Earthquake Engineering Research at UC Berkeley

andRecent Developments at

CSI Berkeley BY

Ed WilsonProfessor Emeritus of Civil Engineering

University of California, Berkeley

October 24 - 25, 2008

Summary of Presentation1. 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

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Copy Papers and Slides

Early Finite Element Research at UC Berkeley by Ray Clough and Ed Wilson

 

The Development of Earthquake Engineering Software at Berkeleyby Ed Wilson - Slides

Dynamic Research at UC BerkeleyRetired 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 Washington1943 - 1946 U. S. Army Air Force1946 - 1949 MIT - D. Science - Bisplinghoff 1949 - 1986 Professor of CE U.C Berkeley1952 and 1953 Summer Work at BoeingNational Academy of EngineeringNational Academy of SciencePresidential 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 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

Professor Joe Penzien – First Director of EERC at UC Berkeley

The Franklin Institute Awards – April 27, 2006

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

New Printing of the Clough and Penzien Book

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

My Book – 23 Chapters

csiberkeley.comAsk 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 HughesProfessor, University of Texas

S A P

STRUCTURAL ANALYSIS

PROGRAM

ALSO A PERSON

“ Who Is Easily Deceived Or Fooled”

“ Who Unquestioningly Serves Another”

"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

From The Foreword Of The First SAP Manual

The SAP Series of Programs1969 - 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 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

FIELD MEASUREMENTS REQUIRED TO VERIFY

1. MODELING ASSUMPTIONS

2. SOIL-STRUCTURE MODEL

3. COMPUTER PROGRAM

4. COMPUTER USER

MECHANICAL VIBRATION DEVICES

CHECK OF RIGID DIAPHRAGM APPROXIMATION

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 0.32 0.32 0.2

- - - -

11 0.23 0.32 2.3

15 th Period

TFIELD = 0.16 Sec.

FIRST DIAPHRAGM MODE SHAPE

Load-Dependent Ritz Vectors

LDR Vectors - 1980

DYNAMIC EQUILIBRIUM EQUATIONS

M a + C v + K u = F(t)

a = Node Accelerationsv = Node Velocitiesu = Node DisplacementsM = Node Mass MatrixC = Damping MatrixK = Stiffness MatrixF(t) = Time-Dependent Forces

PROBLEM TO BE SOLVED

M a + C v + K u = fi g(t)i

For 3D Earthquake Loading

THE OBJECTIVE OF THE ANALYSISIS TO SOLVE FOR ACCURATE

DISPLACEMENTS and MEMBER FORCES

= - Mx ax - My ay - Mz az

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 FREQUENCYDOMAIN 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 ProblemsCalculate Member Forces For Each Time Step and Iterate for Equilibrium - Brute Force Method

MODE SUPERPOSITION METHOD1. Generate Orthogonal Dependent

Vectors And Frequencies

2. Form Uncoupled Modal EquationsAnd Solve Using An Exact MethodFor Each Time Increment.

3. Recover Node DisplacementsAs a Function of Time

4. Calculate Member Forces As a Function of Time

GENERATION OF LOAD

DEPENDENT RITZ VECTORS1. 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,

K u = f

C. MAKE u STIFFNESS AND MASS ORTHOGONAL TO

FORM 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, toMake Block of Vectors, Yi , Orthogonalto all Previously Calculated Vectors - Vi

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 =

10 AT 12" = 240"

100 pounds

FORCE

TIME

DYNAMIC RESPONSE OF BEAM

MAXIMUM DISPLACEMENTNumber 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)

MAXIMUM MOMENTNumber 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 )

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 WITHNONLINEAR ELEMENTS REMOVED ANDDUMMY ELEMENTS ADDED FOR STABILITY

2. SOLVE ALL MODAL EQUATIONS WITH

NONLINEAR FORCES ON THE RIGHT HAND SIDE

USE EXACT INTEGRATION WITHIN EACH TIME STEP

4. FORCE AND ENERGY EQUILIBRIUM ARESTATISFIED AT EACH TIME STEP BY ITERATION

3.

Isolators

BASE ISOLATION

BUILDINGIMPACTANALYSIS

FRICTIONDEVICE

CONCENTRATEDDAMPER

NONLINEARELEMENT

GAP ELEMENT

TENSION ONLY ELEMENT

BRIDGE DECK ABUTMENT

P L A S T I CH I N G E S

2 ROTATIONAL DOF

ALSO DEGRADING STIFFNESS ARE Possible

Mechanical Damper

Mathematical Model

F = C vN

F = kuF = f (u,v,umax )

LINEAR VISCOUS DAMPINGDOES NOT EXIST IN NORMAL STRUCTURESAND FOUNDATIONS

5 OR 10 PERCENT MODAL DAMPING VALUES ARE OFTEN USED TO JUSTIFY ENERGY DISSIPATION DUE TO NONLINEAR EFFECTS

IF ENERGY DISSIPATION DEVICES ARE USEDTHEN 1 PERCENT MODAL DAMPING SHOULD BE USED FOR THE ELASTIC PART OF THE STRUCTURE - CHECK ENERGY PLOTS

103 FEET DIAMETER - 100 FEET HEIGHT

ELEVATED WATER STORAGE TANK

NONLINEAR DIAGONALS

BASEISOLATION

COMPUTER MODEL

92 NODES

103 ELASTIC FRAME ELEMENTS

56 NONLINEAR DIAGONAL ELEMENTS

600 TIME STEPS @ 0.02 Seconds

COMPUTER TIME REQUIREMENTS

PROGRAM

( 4300 Minutes )ANSYS INTEL 486

3 Days

ANSYS CRAY 3 Hours ( 180 Minutes )

SADSAP INTEL 486 2 Minutes

( B Array was 56 x 20 )

Nonlinear Equilibrium Equations

Summary Of FNA Method

Calculate Load-Dependant Ritz Vectors for Structure With Nonlinear Elements Removed.

