
this
This study
particular,
horizontal
a
vertical and
load.
all
that
In
externally.
in
extended
of
models
two-dimensional
limited
STRUCTURE
TUBULAR OFFSHORE
6.5 CROSS-BRACED
loaded
the equilibrium
similar to
contained updating approach,
producing
developed as a
scheme could be
The
results.
accurate
more
while
user
behavior
structure
of
with fundamental
deck
was
and
offshore
period
modelled by
of
the HFT
problem,
limited to a single-bay
6.56,
platform
consisted
of
a large
301
2
5
3'-0
10 f10 (60')
6
4
elastic beams
.lar brace
a = 5184
y
3.25 + 0.0729
K
(60')
=
CM = 2.0
Taft Earthquake
I1
= 3.5 g
{0.05 @ T
10.50'
T4
T, = 0.41 s, T2
0.12 s, T3 = 0.12 s,
=
5 = 0.0002
s, T6 = 0.0001 s
Fig.
6.56
Single-Bay Offshore Structure
T4 = 0.006 s
0.5
302
linear elastic horizontal beam at
The
tubular
the
using
diagonal
specified
fixed-fixed end conditions.
water was
simulated by the added mass
CM- 2 .0.
coefficient
reasonable
--
mode and 0.50
at
integration
the
Newton
required
Newmark
nonlinear
As a
analysis
response, Fig.
vertical, and
employed
fundamental
the
modes
three
deck beam, and hence
that
(a=0.25,
time
direct
the
time increment of
integrator
is
was
the response.
a
iteration scheme.
freedom
Six degrees of
the stiff
indicated
degree of -freedom 4
lateral
a time
approach
to induce a
The highest
significantly in
Preliminary analyses
modified
of
first 5
the
g,
horizontal,
the highest mode.
participate
when using
effect with an inertia
ratio of 0.05 at
damping
to deformations
correspond
do not
sea
Rayleigh damping
top nodes.
a
specifying
level.
of
sets
two
the
rotational at
by
surrounding
The excitation consisted of
nonlinear response
present
were
with
the Taft earthquake scaled to 3.5
of
seconds
The
modelled
6.1,
in Fig.
shown
model
brace
a fixed base
were
braces
cross
elements.
beam
by specifying
Foundation effects were neglected
for both legs.
elastic and
linear
tubular
elastic
linear
by
modelled
Vertical
deck level.
remain
to
assumed
were
members
leg
the
0.02 s
6 =0.50)
The linear response
and
for
6.57
and
time integration and
HFT
shown
in
Fig.
6.58.
direct comparison
of the
schemes, the HFT solution was also
implemented with
increment of 0.02 s using all 6 linear modes
and
an
303
0.6
C
0
0.2
0.
0.0
-0.2
C.
-0.S
-0 .',
0.0
I
0.5
I
1..0
I
I
I.5 2.0
I
I1
2.5 3.0
I
3.5
C00)
DOF
4
35
+0
tLMO
Fig.
+.0
4.5 5.0
6.57 Linear Response to the Taft Earthquake
(lateral deck displacement)
l.0
4
0.8
t
=
0.02 s
0.60
0.+
C
0.2
-0.2
-0.4..1
-0.0
.
.
.
.
tthMe
.
30
($sC)
4.
DOF 4
Fig. 6.58 Nonlinear Response to the Taft Earthquake
(direct time integration, lateral deck displacement)
304
6.59,
6.60,
and
6.61 depict
freedom 4 after 5,
15,
and 30 iterations.
analyses
Additional
of
solution.
Figure
6.62
iterations
using
and
mode.
fundamental
were run wi th
6.65
and time
These
s
0.02
of
increment
respectivel y.
Notice
essentially
converged by the 25th
iterations
producing
response hi story.
reproduce
the
A more
solution
from
all
of
Table
in
times
and
less than 1%
degrees
of
modes
three
lowest
the
response exactly.
available
displacements
end
the
in
the
expected,
stringent evaluation of
is
responses
As
iteration, with additional
changes
minor
a time
scheme has
solution
HFT
and
iterations,
30
and
25
the
that
Figures 6.64
s.
three linear modes and
after
time
a
and subsequent analyses
increment of 0.02
a time
that
indicated
coarse,
the response using
show
Figure
0.05 s.
increment of
analyses
of 0.05 s was too
increment
30
corresponding response when using only the
the
shows
after
linear modes, an artificial hysteretic
6
damping rat io of 0.35,
6.63
optimum HFT
response
the
time
other
determine the
to
shows
the response of
with
conducted
were
modes
number
steps
iterative
30
after
were
Figures
degree of
attained
Acceptable
0.5.
of
responses
displacement
cycles.
ratio
damping
hysteretic
artificial
agree within
freedom up to
6.7,
the
5%
RFT
listing the maximum
occurrence.
of
error for
the accuracy of the
maximum
The
reasonable accuracy ranging
horizontal
and
rotational
error for the vertical
degrees
305
302
0.8
C
0.6
0
0.4-
CL
0.2
0
0.0
-0.2
-0.6, i
0.0
I
0.5
1.0
- L
1.5
2.0
I
2.5
tMe
I
I
3.0
(eo)
3.5
4.0
I
4.5 5.0
DOF 4
6.59 Nonlinear Response after 5 Iterations
(6 linear modes, artificial hysteretic damping ratio of
0.5, time increment of 0.02 s)
Fig.
0.8
C
0.6
0
oCL 0.4-
0.2
-U
0.0
-0.2
-0.+-0 ,6 I
0.0
I
0.5
I
1.0
I
1.5
2.5
tl IMe
Fig.
'I
I
2.0
3.0
(0,o)
3.5
DOF 4
6.60 Nonlinear Response after 15 Iterations
(time increment of 0.02 s)
+.0
I
4.5
5.0
306
C
+W
0
0.8
-
0.+
V
0.6
0.2
-0.2
-0.4
0.0
0.5
1.0
1.5
2.0
2.5
6.61
3.5
4.0 4.5
5.0
DOF 4
(000)
TIMO
Fig.
3.0
Nonlinear Response after 30 Iterations
(time increment of 0.02 s)
1.0
4-
C
0.6
0
0
0.6
0.40.2
0.0
-0.2
-0.4-0.6
- 0 .8
0.0
Fig. 6.62
'
'
'
'
'
'
'
'
0.5
1.0
1.5
2.0
2.5
3.0
3.5
+.0
DOF 4
)
tiMe (e
Nonlinear Response after 30 Iterations
(6 linear modes, time increment of 0.05 s)
'
'
4.5 5.0
307
1.0
I
I
I
I
I
I
I
I
3.5
+.0
4.5
4.0
4.5
I
0.8
0.6
0
0.4
0.2
0.0
C. -0.2
-0.6
I
-0.8
0.0 0.5
1.0
1.5
2.0
2.5 3.0
tiMe
Fig.
(e0)
DOF 4
Nonlinear Response after 30 Iterations
(1 linear mode, time increment of 0.05 s)
6.63
1.0
0.8
CL 0.6
0.4
0
L
0.2
-'
0.0
C. -0.2
-0.40.0
0.5
1.0
1.
2.0
2.5
time
Fig.
6.64
(eo)
3.0
3.5
DOF 4
Nonlinear Response after 25 Iterations
(3 linear modes, time increment of 0.02 s)
5.0
308
0.8
C
0
*=
0.+
0.4 -
-
0.2 -
0.0
i
i
.- 0.2i
O-0.4
0.0
0.5
1.0
1.5
2.0
Tilm
Fig.
6.65
2.5
3.0 3.5
($so)
DOF 4
4.0
Nonlinear Response after 30 Iterations
(3 linear modes, time increment of 0.02 s)
4.5
5.0
309
DOF
Maximum
Displ.
Maximum
Time
(sec)
Velocity
Maximum
Time
(sec)
Accel.
Time
(sec)
t
I.
1
0.838
3 .86
7 .151
3.76
-58.10
3 .82
2
-0.027
3 .38
0 .581
4.26
-23.00
4 .94
3
-0.001
3 .84
-0 .019
4.28
4
0.838
3 .86
7 .152
3.76
-58.23
3 .82
5
-0.030
3 .84
0 .601
3.96
-25.45
3 .98
6
-0.001
3 .84
-0 .019
4.28
a.
DOF
Maximum
Displ.
0.785
4 .94
4 .94
Direct Time Integration Results
Maximum
Velocity
Time
(sec)
Time
(sec)
I
1
0.785
Maximum
Accel
Time
(sec)
t
0. 837
3.86
7.215
3. 76
-64
14
3 84
-31
22
4 92
38
4 98
2
-0.
028
3.38
-0.860
4. 96
3
-0.
001
3.82
0.034
4. 96
0. 837
3.86
7.215
3. 76
-64
13
3 84
-52
19
4 98
38
4 98
4
5
-0.
031
4.66
1.177
4. 96
6
-0.
