Transcript Slide 1
PAST, PRESENT AND FUTURE OF EARTHQUAKE ANALYSIS OF STRUCTURES By Ed Wilson Draft dated 8/15/14 September 22 and 24 2014 SEAONC Lectures
http://edwilson.org/History/Slides/Past%20Present%20Future%202014.ppt
1964 Gene’s Comment – a true story
.
Ed developed a new program for the Analysis of Complex Rockets Ed talks to Gene ----- Two weeks later Gene calls Ed ----- Ed goes to see Gene ---- The next day, Gene calls Ed and tells him
“Ed, why did you not tell me about this program.
It is the greatest program I ever used.”
Summary of Lecture Topics
1. Fundamental Principles of Mechanics and Nature 2. Example of the Present Problem – Caltrans Criteria 3. The Response Spectrum Method 4. Demand Capacity Calculations 5. Speed of Computers– The Last Fifty Years 6. Terms I do not Understand 7. The Load Dependant Ritz Vectors - LDR Vectors 8. The Fast Nonlinear Analysis Method – FNA Method 9. Recommendations
edwilson.org
10. Questions
Fundamental Equations of Structural Analysis 1. Equilibrium
- Including Inertia Forces -
Must be Satisfied 2. Material Properties
or Stress / Strain or Force / Deformation
3. Displacement Compatibility
Or Equations or Geometry
Methods of Analysis 1. Force – Good for approximate hand methods 2. Displacement - 99 % of programs use this method 3. Mixed
Beam Ex. Plane Sections & V = dM/dz
Check Conservation of Energy
My First Earthquake Engineering Paper October 1-5 1962
THE PRESENT SAB Meeting on August 28, 2013
Comments on the Response Spectrum Analysis Method
As Used in the
CALTRANS SEISMIC DESIGN CRITERIA
Topics 1. Why do most Engineers have Trouble with Dynamics?
Taught by people who love math – No physical examples 2 Who invented the Response Spectrum Method?
Ray Clough and I did ? – by putting it into my computer program 3 Application by CalTrans to “Ordinary Standard Structures” Why 30 ? Why reference to Transverse & Longitudinal directions 4 Physical behavior of Skew Bridges – Failure Mode 5 Equal Displacement Rule?
6 Quote from George W. Housner
Who Developed the Approximate Response Spectrum Method of Seismic Analysis of Bridges and other Structures?
1. Fifty years ago there were only digital acceleration records for 3 earthquakes
.
2. Building codes gave design spectra for a one degree of freedom systems with no guidance of how to combine the response of of the higher modes.
3. At the suggestion of Ray Clough, I programmed the square root of the sum of the square of the modal values for displacements and member forces. However, I required the user to manually combine the results from the two orthogonal spectra. Users demanded that I modify my programs to automatically combine the two directions. I refused because there was no theoretical justification.
4. The user then modify my programs by using the 100%+30% or 100%+40% rules.
5. Starting in 1981 Der Kiureghian and I published papers showing that the CQC method should be used be used for combining modal responses for each spectrum and the two orthogonal spectra be combined by the SRSS method. 6. We now have Thousands or of 3D earthquake records from hundred of seismic events. Therefore, why not use Nonlinear Time-History Analyses that SATISFIES FORCE EQUILIBREUM.
If Equal Spectra are applied to any Global X–Y-Z System
Base Shear Mode 2 Y X Torsion or Mode 1, 2 or Mode 3 Base Shear Mode 1
Member Forces are the same for all Global X–Y Systems, If Calculated from F i
F iX
F iY
F iZ
Nonlinear Failure Mode For Skew Bridges
Tensional Failure F(t) Abutment Force Acting on Bridge u(t)
Contact at Right Abutment
u(t) Tensional Failure F(t) Abutment Force Acting on Bridge
Contact at Left Abutment
Possible Torsional Failure Mode
Design Joint Connectors for Joint Shear Forces?
