CalTrans 2010
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Seism ic Analysis and Design
Of Structu resUsing Response Spectra
OrTime Histo ry Mot ions
BY
Ed WilsonProfessor Emeritus of Civil Engineering
University of California, Berkeley
February 24, 2010
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SUMMARY OF PRESENTATION
On Advanced Numerical Modeling and Analytical Techniques
1. Personal Remarks50 years exper ience of dynamic analysis
2. Seismic Analysis Using Response SpectraCQC3
3. Comparison with Di rect Time H istory Dynamic Analysis
4. Retrofi t of the San Mateo Br idge_-
5. The FastNon-LinearAnalysis MethodFNA Method
6. Retrof i t of the Richmond San Rafael Br idge
7. Near Faul t Seismic Analysis
8. Concluding Remarks
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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 Br idg e 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, Cond uc ted the f i rst Time-Histories
Earthqu ake Response of Bui ld ings Bridges &
Dams. -Fifty Years Ago
1963- 65 Worked on the Apollo Program at Aerojet in
Sacramento - Designed Structures for 10 gLoads
1965 -91 Professor at UC Berkeley
edwilson.org and [email protected]
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NINETEEN SIXTIES IN BERKELEY
1. Cold War - Blast Analysis
2. Earthquake Engineer ing Research
3. State And Federal F reeway System
4. Manned Space Program
5. Offshore Dri l l ing
6. Nuclear Reactors And Cooling Towers
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NINETEEN SIXTIES IN BERKELEY
1. Period Of Very H igh Productivi ty
2. No Formal Research I nsti tute
3. F ree Exchange Of I nformationGaveprograms to profession prior to publication
4. Worked Closely With Mathematics Group
5. Students Were Very Successful
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DYNAM IC ANALYSIS USING RESPONSE
SPECTRUM SEI SM ICLOADING
Before the Existence of Inexpensive Personal Computers, the
Response Spectrum Method was the Standard Approach for Linear
Seismic Analysis
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TIME - seconds
0 1 2 3 4 5 6 7 8 9 10-25
-20
-15
-10
-5
0
5
10
15
20
25
F igure 15.1a Typical Earthquake Ground Acceleration -
Percent of Gravity
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0 1 2 3 4 5 6 7 8 9 10
TIME - seconds
- 12
- 10
- 8
- 6
- 4
- 2
0
2
F igure 15.1b Absolute Earthquake Ground Displacements - Inches
MAXy )(
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0 1 2 3 4 5
PERIOD - Seconds
0
2
4
6
8
10
12
14
16
18
20
1.0 Percent Damping
5.0 Percent Damping
Figure 15.2b Pseudo-Acceleration Spectrum,
-
Percent of Gravity
F igure 15.2a Relative Displacement Spectrum y(T)MAX I nches
MAXy )(
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)()()( tutyty gT
)()()( tutyty gT
Major Approximation
StructuretheofBaseAt the
ntsDisplacemeGroundEarthquakeThe
MotionGroundEarthquake
thetoRalativentDisplacemeThe
ntDisplacemeTotalThe)(
(t)u
y(t)
ty
Where
g
T
The loads are applied directly to the structure;
whereas, the real earthquake displacements are
applied at the foundation of the real structure.
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structuretheofproperti estheoffunctionanotareSpectrum3The
ve numbersAl l posi tiS(t)up=y(t)+(t)y2+(t)y
numbersposi ti veAl lS(t)up=y(t)+(t)y2+(t)y
numberposi ti veAl lS(t)up=y(t)+(t)y2+(t)y
ionhree equatol lowing tn of the fby solutioproducedarespectrumOr, the
(t)up+(t)up+(t)up=y(t)+(t)y2+(t)y
zgznzn2nnnnn
gnn2nnnnn
gnn2nnnnn
gznzgngnn2nnnnn
)(
)(
s)(
:3
222
111
2211
Development of the Three Spectrum
I n Addition, All Spectrum Values Are Maximum Peak Values
The Time H istory Detai ls of the Duration of the Earthquake
Have Been Lost
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Examples of Three-Dimensional Spectra Analyses
0
Y
X
X = Y = 106.065 ft.
X = Y = 70.717 ft.
X = 100 ft. X = 150 ft.
1 2
3
4
3
2
3
2
3 2
3 2
Sym.