001
3.82
0.034
4. 96
b.
HFT
Table
6.7
Solution Scheme Results,
Maximum Response Values
Offshore Structure
30
for
-1
-1
Iterations
Single-Bay
310
of
freedom.
exhibit
a
The
maximum
larger
error,
velocities
particularly for
freedom associated with no external
lateral
and
degree
of
accelerations,
freedom
a
of
in some cases.
the HFT
other
to
producing
resulting
scheme
solution scheme
inaccurate
in a false
increment
of
of 0.02
0.35.
values
analyses,
derives
of
using
direct
6.8,
listing
time
is
decreased
the
the
maximum
well,
it
the velocities
finite
difference
response,
final
possibly
time step and
three
optimum
linear
HFT
modes,
solution
a
time
hysteretic damping ratio
required
for an acceptable
from
of modes
optimum HFT
scheme
compared
to
integration analysis is
detailed in Table
expended
The
four
increases by a
number
of
convergence.
times
integration solution.
is
1%
the maximum response.
s, and artificial
The efficiency of the
the
the
the
Thirty iterations were
displacement
solution
for
impression of
these
consisted
the
of occurrence
reproduced as
the converged displacement
Based on
in
degrees
Although
and accelerations indirectly by applying a
expansion
degrees of
Errors
error with times
and accelerations were not
is noted that
those
load.
Comparisons
larger
differing appreciably
velocities
accelerations
freedom velocities are approximately
10%.
indicate
and
CPU
more
Notice
0.05
factor
of
times.
that
than
the
when the
time
increment
the
(140/66).
from six to three
HFT
expensive
s to 0.02 s,
2.1
optimum
time
computation
time
Decreasing
the
decreases the
computation
311
Solution
Scheme
Time
Increment
No. of
Modes
No. of
Iterations
Hysteretic
Damping
CPU
Time
(sec)
Time
Integration
linear
0.02
6
15
nonlinear
0.02
6
22
HFT
0.02
6
30
0.5
0.05
6
30
0.35
66
0.05
1
30
0.35
32
0.02
3
30
0.35
100
Table 6.8
Computation Times for Single-Bay
Offshore Structure
140
312
time
by a
contributor
main
by
of
portion
first
ability
of
response
of a hypothetical
however,
considerations,
small problem whose critical
the
governed
increment was
time
for such a
its use
numerical
than
rather
by
considerations.
integration
Taft
to
Subjected
Structure
Offshore
Two-Bay
6.5.2
prevent
evaluation,
model
material
Efficiency
structure.
offshore
would
nonlinear
the
reproduce
to
the
demonstrated
study
this
scheme
HFT
the
aggravated
evaluation,
the tubular brace material model.
the complexity of
The
time in this particular
to the computation
the pseudo-force history
problem is
the
consequence,
a
As
(140/100).
of 1.4
factor
Earthquake
The
structure
subjected to
extension
more
a
to
The
(12,27,42).
The
frame
was
analysis
model
Pertinent
is
with
investigated
structural
are
elements.
linear elastic
The
beam
deck was
element.
lumped mass
(MNO).
only
6.66.
given in Fig.
linear
replaced by
All
100'.
of
depth
nonlinearities
parameters
each with a
two bays,
limited to a two-dimensional
material
studies
in other
water
situated in a
an
so-called
the
of
model
realistic
offshore
prompted
excitation
seismic
The vertical legs were modelled by
beam
single-bay
a
of
structure consists of
and
of 60',
a
structure
Southern California
height
solution
the
to
scheme
HFT
the
of
application
successfull, albeit expensive,
elastic
tubular
a heavy horizontal
diagonal
braces
were
313
60'
13
10
11
20$ 10
i
12
100'
100'
Taft Earthquake
E = 0.05 @ T1
U_
Fig.
=0.50 g
and T4
Deck mass = 60 kslugs
at each
top node
Earthquake
Taft
to
Subjected
Structure
Offshore
Two-Bay
6.66
314
with
modelled
end
the
added
were
simplified
by
for the appropriate
leg members.
All
conditions
Foundation
surface.
specifying pinned ends
14
were
there
together
degrees
An
Damping was
freedom.
of
Rayleigh type with a damping ratio of
four.
water
the
below
members
included for
Entrained water was
by
CM=2 .0.
coefficient
inertia
with
effect
mass
was simulated
surrounding sea water
The
conditions.
fixed-fixed
model,
the tubular brace material
and
structural
following
eigenvalue analysis gave the
one
modes
at
5%
periods:
T1=1.16s,
The direct time
the
with
integration
integrator
Newmark
Newton equilibrium iterator,
time increment was
The
only
analysis,
HFT
best
that at
captured.
0.01
To reduce
and HFT analyses.
the
the
lateral
s for
formulation.
the
time
first mode was
integration
cost
the computational
included,
be
would
response
modified
6 =0.50),
incremental
both
conducted
were
analyses
(a=0.25,
and
T 1 4 -0.0001s
,
...
of
the
implying
adequately
or
mode
updating
was
of the Taft earthquake
were
used
as~
the
artificial
No
T 4 =0.'15s,
T 3 -0.24s,
T 2 =0.27s,
damping
admitted.
Ten seconds
input
level
excitation.
earthquake for
magnitude
was
of
0.25
Based on API guidelines
Southern California
has
(4)
an
the
strength
approximate
g, and consequently the Taft earthquake
scaled to the ductility
level magnitude of 0.50 g.
315
the
nonlinear
reproduced the
The actual
6.69.
Fig.
in
established
study
only
the
of
that
1/3
its
in
differs
the nonlinear response
of
rapid for
linear
the
from
magnitude
Convergence is
scheme's
solution
HFT
the
shape
the
when
history.
response
lateral
the
capture
the HFT scheme was
cost of
attractiveness
history
converge rapidly if
integration analysis.
direct time
This
shown
12,
dof
computational
responses
scheme converged within three
than adequately
and more
of
response
HFT
the
Indeed,
iterations
nonlinear
adequately
can
fundamental mode
response.
The close
the HFT solution should
that
indicates
and
linear
the
between
similarity
to
level).
(deck
12
freedom
of
degree
lateral
correspond
figures
Both
response.
the nonlinear
6.68,
response, and Fig.
linear
the
6.67,
Fig.
in
presented
are
analyses
integration
the direct time
for
Results
such cases and
hysteretic damping need not be imposed.
artificial
6.5.3 Two-Bay Offshore
Structure
Centro
El
to
Subjected
Earthquake
This
portion of
nonlinearities.
two-bay structure with more significant
first
Centro
5 seconds
of
earthquake,
shown
in
earthquake was
excitation.
The
appreciable
difference
response.
component
the SE
between
Structural modelling
Fig.
scaled
the
was
of
response
the study examined the
of
to
El
were used as
the
May
3.0 g to
linear
the
The
1940
the
6.70,
a
same
induce an
and nonlinear
as
before,
316
4.'
0.4
C
0
0.3
0
U
0.2
U
a. 0. 1
0.0
4
)
4.'
-0.2
U
-0.3
4
-0.4-0.5
0
1
2
3
+
T IMO
Fig.
5
6
'7
9
8
DOF 12
(00)
6.67 Linear Response to -Taft Earthquake
(lateral deck displacement)
0.+
C
0
0.3
4
(3
0.2
C
U
a.
0.L
0.0
4
)
-'0.1
4-'
U
4
C-
-0.4
0
1
2
6
3
TLMe
Fig.
($0)
7
DOF 12
6.68 Nonlinear Response to Taft Earthquake
(lateral deck displacement, time integration)
9
10
317
0.3
0
0.0
-0.1
-0.2
C.
-0.3
0
1
2
+
T ime
Fig. 6.69
5
'7
CSoo)
DOF 12
Nonlinear Response after 3 Iterations
damping)
(1 linear mode, no artificial
8
9
10
200
150
~C
100
U
50
0
0
E4 0
-50
-100
-150
0
30
20
10
TIME
Fig.
6.70
40
50
(SEC)
El Centro Acceleration History
60
319
The
fixed-fixed.
as
specified
now
was
base
the
except
structural periods are as follows'
T
1
T =0.24s,
3
=l.lls, T =0.27s,
2
of
deformations
the very stiff
(stiffness
modified Newton iteration scheme
6.72 and
The time
step).
time
each
6.73 depict the
lateral
response for
Based
preliminary
on
HFT
the
for
increment
prevent
In
instabilities.
the
linear modes were used throughout
examine
exact
efficiently
how
terminated
after
computational cost.
