Ucsd Modeling Similitude and Simulation Andreas Stavridis
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Transcript of Ucsd Modeling Similitude and Simulation Andreas Stavridis
Design, Scaling, Similitude, and Design, Scaling, Similitude, and Modeling of ShakeModeling of Shake--Table TestTable TestModeling of ShakeModeling of Shake--Table Test Table Test
StructuresStructuresAndreas Stavridis, Benson Shing, and Joel Conte
University of California, San DiegoUniversity of California, San Diego
NEES@UNevada‐RenoNEES@UBuffaloNEES@UC San Diego
Shake Table Training Workshop 2010 – San Diego, CA
Topics CoveredTopics Covered
• Overview of shake-table test considerations
• Dimensional analysis
• Similitude lawSimilitude law
• Scaling and design of test structures
• Modeling of test structures
• Case Study: Shake Table tests of an infilled yframe
Shake Table Training Workshop 2010 – San Diego, CA
Needs for ShakeNeeds for Shake--Table TestsTable Tests
• Study the seismic performance of (non-) t t l t d l tstructural components and complex systems
• Provide data to validate/calibrate analyticalProvide data to validate/calibrate analytical models
• Validate design/construction concepts and details
Shake Table Training Workshop 2010 – San Diego, CA
Specimens Tested on Shake TablesSpecimens Tested on Shake Tables
• Non-Structural Componentsh k– e.g. anchors, racks
• Structural Componentsp– e.g. columns, dampers
S• Substructures– e.g. frames, joints, walls
• Complete Structures– e.g. buildings, bridges, wind turbines
Shake Table Training Workshop 2010 – San Diego, CA
g g , g ,
Advantages of Shake Table Tests Over Advantages of Shake Table Tests Over Other Testing MethodsOther Testing MethodsOther Testing MethodsOther Testing Methods
• More realistic consideration of dynamic effects– inertia forces– damping forces
d t tt h l di d i th t i fl– no need to attach loading devices that may influence the structural performance
• Best / more direct way to simulate earthquake ground motion effectsground motion effects
Shake Table Training Workshop 2010 – San Diego, CA
Dynamic EffectsDynamic Effects
• Quasi-static test • Shake-table testQuasi static test
Shake Table Training Workshop 2010 – San Diego, CA
Constraints of ShakeConstraints of Shake--Table TestsTable Tests
• Cost• Shake table availability• Equipment capacity qu p e t capac ty• Accuracy of certain measurements• Boundary conditionsBoundary conditions• Limited time to react if things go wrong
Shake Table Training Workshop 2010 – San Diego, CA
Common SolutionsCommon Solutions
• Testing portions of structures g p(i.e. substructures)
• Building scaled specimens
• Expanding the platen area
• Redundancy in the instrumentation scheme
Shake Table Training Workshop 2010 – San Diego, CA
Testing Flow ChartTesting Flow ChartStep 1• define need for
Identify structural system concept
Design Prototypedefine need for
researchsystem, concept etc. to be tested
Prototype Structure
Step 2Design
Test Structure
Design Instrumentation
Plan
Design Testing
Program
Step 2• facility/cost
constraintsStructure Plan Program
• similitude law
Analyze Test Data
Validate Analytical Models
Evaluate Concept, System
Step 3• data
processingShake Table Training Workshop 2010 – San Diego, CA
processing
Extraction of Test SubstructuresExtraction of Test Substructures
• Special considerations to be ppaid on
– Boundary conditions– Kinematic constraints existing