Use a Global Modal for all Analyses
+Y -X +X -Y
Figure 5.2.2-1 EDA Modeling Techniques Use Global X-Y-Z System for all Models
Seismic Analysis Advice
by Ed Wilson
1. All Bridges are Three-Dimensional and their Dynamic Behavior is governed by the Mass and Stiffness Properties of the structure. The Longitudinal and Transverse directions are geometric properties. All Structures have Torsional Modes of Vibrations.
2. The Response Spectrum Analysis Method is a very
approximate
method of seismic analysis which only produces positive values of displacements and member forces which are not in equilibrium.
Demand / Capacity Ratios have Very Large Errors
3. A structural engineer may take several days to prepare and verify a linear SAP2000 model of an Ordinary Standard Bridge. It would take less than a day to add Nonlinear Gap Elements to model the joints. If a family of 3D earthquake motions are specified, the program will automatically summarize the maximum demand-capacity ratios and the time they occur in a few minutes of computer time.
Convince Yourself with a simple test problem 1. Select an existing Sap 2000 model of a Ordinary Standard Bridge with several different spans – both straight and curved.
2. Select one earthquake ground acceleration record to be used as the input loading which is approximately 20 seconds long. 3. Create a spectrum from the selected earthquake ground acceleration record.
4. Using a number of modes that captures a least 90 percent of the mass in all three directions. 5. At a 45 degree angle, Run a Linear Time History Analysis and a Response Spectrum Analysis.
6. Compare Demand Capacity Ratios for both SAP 2000 analysis for all members. 7. You decide if the Approximate RSA results are in good agreement with the Linear time History Results
.
Educational Priorities of an Old Professor on Seismic Analysis of Structures Convince Engineers that the Response Spectrum Method Produces very Poor Results
1. Method is only exact for single degree of freedom systems 2. It produces only positive numbers for Displacements and Member Forces. 3. Results are maximum probable values and occur at an “Unknown Time” 4. Short and Long Duration earthquakes are treated the same using “Design Spectra” 5. Demand/Capacity Ratios are always “Over Conservative” for most Members. 6. The Engineer does not gain insight into the “Dynamic Behavior of the Structure” Results are not in equilibrium. More modes and 3D analysis will cause more errors.
7. Nonlinear Spectra Analysis is “Smoke and Mirrors” – Forget it
Convince Engineers that it is easy to conduct “Linear Dynamic Response Analysis” It is a simple extension of Static Analysis – just add mass and time dependent loads 1. Static and Dynamic Equilibrium is satisfied at all points in time if all modes are included 2. Errors in the results can be estimated automatically if modes are truncated 3. Time-dependent plots and animation are impressive and fun to produce 4. Capacity/Demand Ratios are accurate and a function of time – summarized by program. 5. Engineers can gain great insight into the dynamic response of the structure and may help in the redesign of the structural system
.
Terminology commonly used in nonlinear analysis that do not have a unique definition 1.Equal Displacement Rule – can you prove it?
2.Pushover Analysis 3.Equivalent Linear Damping 4. Equivalent Static Analysis 5.Nonlinear Spectrum Analysis 6.Onerous Response History Analysis
Equal Displacement Rule
In 1960 Veletsos and Newmark proposed in a paper presented at the 2nd WCEE For a one DOF System, subjected to the El Centro Earthquake, the Maximum Displacement was approximately the same for both linear and nonlinear analyses. In 1965 Clough and Wilson, at the 3rd WCEE, proved the Equal Displacement Rule did not apply to multi DOF structures.
http://edwilson.org/History/Pushover.pdf
1965 Professor Clough’s Comment
. “
If tall buildings are designed for elastic column behavior and restrict the nonlinear bending behavior to the girders, it appears the danger of total collapse of the building is reduced.” This indicates the strong-column and week beam design is the one of the first statements on
Performance Based Design
The Response Spectrum Method Basic Assumptions
I do not know who first called it a “response spectrum,” but unfortunate the term leads people to think that the characterize the building’s motion, rather than the ground’s motion.