45
iterations
the HFT
response.
nonlinear
was
analyses
damping ratio
artificial hysteretic
Figure
using the
not participate in the
first HFT analysis,
the
and
scheme
an
to
seven
process
to
could reproduce
the
iterative
because
was
solution
the
lowest modes
response).
time
critical
also 0.02 s,
6.74 depicts the
seven
the deck level.
of 0.5 was necessary
However,
iterations
45
the
studies,
Figures
s.
displacement
nonlinear
of freedom 10 at
after
updating
increment was 0.02
linear and
degree
using
conducted
integrator combined with a
numerical
rule
trapezoidal
the
to
deck beam.
horizontal
time integration analyses were
Direct
physical
correspond
modes
five
highest
deformations while the
reasonable
to
correspond
modes
seven
first
the
It is noted that
6.71.
parameters are provided in Fig.
structural
Relevant
seven.
and
one
modes
for
5%
ratio
specifying a dampi'ng
Rayleigh damping was included by
of
=0.003s
T
12
.
of
the excessive
response
after
(other modes
Notice that the
do
response
320
60'
11
8
9
12
20 $10
-I
10
-heavy linear elastic
60' | 6.5 $ 0.0833
460'
| 6.5 $ 0.0729
El Centro Earthquake
1=
ui
gma
F ig .
= 3.0 g
3.O
0.05 @ T
and T7
kslugs
60
=
Deck mass
at each
top node
6. 71 Two-Bay Of f shore Structure Subj ect ed to El Centro Earthquake
321
2.0
40
t0.
V
0.5
0.0
-0.5
-L .0
.- t.5
-2.0
-2.5
0.0
0..5
I
.5
1.0
1.5
2
2.
2.0
2.5 3.0
3
Cee0)
t th
Fig.
C
0
6.72
+
4.0
3
3-
+
5.I
+.5 5.0
DOF 10
Linear Response to El Centro Earthquake
(lateral deck displacement)
L.0
0
0
U
a.
0.0
0
U
0
'S
-1 .5
'
.0' 0.5
I
1.0
2.5 3.0 3.5
time
Fig.
I
I
I
1.5 2.0
($e0)
I
I
I
+.0 4.5 5.0
DOF 10
6.73 Nonlinear Response to El Centro Earthquake
(lateral deck displacment, time integration)
322
At = 0.02 s
*C
C
3
-
0.0
-0.5
:1 -'L.0
L.-
2.0
0.0
I
0.5
1.0
2.5
TLMO
Fig.
I
I
I
L.5 2.0
($s)
3.0 3.M
+.0
4-.5 5.0
DOF 10
6.74 Nonlinear Response after 45 Iterations
viscous damping ratio of 0.50)
(7 linear modes, artificial
323
3.0
time
after
s has
in Fig.
the response
comparing
HFT
the
one mode for
using
of
simply
modes
then adding more linear
this approach, the first
Using
linear mode
fundamental
The next twelve
6.75.
giving
linear modes,
Furthermore,
iterations.
of the
actual nonlinear
modes.
This
three
6.76.
Fig.
in
for 10
one mode
iterations
12
the next
for 45
a
fairly good
HFT
The
for
various
the
ten
after
response
the
even
estimate
response,
of
additional
analyses
are
in Table 6.9.
portion
last
that
demonstrated
analysis
for
response.
times
Computation
tabulated
the lowest
to converge, possibly requiring
yet
has
course,
Fig.
with
obtained
one linear mode is
using
iterations
shown in
than when using seven modes
results
better
produced
employing the
ten iterations
the response depicted
and
iterations.
successive
for
iterations included
then three modes
and
iterations
first ten iterations
produced the response
response
the
that
Notice
the
consists
The algorithm
the convergence process.
accelerate
scheme
developed to
seven modes, a new solution algorithm was
using
6.74.
in Fig.
6.73 with that
of the slow convergence of
As a result
seen by
not yet converged adequately as
of
the
employing all
offshore
significant modes
in
during all iterations may result
process.
An
study
structure
an
alternate
in an UFT
extremely
solution
inefficient
solution
algorithm was
implemented whereby only a few modes were used
during
the
estimate of
initial
the
iterative
dominating
cycles
nonlinear
to
obtain
response,
a
good
and
then
324
1 .!5
C
CL 0.5
0.
0.0
-1.60
C.
-1.5
0.0
0.5
1.0
1 .5
2.0
2.5
3.0
3.5
5.0
+.0
4.5
+.0
4.5 5.0
DOF 10
TIM$ C00)
6.75 Nonlinear Response after 10 Iterations
(1 linear mode)
-
Fig.
1.0
0
CL
V
0.0
0.0
0.5
1.0
1.5 2.0
tiM.
Fig.
2.5
3.0 3.5
($C)
DOF 10
6.76 Nonlinear Response after 10+12 Iterations
(3 linear modes)
325
Modes
CPU
Time
(sec)
250
all
25
nonlinear
250
all
62
HFT
256
of
No.
Points
Solution
Scheme
Iterations
T i me
Time
Integration
linear
45
increment =
time
*
Table
6.9
*
10+12
256
0.02
seconds
350
7 linear
for all
83
analyses
1 linear for first 10 iterations
12 iterations
3 linear for next
Computation Times for Two-Bay Offshore
Structure Subjected to El Centro Earthquake
326
provide
were
modes
additional
a finer
efficient and produced better
algorithm proved more
6.6
EXTREMELY SOFT
This
the
by
portion
dominated
alternate
the sources
Although the HFT
of
the
yield
SDOF
The
obtained
with
of
immediately after
The excitation consisted of
5 seconds
earthquake.
time integration
using
the
and an
was employed.
results,
time increment
of 0.005
time
and stiffness
6.77,
were
s.
increment
artificial hysteretic
The response after
mode
shown in Fig.
rule and Newton equilibrium
trapezoidal
In the HFT analysis a
linear
an
plastic
displacement
plastic behavior almost
load.
iteration with a
specified
systems
Elastic-perfectly
levels.
were employed with a yield
producing
suggest
divergence
the HFT solution.
for implementing
low
the Taft
and another,
of the
introducing the
the
stiffness
a converged solution using the techniques
material models
0.001,
linear
could
scheme
extremely
of
by the
study examined the behavior of
The
softer than
scheme
studies,
other
nonlinear
the
where
study
The actual response history contained
the nonlinear stiffness.
not produce
a
characterized by a stiffness much
linear stiffness.
one
results.
SDOF SYSTEM
section describes
was
response
to
proposed
The
response.
of the
refinement
successive cycles
during
added
damping
10 and 30
is
of
shown
0.01 s
was
ratio of 0.75
iterations using
in Figs.
6.78
and
327
.. 0.02
C
.At
=0.005 s
<0.0l
C
c- 0 . 00 .- 0.03 -L
-0.05 -,--a. 06 --0.071
0.0 0.5
1.0
1.5 2.0 2.5 3.0 3.5 +.0
tine (sec)
Fig.
DOF 1
6. 77 Nonlinear Response Using Direct Time Integration
+.5 5.0
328
to
seconds, whereupon it begins to
3.5
time integration results.
This divergence
is attributed
a
pseudo-force
The
stabilized
during
artificial
damping.
the
example appears
This particular
be
pseudo-force
significant
implying
stress,
yield
to
extremely
of the
stabilized insufficiently, probably because
low
including
by
process
iterative
were
studies
case
Previous
iterative process.
the
during
converged form
its
from
differences
significant
exhibit
may
hence
and
fashion,
exact
to its
pseudo-force history converges
forward time progressing
form in a
on
based
the HFT solution scheme being
approach.
in
present
displays an oscillatory behavior not
and
diverge
fairly
is
response
the
indicate that
time
to
up
accurate
the
figures
These
6.79.
amplitudes.
nonlinear mode updating scheme was
A
a means
of
5 cycles with the
iterating
stiffness,
iterating
another
stiffness,
repeating
this
The
iterations.
1264,
and the
15 was 206,
initial
procedure consisted
The
the problem.
of stabilizing
26,
and 15,
linear
represented
seconds,
behavior.
a
for
process
total
fairly
however,
This
well.
The
after
10,
5,
Figure 6.80 shows
procedure.
time
response
is incorrect and
of
stiffness had a value
respectively.
response
the
5 cycles with the new secant
secant stiffness after iterations
nonlinear
the
updating
mode,
linear
response after 20 iterations using this
that
considered next as
2.8
before
20
of
and
the
Notice
seconds is
time
2.8
displays an oscillatory
result contrasts directly with the
previous
329
0.08
4",0 90 6
I
_
c
I
t0.
I
I
I
I
I
I
I
01S
0.02 --
02
*- 0.0+
2
0.0 0.5
1.0
1.5 2.0 2.5 3.0 3.5 +.0 +.5
t ine (sec)
5.0
DOF 1
Fig. 6.78 Nonlinear Response after 10 Iterations
(linear stiffness)
0.08
00.02
0 -0. 02
*-0. 0+
0.0 0.5
1.0 1.5 2.0 2.5 3.0 3.5 +.0 4.5 5.0
t ine (see)
Fig. 6.79
DOF 1
Nonlinear Response after 30 Iterations
(linear stiffness)
330
0.05
,
... 0.0+
4~
t
At=
C
0.03
0.01.
