Earthquake Engineering
Research at UC Berkeley
and
Recent Developments at
CSI Berkeley
BY
Ed Wilson
Professor Emeritus of Civil Engineering
University of California, Berkeley
October 24 - 25, 2008
Summary of Presentation
1. UC Berkeley in the in the period of 1953 to 1991
2. The Faculty
3. The SAP Series of Computer Programs
4. Dynamic Field Testing of Structures
5. The Load-Dependent Ritz Vectors – LDR Vectors - 1980
6. The Fast Nonlinear Analysis Method – FNA Method - 1990
7. A New Efficient Algorithm for the Evaluation of All
Static and Dynamic Eigenvalues of any Structure - 2002
9. Final Remarks and Recommendations
All Slides can be copied from we site
edwilson.org
“edwilson.org”
Copy Papers and Slides
Early Finite Element Research at UC Berkeley
by Ray Clough and Ed Wilson
The Development of Earthquake Engineering Software at Berkeley
by Ed Wilson - Slides
Dynamic Research at UC Berkeley
Retired Faculty Members by Date Hired
1946 Bob Wiegel – Coastal Engineering - Tsunamis
1949
Ray Clough – Computational and Experimental Dynamics
1950
Harry Seed – Soil Mechanics - Liquefaction
1953
Joseph Penzien – Random Vibrations – Wind, Waves & Earthquake
1957
Jack Bouwkamp – Dynamic Field Testing of Real Structures
1963 Robert Taylor – Computational Solid and Fluid Dynamics
1965
James Kelly – Base Isolation and Energy Dissipation
1965
Ed Wilson – Numerical Algorithms for Dynamic Analysis
196?
Beresford Parlett – Mathematics - Numerical Methods
196?
Bruce Bolt – Seismology – Earthquake Ground Motions
Professor Ray W. Clough
1942
BS University of Washington
1943 - 1946 U. S. Army Air Force
1946 - 1949 MIT - D. Science - Bisplinghoff
1949 - 1986 Professor of CE U.C Berkeley
1952 and 1953 Summer Work at Boeing
National Academy of Engineering
National Academy of Science
Presidential Medal of Science
The Franklin Institute Medal – April 27, 2006
Doug, Shirley and Ray Clough
The Franklin Institute Awards – April 27, 2006
Joe Penzien
1945
1945
1946
1953
1953 - 88
1990 - 2006
BS University of Washington
US Army Corps of Engineers
Instructor - University of Washington
MIT - D. Science
Professor UCB
International Civil Engineering
Consultants – Principal with
Dr. Wen Tseng
Professor Joe Penzien – First Director of EERC at UC Berkeley
The Franklin Institute Awards – April 27, 2006
New Printing of the Clough and Penzien Book
Berkeley, CA, February 26, 2004 –
Computers and Structures, Inc., is pleased
to release the latest revision to Dynamics
of Structures, 2nd Edition by Professors
Clough and Penzien. A classic, this
definitive textbook has been popular with
educators worldwide for nearly 30 years.
This release has been updated by the
original authors to reflect the latest
approaches and techniques in the field of
structural dynamics for civil engineers.
csiberkeley.com
Ask for Educational Discount
Ed Wilson - edwilson.org
1955
1955 - 57
1958
1957 - 59
1960
1963
1963 - 1965
1965 - 1991
1991 - 2008
BS University of California
US Army – 15 months in Korea
MS UCB
Oroville Dam Experimental Project
First Automated Finite Element Program
D Eng UCB
Research Engineer, Aerojet - 10g Loading
Professor UCB – 29 PhD Students
Senior Consultant To CSI Berkeley –
where 95% of my work is in Earthquake Engineering
My Book – 23 Chapters
csiberkeley.com
Ask for Educational Discount
NINETEEN SIXTIES IN BERKELEY
1. Cold War - Blast Analysis
2. Earthquake Engineering Research
3. State And Federal Freeway System
4. Manned Space Program
5. Offshore Drilling
6. Nuclear Reactors And Cooling Towers
NINETEEN SIXTIES IN BERKELEY
1.
Period Of Very High Productivity
2.
No Formal Research Institute
3.
Free Exchange Of Information – Gave
programs to profession prior to publication
4.
Worked Closely With Mathematics Group
5.
Students Were Very Successful
UC Students
“Berkeley During The Late 1960’s And
