Seismic Analysis and Design
Of Structures
Using Response Spectra
Or
Time History Motions
BY
Ed Wilson
Professor Emeritus of Civil Engineering
University of California, Berkeley
February 24, 2010
SUMMARY OF PRESENTATION
On Advanced Numerical Modeling and Analytical Techniques
1. Personal Remarks – 50 years experience of dynamic analysis
2. Seismic Analysis Using Response Spectra – CQC3
3. Comparison with Direct Time History Dynamic Analysis
4. Retrofit of the San Mateo Bridge
_-
5. The Fast Non-Linear Analysis Method – FNA Method
6. Retrofit of the Richmond San Rafael Bridge
7. Near Fault Seismic Analysis
8. Concluding Remarks
edwilson.org and ed-wilson1@juno.com
1882
Father Born In San Francisco – Carpenter and
Walked Guard in S.F. after 1906 Earthquake
1931
Ed born in Ferndale CA – Earthquake Capitol of USA
1950
Graduated - Christian Brothers HS in SAC.
1950 - 52
Sacramento Jr. College
1953 - 54
BS in Civil Eng. – UC Berkeley
1953 - 54
DOT CA Bridge Dept. – Ten Mile River Bridge
1955 - 57
US Army – Korea – Radio Repairman
1957 - 63
M.S. and D. Eng. With Prof. Ray Clough
1960
With Ray, Conducted the first Time-Histories
Earthquake Response of Buildings Bridges &
Dams. - Fifty Years Ago
1963- 65
Worked on the Apollo Program at Aerojet in
Sacramento - Designed Structures for
1965 -91
Professor at UC Berkeley
10 g
Loads
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
DYNAMIC ANALYSIS USING RESPONSE
SPECTRUM SEISMIC LOADING
Before the Existence of Inexpensive Personal Computers, the
Response Spectrum Method was the Standard Approach for Linear
Seismic Analysis
25
20
15
10
5
0
-5
-10
-15
-20
-25
0
1
2
3
4
5
6
7
8
9
TIME - seconds
Figure 15.1a Typical Earthquake Ground Acceleration Percent of Gravity
10
2
0
-2
-4
-6
-8
- 10
- 12
0
1
2
3
4
5
6
TIME - seconds
7
8
9
10
Figure 15.1b Absolute Earthquake Ground Displacements - Inches
y () MAX
20
18
16
14
12
10
Figure 15.2b Pseudo-Acceleration Spectrum,
8
1.0 Percent Damping
5.0 Percent
Damping
Percent of Gravity
6
4
2
0
0
1
2
3
4
5
PERIOD - Seconds
Figure 15.2a Relative Displacement Spectrum y (T)MAX Inches
100
90
80
1.0 Percent Damping
5.0 Percent Damping
70
60
50
40
30
20
10
0
0
1
Sa   y () MAX
2
2
3
4
5
PERIOD - Seconds
Figure 15.2b Pseudo-Acceleration Spectrum Percent of Gravity
Major Approximation
yT (t )  y(t )  ug (t )
yT (t )  y(t )  ug (t )
Where
yT ( t )  The Total Displaceme nt
y(t)  The Displaceme nt Ralative to the
Earthquake Ground Motion
u g (t)  The Earthquake Ground Displaceme nts
At the Base of the Structure
The loads are applied directly to the structure;
whereas, the real earthquake displacements are
applied at the foundation of the real structure.
Development of the Three Spectrum
y(t)n + 2  n  n y (t)n +  n2 y(t)n = p n1 u(t)g1 + p n 2 u(t)g 2 + p nz u(t)gz
Or, the 3 spectrum are produced by solution of the following three equation :
y(t)n + 2  n  n y (t)n +  n2 y(t)n = p n1 u(t)g1
 S1 ( ) All positive numbers
y(t)n + 2  n  n y (t)n +  n2 y(t)n = p n 2 u(t)g 2
 S2 ( ) All positive numbers
y(t)n + 2  n  n y (t)n +  n2 y(t)n = p nz u(t)gz
 S z ( ) All positive numbers
The 3 Spectrum are not a function of the properties of the structure
In Addition, All Spectrum Values Are Maximum Peak Values
The Time History Details of the Duration of the Earthquake
Have Been Lost
Examples of Three-Dimensional Spectra Analyses
Y
2
3
3
4
2
X = Y = 106.065 ft.
Sym.
3
X = Y = 70.717 ft.
3
3
1 2
0
X = 100 ft.
2
2
X = 150 ft.
X
Definition of Earthquake Spectra Input
90
S2
90
S1

Plan View
0
Three-Dimensional Spectra Analyses
Equal Spectrum from any direction – CQC3 Method
Y
2.705
2.703
4
2.705
1.901
3
2.705
2.705
2.703
1.901
0
1
2
X
Maximum Peak Column Moments - Symmetrical
All Values are Positive
Three-Dimensional Spectra Analyses
100/30 Spectrum Method
Y
2.794
2.493
-7.8 % Error
4
2.493
1.934
3
2.743
0
2.743
1.973
01
1
2.797
X
2
Maximum Peak Column Moments - Not Symmetrical
All Values are Positive
Summary of Multi-Component
Combination Rules
1. The 100/30 and 100/40 percent rules
have no theoretical basis.
2. The SRSS combination rule, applied
to equal spectra, produces identical
results for all reference systems and
requires only one analysis to produce
all design forces and displacements.
3. The CQC3 method should be used
where the horizontal orthogonal
components of the seismic input are
not equal.
4. In case of the seismic analysis of
structures near a fault, the fault
normal and parallel motions are not
equal.
In 1996 The CQC3 was Proposed
by
Professor Armen Der Kiureghian
As a Replacement for the
30%, 40% & SRSS Rules
For Multi-Component Seismic Analysis
2
Fpeak  [ F0
a
2
2
F90
 (1  a )
2
 2(1  a ) F090 sin  cos 
2
2
( F0