These Vectors Satisfy the FollowingOrthogonality Properties

2 KT T M I

The Solution Is Assumed to Be a Linear Combination of the LDR Vectors. Or,

Which Is the Standard Mode Superposition Equation

n

nn tytYtu )()()(

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 theunknown nonlinear modal forces whichare of the following form:

The deformations in the nonlinear elementscan be calculated from the followingdisplacement transformation equation:

f Fn n n

Au

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

)()( tYtu

( ) ( ) ( )t A Y t B Y t

B

B

B

The nonlinear element forces are calculated, for iteration i , at the end

of each time step t

Equation Modal of SolutionNew

Loads Modal Nonlinear

History Element of Function

Elements Nonlinear

in nsDeformatio

)(

)(t

T)(N

)(

)(t

Y

YBf

P

BY

1

)(

it

ii

it

iit

t

FRAME WITH UPLIFTING ALLOWED

UPLIFTING ALLOWED

Four Static Load Conditions Are Used To Start The

Generation of LDR Vectors

EQ DL Left Right

TIME - Seconds

DEAD LOAD

LATERAL LOAD

LOAD

0 1.0 2.0 3.0 4.0 5.0

NONLINEAR STATIC ANALYSIS50 STEPS AT dT = 0.10 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

11 0 T

( a ) B e a m M o d e l

( b ) R i g i d B o d y M o d e

( c ) R i g i d B o d y M o d e

( d ) D y n a m i c M o d e

( e ) S t a t i c M o d e

( f ) S t a t i c M o d e

( g ) S t a t i c M o d e

1 0 0 1 0 0

I = 1 . 0 E = 1 0 , 0 0 0

M = 0 . 0 5 M = 0 . 1 0 M = 0 . 0 5

22 0 T

31.6995.0 33 T

044 T

055 T

066 T

EXAMPLE OF SIX DEGREE OF FREEDOM SYSTEM

How Do We Solve a System That Has Both Zero and Infinite Frequencies?

definite positive and rnonsingula now is where

Or,

Equation of SidesBoth To Add

definite - semipositive

and singularbemay K and Mboth where

Equation mEquilibriu Dynamic and Static

K

Mu(t)R(t)u(t)K(t)uM

Mu(t)R(t)u(t)MK(t)uM

Mu(t)

FG(t)R(t)Ku(t)(t)uM

][

Solve Static and Dynamic Equilibrium Equations by Mode Superposition

ΨMΦΦ and IΦKΦ

that sodNormallize are and Orthogonal Mass

and Stiffnessare ModesAll Modes.Dynamic

and StaticBody,-Rigid Contain Can Φ Where

(t)YΦ(t)u and ΦY(t)u(t) Let

TT

The Modal Equations Can Now Be Written As

R(t)ΦY(t)ΨI(t)YΨ T ][ ρ

SOLUTION OF TYPICAL MODAL EQUATION

)( =][1 + t(t)Y(t) Y Tnnnnn Fg

FOR DYNAMIC MODES, USE PIECE-WISE EXACT SOLUTION

FOR RIGID-BODY MODES, Direct INTEGRATION

FOR STATIC MODES, THE SOLUTION IS

)( = t(t)Y Tnn Fg

)( = t(t)Y Tnn Fg

Calculation of Frequencies from Natural Eigenvalues

nn

nn T

2

and 1

Eigenvalues for Simple Beam

Mode Natural Eigenvalue Frequency Period

1 100 02 100 03 0.826 0.995 6.314 0 05 0 06 0 0

n 2

nnT

1

n

n

01.0

CALCULATION OF NATURAL EIGENVECTORS

MΦΦKΨ Use Recurrence Equation Of Following Form

Blocks ,1 )1()( N iii FVKThe First Load Block Must Be The Static Load Patterns Acting on The Structure

FF )0( Subsequent Load Blocks Are Calculated From

)1()( ii MVFIteration Is Not Required

The Natural Eigenvector Algorithm (2)

:tsRequiremen Following TheSatisfy To Modified

beMust Block in Vectors Candidate All (i)V

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

as Designated isBlock Vectors New The (i)V

The Natural Eigenvector Algorithm (3)

VZ

ZM

V

Where

:Method Jocobi TheBy Probem Eigenvalue

Subspace Following The SolvingBy

Orthogomal Mass MadeThen Are Vectors These

by Defined Are Vectors Candidate All

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

r j

Tj

N

nj

Tnn

dj fMf

f

11

2)(

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,

r j

Tj

N

nj

Tn

sj fu

f 1

2)(

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 PushoverAnalysis 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

COMPUTERS1958 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

1957 1997time

S = MONTHLY SALARY OF ENGINEER

C/S = 5,000

C/S = 0.5

A FACTOR OF 10,000 REDUCTION IN 40 YEARS

$

$1,500

$7,500

$4,000,000

$800

Floating-Point Speeds of Computer SystemsDefinition of one Operation A = B + C*D 64 bits - REAL*8

YearComputer

or CPUOperationsPer 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

YEAR CPUSpeed 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

Cost of Personal Computer Systems

Ed Wilson at

UCLA Meet

April 17, 1954

Ed set a 880 yard record of 1 Minute and 54 Seconds.

PresidentRobert .Sproul

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