i t t t tin prototype structure– Gravity loading conditions– Seismic loading conditions– Seismic loading conditions
Shake Table Training Workshop 2010 – San Diego, CA
Mismatch Between Gravity and Inertia Mismatch Between Gravity and Inertia MMMassesMasses
Possible solutions– Gravity columns
• may influence the structural performance
– Secondary structure for inertia loads (e.g. Buffalo)• does not apply gravity loads
– Scaling up the accelerations• strain rate effects may become important• strain-rate effects may become important
Shake Table Training Workshop 2010 – San Diego, CA
BackgroundBackground
• Scale models – should satisfy similitude requirements so that
they can be used to study the response of full-scale structuresscale structures
Si ilit d i t• Similitude requirements– based on dimensional analysis
Shake Table Training Workshop 2010 – San Diego, CA
BackgroundBackground
• Dimensional analysisy– a mathematical technique to deduce the theoretical
relation of variables describing a physical phenomenonphenomenon
• Dimensionally homogeneous relations• Dimensionally homogeneous relations– relations valid regardless of the units used for the
physical variablesp y
Shake Table Training Workshop 2010 – San Diego, CA
Fundamental Dimensions in Physical Fundamental Dimensions in Physical P blP blProblemsProblems
• Length (L) Most important for• Force (F) or Mass (M)• Time (T)
Most important for problems in structural engineering( )
• Temperature (θ)• Electrical chargeElectrical charge• …
Any equation describing a physical phenomenon should be in dimensionally homogeneous form
Shake Table Training Workshop 2010 – San Diego, CA
ExampleExample
Deflection of a beamw(x)
Deflection of a beam
Governing Differential Equation
xwudEI 4
4Governing Differential Equation
dx4
44
2 LLLL
FL
F 42 LL L
Shake Table Training Workshop 2010 – San Diego, CA
Buckingham’s Buckingham’s ππ TheoremTheorem
• A general approach for dimensional analysisg pp y
• Any dimensionally homogeneous equationAny dimensionally homogeneous equation involving physical quantities can be expressed as an equivalent equation involving a set of dimensionless parameters
Shake Table Training Workshop 2010 – San Diego, CA
Buckingham’s Buckingham’s ππ TheoremTheorem
• Initial equation
• Equivalent equation of
nXXXXf ,...,,, 321withq q
dimensional parameters
mg ,...,, 21
rnm
in which:
mg , ,, 21
X physical variableiX physical variabledimensionless product of the physical variables
cm
bl
aki XXX ...r number of fundamental dimensions
Shake Table Training Workshop 2010 – San Diego, CA
r number of fundamental dimensions
Properties of Properties of ππii’s’s
• All variables must be included
• The m terms must be independent
• There is no unique set of πi’s
Shake Table Training Workshop 2010 – San Diego, CA
Example 1: Free Falling ObjectExample 1: Free Falling Object
initial assumptionbatkgS 0, tgFor
initial assumption
baTMTKL 2
in dimensional terms
TMTKLfrom dimensional homogeneity
aM 1: 2 0
SG
baT 20:2tkgS 02
gt
Gor
K can be determined experimentallyShake Table Training Workshop 2010 – San Diego, CA
K can be determined experimentally
Application of Similitude TheoryApplication of Similitude Theory
• The π terms are general, non-dimensional,The π terms are general, non dimensional, and independent; hence they apply to any system. In this case the prototype syste t s case t e p ototypestructure (p) and the scaled model (m).