Early 1970’s Graduate Study Was Like
Visiting An Intellectual Candy Store”
Thomas Hughes
Professor, University of Texas
SAP
STRUCTURAL ANALYSIS
PROGRAM
ALSO A PERSON
“ Who Is Easily Deceived Or Fooled”
“ Who Unquestioningly Serves Another”
From The Foreword Of
The First SAP Manual
"The slang name S A P was selected to
remind the user that this program, like
all programs, lacks intelligence.
It is the responsibility of the engineer to
idealize the structure correctly and
assume responsibility for the results.”
Ed Wilson 1970
The SAP Series of Programs
1969 - 70
SAP
Used Static Loads to Generate Ritz Vectors
1971 - 72
Solid-Sap
Rewritten by Ed Wilson
1972 -73
SAP IV
Subspace Iteration – Dr. Jűgen
1973 – 74
NON SAP
New Program – The Start of ADINA
Bathe
Lost All Research and Development Funding
1979 – 80
SAP 80
New Linear Program for Personal Computers
1983 – 1987 SAP 80
CSI added Pre and Post Processing
1987 - 1990
Significant Modification and Documentation
SAP 90
1997 – Present SAP 2000
Nonlinear Elements – More Options –
With Windows Interface
FIELD MEASUREMENTS
REQUIRED TO VERIFY
1. MODELING ASSUMPTIONS
2. SOIL-STRUCTURE MODEL
3. COMPUTER PROGRAM
4. COMPUTER USER
CHECK OF RIGID
DIAPHRAGM
APPROXIMATION
MECHANICAL
VIBRATION
DEVICES
FIELD MEASUREMENTS OF
PERIODS AND MODE SHAPES
MODE
TFIELD
TANALYSIS
Diff. - %
1
1.77 Sec.
1.78 Sec.
0.5
2
3
4
1.69
1.68
0.60
1.68
1.68
0.61
0.6
0.0
0.9
5
6
0.60
0.59
0.61
0.59
0.9
0.8
7
-
0.32
-
0.32
-
0.2
-
11
0.23
0.32
2.3
FIRST DIAPHRAGM
MODE SHAPE
15 th Period
TFIELD = 0.16 Sec.
Load-Dependent Ritz Vectors
LDR Vectors - 1980
DYNAMIC EQUILIBRIUM
EQUATIONS
M a + C v + K u = F(t)
a
v
u
M
C
K
F(t)
=
=
=
=
=
=
=
Node Accelerations
Node Velocities
Node Displacements
Node Mass Matrix
Damping Matrix
Stiffness Matrix
Time-Dependent Forces
PROBLEM TO BE SOLVED
Ma + Cv+ Ku =
 f g(t)
i
i
= - Mx ax - My ay - Mz az
For 3D Earthquake Loading
THE OBJECTIVE OF THE ANALYSIS
IS TO SOLVE FOR ACCURATE
DISPLACEMENTS and MEMBER FORCES
METHODS OF DYNAMIC ANALYSIS
For Both Linear and Nonlinear Systems
÷
STEP
BY STEP INTEGRATION - 0, dt, 2 dt ... N dt
USE OF MODE SUPERPOSITION WITH EIGEN OR
LOAD-DEPENDENT RITZ VECTORS FOR
FNA
For Linear Systems Only
÷
TRANSFORMATION
TO THE FREQUENCY
DOMAIN and FFT METHODS
RESPONSE SPECTRUM METHOD - CQC - SRSS
STEP BY STEP SOLUTION METHOD
1. Form Effective Stiffness Matrix
2. Solve Set Of Dynamic Equilibrium
Equations For Displacements At
Each Time Step
3. For Non Linear Problems
Calculate Member Forces For
Each Time Step and Iterate for
Equilibrium - Brute Force Method
MODE SUPERPOSITION METHOD
1. Generate Orthogonal Dependent
Vectors And Frequencies
2. Form Uncoupled Modal Equations
And Solve Using An Exact Method
For Each Time Increment.
3.
Recover Node Displacements
As a Function of Time
4.
Calculate Member Forces
As a Function of Time
GENERATION
DEPENDENT
LOAD
VECTORS
OF
RITZ
1.
Approximately Three Times Faster Than
The Calculation Of Exact Eigenvectors
2.
Results In Improved Accuracy Using A
Smaller Number Of LDR Vectors
3.
Computer Storage Requirements
Reduced
4.
Can Be Used For Nonlinear Analysis To
Capture Local Static Response
STEP 1. INITIAL CALCULATION
A.
TRIANGULARIZE STIFFNESS MATRIX
B.
DUE TO A BLOCK OF STATIC LOAD VECTORS, f,
SOLVE FOR A BLOCK OF DISPLACEMENTS, u,
Ku=f
C.
MAKE
FORM
u STIFFNESS AND MASS ORTHOGONAL TO
FIRST BLOCK OF LDL VECTORS V 1
V1T M V1 = I
STEP 2.
VECTOR GENERATION
i = 2 . . . . N Blocks
A.
Solve for Block of Vectors, K Xi = M Vi-1
B.
Make Vector Block, Xi , Stiffness and
Mass Orthogonal - Yi
C.
Use Modified Gram-Schmidt, Twice, to
Make Block of Vectors, Yi , Orthogonal
to all Previously Calculated Vectors - Vi
STEP 3. MAKE VECTORS
STIFFNESS ORTHOGONAL
A.
SOLVE Nb x Nb Eigenvalue Problem
[ VT K V ] Z = [ w 2 ] Z
B.
CALCULATE MASS AND STIFFNESS
ORTHOGONAL LDR VECTORS
VR = V Z =