George W. Housner EERI Oral History, 1996
Typical Earthquake Ground Acceleration – percent of gravity
Integration will produce Earthquake Ground Displacement – inches These real Eq. Displacement can be used as Computer Input
Relative Displacement Spectrum for a unit mass with different periods
1. These displacements Y max are maximum (+ or -) values versus period for a structure or mode.
2. Note: we do not know the time these maximum took place.
Pseudo Acceleration Spectrum
Note: S = w 2 Y max has the same properties as the Displacement Spectrum. Therefore, how can anyone justify combining values, which occur at different times, and expect to obtain accurate results.
CASE CLOSED
General Horizontal Response Spectrum from ASCE 41- 06 Simple User Defined Data
effective viscous ratio S xs
peak base accelerati on S x 1
value at 1 .
0 sec period
Constants to be Calculated
B 1
4 [ 5 T s T
0
S x 1 / 0 .
2 T s .
6 S
sx ln( 100
)
Duration has been Neglected Shape is not Realistic NonLinear Structures No Where did the hat go?
Where did the Hat go - on the Response Spectrum ? As I Recall -------
Demand-Capacity Ratios
The Demand-Capacity ratio for a linear elastic, compression member is given by an equation of the following general form: R ( t )
P ( t
c P cr )
b M M 2 ( t ) C 2 P ( t ) c 2 ( 1
P e 2 )
b M M 3 ( t ) C 3 P ( t ) c 3 ( 1
P e 3 )
1 .
0 If the axial force and the two moments are a function of time, the Demand-Capacity ratio will be a function of time and a smart computer program will produce R(max) and the time it occurred. A smart engineer will hand check several of these values.
RSM Demand-Capacity Ratios
If the axial force and the two moments are produced by the Response Spectrum Method the Demand-Capacity ratio may be computed by an equation of the following general form: R (max)
P (max)
c P cr
b M M 2 (max) C 2 c 2 ( 1
P (max) P e 2 )
b M 3 (max) C 3 P (max) M c 3 ( 1
P e 3 )
1 .
0 A smart computer program can compute this Demand Capacity Ratio. However, only an idiot would believe it.
SPEED and COST of COMPUTERS 1957 to 2014 to the Cloud You can now buy a very powerful small computer for less than $1,000 However, it may cost you several thousand dollars of your time to learn how to use all the new options.
If it has a new operating system
1957 My First Computer in Cory Hall IBM 701 Vacuum Tube Digital Computer Could solve 40 equations in 30 minutes
1981 My First Computer Assembled at Home
Paid $6000 for a 8 bit CPM Operating System with FORTRAN. Used it to move programs from the CDC 6400 to the VAX on Campus. Developed a new program called SAP 80 without using any Statements from previous versions of SAP.
After two years, system became obsolete when IBM released DOS with a floating point chip.
In 1984, CSI developed Graphics and Design Post-Processor and started distribution of the Professional Version of Sap 80
Floating-Point Speeds of Computer Systems
Definition of one Operation A = B + C*D 64 bits - REAL*8
Year
1962 1964 1974 1981 1981 1994 1995 1995 1998 1999
Computer or CPU
CDC-6400 CDC-6600 CRAY-1 IBM-3090 CRAY-XMP Pentium-90 Pentium-133 DEC-5000 upgrade Pentium II - 333 Pentium III - 450
Operations Per Second
50,000 100,000 3,000,000 20,000,000 40,000,000 3,500,000 5,200,000 14,000,000 37,500,000 69,000,000
Relative Speed
1 2 60 400 800 70 104 280 750 1,380
YEAR 1980 1984 1988 1991 1994 1996 1997 1998 1999 2003 2006
Cost of Personal Computer Systems