• If we have complete similarity between the prototype and the model
mi
pi
between the prototype and the model– true model
Shake Table Training Workshop 2010 – San Diego, CA
IfIf mi
pi
• In case πi‘s are not importantIn case πi s are not important – the model maintains ‘first-order’ similarity– adequate modeladequate model
• In case π ‘s are important• In case πi s are important – the model does not maintain ‘first-order’
similaritysimilarity– distorted model
Shake Table Training Workshop 2010 – San Diego, CA
Example of Adequate/Distorted (?) ModelExample of Adequate/Distorted (?) ModelLarge-scale specimenSmall-scale specimen
300
350
150
200
250
300
eral
forc
e, k
ips
1/5-scale specimen
0
50
100
0 0.5 1 1.5 2
Late
p
2/3-scale specimen
Shake Table Training Workshop 2010 – San Diego, CA
Drift, %
Example of Adequate/Distorted (?) ModelExample of Adequate/Distorted (?) ModelLarge-scale specimenSmall-scale specimen
δ = 1 %
Shake Table Training Workshop 2010 – San Diego, CA
Application of Similitude TheoryApplication of Similitude Theory
• Rewriting the equations for the prototype and g q p ypmodel structures
and pn
pl
pk
pi ,...,, m
nml
mk
mi ,...,,
• Scale factors:
nlki , ,, nlki
ll dititi dprototypeinquantityi
elscaledinquantityiS imod
• Obtained by equating the π-terms and solving for the ratio
mi
pi
iSShake Table Training Workshop 2010 – San Diego, CA
giS
Example of Scale Factor DerivationExample of Scale Factor Derivation
mAF
SaLaLA
aVF
Vm
S
3
lp SS
aLaLA
aVF
Vm
S
3 aLaLaVV
Shake Table Training Workshop 2010 – San Diego, CA
Similitude RequirementsSimilitude Requirements
In structural problems we have in general • 3 fundamental dimensions:
– F (or M), L, T
• 3 dimensionally independent variables
• n-3 π terms involving f– one of the remaining variables
– the dimensionally independent variables
Shake Table Training Workshop 2010 – San Diego, CA
Calculating the Scale FactorsCalculating the Scale Factors
• Select scale factors for 3 dimensionally yindependent quantities
• Express remaining variables in terms of the selected scale factors
• Except for dimensionless variables (e.g. ν, ε)which have a scale factor of 1
Shake Table Training Workshop 2010 – San Diego, CA
Infill ExampleInfill Example
• 2/3-scale, three-,story, masonry-infilled, non-ductile RC fRC frame
• tested in Fall 2008 @ UCSD
Shake Table Training Workshop 2010 – San Diego, CA
Prototype StructurePrototype Structure• Represents structures built in California 1920’s
E li t b ildi d f d 1936• Earliest building code we found: 1936
• Design considerationsg– Currently available materials used– Only gravity loads considered– Allowable stress design procedure– Contribution of infills ignored– No shear reinforcement in beams
Th th ll th i t• Three-wythe masonry walls on the perimeter
Shake Table Training Workshop 2010 – San Diego, CA29
Design of Prototype StructureDesign of Prototype Structure.30*L = 5’ 5’’
90o
bend
0.25*L = 4’ 6’’0.20*L = 3’ 8’’
Design of beams Story level
Width Depth Bent bars
Straight bars
Stirrups
Roof 16” 18” 2#8 2#6 no stirrups
2nd Story 16” 22” 3#8 3#7 no stirrups
1st Story 16” 22” 3#8 3#7 no stirrups
Story level Size ρ Vertical bars StirrupsDesign of columns
Story level Size ρ Vertical bars Stirrups
Roof 16” sq 1.0% 8#5 #3@16”
2nd Story 16” sq 1.5% 8#6 #3@16”
1st Story 16” sq 2% 8#7 #3@16”
Shake Table Training Workshop 2010 – San Diego, CA30
1 Story 16 sq 2% 8#7 #3@16
