1. PAST, PRESENT AND FUTURE OF EARTHQUAKE ANALYSIS OF STRUCTURES By Ed Wilson Draft dated 8/15/14 September 22 and SEAONC Lectures

2. 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.”

3. Summary of Lecture Topics
Fundamental Principles of Mechanics and Nature
Example of the Present Problem – Caltrans Criteria
The Response Spectrum Method
Demand Capacity Calculations
Speed of Computers– The Last Fifty Years
Terms I do not Understand
The Load Dependant Ritz Vectors - LDR Vectors
The Fast Nonlinear Analysis Method – FNA Method
Recommendations edwilson.org
Questions

4. Fundamental Equations of Structural Analysis
Equilibrium - Including Inertia Forces - Must be Satisfied
Material Properties or Stress / Strain or Force / Deformation
Displacement Compatibility Or Equations or Geometry
Methods of Analysis
Force – Good for approximate hand methods
Displacement % of programs use this method
Mixed - Beam Ex. Plane Sections & V = dM/dz
Check Conservation of Energy

5. My First Earthquake Engineering Paper
October

6. THE PRESENT SAB Meeting on August 28, Comments on the Response Spectrum Analysis Method As Used in the CALTRANS SEISMIC DESIGN CRITERIA

7. Topics Why do most Engineers have Trouble with Dynamics?
Taught by people who love math – No physical examples
Who invented the Response Spectrum Method?
Ray Clough and I did ? – by putting it into my computer program
Application by CalTrans to “Ordinary Standard Structures”
Why 30 ? Why reference to Transverse & Longitudinal directions
Physical behavior of Skew Bridges – Failure Mode
Equal Displacement Rule?
6 Quote from George W. Housner

8. 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.
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.
The user then modify my programs by using the 100%+30% or 100%+40% rules.
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.
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.

9. Torsion or Mode 1, 2 or Mode 3

10. Nonlinear Failure Mode For Skew Bridges
F(t)
F(t)
Abutment Force Acting on Bridge
Tensional Failure
u(t)
Contact at Right Abutment
u(t)
Tensional Failure
F(t)
Abutment Force Acting on Bridge
Contact at Left Abutment

11. Possible Torsional Failure Mode
Design Joint Connectors for Joint Shear Forces?

12. Use a Global Modal for all Analyses

13. Seismic Analysis Advice by Ed Wilson
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.
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
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.

14. Convince Yourself with a simple test problem
Select an existing Sap 2000 model of a Ordinary Standard Bridge with several different spans – both straight and curved.
Select one earthquake ground acceleration record to be used as the input loading which is approximately 20 seconds long.
Create a spectrum from the selected earthquake ground acceleration record.
Using a number of modes that captures a least 90 percent of the mass in all three directions.
At a 45 degree angle, Run a Linear Time History Analysis and a Response Spectrum Analysis.
Compare Demand Capacity Ratios for both SAP 2000 analysis for all members.
You decide if the Approximate RSA results are in good agreement with the Linear time History Results.

15. Educational Priorities of an Old Professor
on Seismic Analysis of Structures
Convince Engineers that the Response Spectrum Method
Produces very Poor Results
Method is only exact for single degree of freedom systems
It produces only positive numbers for Displacements and Member Forces.
Results are maximum probable values and occur at an “Unknown Time”
Short and Long Duration earthquakes are treated the same using “Design Spectra”
Demand/Capacity Ratios are always “Over Conservative” for most Members.
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.
Nonlinear Spectra Analysis is “Smoke and Mirrors” – Forget it

16. 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
Static and Dynamic Equilibrium is satisfied at all points in time if all modes are included
Errors in the results can be estimated automatically if modes are truncated
Time-dependent plots and animation are impressive and fun to produce
Capacity/Demand Ratios are accurate and a function of time – summarized by program.
Engineers can gain great insight into the dynamic response of the structure and may help in the redesign of the structural system.

17. Terminology commonly used in nonlinear analysis that do not have a unique definition
Equal Displacement Rule – can you prove it?
Pushover Analysis
Equivalent Linear Damping
Equivalent Static Analysis
Nonlinear Spectrum Analysis
Onerous Response History Analysis

18. 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.

19. 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

20. 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

21. Typical Earthquake Ground Acceleration – percent of gravity

22. Integration will produce Earthquake Ground Displacement – inches
These real Eq. Displacement can be used as Computer Input

23. Relative Displacement Spectrum for a unit mass with different periods
These displacements Ymax are maximum (+ or -) values versus period for a structure or mode.
Note: we do not know the time these maximum took place.
Pseudo Acceleration Spectrum
Note: S = w2 Ymax 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

24. General Horizontal Response Spectrum
from ASCE

25. Where did the Hat go - on the Response Spectrum ?
As I Recall

26. Demand-Capacity Ratios
The Demand-Capacity ratio for a linear elastic, compression member is given by an equation of the following general form:
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.