CPU 8080 8087 80387 80486 80486 Pentium Pentium II Pentium II Pentium III Pentium IV AMD - Athlon 20 33 66 233 233 333 450 2000 2000 Speed MHz 4 10 Operations Per Second 200 13,000 Relative Speed 1 65 93,000 605,000 1,210,000 10,300,000 465 3,025 6,050 52,000 11,500,000 58,000 37,500,000 198,000 69,000,000 345,000 220,000,000 1.100,000 440,000,000 2,200,000 COST $6,000 $2,500 $8,000 $10,000 $5,000 $4,000 $3,000 $2,500 $1,500 $2.000
$950
Year 1962
1974 1981 1994 1999 2006 2009
2010
2013
Computer or CPU CDC-6400
CRAY-1 VAX or Prime Pentium-90 Intel Pentium III-450 AMD 64 Laptop Min Laptop 2.4 GHz Intel Core i3 64 bit Win 7 Laptop 2.80 GHz 2 Quad Core 64 bit Win 7
Cost $1,000,000
$4,000,000 $100,000 $5,000 $1,500
Operations Per Second 50,000 Relative Speed 1
3,000,000 60 100,000 2 4,000,000 70 69,000,000 1,380 $2,000 400,000,000 8,000 $300 200,000,000 4,000
$1,000
1.35 Billion Intel Fortran 27,000 $1,000 2.80 Billion Parallelized Fortran 56,000
The cost of one operation has been reduced by 56,000,000 in the last 50 years
Computer Cost versus Engineer’s Monthly Salary
$1,000,000 c / s = 1,000 c / s = 0.10
$10,000 $1,000 $1,000 1963 Time 2013
Fast Nonlinear Analysis
With Emphasis On Earthquake Engineering
BY
Ed Wilson
Professor Emeritus of Civil Engineering University of California, Berkeley edwilson.org
May 25, 2006
1.
Summary Of Presentation
History of the Finite Element Method
2.
History Of The Development of SAP 3.
4.
5.
Computer Hardware Developments Methods For Linear and Nonlinear Analysis Generation And Use Of
LDR
Fast Nonlinear Analysis Vectors and
FNA
Method 6. Example Of Parallel Engineering Analysis of the Richmond - San Rafael Bridge
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 Bathe 1973 – 74 NON SAP New Program – The Start of ADINA 1979 – 80 SAP 80 New Linear Program for Personal Computers Lost All Research and Development Funding 1983 – 1987 SAP 80 CSI added Pre and Post Processing 1987 - 1990 SAP 90 1997 – Present SAP 2000 Significant Modification and Documentation Nonlinear Elements – More Options – With Windows Interface
Load-Dependent Ritz Vectors LDR Vectors – 1980 - 2000
MOTAVATION – 3D Reactor on Soft Foundation Dynamic Analysis - 1979 by Bechtel using SAP IV
200 Exact Eigenvalues were Calculated and all of the Modes were in the foundation – No Stresses in the Reactor.
The cost for the analysis on the CLAY Computer was
$10,000
3 D Concrete Reactor 3 D Soft Soil Elements 360 degrees
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
M a + C v + K u = f i g(t) i = - M x a x - M y a y - M z a z 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
USE OF MODE SUPERPOSITION WITH EIGEN OR LOAD-DEPENDENT RITZ VECTORS FOR FNA
For Linear Systems Only
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
1.
G
ENERATION OF
L
OAD
D
EPENDENT RITZ
V
ECTORS
Approximately Three Times Faster Than The Calculation Of Exact Eigenvectors 2.
3.
4.
Results In Improved Accuracy Using A Smaller Number Of LDR Vectors Computer Storage Requirements Reduced Can Be Used For Nonlinear Analysis To Capture Local Static Response
STEP 1. INITIAL CALCULATION
A.
TRIANGULARIZE STIFFNESS MATRIX B.
C.
DUE TO A BLOCK OF STATIC LOAD VECTORS, f, SOLVE FOR A BLOCK OF DISPLACEMENTS, u, K u = f MAKE u STIFFNESS AND MASS ORTHOGONAL TO FORM FIRST BLOCK OF LDL VECTORS
V 1 V 1 T M V 1 = I
STEP 2. VECTOR GENERATION
i = 2 . . . . N Blocks
A.
Solve for Block of Vectors, K X i = M V i-1 B.
Make Vector Block, X i Mass Orthogonal - Y i , Stiffness and C.
