SAP 2000 SEMINAR

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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
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• Summary of Presentation
• UC Berkeley in the in the period of 1953 to 1991
• The Faculty
• The SAP Series of Computer Programs
• Dynamic Field Testing of Structures
• The Load-Dependent Ritz Vectors LDR Vectors -
  1980
• The Fast Nonlinear Analysis Method FNA Method -
  1990
• A New Efficient Algorithm for the Evaluation of
  All
• Static and Dynamic Eigenvalues of any Structure
  - 2002
• Final Remarks and Recommendations
• All Slides can be copied from we site
  edwilson.org

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Early Finite Element Research at Berkeley by Ray Clough and Ed Wilson The Development of Earthquake Engineering Software at Berkeley  by Ed Wilson - Slides
Early Finite Element Research at Berkeley by Ray Clough and Ed Wilson The Development of Earthquake Engineering Software at Berkeley  by Ed Wilson - Slides
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
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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
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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
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Doug, Shirley and Ray Clough The Franklin
Institute Awards April 27, 2006
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Joe Penzien 1945 BS University of
Washington 1945 US Army Corps of
Engineers 1946 Instructor - University of
Washington 1953 MIT - D. Science 1953 - 88
Professor UCB 1990 - 2006 International Civil
Engineering Consultants Principal
with Dr. Wen Tseng
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Professor Joe Penzien First Director of EERC at
UC Berkeley The Franklin Institute Awards April
27, 2006
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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
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Ed Wilson - edwilson.org 1955 BS University
of California 1955 - 57 US Army 15 months in
Korea 1958 MS UCB 1957 - 59 Oroville Dam
Experimental Project 1960 First Automated
Finite Element Program 1963 D Eng UCB 1963 -
1965 Research Engineer, Aerojet - 10g
Loading 1965 - 1991 Professor UCB 29 PhD
Students 1991 - 2008 Senior Consultant To CSI
Berkeley where 95 of my work is in
Earthquake Engineering
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My Book 23 Chapters
csiberkeley.com Ask for Educational Discount
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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
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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
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UC Students Berkeley During The Late 1960s
And Early 1970s Graduate Study Was Like
Visiting An Intellectual Candy Store Thomas
Hughes Professor, University of Texas
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S A P STRUCTURAL ANALYSIS PROGRAM
ALSO A PERSON Who Is Easily Deceived
Or Fooled Who Unquestioningly Serves
Another
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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
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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. Jugen
Bathe 1973 74 NON SAP New Program The
Start of ADINA 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 SAP
90 Significant Modification and
Documentation 1997 Present SAP 2000
Nonlinear Elements More Options With
Windows Interface
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FIELD MEASUREMENTS REQUIRED TO VERIFY 1. MODELING
ASSUMPTIONS 2. SOIL-STRUCTURE MODEL 3. COMPUTER
PROGRAM 4. COMPUTER USER
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CHECK OF RIGID DIAPHRAGM APPROXIMATION
MECHANICAL VIBRATION DEVICES
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FIELD MEASUREMENTS OF PERIODS AND MODE
SHAPES MODE TFIELD TANALYSIS Diff. -
1 1.77 Sec. 1.78 Sec. 0.5 2 1.69 1.68 0.
6 3 1.68 1.68 0.0 4 0.60 0.61 0.
9 5 0.60 0.61 0.9 6 0.59 0.59 0.8 7 0.3
2 0.32 0.2 - - - - 11 0.23 0.32 2.3
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FIRST DIAPHRAGM MODE SHAPE
15 th Period TFIELD 0.16 Sec.
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Load-Dependent Ritz Vectors LDR Vectors - 1980
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DYNAMIC EQUILIBRIUM EQUATIONS M a C v
K u F(t) a Node Accelerations v
Node Velocities u Node
Displacements M Node Mass Matrix C
Damping Matrix K Stiffness Matrix F(t)
Time-Dependent Forces
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PROBLEM TO BE SOLVED
M a C v K u fi g(t)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
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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
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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
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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
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GENERATION OF LOAD DEPENDENT RITZ VECTORS
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
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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, K u f
C. MAKE u STIFFNESS AND MASS ORTHOGONAL TO
FORM FIRST BLOCK OF LDL VECTORS V 1 V1T
M V1 I
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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
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STEP 3. MAKE VECTORS STIFFNESS ORTHOGONAL
A. SOLVE Nb x Nb Eigenvalue Problem VT
K V Z w2 Z B. CALCULATE MASS AND
STIFFNESS ORTHOGONAL LDR VECTORS VR V Z