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Pla n View
9 0
0
9 0
S1
S2
Defini tion of Earthquake Spectra I nput
Th Di i l S t A l
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Three-Dimensional Spectra Analyses
Equal Spectrum from any directionCQC3 Method
0
Y
X
1 2
3
4
2.705
1.901
2.705
2.703
2.705
1.901
2.7052.703
Maximum Peak Column Moments - Symmetr ical
All Values are Positive
Three Dimensional Spectra Analyses
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Three-Dimensional Spectra Analyses
100/30Spectrum Method
2.493
2.493
-7.8 % Erro r
0
Y
X
1 2
3
4
2.743
1.973
2.797
1.934
2.794
2.743
Maximum Peak Column Moments - Not Symmetr ical
All Values are Positive
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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.
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3. The CQC3 method should be used
where the horizontal orthogonalcomponents 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 notequal.
I 1996 Th CQC3 P d
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I n 1996 The CQC3 was Proposed
by
Professor Armen Der Kiureghian
As a Replacement for the
30%, 40% & SRSS Rules
For Multi-Component Seismic Analysis
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ruleSRSSthetoreducesmethodCQC3The1.0aIf
spectrumhorizontalotherthedefinetoused
constantalproportiontheis""Where
]cossin)1(2
sin)()1([
12
2
1
2900
2
22
90
2
0
22
90
22
0
SaS
a
FFa
FFaFaFF
z
peak
D i Ch k f Th Di i l
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Design Checks of Three-Dimensional
F rame Members for Seismic Forces
I n order to stratify var ious bui lding codes, every
one-dimensionalcompressionmember within astructure must satisfy the fol lowing
Demand/Capacity Ratio at all points in time:
t = 0 = Static Loads Only
0.1)
)(1(
)(
))(
1(
)()(
)(
3
3
33
2
2
22
e
cb
e
cbcrc
P
tPM
CtM
P
tPM
CtM
P
tP
tR
Wh h f h f l
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Where the forces acting on the frame element cross-
section at time t are
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
.
)(and)(),( 32 tMtMtP
ed.approximatlengthseffectivewithaxisan32about thecapacitiesloadbuckingEulerand
capacityloadAxial
capacitiesMomentand
factorsreductionMomentand
factorsResistanceand
32
32
32
ee
cr
cc
bc
PP
P
MM
CC
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Design Checks of Three-Dimensional
F rame Members for Spectra Forces
For the case maximum peak spectra forces,
compressionmembers within a structure must
satisfy the following Demand/Capacity Ratio
0.1
)
(max)
1(
(max)
)
(max)
1(
(max)(max))(
3
3
33
2
2
22
e
cb
e
cbcrc
P
P
M
CM
P
P
M
CM
P
PtR
Where P(max), M2(max) and M3(max) have been
Calculated by the CQC Method
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The Retrof i t of the San Mateo Bridge
Demand/Capacity Ratios were calculated using COCforces using spectrum calculated from several three-
dimensional sets of earthquake motions.
Time-dependent Demand/Capacity Ratioswerecalculateddirectlyf rom the same set of earthquake
motions.
I n general, the time-dependent Demand/Capacity Ratios
were approximately 50 percentof the ratios usingthe CQC forces.
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1. All forces and displacements obtained from a
Response Spectrum Analysis are Maximum PeakValues and are all positive numbers.
2. The specif ic time the Maximum Peak Values occur
is different for every per iod.
3. Nonl inear 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
L imitations of Response Spectrum Analysis
S A P
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S A P
STRUCTURAL ANALYSIS
PROGRAM
ALSO A PERSON
Who Is Easily Deceived Or Fooled
Who Unquestioningly Serves Another
From The Foreword Of
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" The slang name S A P was selected to
remind the user that this program, like
all programs, lacks intel l igence.
I t is the responsibil i ty of the engineer to
idealize the structure correctly and
assume responsibil i tyfor the results.
Ed Wilson 1970
From The Foreword Of
The F irst SAP Manual
Th SAP S i f P
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The SAP Ser ies of Programs
1969 - 70 SAP Used Stati c Loads to Generate Ritz Vectors
1971 - 72 Solid-Sap Rewritten by Ed Wilson
1972 -73 SAP IV Subspace I terationDr.Jgen Bathe
197374 NON SAPNew Program
The Start of ADI NA
1979 Lost All Research and Development Funding
197980 SAP 80 New L inear Program for Personal Computers
19831987 SAP 80 CSI added Pre and Post Processing
1987 - 1990 SAP 90 Signif icant Modif ication and Documentation
1997PresentSAP 2000 Nonl inear ElementsMore Options
With Windows I nterface
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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
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MECHANICAL
VIBRATIONDEVICES
CHECK OF RIGID
DIAPHRAGM
APPROXIMATION
FIELD MEASUREMENTS OF
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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 0.32 0.32 0.2
- - - -
11 0.23 0.32 2.3
FIRST DIAPHRAGM
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15 th Period
TFIELD = 0.16 Sec.