2
2
F90 ) sin

1
2 2
Fz ]
Where " a" is the proportion al constant
used to define the other horizontal spectrum
S 2  a S1
If a  1.0 The CQC3 method reduces to the SRSS rule
Design Checks of Three-Dimensional
Frame Members for Seismic Forces
In order to stratify various building codes, every
one-dimensional compression member within a
structure must satisfy the following
Demand/Capacity Ratio at all points in time:
P(t )
M 3 ( t ) C3
M 2 (t ) C2
R(t ) 


 1.0
P(t )
P(t )
c Pcr
b M c 2 (1 
) b M c 3 (1 
)
Pe 2
Pe3
t = 0 = Static Loads Only
Where the forces acting on the frame element crosssection at time “t” are P(t ), M 2 (t ) and M 3 (t )
including the static forces prior to the application of
the dynamic loads. The empirical constants are code
and material dependent and are normally defined as
c and φ. b
 Resistance factors
C 2 and C 3
 Moment reduction factors
M c 2 and M c 3  Moment capacities
Pcr
 Axial load capacity
Pe 2 and Pe 3  Euler bucking load capacities about the 2 an3 axis
with effective lengths approximat ed.
Design Checks of Three-Dimensional
Frame Members for Spectra Forces
For the case maximum peak spectra forces,
compression members within a structure must
satisfy the following Demand/Capacity Ratio
P(max)
R( t ) 

c Pcr
M 2 (max) C2
M 3 (max) C3

 1.0
P(max)
P (max)
b M c 2 (1 
) b M c 3 (1 
)
Pe 2
Pe3
Where P(max), M2(max) and M3(max) have been
Calculated by the CQC Method
The Retrofit of the San Mateo Bridge
Demand/Capacity Ratios were calculated using COC
forces using spectrum calculated from several threedimensional sets of earthquake motions.
Time-dependent Demand/Capacity Ratios were
calculated directly from the same set of earthquake
motions.
In general, the time-dependent Demand/Capacity Ratios
were approximately
the CQC forces.
50 percent
of the ratios using
Limitations of Response Spectrum Analysis
1. All forces and displacements obtained from a
Response Spectrum Analysis are Maximum Peak
Values and are all positive numbers.
2. The specific time the Maximum Peak Values occur
is different for every period.
3. Nonlinear Behavior CANNOT be considered in a
Response Spectrum Analysis.
4. Except for a single degree of freedom, a Response
Spectrum Analysis is an APPROXIMATE
METHOD
5. This is not Performance Based Design
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
1979 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.
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
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
Degrading Stiffness Elements are in SAP 2000
Mechanical Damper
F = f (u,v,umax )
F = ku
F = C vN
Mathematical Model
First Application of the FNA Method - 1994
103 FEET DIAMETER - 100 FEET HEIGHT
NONLINEAR
DIAGONALS
BASE
ISOLATION
Nonlinear Seismic Analysis of
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
EXAMPLE OF
FRAME WITH
UPLIFTING
ALLOWED
UPLIFTING
ALLOWED
Four Static Load Conditions
Are Used To Start The
Generation of LDR Vectors
EQ
DL
Left
Right
Column Axial Forces
600
400
LEFT
RIGHT
200
0
-200
-400
-600
0
1
2
3
4
5
6
TIME - seconds
7
8
9
10
Summary of Results for Building Uplifting Example
from Two Times the Loma Prieta Earthquake
Max.
Max.
Displace- Column
  0.05
ment
Force
Computer (inches)
(kips)
Uplift
Time
Without
Max.
Base
Shear
(kips)
Max.
Base
Moment
(k-in)
Max.
Strain
Energy
(k-in)
Max. Uplift
(inches)
7.76
924
494
424,000
1,547
0.0
15.0 Sec
5.88
620
255
197,000
489
1.16
Percent
Diff.
-24%
-33%
-40%
-53%
-68%
14.6 Sec
With
Confirmed by Shaking Table Tests
By Ray Clough on Three Story Frame
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.
MULTISUPPORT SEISMIC ANALYSIS
(Earthquake Displacements Input )
ANCHOR PIERS
Hayward Fault
East
San Andreas Fault
West
Eccentrically Braced Towers
Analysis and Design of Structures for
Near Fault Earthquake Motions
On the UC Berkeley Campus
Fault Normal and Parallel
Foundation Displacements are
Significantly Different
Used six different Time-History Earthquake
Motions for Nonlinear Dynamic Analyses
Hearst Mining Building – Built in 1905 to 07
50 Yards from the Hayward Fault
Base Isolated in 2004
Near Fault Analysis and Design - SRC
Concluding Remarks
1. The 100/30 percent Rule should replaced by the SRSS
Rule - Until the CQC3 is implemented in SAP 2000.
2. Response Spectra Seismic Analysis is an Approximate
Method and is restricted to linear structural behavior
and may satisfy a design code. However, it may not
produce a Performance Based Design
3. In general, Nonlinear Time-History Analyses produce
more realistic results and can produce Performance
Based Design
4. Performance Based Design is using all the
information about the seismic displacement
loading on the structure and to the accurately
predict the nonlinear behavior and damage to the
structure.
5. All Code Based Designed Structures appear to be
based on Linear Analysis.
6. Nonlinear Seismic Analyses are possible due to:
• New Methods of nonlinear analysis have been developed.
• New Nonlinear Energy Dissipation and Simple Isolation
Device can be used.
• The new inexpensive personal computer can easily
conduct the required calculations.
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
1981
CRAY-XMP
40,000,000
800
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
2009
Intel – Core 2 Duo
1,200,000,000
25,000