0-0.01-
Z-0. 02 -- 0.03
-0.05
0.0 0.5
1.0
1.5 2.0 2.5 3.0 3.5 +.0 +.5
line (sec)
DOF 1
Fig. 6.80 Nonlinear Response after 20 Iterations
(least-squares updated secant stiffness)
5.0
331
results,
the
iterations
4,
ksec
=
in
provided
converge while
that before
to
appears
to
cycles.
history.
In
seconds appears
900
the 49th
and
response
even for
cycles,
phase for consecutive
and, moreover,
iterative
time 3.0
the response before
particular,
of
the
and 50
the
tends to oscillate
applies
remark
similar
49,
Notice that
seconds
3.0
time
after
5,
4,
6.89.
significantly between consecutive
varies
out
to
6.85
Figs.
pseudo-force history
A
iterations 3,
the response after
to 6.84.
6.81
are given in Figs.
to
case with
6, and 50 of an updated stiffness
15 were plotted and
Similarly,
are
5,
corresponding
histories
pseudo-force
poor
the
of
source
the
diagnosing
of
means
a
As
linear stiffness.
the
obtained with
results
50th iterations.
on these
Based
solution
scheme
stiffness
employed
significantly
pseudo-forces
the
a
stiffness
material
very
on
actual nonlinear
can
during the
soft
the
material
LHS
behavior.
This
when
the
differs
stiffness.
These
large
excessively
from
theoretically, but
induce
The
large
iterative
equation
model
combined
HFT
process
process.
stem from an excessively large
the
the
that
problems
iterative
the
originate
left hand side of
on the
concluded
that may be correct
inaccuracies
pseudo-forces
with
during
from
it is
convergence
exhibits
problems
convergence
large
results
with
of
motion
stiffness
combined
or an excessively soft
a
problem
reasonably
is
stiff
amplified
332
2.01.5 1.0 cD
C)
0
(A
0
0.5
0.0
-0.5
-1.0 -1.5
-2.0
-2.5 -3.0 0.0 0.5
1.0
1.5 2.0 2.5 3.0 3.5 +.0 4.5 5.0
t ine (sec)
Fig.
cp
0
6.81 Pseudo-Force History after 4 Iterations
2.52.0 1.5 1.0 0.5
0.0
-0.5
-1.0-1.5
-2.0 -2.5 0.0 0.5
1.0
1.5 2.0 2.5 3.0 3.5 +.0
tine (3ec)
Fig. 6.82 Pseudo-Force History
after 5 Iterations
+.5 5.0
333
2.0
1.5
1.0
0
0.5
C0
0.0
0
V -0.5
CD
0 -1.0
0.
-1.5
-2.0
-2.5
0 .0 0.5
(j~
1.0 1.5 2.0 2.5 3.0 3.5 +.0 +.5
5.0
tine (sec)
Fig.
Pseudo-Force History after 6 Iterations
6.83
2.0
1.5
1.0
0
06
0.0
-1.0
-1.5
I
-2 .0 'I
0.0 0.5
1.0
I
I
I
Fig.
I
1.5 2.0 2.5 3.0 3.5 +.0 +.5
time (sec)
6.84
I
Pseudo-Force History after 50 Iterations
5.0
334
0.0+
00.00
10
-0.08
-0.08
\fv
-0.1.0 '
0.0 0.5
1.0 1.5 2.0 2.5 3.0 3.5 +.0
+.5
5.0
tjie (3ec)
Fig.
6.85
Nonlinear Response after 3 Iterations
(updated secant stiffness)
0.08
0.00
0-0.02 -
-0.06 -
-0.08 0.0 0.5 1.0
1.5 2.0 2.5 3.0 3.5 +.0 +.5 5.0
time (sec)
Fig.
6.86
Non linear Response after 4 Iterations
335
0.0
I
I
At = 0.01
0.00
-0. 02-
-0.06
0.0 0.5
1.0 1.5 2.0 2.5 3.0 3.5 +.0 4.5 5.0
time (3ec3
Fig.
6.87
Nonlinear Response after 5 Iterations
0.05
0.02
0.03
c -0.02
~-0.01
-O0.00
-0.05
0.0 0.5
1.0
1.5 2.0 2.5 3.0 3.5 +.0 4.5 5.0
t ine (sec)
Fig. 6.88
Nonlinear Response after 49 Iterations
336
0.0
CL
0.0
-0.00
0-0.0+
0.06
0.0 0.5
1.0 1.5 2.0 2.5 3.0 3.5 +.0 +.5 5.0
t ine (sec)
Fig.
DOF 1
6.89 Nonlinear Response after 50 Iterations
337
unrealistically
in a SDOF
problem -because it
is not
possible
stiff
and soft
to create a stiffness matrix representing the
the structure with a SDOF model.
of
portions
These conclusions
is
excitation
determined with a different K sec.
section is
in
interval before 3.0 s and k sec=15
for the
An
overlap-save
link the
two
response histories
would
history
cycles and
iterative
excitation
procedure,
to
of
implementing the HFT solution scheme.
following section examines
the
initial
nonlinear response.
section,
stiffness
In contrast
system with period 0.063 s.
in
Fig.
6.90.
stopped
few
and
This
segmenting
attractiveness
problem
SDOF
does
not
govern the
to
the
problem
of
actual
this
rapid convergence.
SYSTEM
The HFT scheme was applied
shown
first
the
determine how the
a similar
the following problem displays
6.7 BILINEAR ELASTIC SDOF
general,
the
and decrease the
the
where
interval
the
a substantial participation
on
The
In
sectioned.
however, may involve
the analyst,
for
during
examined
be
history should
part of
k, would be used
solution process
then the
histories
pseudo-force
the
together.
analyzed
be
for each
overlap-add method would then
or
after.
entire
the
example,
For
the very soft structure,
problem of
the
the response
time and
in
sectioned
where
solution scheme
suggest a
The
to a bilinear
elastic
SDOF
Relevant problem attributes are
loading
consisted
of
two
338
m= 1
T
0.063 seconds
=
0
&=
kl= 10,
=
F
sin w1 t + F2 sin w2
F,
=
50
F 2= 100
W = 1.5
w = 0.005
100
Bilinear Elastic Material Model
q
k = 10,000
k2 = 10
'a
Fig.
ilkk
6.90 Bilinear Elastic SDOF System
-100
339
The
second had amplitude
the
s and
4.2
100
low period
the
first
had
model
The bilinear elastic material
component.
period
1260 s.
period
and
generated mainly by
response was
dynamic
50 and
had amplitude
first
the
waves;
sinusoidal
slope k 1 =10,000 and second slope k 2 =10.
All
difference
critical
time increment for
At
- T /i
n
cr
captured,
was
s
used
in the
The
this
proved
shows
6.92
Fig.
time integration
the
differences
significant
responses indicate
analysis
essential
is
in
study.
The
HFT
damping.
k1
LHS
of
k2
was
the
stiffness
scheme
solution
artificial
on
The
bilinear static and dynamic
that a dynamic
and
response, obtained by increasing
the load history.
the
between
also
responses are
bilinear dynamic
Similarly,
therefore
0.02 s was
of
integration analysis
and
linear
static
bilinear
of
the maximum response
time increment
A
6.91 and 6.93.
shown in Figs.
periods
by the high period
the minimum interval of analysis was
nonlinear time
adequate.
the
To ensure that
cycle).
(one
1260
(6.2)
interval was determined
(T-1260 s).
component
linear analysis was
the
0.063/t = 0.020 s
=
time
The analysis
and therefore the
numerical integrator,
central
the
with
integration responses were obtained
time
was
Rather than using the
the
governing
employed.
with
conducted
linear
equations,
The
time
no
stiffness
the
second
increment
was
340
20E-3
.
.
200
400
,
,
,
800
B00
1000
15
1.0
C
a
0
0
-5
-1o
1ME-3
-
..........
0
TLMe
Fig.
6.91
1200
1400
(se0)
Linear Dynamic Response
lo
4-,
C
0
5
0
0
U
0
0.
0
V
-10
-lt5
500
0
1500
tLMe
Fig.
6.92
2000
[00)
Bilinear Elastic Static Response
250
WOO
341
20
15
-
10
+0
0E
0
-5
-20
200
0
+00
6O
tithe
800
1000
1200
1400
[00)
Response
Bilinear Elastic Dynamic
(time integration, time increment of 0.02 s)
Fig .6.93
20
15
.4-,
1.0
C
0
0
0
0
a.
-5
.3
-10
-15
-20
0
200
+00
6OO
800
1000
1200
T ime ($00)
Fig.