DYNAMIC RESPONSE OF BEAM
100 pounds
10 AT 12" = 240"
FORCE
TIME
MAXIMUM DISPLACEMENT
Eigen Vectors
Load Dependent
1
0.004572 (-2.41)
0.004726 (+0.88)
2
0.004572 (-2.41)
0.004591 ( -2.00)
3
0.004664 (-0.46)
0.004689 (+0.08)
4
0.004664 (-0.46)
0.004685 (+0.06)
5
0.004681 (-0.08)
0.004685 ( 0.00)
7
0.004683 (-0.04)
9
0.004685 (0.00)
Number of Vectors
Vectors
( Error in Percent)
MAXIMUM MOMENT
Number of Vectors
Vectors
1
Eigen Vectors
Load Dependent
4178
( - 22.8 %)
5907
( + 9.2 )
2
4178
( - 22.8 )
5563
( + 2.8 )
3
4946
( - 8.5 )
5603
( + 3.5 )
4
4946
( - 8.5 )
5507
( + 1.8)
5
5188
( - 4.1 )
5411
( 0.0 )
7
5304
( - .0 )
9
5411
( 0.0 )
( Error in Percent )
LDR Vector Summary
After Over 20 Years Experience Using the
LDR Vector Algorithm
We Have Always Obtained More Accurate
Displacements and Stresses
Compared to Using the Same Number of
Exact Dynamic Eigenvectors.
SAP 2000 has Both Options
The Fast Nonlinear Analysis Method
The FNA Method was Named in 1996
Designed for the Dynamic Analysis of
Structures with a Limited Number of Predefined
Nonlinear Elements
FAST NONLINEAR ANALYSIS
1. EVALUATE LDR VECTORS WITH
NONLINEAR ELEMENTS REMOVED AND
DUMMY ELEMENTS ADDED FOR STABILITY
2. SOLVE ALL MODAL EQUATIONS WITH
NONLINEAR FORCES ON THE RIGHT HAND SIDE
3. USE EXACT INTEGRATION WITHIN EACH TIME
STEP
4. FORCE AND ENERGY EQUILIBRIUM ARE
STATISFIED AT EACH TIME STEP BY ITERATION
BASE ISOLATION
Isolators
BUILDING
IMPACT
ANALYSIS
FRICTION
DEVICE
CONCENTRATED
DAMPER
NONLINEAR
ELEMENT
GAP ELEMENT
BRIDGE DECK
ABUTMENT
TENSION ONLY ELEMENT
PLASTIC
HINGES
2 ROTATIONAL DOF
ALSO DEGRADING STIFFNESS ARE Possible
Mechanical Damper
F = f (u,v,umax )
F = ku
F = C vN
Mathematical Model
LINEAR VISCOUS DAMPING
DOES NOT EXIST IN NORMAL STRUCTURES
AND FOUNDATIONS
5 OR 10 PERCENT MODAL DAMPING
VALUES ARE OFTEN USED TO JUSTIFY
ENERGY DISSIPATION DUE TO NONLINEAR
EFFECTS
IF ENERGY DISSIPATION DEVICES ARE USED
THEN 1 PERCENT MODAL DAMPING SHOULD
BE USED FOR THE ELASTIC PART OF
THE STRUCTURE - CHECK ENERGY
PLOTS
103 FEET DIAMETER - 100 FEET HEIGHT
NONLINEAR
DIAGONALS
BASE
ISOLATION
ELEVATED WATER STORAGE TANK
COMPUTER MODEL
92 NODES
103 ELASTIC FRAME ELEMENTS
56 NONLINEAR DIAGONAL ELEMENTS
600 TIME STEPS @ 0.02 Seconds
COMPUTER TIME
REQUIREMENTS
PROGRAM
ANSYS
INTEL 486
3 Days
ANSYS
CRAY
3 Hours
SADSAP
INTEL 486
( B Array was 56 x 20 )
( 4300 Minutes )
( 180 Minutes )
2 Minutes
Nonlinear Equilibrium Equations
Summary Of FNA Method
Calculate Load-Dependant Ritz Vectors for