27. 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:
A smart computer program can compute this Demand-Capacity Ratio. However, only an idiot would believe it.

28. 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

29. 1957 My First Computer in Cory Hall
IBM 701 Vacuum Tube Digital Computer
Could solve 40 equations in 30 minutes

30. 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

31. 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
1962
CDC-6400
50,000 1
1964
CDC-6600
100,000 2
1974
CRAY-1
3,000,000 60
1981
IBM-3090
20,000,000
400
CRAY-XMP
40,000,000
800
1994
Pentium-90
3,500,000 70
1995
Pentium-133
5,200,000
104
DEC-5000 upgrade
14,000,000
280
1998
Pentium II - 333
37,500,000
750
1999
Pentium III - 450
69,000,000
1,380

32. 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 66
1,210,000
6,050
$5,000
1996
Pentium
233
10,300,000
52,000
$4,000
1997
Pentium II
11,500,000
58,000
$3,000
1998
333
37,500,000
198,000
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
440,000,000
2,200,000
$950

34. 2.4 GHz Intel Core i3 64 bit Win 7 Laptop
Year
Computer or CPU
Cost
Operations Per Second
Relative Speed
1962
CDC-6400
$1,000,000
50,000 1
1974
CRAY-1
$4,000,000
3,000,000 60
1981
VAX or Prime
$100,000
100,000 2
1994
Pentium-90
$5,000
4,000,000 70
1999
Intel Pentium III-450
$1,500
69,000,000
1,380
2006
AMD 64 Laptop
$2,000
400,000,000
8,000
2009
Min Laptop
$300
200,000,000
4,000
2010
2.4 GHz Intel Core i3 64 bit Win 7 Laptop
$1,000
1.35 Billion
Intel Fortran
27,000
2013
2.80 GHz 2 Quad Core 64 bit Win 7
2.80 Billion Parallelized Fortran
56,000
The cost of one operation has been reduced by 56,000,000 in the last 50 years

35. Computer Cost versus Engineer’s Monthly Salary
$1,000,000
c/s = 1,000
c/s = 0.10
$10,000
$1,000
$1,000
Time

36. Fast Nonlinear Analysis On Earthquake Engineering
With Emphasis
On Earthquake Engineering BY
Ed Wilson
Professor Emeritus of Civil Engineering
University of California, Berkeley
edwilson.org
May 25, 2006 1

37. Summary Of Presentation
1. History of the Finite Element Method
2. History Of The Development of SAP
3. Computer Hardware Developments
4. Methods For Linear and Nonlinear Analysis
5. Generation And Use Of LDR Vectors and
Fast Nonlinear Analysis - FNA Method
6. Example Of Parallel Engineering
Analysis of the Richmond - San Rafael Bridge 2

38. 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 6

39. The SAP Series of Programs
SAP Used Static Loads to Generate Ritz Vectors
Solid-Sap Rewritten by Ed Wilson
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
SAP 90 Significant Modification and Documentation
1997 – Present SAP Nonlinear Elements – More Options –
With Windows Interface

40. Load-Dependent Ritz Vectors
LDR Vectors – 11

41. MOTAVATION – 3D Reactor on Soft Foundation
Dynamic Analysis
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

42. 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 12

43. 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 13

44. 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 14

45. 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 15

46. 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 15

47. 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 16

48. 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 17

49. STEP 2. VECTOR GENERATION
i = 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 18

50. 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 = 19

51. DYNAMIC RESPONSE OF BEAM
100 pounds
10 AT 12" = 120"
FORCE = Step Function
TIME 20

52. MAXIMUM DISPLACEMENT 1 0.004572 (-2.41) 0.004726 (+0.88) 2 0.004572
Number of Vectors Eigen Vectors Load Dependent Vectors 1
(-2.41)
(+0.88) 2
(-2.41)
( -2.00) 3
(-0.46)
(+0.08) 4
(-0.46)
(+0.06) 5
(-0.08)
( 0.00) 7
(-0.04) 9
(0.00)
( Error in Percent) 21

53. MAXIMUM MOMENT ( Error in Percent )
Number of Vectors Eigen Vectors Load Dependent Vectors 1
4178
( %)
5907
( ) 2
4178
( )
5563
( ) 3
4946
( )
5603
( ) 4
4946
( )
5507
( ) 5
5188
( )
5411
( ) 7
5304
( - .0 ) 9
5411
( )
( Error in Percent ) 22