6.94
Dynamic Response
Bilinear Elastic
(HFT, 3 iterations, time increment of 1 s)
1400
342
determined by
and
the low period component
increment was
the time
hence
HFT
scheme are
by
the
given in Fig.
third
of
outline
general
response
converged
the HFT
scheme has
that
and
envelops
the
to
due
the
can
the HFT
nonlinearly
higher
capture
components
frequency
results
plotting increment for
to
the inability
response and
for the
Results
time integration response history.
the
to the coarser
attributed
induced,
Notice
excitation,
the time integration and HFT
Deviations between
be
The
the maximum response
accurately captured
the
1 second.
6.94.
iteration.
of
time
larger
increment.
study
This
when
scheme
than a factor of
Rapid
convergence
nonlinear stiffness
actual behavior.
gained
in
MDOF
such problems
reformulate
behavior
scheme
than that of
that
Notice
for the
was
attained
a
the
the time
scheme
more
was
time integration
by
amount
the stiffness
was
the
using
the
of the
be
integration analyses, since
significant
and refactorize
HFT
were well
further efficiencies would
time
implicit
require
because
that
but
(k 2 ) that was more representative
Finally, the HFT
problem
that
two less
the
excitation
captured exactly,
Computation time for the HFT
reproduced.
analysis.
long
with
oscillatory
general
and
values
of
advantages
problems
to
applied
the
The response was not
intervals.
peak
demonstrated
more
time increment was
of
time
to
matrix.
efficient
fifty times
integration analysis.
this
in
larger
In general,
the
343
time
by
increment
the
of
the
impose considerable
increment
demanding,
domain
the
load
however,
or response history.
increment
is
selected
As a result,
the
domain analysis can usually be
than that
allowable
although
constraints.
In
the time
that
greater than the highest frequency
histories.
the
increment is
Stated differently,
such
methods
methods,
implicit
also place accuracy
solutions,
on
stability constraints
while
physical
Explicit
system.
structural
determined
the
free vibration problem, in other words,
properties
time
analysis is
for a time integration
the Nyquist
in the
five
for a time domain analysis.
to
ten
as
frequency
based on
the
time
frequency
load and
time increment
not
is
response
for a frequency
times
greater
344
6.8
OF
SUMMARY
STUDIES
in sections
The studies presented
three
studies, sections
highlights
the
summarizes
four studies,
first
the
purpose
handle
the studies
El
for
of the RFT scheme, while
attributes column lists the
the
the
that
indicated
scheme
HFT
many strain rate independent material models
elastic-perfectly plastic,
bilinear
and
identifies the
each study.
of
In general
can
studies,
three
last
the last column
a new attribute
introduction of
the
The description,
study.
of each
6.10
and excitation columns are self explanatory.
material model,
For
for
Table
convergence.
ensuring
and
accelerating
its
scheme in
suggested alternate approaches
form and
developed
fully
examined the
6.5-6.7,
final
the
solution scheme while
of the HFT
dev-elopment
the
illustrated
6.1-6.4
Centro, and
elastic.
Taft),
Three
Ramberg-Osgood,
brace,
tubular
earthquake records
a single cycle
--
(Bucarest,
sine wave, and
dual
a
frequency component harmonic loading were used.
Table
6.10) were
2,
(studies
The highly nonlinear studies
tenth iteration.
type artificial damping proved adequate for
its use
in
the MDOF
in a low frequency drift appearing
a
high
frequency
6,
Without the artificial damping,
response usually diverged by the
However,
5,
7
of
stabilized by including artificial viscous
or hysteretic damping.
system.
4,
contamination
A viscous
SDOF
soil
soil problem resulted
by the
by
the
the
20th iteration and
the 30th cycle.
As a
Study
1
1
Material
Model
Description
Excitation
4
General
SDOF, QL
I
Elastic-Plastic
SDOF Soil,
3
MDOF Shear
Beam,
QL
4
MDOF
5
Offshore
Structure,
QL and HN
Soil,
HN
6
Soft
7
Bilinear SDOF
HN
QL
HN
-
Taft
Ramberg-Osgood
Elastic-Plastic
HN
SDOF, HN
Single Cycle
Sine
EQ
Single Cycle
Sine
Ramberg-Osgood
Taft
EQ
Tubular
Taft
EQ
El
Bilinear Elastic
(2
Harmonic
components)
response
Table 6.10
Total Displacement,
Artificial Viscous Damping
Mode Updating Scheme
Summary of Studies
1
Artificial Hysteretic Damping,
Mode Updating Scheme 2
Culminating Study
Centro EQ
Taft EQ
Elastic-Plastic
quasi-linear
highly nonlinear
Zero Minimizat.ion, Relaxation,
Dual Displacement
Bucarest EQ
Tubular
2
Attributes
Using a Segmenting
Scheme
Selecting a Time
Increment
t.n
346
type
Although
also
excessively attenuating
to accelerate the
Schemes
the
employed
1
Study
examined.
from 0.35
inability
evaluated
a
As
the
since
result,
entire history may accelerate
the
the
inaccuracies
by
forward time converging behavior
the
the time domain, the
of
initial
was
converged.
beginning
the convergence
relaxation
toward the end
are
of
applied to the
convergence
response history,
the
response
the prior
relaxation
the
of
nonlinearities
response
the history usually cannot converge until
has
Results
history.
convergence in
convergence
In other words,
in
to
The
to
RFT scheme.
also
eventually produced a diverging solution.
to accelerate the
attributed
history.
were
rate
that under-relaxation decelerated
but
by
relaxation schemes applied
while over-relaxation accelerated the
cycles,
some cases
to 0.75.
convergence
entire converging nonlinear response
indicated
it
response
converging
the
damping ratios ranged
Artificial
in
rate
convergence
the
problems.
studies,
all
stabilized
the artificial damping
decelerated
artificial hysteretic
convergence
no
produced
that
damping
employed an
studies
subsequent
result,
toward
but may also
in the pseudo-force toward the
the
aggravate
end
of
the
history.
The forward time progressing behavior of
suggested
a segmented approach outlined in
than evaluate the response
excitation
would
be
to the
the HFT
study 6.
scheme
Rather
entire force history,
sectioned in
time and
the response
the
to
347
different
structural matrices
response
history
of points in the
The number
a
by
reduced
technique
factor
for minimizing
in the
few dominant modes
the
7
Study 5 also
system
a
creating
presented a
only
the accuracy of
on
stiffness
the
of the
LHS
convergence
the
to
results.
the
the
a
and later
initial iterative cycles
corresponding
easily
employing
governing equations, and hence accelerated
by
was
spectrum
iterations by
the nonlinear
employed
efficiency
zero minimization technique.
four.
adding more modes to increase
Study
the
the divergence
schemes,
frequency
of
of
relaxation schemes.
the
increased by employing
was
allows
to model each portion
addition to these acceleration
In
time
approach
and may possibly alleviate
associated with the
problems
forward
a
segmented
This
fashion.
progressing
in
separately
obtained
section
each
dominating
nonlinear response.
The
studies
MDOF
efficiency
additional
(studies
a
system with spatially
simple temporal pseudo-force
updating
scheme
appropriate.
by
study
4,
(static
entailed
an
of
the
terms
in
Study 3 demonstrated
distribution,
that
secant
mode
approach) was
problems,
however were more conducive
a nonlinear
first
the
distribution
load
Significantly nonlinear
produced
4)
concentrated nonlinearities and
updating approach (least squares
approach
and
consideration
nonlinear mode updating schemes.
for
3
to the
exemplified
second mode
stiffness).
stiffness and mode
This
shape more
348
appropriate
to
the
nonlinear
response,
in
resulting
-an
accelerated convergence.
In general
of
producing
of the
nature
of
behavior
for all
studies,
accurate results.
loading,
the
complexity
material
governed more by numerical
and
4,
7)
5).
model.
of
the
response,
the
time
integration restrictions
Furthermore, study 7 demonstrated the
structure.
significantly
higher
period
was
(studies
(studies
2
advantages
of
loading and
than
and
proved
increment
than material modelling considerations
of
capable
The HFT approach
the frequency domain approach when the
are
scheme was
Efficiency varied with the
efficiency in problems where
higher
3,
the HFT
that
response
of
the
349
CHAPTER 7
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
thesis
This
scheme,
solution
The
approach.
the hybrid
frequency-time
principal
objectives
accomplishments
section
developed
newly
7.1
for
RESEARCH
Initial
review
of
study
7.3
Section
7.2.
suggestions
this
of
further
solution
chapters
nonlinear
schemes
systems
technique.
concisely in
with
chapter
to expand and adapt
the
SUMMARY
of
thesis
this
The
were
continuum mechanics
An
and their
and artificial damping errors
linear
the
main
the
scheme.
solution of nonlinear systems.
integration
are discussed
research
topics
primary
and
concludes
analysis
domain
section 7.1, while
in
investigated are summarized
introduced a new
systems and
structural
dynamic
nonlinear
of
solution
the
presented an overview of
theory
of
to
and
technique
a
numerical
numerical
frequency distortion
lead to the presentation of
stability
alternate
and the
overview
related
devoted
accuracy
a
analysis
substitutes
the
350
and
Accuracy
the
hybrid
a
superposition approach with
transferred
pseudo-force
force vector
using
the
transfer
response
using
the
of
the
of
The
vector.
then transferred
FFT.