Use Modified Gram-Schmidt, Twice, to Make Block of Vectors, Y i , Orthogonal to all Previously Calculated Vectors - V i
STEP 3. MAKE VECTORS STIFFNESS ORTHOGONAL A.
SOLVE Nb x Nb Eigenvalue Problem [ V T K V ] Z = [
w 2
] Z B.
CALCULATE MASS AND STIFFNESS ORTHOGONAL LDR VECTORS
V R
= V Z =
DYNAMIC RESPONSE OF BEAM
100 pounds 10 AT 12" = 120" FORCE = Step Function TIME
MAXIMUM DISPLACEMENT
Number of Vectors Eigen Vectors Load Dependent Vectors 1 0.004572
(-2.41) 0.004726
(+0.88) 2 3 4 0.004572
0.004664
0.004664
(-2.41) (-0.46) (-0.46) 0.004591
0.004689
0.004685
( -2.00) (+0.08) (+0.06) 5 7 9 0.004681
(-0.08) 0.004685
( 0.00) 0.004683
(-0.04) 0.004685
(0.00) ( Error in Percent)
MAXIMUM MOMENT
Number of Vectors Eigen Vectors Load Dependent Vectors 1 4178 ( - 22.8 %) 5907 ( + 9.2 ) 2 3 4178 4946 ( - 22.8 ) ( - 8.5 ) 5563 5603 ( + 2.8 ) ( + 3.5 ) 4 5 7 9 4946 5188 5304 5411 ( - 8.5 ) ( - 4.1 ) 5507 5411 ( + 1.8) ( 0.0 ) ( - .0 ) ( 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
F
AST
N
ONLINEAR
A
NALYSIS
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
P L A S T I C H I N G E S
2 ROTATIONAL DOF
DEGRADING STIFFNESS ?
Mechanical Damper
F = f (u,v,u max ) F = ku F = C v N
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 ( 4300 Minutes ) ANSYS CRAY SADSAP INTEL 486 ( B Array was 56 x 20 ) 3 Hours ( 180 Minutes ) (2 Minutes )
Nonlinear Equilibrium Equations
Nonlinear Equilibrium Equations
Summary Of FNA Method
1.
Calculate LDR Vectors for Structure With the Nonlinear Elements Removed
.
2.
These Vectors Satisfy the Following Orthogonality Properties
T
K
2
T
M
I
3. The Solution Is Assumed to Be a Linear Combination of the LDR Vectors. Or ,
u
(
t
)
Y
(
t
)
n y
(
t
Which Is the Standard
n
)
n
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
.
4.
A typical modal equation is uncoupled. However, the modes are coupled by the unknown nonlinear modal forces which are of the following form:
f
n
n F n
5. The deformations in the nonlinear elements can be calculated from the following displacement transformation equation:
A u
6.
the nonlinear elements can be expressed in terms of the modal response by
A
the number of deformations times the number of LDR vectors.
to the start of mode integration.
7.
7. The nonlinear element forces are calculated, for iteration
i
, at the end of each time step
t ( i )
BY t (
i
)
Deformatio ns in Nonlinear Elements
P
t
(
i
)
Function of Element History
(
i
) f N
t
B T Y t (
i
)
Nonlinear Modal Loads
Y
t
(
i
1
)
New Solution of Modal Equation
8.
Calculate error for iteration i , at the end of each time step, for the N Nonlinear elements – given Tol
If If Err = N
n =1 | Err f (t
Tol ) i n N
n =1 | f | N
n =1 (t | ) i n | f (t Continue to ) i n 1 | iterate with i
i
1 Err
Tol go to next time step with t
t
t
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 LOAD DEAD 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.
PARALLEL ENGINEERING AND PARALLEL COMPUTERS
ONE PROCESSOR ASSIGNED TO EACH JOINT 3 2 1 3 1 2 ONE PROCESSOR ASSIGNEDTO EACH MEMBER
PARALLEL STRUCTURAL ANALYSIS DIVIDE STRUCTURE INTO "N" DOMAINS FORM ELEMENT STIFFNESS IN PARALLEL FOR "N" SUBSTRUCTURES FORM AND SOLVE EQUILIBRIUM EQ.