54. 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

55. 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

56. FAST NONLINEAR ANALYSIS1.
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 23

57. BASE ISOLATION
Isolators 25

58. BUILDING
IMPACT
ANALYSIS 26

59. FRICTION
DEVICE
CONCENTRATED
DAMPER
NONLINEAR
ELEMENT 27

60. GAP ELEMENT
BRIDGE DECK ABUTMENT
TENSION ONLY ELEMENT 28

61. P L A S T I C
H I N G E S
2 ROTATIONAL DOF
DEGRADING STIFFNESS ? 29

62. Mechanical Damper
F = f (u,v,umax )
F = ku
F = C vN
Mathematical Model

63. 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

64. ELEVATED WATER STORAGE TANK
103 FEET DIAMETER FEET HEIGHT
NONLINEAR
DIAGONALS
BASE
ISOLATION
ELEVATED WATER STORAGE TANK 30

65. COMPUTER MODEL 92 NODES 103 ELASTIC FRAME ELEMENTS 56
NONLINEAR DIAGONAL ELEMENTS
600
TIME Seconds 31

66. COMPUTER TIME REQUIREMENTS PROGRAM ANSYS INTEL 486 3 Days
( 4300 Minutes )
ANSYS
CRAY
3 Hours
( 180 Minutes )
SADSAP
INTEL 486
(2 Minutes )
( B Array was 56 x 20 ) 32

67. Nonlinear Equilibrium Equations33

68. Nonlinear Equilibrium Equations33

69. Summary Of FNA Method 1. Calculate LDR Vectors for Structure
With the Nonlinear Elements Removed.
2. These Vectors Satisfy the Following
Orthogonality Properties 35

70. No Additional Approximations Are Made.3.
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. 36

71. 4. A typical modal equation is uncoupled.
However, the modes are coupled by the
unknown nonlinear modal forces which
are of the following form:
5. The deformations in the nonlinear elements
can be calculated from the following
displacement transformation equation: 37

72. THE ARRAY CAN BE STORED IN RAM
6. 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 38

73. 7. The nonlinear element forces are calculated, for iteration i , at the end of each time step39

74. Nonlinear elements – given Tol8.
Calculate error for iteration i , at the end of each time step, for the N
Nonlinear elements – given Tol 39

75. FRAME WITH UPLIFTING ALLOWED40

76. Four Static Load Conditions Generation of LDR Vectors
Are Used To Start The
Generation of LDR Vectors
EQ DL Left Right 41

77. NONLINEAR STATIC ANALYSIS
50 STEPS AT dT = SECONDS
DEAD LOAD
LOAD
LATERAL LOAD
TIME - Seconds 42

78. 43

79. 44

80. 45

81. 46

82. 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. 47

83. PARALLEL ENGINEERING
AND
PARALLEL COMPUTERS 73

84. ONE PROCESSOR ASSIGNED TO EACH JOINT3 2 1 3 1 2
ONE PROCESSOR ASSIGNEDTO EACH MEMBER 74

85. 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 75

86. FIRST PRACTICAL APPLICTION RICHMOND - SAN RAFAEL BRIDGEOF
THE FNA METHOD
Retrofit of the
RICHMOND - SAN RAFAEL BRIDGE
1997 to 2000
Using SADSAP 49

87. 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 8

88. 50

89. 51

90. 52

91. TYPICAL ANCHOR PIER 53

92. MULTISUPPORT ANALYSIS
( Displacements )
ANCHOR PIERS
RITZ VECTOR
LOAD PATTERNS 54

93. 55

94. 56

95. 57

96. SUBSTRUCTURE PHYSICS Stiffness Matrix Size = 3 x 16 = 48 "a"
MASSLESS JOINT
( Eliminated DOF )
"b"
MASS POINTS and
JOINT REACTIONS
( Retained DOF ) 58

97. SUBSTRUCTURE STIFFNESS REDUCE IN SIZE BY LUMPING MASSES
OR BY ADDING INTERNAL MODES 60

98. 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 62

99. 64

100. 65

101. ECCENTRICALLY BRACED FRAME66

102. FIELD MEASUREMENTS REQUIRED TO VERIFY
1. MODELING ASSUMPTIONS
2. SOIL-STRUCTURE MODEL
3. COMPUTER PROGRAM
4. COMPUTER USER

104. CHECK OF RIGID DIAPHRAGM APPROXIMATION
MECHANICAL VIBRATION DEVICES

105. FIELD MEASUREMENTS OF PERIODS AND MODE SHAPES
MODE TFIELD TANALYSIS Diff. - %
Sec Sec. 0.5

106. FIRST DIAPHRAGM MODE SHAPE
15 th Period
TFIELD = 0.16 Sec.

107. Can be directly connected to a 3D Acceleration Seismic Box
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

108. Final Remark Geotechnical Engineers must
produce realistic Earthquake records
for use by Structural Engineers

109. 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

110. Comparison of Relative and Absolute Displacement Seismic Analysis

111. Shear at Second Level Vs. Time With Zero Damping
Time Step = 0.01

112. Illustration of Mass-Proportional Component in Classical Damping.