ConsiderationsConsiderations– Amount of gravity
mass to be added
– Scaling issues
– Attachment of mass
– Out-of-plane stability
M t f fl– Measurement of floor displacements
– Loading protocolLoading protocol
Shake Table Training Workshop 2010 – San Diego, CA31
Layout of Prototype StructureLayout of Prototype Structure
333
5.50
3
2
5.50
3
2
5.50
3
2
5.50
2
1
5.50
2
1
5.50
2
1
A B C D6.70 6.70 6.70
1
Exteriorframe
T ib t f i iA B C D
6.70 6.70 6.70
1
A B C D6.70 6.70 6.70
1
Exteriorframe
T ib t f i iTributary area for seismic mass
Tributary area for gravity massMasonry-infilled bays
Tributary area for seismic mass
Tributary area for gravity massMasonry-infilled bays
Shake Table Training Workshop 2010 – San Diego, CA
Gravity LoadsGravity Loads
`
`
`
`
`
`
`
`
3#5 bars3#5 bars27.9
Transverse Beam Transverse BeamSlab
28.374.775.1 61.8 75.1
15.4
Shake Table Training Workshop 2010 – San Diego, CA33
Transverse Beam Transverse BeamSlab
Mismatch Between Gravity and Inertia Mismatch Between Gravity and Inertia MM Gravity Mass MassesMasses
agravity
Inertia Mass
Shake Table Training Workshop 2010 – San Diego, CA34
aseismic
Derivation of Scale Factors: GravityDerivation of Scale Factors: Gravityyy
• Length: 32LSg
• Stress: • Acceleration:
3
1S1aScce e at o
1S9
4 SSS AF•Strain: •Force:
231
LSS
94 LLA SSS
278 LFM SSS
94
a
Fm S
SS
•Curvature:
•Area:
•Moment:
•Mass:
278 LLLV SSSS
8116 LLLLI SSSSS
a
816.032
a
Lt S
SS
224.11 f SS
•Volume:
•Moment of inertia:•Time:
•Frequency:
Shake Table Training Workshop 2010 – San Diego, CA
tf S
Derivation of Scale Factors: InertiaDerivation of Scale Factors: Inertia
Mismatch of gravity and inertia masses grav
seisspec
grav
seisprot
M MM
MM
and inertia masses specprot MM
Scaling of the inertia mass 20.0gravmseis
mSS
The force scale factor needs to be preserved 9
4 fgravf
seisf SSS
M
273.2 gravaM
seisa SS •Seismic acceleration:
needs to be preserved 9fff
542.01 gravt
MgravaM
seisLseis
t SSSS
846.11 gravMseisf SS
•Time
•Frequency:
Shake Table Training Workshop 2010 – San Diego, CA
846.1gravt
Mf SS q y
Alternative DerivationAlternative DerivationAlternative DerivationAlternative Derivation
gravi M
seisprot
specseism M
MS
i
•Seismic mass:
seisM
Fseism
seisFseis
a SS
SSS
i
•Seismic acceleration:
seisa
Lseisa
seisLseis
t SS
SSS •Time:
Fseist
seisf SS 1•Frequency:
Shake Table Training Workshop 2010 – San Diego, CA
InstrumentationInstrumentation• Instrumentation
– 135 strain-gauges– 66 accelerometers– 79 displacement
transducers
• Story displacements– Mass-less poles
Deformation of– Deformation of triangles attached on the RC frame
• 8 GB of raw data
Shake Table Training Workshop 2010 – San Diego, CA38
Seismic LoadingSeismic Loading Elastic range
6 low-level earthquakes 10%-40% 2
2.5DBEMCE67% of Gilroy
Structural Period
Mild nonlinearity 67% of Gilroy 67% of Gilroy 0
1
1.5
Sa, g
y100% of Gilroy
67% of Gilroy 83% of Gilroy
Significant nonlinearity0
0.5
0 0.5 1 1.5 2
91% of Gilroy 100% of Gilroy
“Collapse” of structure
Period, sec
Before and after each earthquake test
Collapse of structure 120% of Gilroy 250% El Centro 1940
Ambient vibration was recorded White noise tests were
performed
Shake Table Training Workshop 2010 – San Diego, CA39
Test SummaryTest SummaryFrequency Damage Max Drift V1 / W V1 / W
Hz % Specimen Prototype– Initial Structure 18 - 0.01 0.97 0.43
MCEa
ia
SS
recorded