Structure With Nonlinear Elements Removed.
These Vectors Satisfy the Following
Orthogonality Properties
 K  
T
2
 M  I
T
The Solution Is Assumed to Be a Linear
Combination of the LDR Vectors. Or,
u (t )  Y (t )    n y (t ) n
n
Which Is the Standard
Mode Superposition Equation
Remember the LDR Vectors Are a Linear
Combination of the Exact Eigenvectors;
Plus, the Static Displacement Vectors.
No Additional Approximations Are Made.
A typical modal equation is uncoupled.
However, the modes are coupled by the
unknown nonlinear modal forces which
are of the following form:
fn  
F
n
n
The deformations in the nonlinear elements
can be calculated from the following
displacement transformation equation:
  Au
Since u (t )  Y (t ) the deformations in
the nonlinear elements can be expressed
in terms of the modal response by
 (t )  A  Y(t )  B Y(t )
B
Where the size of the
array is equal to
the number of deformations times the
number of LDR vectors.
The B array is calculated only once prior
to the start of mode integration.
THE
B
ARRAY CAN BE STORED IN RAM
The nonlinear element forces are
calculated, for iteration i , at the end
of each time step t
  BY  Deformatio ns in
(i )
t
(i )
t
Nonlinear Elements
P  Function of Element History
(i )
t
f
(i )
Nt
B Y
( i 1)
t
Y
T
(i )
t
 Nonlinear Modal Loads
 New Solution of Modal Equation
FRAME WITH
UPLIFTING
ALLOWED
UPLIFTING
ALLOWED
Four Static Load Conditions
Are Used To Start The
Generation of LDR Vectors
EQ
DL
Left
Right
NONLINEAR STATIC ANALYSIS
50 STEPS AT dT = 0.10 SECONDS
DEAD LOAD
LOAD
LATERAL LOAD
0
1.0
2.0
3.0
4.0
5.0
TIME - Seconds
Advantages Of The FNA Method
1.
The Method Can Be Used For Both
Static And Dynamic Nonlinear Analyses
2.
The Method Is Very Efficient And
Requires A Small Amount Of
Additional Computer Time As
Compared To Linear Analysis
2.
The Method Can Easily Be Incorporated
Into Existing Computer Programs For
LINEAR DYNAMIC ANALYSIS.
A COMPLETE EIGENVECTOR
SUBSPACE FOR THE LINEAR
AND NONLINEAR DYNAMIC
ANALYSIS OF STRUCTURES
Definition Of Natural Eigenvectors
The total number of Natural Eigenvectors
that exist is always equal to the total number
of displacement degrees-of-freedom of the
structural system. The following three types
of Natural Eigenvectors are possible:
Rigid Body Vectors
Dynamic Vectors
Static Vectors
EXAMPLE OF SIX DEGREE OF FREEDOM SYSTEM
M=0.05
100
M=0.05
M=0.10
100
(a) Beam Model
I = 1.0
E=10,000
(b) Rigid Body Mode
10
T1  
(c) Rigid Body Mode