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DYNAMIC RESPONSE OF BEAM
100 pounds
10 AT 12" 240"
FORCE
TIME
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MAXIMUM DISPLACEMENT
Number of Vectors Eigen Vectors
Load Dependent Vectors
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)
( Error in Percent)
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MAXIMUM MOMENT
Number of Vectors Eigen Vectors
Load Dependent Vectors
1
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 )
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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
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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
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FAST NONLINEAR ANALYSIS
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
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BASE ISOLATION
Isolators
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BUILDING
IMPACT
ANALYSIS
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FRICTION DEVICE
CONCENTRATED DAMPER
NONLINEAR ELEMENT
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GAP ELEMENT
BRIDGE DECK ABUTMENT
TENSION ONLY ELEMENT
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P L A S T I C H I N G E S
2 ROTATIONAL DOF
ALSO DEGRADING STIFFNESS ARE Possible
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Mechanical Damper
F ku
F f (u,v,umax )
F C vN
Mathematical Model
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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
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103 FEET DIAMETER - 100 FEET HEIGHT
NONLINEAR DIAGONALS
BASE ISOLATION
ELEVATED WATER STORAGE TANK
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COMPUTER MODEL
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COMPUTER TIME REQUIREMENTS
PROGRAM
( 4300 Minutes )
ANSYS
INTEL 486
3 Days
ANSYS
CRAY
3 Hours
( 180 Minutes )
2 Minutes
SADSAP
INTEL 486
( B Array was 56 x 20 )
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Nonlinear Equilibrium Equations
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Summary Of FNA Method Calculate
Load-Dependant Ritz Vectors for Structure With
Nonlinear Elements Removed. These Vectors
Satisfy the Following Orthogonality Properties
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The Solution Is Assumed to Be a Linear
Combination of the LDR Vectors. Or, 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.
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A typical modal equation is uncoupled.
However, the modes are coupled by the unknown
nonlinear modal forces which are of the
following form The deformations in the
nonlinear elements can be calculated from the
following displacement transformation equation
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Since the
deformations in the nonlinear elements can be
expressed in terms of the modal response
by Where the size of the array is
equal to the number of deformations times the
number of LDR vectors. The array is
calculated only once prior to the start of
mode integration. THE ARRAY CAN BE
STORED IN RAM
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The nonlinear element forces are calculated,
for iteration i , at the end of each time step t
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FRAME WITH UPLIFTING ALLOWED
UPLIFTING ALLOWED
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Four Static Load Conditions Are Used To Start
The Generation of LDR Vectors
EQ DL Left
Right
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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
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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.
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A COMPLETE EIGENVECTOR SUBSPACE FOR THE LINEAR
AND NONLINEAR DYNAMIC ANALYSIS OF STRUCTURES
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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
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EXAMPLE OF SIX DEGREE OF FREEDOM SYSTEM
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How Do We Solve a System That Has Both Zero and
Infinite Frequencies?
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Solve Static and Dynamic Equilibrium Equations by
Mode Superposition
The Modal Equations Can Now Be Written As
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SOLUTION OF TYPICAL MODAL EQUATION
FOR DYNAMIC MODES, USE PIECE-WISE EXACT
SOLUTION FOR RIGID-BODY MODES, Direct
INTEGRATION FOR STATIC MODES, THE SOLUTION IS
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Calculation of Frequencies from Natural
Eigenvalues
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Eigenvalues for Simple Beam Mode Natural
Eigenvalue Frequency
Period 1 100 0 2 100
0 3 0.826 0.995 6.31 4 0 0 5 0
0 6 0 0
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CALCULATION OF NATURAL EIGENVECTORS
Use Recurrence Equation Of Following Form
The First Load Block Must Be The Static Load
Patterns Acting on The Structure
Subsequent Load Blocks Are Calculated From
Iteration Is Not Required
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The Natural Eigenvector Algorithm (2)
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
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The Natural Eigenvector Algorithm (3)
A Static Vector Has A Zero Eigenvalue And An
Infinite Frequency
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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
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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
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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
For earthquake loading, this is identical to the
mass participation factor in the three different
directions A minimum of 90 percent is recommended
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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,
Always equal to 1.0 for LDR vectors
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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.
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In a large earthquake the safest place to be is
on the top of a high-rise building Over 25
Stories
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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
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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
time
1957
1997
A FACTOR OF 10,000 REDUCTION IN 40 YEARS
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Floating-Point Speeds of Computer
Systems Definition of one Operation A B
CD 64 bits - REAL8
Year Computeror CPU OperationsPer 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
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Cost of Personal Computer Systems
YEAR CPU Speed MHz Operations Per Second Relative Speed COST
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
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Ed Wilson at UCLA Meet April 17, 1954 Ed set a
880 yard record of 1 Minute and 54 Seconds.
President Robert . Sproul
In the last 50 years, Ed is getting Slower and
Computer are getting Faster
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The Future Of Personal Computers Multi-Processors
Will Require New Numerical Methods and Modificatio
n of Existing Programs Speed and Accuracy are
Important