Multiplying
the
in
During the
FFT
as
external
the
problems.
the mode
solution.
motion
plus
force
as
spectrum
domain
by
from the structural matrices
the
transferred back to
solution parameters such as
the
on
frequency domain
the
to provide an updated
iterative process,
a
nonlinear
frequency
the
and
domain,
of
equations
to
approach, was
time
the
governing equations,
inverse
to
domain
frequency
total
is
is obtained and
the
history.
to the RHS
function, evaluated
LHS
the
evaluated
are
Nonlinearities
an
by combining
economy
its
gains
scheme
HFT
The
particular,
response
to analyze nonlinear transient
developed
is
analysis
domain
frequency-time
frequency
referred
domain solution scheme,
iterative frequency
the
In
solution.
the
derive
to
employed
an
integrator
numerical
in essence an exact
domain, where
the
in
solution techniques
examine
to
incentive
provided
integration
numerical
on
The background
exact
the
with
it
comparing
and
function.
transfer
theoretical
the response
transfer function, defined as
force,
the
divided by
deriving
by
is then easily and elegantly analyzed
approximate
concepts.
transform
Fourier
using
by
domain
frequency
the
to
equation
approximate
resulting
the
transfers
governing equation of motion
the
into
integrators
numerical
simple
time domain
pseudo-force
changes
in
the frequency discretization
351
increment and number
the
mode
shapes
the
developing
not
examining
small
Of
the numerous
transform
actual
numerical
plastic shear
structures,
level
applied to
a
was
problems could
followed.
from
Techniques
and a
general
conducted,
studies
the
its
steps
seven
to
necessary
basis
conceptual
to
The studies consisted
application.
be
scheme was developed.
emphasized
scheme
systems
approach
scientific
problems were proposed,
the HFT
seismically
large scale
to
to
mainly
its
of a
system, a soil amplification problem using SDOF
models,
MDOF
identified
developmental
that
the HFT
general SDOF
bilinear
Rather,
implementing
presented
were
scheme
systems whereby numerical
these
for
level of
accuracy
limited
Actual applications
economically
eliminating
framework
were
solution
implemented.
and
easily
RFT
systems.
excited
Any
allow
less efficiency.
considerations
Research
were
obtained, although a higher
implies
usually
and
frequencies and
solution accuracy to be easily monitored.
accuracy can be
for
of
a
beam model,
a hypothetical
elastic SDOF
freedom elastic-perfectly
10 degree of
a
set
of
steel
extremely soft
problem.
All
jacket
SDOF system, and
studies
various accuracy and efficiency problems.
offshore
a
demonstrated
352
7.2
CONCLUSIONS:
PRINCIPAL RESULTS OF
Conclusive comments
regarding the
stability and accuracy analysis
1.
by
exact and
artificial
transfer
damping
parameters
obtained,
and
entire
the
integrator reproduces
the numerical
the
component.
are
possible
by
the z-transform.
The effect of
vector
the finite difference
upon the accuracy and
easily
transferring
by
evaluated
then
multiplying
by
expanded
load
stability characteristics
difference expanded load vector to
and
by
providing a measure of
Extensions to stability analyses
using
is
exact
the
for each individual frequency
response
4.
evaluated
be
of
of the
functions.
spectrum is
how accurately
3.
can
be
Similarly,
Amplitude amplification information for
frequency
can
frequencies
functions.
approximate
approximate transfer
2.
parameters
the imaginar.y components
comparing
theory
approach are as follows:
the resonant
comparing
STUDY
linear systems
distortion
Standard frequency
derived
THE PRESENT
the
the
finite
the frequency domain
approximate
transfer
function.
5.
and
The conventional
the
for the
linear
stability
systems
and
accuracy
analysis
theory approach are equivalent
free vibration problem.
353
Results
the HFT
of
scheme
HFT
The
1.
the HFT
regarding
conclusions
solution scheme were deduced:
numeri-cal
integrator
sense
The
frequency
peaks
at
particularly
Due
exact
and
stability
the
When
spectrum.
selected
dominating
numerical
accuracy
by
integration
considerations
analysis,
3.
in
increment
time
response
restrictions on the time increment
five
theoretically
integration
numerical
in the
a
but
rather by
or
loading.
are
the
time
the time increment for an HFT analysis
to ten times
only
domain
can be
larger.
The HFT scheme displays
a forward time
convergence behavior, attributed
being based
that
is negligible.
considerations,
frequencies
Furthermore,
properly such
the
integrator,
restricted
selection is not
exact
important,
to the transfer function being a
numerical
is
become
the Nyquist frequency
the component at
numerical
Resolution
in the
time increment must be
a
frequencies.
therefore,
considerations,
the
for
function
transfer
discrete
at
only
From
spectrum.
perspective, however, the
2.
theoretical
exact in a
is
transfer
the
by
because an analytical expression exists
continuous
the
a theoretically exact
represented
integrator
the
in
solution
the
derives
domain, and hence uses
H.
following
the
studies,
frequency
function
in
summarized
were
studies
these
on
Based
6.
Chapter
case
to
progressing
the solution scheme
on the pseudo-force formulation.
Since
the
354
the
of
latter portion
has
portion
initial
the
usually
history
response
the
time domain,
in the
evaluated
until
converge
cannot
is
history
pseudo-force
converged.
4.
by adding
Highly nonlinear problems were stabilized
a
employed
studies
Subsequent
MDOF'
the
for
inaccuracies
solution
produced
but
model,
soil
SDOF
for the
was adequate
type artificial damping
Viscous
damping.
artificial
model.
soil
type
hysteretic
artificial damping that displayed no accuracy problems.
Artificial damping ratios ranged
5.
The
from 0.35
zero minimization scheme optimized
by
efficiency
frequency
spectrum
(compared
to
conventional
by
number
the
in the
four
in
required
points
of
solution
of
factor
a
least
at
the
of points
number
the
reducing
to 0.75.
frequency domain solution without
the
a
zero
minimization technique).
6.
because
solution
attractive
than
are
easily
derived
in
the
corrected
better initial
guess
the
entire
the
diverging
a
time
progressing
the HFT scheme.
of
first
displacement
in
in
the
formulation
solution
later cycles and
by
history
a possibly
the nonlinear response
cycle
more
is
formulation
dual
inaccuracies
because initial
to
produced
forward
displacement
total
The
the
of
nature of
converging
7.
history
response
converging
applied
techniques
Relaxation
can
be
employing artificial
355
damping.
8.
load
MDOF
distribution)
adequate for a quasi-linear
updating
second
The
problem.
proved
is
stiffness)
squares
secant
systems
exhibiting
significant
the
from
derived
nonlinearities.
The
contained algorithm,
analyst.
the
cost is
since
problem,
eigenvalue
evaluating
for
operations
with
only additional computational
the
Furthermore,
compatible
from
participation
minimal
requiring
(least
technique
more
can be developed as a self
scheme
(static
scheme
updating
first nonlinear mode
The
the
secant stiffness matrix
the
must also be executed when evaluating
the
pseudo-force
history.
9.
If
only the peak response
history is
response
the
lower number
of
iterations
outline
or a general
desired,
then a significantly
required
is
of
for
the
HFT
analysis.
10.
The linear stiffness
the
transfer function H.
better
the
representing
employed,
to
evaluate
Rather a nonlinear
stiffness
need
not
actual
be used
such that the pseudo-forces are
convergence
rate
The HFT scheme
should
be
small and
the
response
rapid.
is
particularly suited to the
following
systems:
1.
A
few
linear
or
nonlinear
modes
dominate
response
does
not
the
response.
2.
The
nonlinear
deviate
356
response.
significantly from the linear
3.
The
time
severely
and
increment in a
by
restricted
domain
time
analysis
is
numerical integration accuracy
stability considerations.
The
4.
function
forcing
the lowest
than that of
significantly higher
period
dominant
a
has
structural
periods.
5.
Frequency dependent
such as
used,
6.
7.
must
be
in soil-structure interaction problems.
The response must be
spectrums,
stiffness and damping
rather
of
in terms
viewed
frequency
than time histories.
The excitation history is
and
is
inefficient
or
of long duration
dominated by low frequency components.
Examples when
HFT
the
technique
is
inapplicable are as follows:
1.
Excitation is
almost
all
an impulsive
excites
significant spatial variation.
The loading has a
3.
Material model evaluation
time
that
modes.
structural
2.
the
type load
step size,
such as
considerations
for
restrict
strain rate dependent
material models.
4.
Each segment of
by
a
words,
different
the
the response history
is
nonlinear stiffness matrix.
structural properties
dominated
In
other
continuously change at
a rapid rate as the excitation is applied.
5.