T2  
(d) Dynamic Mode

3
 0 .995 T3  6 .31

4

T4  0

5

T5  0

6

T6  0
(e) Static Mode
(f) Static Mode
(g) Static Mode
0
2
How Do We Solve a System That Has
Both Zero and Infinite Frequencies?
Static and Dynamic Equilibriu m Equation
(t)  Ku(t)  R(t)  FG(t)
Mu
where both M and K may be singular and
positive semi - definite
Add  Mu(t) To Both Sides of Equation
(t)  [K  M ]u(t)  R(t)  Mu(t)
Mu
(t)  Ku(t)  R(t)  Mu(t)
Or, Mu
where K is now nonsingula r and positive definite
Solve Static and Dynamic Equilibrium
Equations by Mode Superposition
u(t)  ΦY(t)
Let u(t)  ΦY(t) and
Where Φ Can Contain Rigid - Body, Static and
Dynamic Modes. All Modes are Stiffness and
Mass Orthogonal and are Normallize d so that
Φ K Φ  I and Φ MΦ  Ψ
T
T
The Modal Equations Can Now Be Written As
T


ΨY(t)  [I  ρΨ]Y(t)  Φ R(t)
SOLUTION OF TYPICAL MODAL EQUATION
T


nYn(t) + [1   n ]Yn(t) = n Fg (t )
FOR DYNAMIC MODES, USE PIECE-WISE EXACT SOLUTION
FOR RIGID-BODY MODES, Direct INTEGRATION
T


Yn(t) =  n Fg (t )
FOR STATIC MODES, THE SOLUTION IS
T
Yn(t) = n Fg (t )
Calculation of Frequencies from
Natural Eigenvalues
1
n 

n
and
Tn 
2
n
Eigenvalues for Simple Beam
  0.01
Mode Natural Eigenvalue
n
1
2
3
4
5
6
100
100
0.826
0
0
0
Frequency
1
n 