FIRST DIAPHRAGM
MODE SHAPE
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The FastNonlinearAnalysis 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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Isolators
BASE ISOLATION
BUILDING
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BUILDING
IMPACT
ANALYSIS
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FRICTION
DEVICE
CONCENTRATED
DAMPER
NONLINEARELEMENT
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GAP ELEMENT
TENSION ONLY ELEMENT
BRIDGE DECK ABUTMENT
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P L A S T I C
H I N G E S
2 ROTATIONAL DOF
Degrading Stiffness Elements are in SAP 2000
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Mechanical Damper
Mathematical Model
F = C vN
F = kuF = f (u,v,umax )
F irst Application of the FNA Method 1994
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103 FEET DIAMETER - 100 FEET HEIGHT
Nonlinear Seismic Analysis of
ELEVATED WATER STORAGE TANK
NONLINEAR
DIAGONALS
BASEISOLATION
F irst Application of the FNA Method - 1994
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COMPUTER MODEL
92 NODES
103 ELASTIC FRAME ELEMENTS
56 NONLINEAR DIAGONAL ELEMENTS
600 TIME STEPS @ 0.02 Seconds
COMPUTER TIME
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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 )
EXAMPLE OF
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EXAMPLE OF
FRAME WITH
UPLIFTINGALLOWED
UPLIFTING
ALLOWED
Fou r Stat ic Load Condi t ions
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Fou r Stat ic Load Condi t ions
Are Used To Start The
Generat ion o fLDR Vecto rs
EQ DL Left Righ t
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Summary of Resul ts for Building Uplif ting Example
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0 05.
f rom Two Times the Loma Prieta Earthquake
Uplift
Computer
Time
Max.Displace-
ment(inches)
Max.ColumnForce(kips)
Max.BaseShear(kips)
Max.Base
Moment(k-in)
Max.StrainEnergy(k-in) Max. Uplift
(inches)
Without14.6 Sec
7.76
924
494
424,000
1,547
0.0
With15.0 Sec 5.88 620 255 197,000 489 1.16
PercentDiff. -24% -33% -40% -53% -68%
Confirmed by Shaking Table Tests
By Ray Clough on Three Story F rame
Advantages Of The FNA Method
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Advantages Of The FNA Method
1. The Method Can Be Used For BothStat ic And Dynamic Nonl inear Analyses
2. The Method Is Very Eff icient And
Requ ires A Smal l Amount OfAdd i t ional Compu ter Time As
Compared To L inear Analys is
2. The Method Can Easily Be Inco rporated
Into Exist ing Compu ter Programs For
L INEAR DYNAMIC ANALYSIS.
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MULTISUPPORT SEISM IC ANALYSIS
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(Earthquake Displacements I nput )
ANCHOR PIERS
Hayward Fault San Andreas Fault
East West
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Eccen tr ical ly Braced Towers
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Analysis and Design of Structures for
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Near Faul t Earthquake MotionsOn the UC Berkeley Campus
Faul t Normal and ParallelFoundation Displacements are
Signi f icantly Di fferentUsed six different Time-H istory Earthquake
Motions for Nonl inear Dynamic Analyses
Hearst M ining Bui ldingBui l t in 1905 to 07
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50 Yards from the Hayward Fault
Base Isolated in 2004
Near Fault Analysis and Design - SRC
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Concluding Remarks
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1. The 100/30 percent Rule should replaced by the SRSS
Rule - Unti l the CQC3 is implemented in SAP 2000.
2. Response Spectra Seismic Analysis is an Approximate
Method and is restr icted to linear structural behavior
and may satisfy a design code. However, it may not
produce a Performance Based Design
3. I n general, Nonl inear Time-H istory Analyses produce
more realistic results and can produce Performance
Based Design
4. Performance Based Design is using all theinformation about the seismic displacement
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loading on the structure and to the accurately
predict the nonl inear behavior and damage to thestructure.
5. All Code Based Designed Structures appear to be
based on L inear Analysis.6. Nonl inear Seismic Analyses are possible due to:
New Methods of nonlinear analysis have been developed.
New Nonlinear Energy Dissipation and Simple I solation
Device can be used.
The new inexpensive personal computer can easily
conduct the required calculations.
F loating-Point Speeds of Computer SystemsDefinition of one Operation A = B + C*D 64 bits - REAL*8
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Year
Computer
or CPU
Operations
Per Second
Relative
Speed
1962 CDC-6400 50,000 11964 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 8001994 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