A
response history with a high level of accuracy
desired.
is
357
FOR FUTURE RESEARCH
7.3 RECOMMENDATIONS
explore
and
future
recommended for
are
following topics
investigation to
of
effectiveness
and
expand the applicability
the
study,
this
from
experience gained
on the
Based
the HFT scheme:
1.
behavior,
2.
as well
approach
HFT
the
of
with
combined
operators
such as the
iterators
quasi-Newton
of
properties
integration schemes consisting
difference
finite
other
time
with
material
behavior.
rate dependent
convergence
the
Compare
as
nonlinear
severe
Apply to problems with
Davidon, BFGS, and
DFP.
3.
Consider the solution of large
interaction
soil-structure
dependent
improved
for
5.
segmented
efficiency.
to
the
different
time
when applied
using
the
problems
as
frequency
analysis
Evaluate the
segmented
for
procedure
effect of
relaxation
histories.
Consider
increments and stiffness matrices
each segment.
an
As
approach,
alternative
to
segmented
the
consider evaluating
obtaining the entire
response,
response simultaneously.
pseudo-force history has
interval,
time
store
the
converged
history and
interval.
This
analysis
the pseudo-force history
for only certain portions of the
next
with
well
damping.
stiffness and
Develop
4.
systems as
for a
but
Once
certain
proceed forward
approach
still
the
time
to the
alleviates
358
to
due
instabilities
history and also
inaccuracies in
improves
the
the pseudo-force
efficiency by
minimizing
the pseudo-force history evaluation.
6.
using
Consider
(68)
differentiators
of
the
initial
minimization
velocity and
response
impulse
finite
to obtain more accurate
velocity
technique
for
and
in
use
also
to
estimates
the
derive
acceleration response histories.
(FIR)
zero
better
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368
APPENDIX A
ALTERNATE ACCURACY ANALYSIS OF
THE PARK
STIFFLY-STABLE METHOD
accuracy analysis of
This appendix examines the
using
(59)
method
stiffly-stable
a
linear systems
a
with
Rather than analyzing accuracy
approach.
the Park
theory
standard
domain approach, the equation of motion is transferred
time
to the
then the exact
frequency domain, and
corresponding
functions
transfer
are
compared
frequency distortion and artificial damping
Consider the expressions
the velocity.
(Al)
+ 1 - 15u n + 6u n-i - un)/6At
n-2
n+(loun1
- 15
The acceleration is
(2
10u n+ - 300u
- 12u
4 + u
n-iAn-5
Substituting Eqs.
+
34
(A2)
)/6&t
rewritten in
terms of
displacements
by
A2.
Al into Eq.
substituting Eq.
Sn+=
+ 6ni -
of
terms
the velocity in
=
Un+l= (106 +
to extract
characteristics.
and acceleration in terms of
the displacement
n+
for
approximate
and
5un-
)/36At
- 200u
2 + 66u
3
2
Al and A3 into the governing equation
of
369
motion
in+1
Un+l
+2 Ew1 n++-2
n+1 + WU n+1
we obtain
(A4)
fn+1
the numerical integration
form of
the equation
of
motion
(10Oun+1
- 300un + 345u n-1 - 200un-2 + 66un- 3 - 12un-4
-2
+ W un+l
is
Equation A5
- 15u
+ 2E(10un+
+ u n-5) /36At2
- un-2) /6At
+ 6un-1
(A5)
n+1
simplified by
similar
collecting
terms
to
yield
9AtL
U n+1+
+3+t
+
-
Equation A6
36t4 -n-5
is
6&t
un- 2 +
1
+
5_
-50
-2
10
25
+[
2
+2
un-3
~3 1
-3AtZ
u
At _n-1
2
~At
6t
115
(A6)
-n-4
Sn+1
transferred
discrete-time Fourier
2
to
the frequency
domain using
the
transform
=O
Un = (1/21 ) f00U(e
resulting
in
its
iQ)eiMn dQ
frequency
(A7)
domain
equivalent
for
any
370
frequency
10
25
9&tZ + 3 t+
-50
9 2
9At
+
2
)
The
e
3Lt
1 z e-6j
+3 6At
5_
A
-50
6Atz
(
+
-w
-3jQ
U(2)
11 2
+ 6At
,t&2
\-jQ
e
-4jQ
2
+ ( 115 +
12Ztl
1t
-
1
2
At
e-22
-5j 2
(A 8)
= F(Q)
function is defined as
transfer
H()=
U(Q) /F(G)
(A
Therefore,
H(2)
=(25a/9 + 10b/3 + c + (-50a/6
5b)e
-
+ (115a/12 + 2b)e-2jQ + (-50a/9 - b/3)e-3jo
+ lla/6 e~-4 i
(A1 0)
+ a/36 e-61j
- a/3 e-5j
where
-2--2
a
H(2)
=1/(2Lt 2)
oac
multiplied
is
nondimensionalized
redefined
b =
form.
=
(Al
W
4Tr2 /At
by
The
2
constants a,
its
obtain
to
b,
and
L)
c are now
as
a = 1/(47r 2
b = E(At/T )/27
(A12)
C = (At/T2n
Plots of
the nondimensionalized
shown in Fig.
10%
4.11 for 5%
physical damping.
Park transfer
physical damping and
function
Fig.
are
4.12 for
371
Frequency distortion and
estimated
?
next.
The
IH(Q)1'
such that
artificial damping effects are
peak of H(Q) occurs at
=
the
frequency
0 where
p
H(Q)
= -
(A + B cos2 + Ccos2Q + Dcos3Q +lla/6 cos4Q
-a/3 cos50 + a/36 cos62)2 + (-BsinQ
- Csin22
4
- Dsin3Q-lla/6 sin Q+ a/3 sin5Q- a/36 sin6Q)2] -1.5
-[(A + Bcoso + Ccos22 + Dcos3Q +lla/6 cos4Q
- a/3 cos50 + a/36 cos6Q)-(-Bsin2 - 2Csin2Q
- 3Dsin3Q - 22a/3 sin4O + 5a/3 sin5Q -a/6 sin6Q
)
+ (-BsinQ - Csin2Q - Dsin3Q - lla/6 sin4Q
+a/3 sin5Q - a/36sin6O)*(-BcosQ - 2Ccos2O
- 3Dcos3Q -22a/3
- a/6 cos62)I
cos4Q + 5a/3 cos5Q
(A13)
372
and
A = 25a/9 + lOb/3 + c
B =
- 5b
-50a/6
(A14)
C = 115a/12 + 2b
D = -50a/9
The
peak of
- b/3
the
exact
e
transfer
function is located at
ont
=
/1-T
217 ---At
T
n
-
(A15)
The period elongation is therefore given as
P
=
T
(A16)
p
Similarly,
equating
terms
.to the
the
in the
In
the artificial damping ratio is obtained
imaginary
terms
exact transfer
particular,
are given as
2a 2 Q=
H(Q) with the
function and
from Eq.
imaginary component
in
4.40
of the
the
exact
then
by
imaginary
solving
for
terms corresponding
transfer
function
follows:
- Q
.A
T
n
(A17)
Tr
The terms corresponding
to the
imaginary
component
of
the
Park transfer function are
-BsinE2-Csin2Q-Dsin3E-(lla/6) sin4E+(a/3) sin5-(a/36) sin6&l
(A18)
373
Equating
and
A17
and
A18
yields
for
solving
the
algorithmic damping ratio
-BsinG -Csin2_2p-Dsin3Gg-lla/6
sin4 P+a/3 sin5Q -a/36
sin6Q,
(At/T )(.e /IT)
n
e
(A19)
where
iT
and
=
WAt
T
p
(A20)
and
frequencies
respectively.
f
for
e
are
the
Using
period elongation and
were derived and
the
Park
the
nondimensionalized
and
exact
procedure
algorithmic
are shown
in Figs.
resonant
transfer functions,
above,
damping
plots
ratio
4.13-4.16.
of
versus
the
Q
374
APPENDIX B
ALTERNATE STABILITY ANALYSIS OF THE CENTRAL
DIFFERENCE METHOD USING THE Z-TRANSFORM CONCEPT
stability characteristics
The
are examined
method
concept.
The
in the
presentation
of
following using the
is
difference
the central
limited
to
z-transform
the
undamped
problem
= f
+2u
(B1)
Substituting the central difference
the
into
acceleration
motion
given in terms
un+l - 2un + un-1
Eq.