n
0
0
0.995



Period
Tn 
2
n


6.31
0
0
0
CALCULATION OF NATURAL
EIGENVECTORS
ΨKΦ  MΦ
Use Recurrence Equation Of Following Form
KV
(i )
F
(i 1)
i  1,     N Blocks
The First Load Block Must Be The Static Load
Patterns Acting on The Structure
F
( 0)
F
Subsequent Load Blocks Are Calculated From
F
(i )
 MV
(i 1)
Iteration Is Not Required
The Natural Eigenvector Algorithm (2)
(i)
All Candidate Vectors in Block V Must be
Modified To Satisfy The Following Requiremen ts :
Must be made Stiffness Orthogonal to All
Previously Calculated Vectors By the Modified
Gram-Schmidt Algorithm
If a Vector Is the Same As a Previously
Calculated Vector It Must Be Rejected
The New Vectors Block is Designated as V
(i)
The Natural Eigenvector Algorithm (3)
All Candidate Vectors Are Defined by V
These Vectors Are Then Made Mass Orthogomal
By Solving The Following Subspace
Eigenvalue Probem By The Jocobi Method :
MZ  
Where   VZ
A Static Vector Has A Zero Eigenvalue And
An Infinite Frequency
A Truncated Set of Natural
Eigenvalues Contains Linear
Combinations of the Dynamic and
Static Eigen Vectors That Are
Excited by the Loading
Therefore, They Are a Set of
Load Dependent Ritz Vectors
Error Estimation
1. Dynamic Load Participation Ratio
2. Static Load Participation Ratio
Therefore, this allows the LDR Algorithm
to Automatic Terminate Generation when
Error Limits are Satisfied
The dynamic load participation ratio for load
case Fj is defined as the ratio of the kinetic
energy captured by the truncated set of
vectors to the total kinetic energy. Or
N
rdj 
 ( 
T
2
n n j
n 1
T
j
f )
1
f M fj
For earthquake loading, this is identical to
the mass participation factor in the three
different directions A minimum of 90
percent is recommended
The static load participation ratio for load
case Fj is defined as the ratio of the strain
energy captured by the truncated set of
vectors to the total strain energy due to the
static load vectors. Or,
N
rsj 
 (
n 1
T
2
n j
f )
T
j j
u f
Always equal to 1.0 for LDR vectors
FINAL REMARKS
Existing Dynamic Analysis Technology allows us to
design earthquake resistant structures economically .
However, many engineers are using Static Pushover
Analysis to approximate earthquake forces.
Advances in Computational Aero and Fluid Dynamics are
not being used by the Civil Engineering Profession to
Design Safe Structures for wind and wave loads.
Many engineers are still using approximate wind tunnel
results to generate Static Wind Loads.
In a large earthquake the
safest place to be is on the top
of a high-rise building
Over 25 Stories
COMPUTERS
1958 TO 2008
IBM 701 - Multi-Processors
The Current Speed of a $1,000 Personal Computer
is 1,500 Times Faster than the
$10,000,000 Cray Computer of 1975
$
C = COST OF THE COMPUTER
S = MONTHLY SALARY OF ENGINEER
$4,000,000
C/S = 5,000
C/S = 0.5
$7,500
$1,500
$800
1957
time
1997
A FACTOR OF 10,000 REDUCTION IN 40 YEARS
Floating-Point Speeds of Computer Systems
Definition of one Operation A = B + C*D 64 bits - REAL*8
Year
Computer
or CPU
Operations
Per Second
Relative
Speed
1963
CDC-6400
50,000
1
1967
CDC-6600
100,000
2
1974
CRAY-1
3,000,000
60
1981
IBM-3090
20,000,000
400
1981
CRAY-XMP
40,000,000
800
1990
DEC-5000
3,500,000
70
1994
Pentium-90
3,500,000
70
1995
Pentium-133
5,200,000
104
1995
DEC-5000 upgrade
14,000,000
280
1998
Pentium II - 333
37,500,000
750
1999
Pentium III - 450
69,000,000
1,380
2003
Pentium IV – 2,000
220,000,000
4,400
2006
AMD - Athlon
440,000,000
8,800
Cost of Personal Computer Systems
YEAR
CPU
Speed
MHz
Operations
Per Second
Relative
Speed
1980
8080
4
200
1
$6,000
1984
8087
10
13,000
65
$2,500
1988
80387
20
93,000
465
$8,000
1991
80486
33
605,000
3,025
$10,000
1994
80486
66
1,210,000
6,050
$5,000
1996
Pentium
233
10,300,000
52,000
$4,000
1997
Pentium II
233
11,500,000
58,000
$3,000
1998
Pentium II
333
37,500,000
198,000
$2,500
1999
Pentium III
450
69,000,000
345,000
$1,500
2003
Pentium IV
2000
220,000,000
1.100,000
$2.000
2006
AMD - Athlon
2000
440,000,000
2,200,000
$950
COST
Ed Wilson at
UCLA Meet
April 17, 1954
President
Robert .
Sproul
Ed set a 880
yard record of
1 Minute and
54 Seconds.
In the last 50 years, Ed is getting Slower and
Computer are getting Faster
The Future Of Personal Computers
Multi-Processors Will Require
New Numerical Methods
and
Modification of Existing Programs
Speed and Accuracy are Important