EVALUATE ELEMENT FORCES IN PARALLEL IN "N" SUBSTRUCTURES TYPICAL COMPUTER NONLINEAR LOOP
FIRST PRACTICAL APPLICTION OF THE FNA METHOD
Retrofit of the RICHMOND - SAN RAFAEL BRIDGE 1997 to 2000 Using SADSAP
S A D S A P
T A T I C N D Y N A M I C T R U C T U R A L N A L Y S I S R O G R A M
TYPICAL ANCHOR PIER
MULTISUPPORT ANALYSIS
( Displacements ) ANCHOR PIERS RITZ VECTOR LOAD PATTERNS
SUBSTRUCTURE PHYSICS
Stiffness Matrix Size = 3 x 16 = 48 "a" MASSLESS JOINT ( Eliminated DOF ) "b" MASS POINTS and JOINT REACTIONS ( Retained DOF )
SUBSTRUCTURE STIFFNESS
k k a a b a k a b k b b
REDUCE IN SIZE BY LUMPING MASSES OR BY ADDING INTERNAL MODES
ADVANTAGES IN THE USE OF SUBSTRUCTURES 1.
FORM OF MESH GENERATION 2.
LOGICAL SUBDIVISION OF WORK 3.
MANY SHORT COMPUTER RUNS 4.
RERUN ONLY SUBSTRUCTURES WHICH WERE REDESIGNED 5.
PARALLEL POST PROCESSING USING NETWORKING
ECCENTRICALLY BRACED FRAME
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 1 2 3 4 5 6 7 11 T FIELD 1.77 Sec.
1.69
1.68
0.60
0.60
0.59
0.32
0.23
T ANALYSIS 1.78 Sec.
1.68
1.68
0.61
0.61
0.59
0.32
0.32
Diff. - % 0.5
0.6
0.0
0.9
0.9
0.8
0.2
2.3
FIRST DIAPHRAGM MODE SHAPE
15 th Period T FIELD = 0.16 Sec .
At the Present Time Most Laptop Computers Can be directly connected to a 3D Acceleration Seismic Box Therefore, Every Earthquake Engineer C an verify Computed Frequencies If software has been developed
Final Remark
Geotechnical Engineers must produce realistic Earthquake records for use by Structural Engineers
Errors Associated with the use of Relative Displacements Compared with the use of real Physical Earthquake Displacements Classical Viscous Damping does not exist In the Real Physical World
Comparison of Relative and Absolute Displacement Seismic Analysis
z
Properties:
Thickness = 2.0 ft Width =20.0 ft
Typical Story Load M
b
(
t
)
I = 27,648,000 in 4 E = 4,000 ksi W = 20 kips /story M x = 20/g = 0.05176 kip-sec 2 /in M yy = 517.6 kip-sec 2 -in Total Mass = 400 /g Typical Story Height h = 15 ft = 180 in.
x
A. 20 Story Shear Wall With Story Mass First Story Load First Story Moment
b
(
t
) B. Base Acceleration Loads Relative Formulation 12
EI h
3
u b
(
t
) 6
EI h
2
u b
(
t
) C. Displacement Loads Absolute Formulation
Shear at Second Level Vs. Time With Zero Damping Time Step = 0.01
140 120 Linear Acceleration Loads, or Cubic Displacement Loads - Zero Damping - 40 Modes Linear Displacement Loads - Zero Damping - 40 Modes 100 80 60 40 20 0 -20 -40 -60 -80 -100 0.00
0.20
0.40
0.60
0.80
1.00
1.20
TIME - Seconds
1.40
1.60
1.80
2.00
Illustration of Mass-Proportional Component in Classical Damping .
u r u s m
x
C ss I x
u
x
0 C ss I x
u
x
0
u x u s
u x
u r
RELATIVE DISPLACEMENT FORMULATION ABSOLUTE DISPLACEMENT FORMULATION