B1,
obtain the
equation of
of the numerical integrator
+
-2
= f
(B2)
nl
2n
At2
we
for
approximation
or equivalently
un+l
-2
2
2At-2)u
+un-1
Equation B3 is transferred
=At
2
to the
n
(B3)
frequency domain using
the
z-transform
X(z)
=
x(n)z-n
(B4)
375
where z is
z + (w At
The transfer
H(z)
=
to obtain
a complex variable,
- 2) + z
U(z)
is then given as
function H(z)
U(z)/F(z)
=
CB 5)
=At2F(z)
2
-2
At /(z+(w At
2 -
+
2) + z
)
(B 6)
= At 2 z/(z
We
next
2
+ (W2At 2 - 2)z + 1)
evaluate
difference
central
the
convergence
therefore
outside
is
=
system, the
of a causal
outside
the
unit
unstable when a pole
such a system is
The roots
z
defined
the unit
by examining the poles
method
right-sided sequence
For a
properties
stability
of
of H(z).
region of
circle,
is
the
and
located
circle.
of the numerator are given by
(-(Er2At 2 - 2) + /r.At
4
-
4 2At 2 + 4
-
4
)/2.
(B7)
=
(-(w At
- 2) +
wat
At
-
4 )/2.
376
Consider
1.
three cases:
WAt
2
=
From Eq.
4
B7 we have
z = -(4
- 2)/2
=
-1
Since both poles are
(B8)
located at
z
-1,
the
system
is
stable.
2.
WAt
2
> 4
Implies
- 2)/2 < -1
-2At
(B9)
and
w2 At
o6t
4
-
/2.
Therefore, one value
the
other,
because of
3.
between
>0
(B10)
of z is always
-1
and 0.
the pole located
The
less
than
-1
and
system is unstable
outside of
the unit
circle.
-At 2 < 4
From Eq.
z
B7
-
<-(w At
2 2-2
-
2)1/2. + wTt j
4
t 2 /2.(B
(B11))
377
The modulus
z
2
<
+
The
w2At 2 ) 4
(B12)
2 At )/4.
1.
the pole is always
circle,
and
the
system is
critical time increment
and given as
next
-4
4
-2
2
G At - ka2 t + 4)/4.
Therefore,
unit
of z is evaluated
located on
or inside
the
stable.
is determined
from
case
2
follows:
-2
2
=4
w At
(B13)
Implying
Atcr = 2/ w= Tnl
(4
(B14)
378
APPENDIX C
FOURIER TRANSFORM PAIR OF x3
The Fourier
appendix.
Let us
f
A()
transform pair
-
X(w
The inverse
B()
evaluated
We
in
this
define
(C1)
transform of A is
Aeiwitd
given by
= (1/21T)ffX(W1 - W 2 )X(w 2 )dw 2 eiwi t dwi
-
= (1/27)Jo
Now
is
X(w 2 ) dw 2
2
-00
which shows
3
first begin with x 2 .
Fourier
(1/2) f
of x
X(wi - W2 )eiCw1
X(w 2 )f
=
fo
=
21x(t)x(t) = 2iTx
CO- 00
X(w2)x(t)eiw2tdw2
- w2)tdwieiw2tdo
by definition of x(t)
2
(C2)
2
that the Fourier
transform of x 2
is A/2r.
consider
=f00
-00
The inverse
-CO
X(Wi - W 2 ) X(W2 - W 3 ) X(W3 ) dw2dw 3
Fourier
transform of B is
(.C3)
379
(1/2T) \1
Be
dw
= (1/2Tr)f f/f'X(wi-w2 ) X(w
O-
-CO
0-
2
-w 3 ) X(w
3
)
00~
dw 2 dw 3 e iitdw
= (1/2)f/*
-00
fc
X(w2 -w 3 )X(w 3 )f
X(wi-w 2 )ei(w-w2)t
-00-0
dwieiw2tdw2 dw 3
= /*jj*
-00
X(w 2 -w 3 )X(w
3 )x(t)eio2tdw2
dwo
-CO
= 21Tx(t)f
X(w
3 )(1/27T)fc
X(w2-w
3 )ei(w2-w3)
dw 2 eiw3td
=
2nx(t)2 f
X(w
3 )eiw3tdw 3
(C4)
= 472 X(t)3
Therefore,
the Fourier
transform
of x3
is
B/47T2 .
380
APPENDIX D
ZERO MINIMIZATION TECHNIQUE
A
solution in
to
an
aperiodic
force
necessary to
is
consequently,
is
the
a
approach, however, may
particularly
unfeasible,
or the natural
specified
An
render
alternate
technique eliminates
analytical
solution
the
was
zeroes.
The
is
This
technique
damping
the
as
by
subtracting
free vibration problem
response obtained with an
(N
zeroes).
the next power
of
conducting
2
is
zero
This
the
from the
number
insufficient
consists of
of
the
of
end
consequently developed.
use of zeroes
of
is large.
to
referred
incorrect
with N points
the end
frequency domain
structural period
of the
procedure
at
when very low physical
approach,
technique,
mimimization
the
the end
sufficient
of
the
appended
with
(history
history
extended
zeroes to
history
new
The
approach.
domain
load history are negligible by
the actual
gimmick,
obtain an apparent aperiodic
free vibration effects
the
that
such
solution
seek the
numerical
to append additional
load history and create
length
A
frequency
the
response when using
usual procedure
history.
In
function.
load
problems, however, we
response
transient
periodic
a
the response to
evaluates
therefore
and
concept,
transform
Fourier
discrete
the
of
the use
the frequency domain involves
of
the analysis
greater
than
the
381
obtaining
FFT),
the
in the
points
of
number
free vibration response.
the
subtracting out
solution by
to
and
of
least
two
are
technique
the
the
The mathematical basis of
available in section 5.2.2.2.
in
is presented
technique
minimization
of
details
Additional
efficiency
solution
the
increased
hence
considerably.
zero
number
frequency spectrum by a factor of at
in the
four,
the
reduced
Actual applications of the procedure
points
correcting
then
solution, and
an incorrect
for a radix-2
load history
actual
following
the
paragraphs.
The proof
mathematically
Fourier
derived
the
the
evaluating
the
axis,
shifting
h
to
Consider next a
A
implies
zero
time
t,
and
then
to
right
solution
q(T)
and
linear
in Fig.
to
yield
convolution concept
the
is
D.l.
obtained
in
the
frequency
frequency domain strictly implies
that we are concerned with a
periodic system since we employ
the discrete Fourier transform.
solution in the time
by
obtained
is
time
The
solution in the
concept
the
the
at time t (32).
illustrated conceptually
domain.
solution
function h about
integrating the product of h(t-T)
response
the
linear convolution, which in essence
impulse response
folding
of
the convolution
solution
The exact
(Duhamel's integral).
is
first consider a
Let us
domain using
time
technique
periodic nature
sound is based on the
transform concept.
in
minimization
zero
the
that
As
a result,
domain corresponds
to a
the equivalent
periodic rather
382
p(r)
r
h(-r)
h(-r)
r
h(t -r)
r
t
p(r)-h(t-r)j
Fig,
D. 1
I '
)(
)=
r)
Convolution with continuous functions
h
-
r)
dT
(32)
383
than
a
function is
periodic, and
periodic.
However,
creating
thus
impulse
if
M
response
zeroes
are added
correct response
the
load history has N
in particular
for another
problems, especially when a significant number
evaluated.
load history has
we use N+K, less
integrating
an
we
a
free
correct
essence
response is
history (M-K points)
by
in
where
the error
vibration component
frequency
evaluating
obtained
response, therefore,
the
in
D.2e.
in Fig.
the product shown
incorrect
D.2.
Fig.
If
N points and convolution, M points, but
than N+M, points
are
this case
except in
D.1 again,
Fig.
Consider
minimization
zero
the
of
redrawn for a periodic convolution, shown in
solution,
cycle.
inefficient
be
previously, may
Let us now examine the basis
the
from one
the response
the response
Then,
load history, the
the
is calculated because
transforms must be
technique.
points and
duration M points.
to the end of
approach, as mentioned
This
of
apparent
not contaminate
does
cycle
system.
aperiodic
an
function has
forcing
the
the end of
zeroes to
solution
correct
the
derive
to
Stated differently, assume
the
also
the solution corresponding to only one
adding additional
function,
forcing
is
response
the
therefore
approach
The usual
involves
the
other words,
forcing function must be obtained.
the
of
cycle
In
convolution.
linear
for
is easily
As
response by
result,
a
the beginning
is due
from the
the
domain
of the
to contamination
previous cycle.
The
from
the
extracted
384
f orce
- N
a.
N4
K
N
l'K
TT
h(T)
b.
M
W
h(-T)
C.
h(t-Tr)
h
(t
\.t
d.~
p(T) h(t-T)
e.
integrand giving
additional
free
vibration response
Fig.
D.2
integrand for
correct response
Zero Minimization Technique Viewed
the Time Domain
in
385
response
incorrect
Of
component.
extracted
rather by
by
subtracting
course, the
once
response.
component
could arise
very large),
and
out
the
free vibration
is
free vibration component
again
the
evaluating
imposing the known
incorrect
limit we
by
Notice
that
from more than one
the technique would
could have K equal to
zero.
convolution, but
conditions
initial
the
not
free
previous
upon
the
vibration
cycle
(M
In
the
still be valid.