– Gilroy 67% 0.64 16.7 minor 0.10 1.41 0.62– Gilroy 67% 0.69 15.9 minor 0.17 1.75 0.77– Gilroy 83% 0.77 14.8 some 0.28 1.77 0.78– Gilroy 91% 0 96 13 5 some 0 40 1 76 0 78– Gilroy 91% 0.96 13.5 some 0.40 1.76 0.78– Gilroy 100% 1.43 8.5 significant 0.55 1.68 0.74– Gilroy 120% 1.55 5.3 severe 1.06 1.68 9.74– El Centro 250% 1.04 c o l l a p s e
Shake Table Training Workshop 2010 – San Diego, CA41
Limit analysis methodsAnalytical Methods for Infilled FramesAnalytical Methods for Infilled Frames
• Limit analysis methods– Predefined failure modes– Limited information on the behavior
Smeared +
Discrete CrackSmeared Crack Only
• Strut models– Not all failure modes captured
E i i l f l b d– Empirical formulas based on case-specific experimental data
– A variety of proposed implementation schemes
• Finite element analysis– Frame elements– Shear panel element– Smeared crack elements– Interface elements
Shi d S (1999)Shake Table Training Workshop 2010 – San Diego, CA
– Bond slip elements Shing and Spencer (1999)
Simplified ModelingSimplified Modeling Consider single-bay w/
diagonal struts400
500
600
700
orce
, kN
80
120
rce,
kip
s
OpenSEES modelSimplified curveBare Frame
Obtain response of frame w/ solid infill
0
100
200
300
0 0 2 0 4 0 6 0 8 1 1 2
Late
ral f
o
0
40
80
Late
ral f
or
Obtain response of bare frame
M dif f l ith 3 96
10,33 12,3611,34,35
151425 26
3 96
10,33 12,3611,34,35
151425 26
0 0.2 0.4 0.6 0.8 1 1.2
Drift ratio, %
Modify for panels with openings
Calibrate struts to simulate2
3
8
9
5
6
7,25,29 9,28,328,26,27,31
5 18 19 23
131221
20
22
23
24 27
2
3
8
9
5
6
7,25,29 9,28,328,26,27,31
5 18 19 23
131221
20
22
23
24 27
Calibrate struts to simulate failure of the RC columns
Assemble multi-bay, multi-t d l
1 74
1,13
4,17,21,
3,16
6,20,245,18,19,23
2,14,15
111017
16
18
19
k Strut Elements
1 74
1,13
4,17,21,
3,16
6,20,245,18,19,23
2,14,15
111017
16
18
19
k Strut Elements
Shake Table Training Workshop 2010 – San Diego, CA43
story model i,jkNodes (with bold letters the
master nodes for the RC frame)
Strut Elements
k RC elementsi,j
kNodes (with bold letters the master nodes for the RC frame)
Strut Elements
k RC elements
Simplified ModelSimplified Model
900
1350
1800
200
300
400
s
Shake-Table Tests
Strut model
-450
0
450
ase
shea
r, kN
-100
0
100
se s
hear
, kip
sSt ut ode
-1800
-1350
-900
50
Ba
-400
-300
-200
00
Bas
-1800-1.5 -1 -0.5 0 0.5 1
1st Story drift, %
-400
Shake Table Training Workshop 2010 – San Diego, CA44
Behavior of Physical SpecimenBehavior of Physical SpecimenConcrete
Shear Tensile failureCrack Tensile failure of head joint
Brick Sliding of
j
CrushingSliding of
bed joint
Tensile Splitting Concrete
Flexural
Shake Table Training Workshop 2010 – San Diego, CA45
of a BrickCrack
Modeling Scheme for Masonry Modeling Scheme for Masonry El tEl t
• Brick units– Split into two smeared-crack
elements Half Brick
ElementsElements
elements
– Interface element allows tensile splitting
½ Brick to ½ Brick joints
• Mortar joints– Interface elements
Mortar Joint
Interface element for brick interface
Interface elements for mortar joints
Interface elements
Interface elements for mortar joints
Smeared crack brick element
Shake Table Training Workshop 2010 – San Diego, CA46
Modeling Scheme for ConcreteModeling Scheme for Concrete• Concrete members
– Smeared crack elements– Interface elements allow for
diagonal cracksdiagonal cracks
Shear Reinforcement Distributed in 2 bars per x-section
Longitudinal reinforcement Distributed in 8 bars p
Zig-zag pattern
Flexural steel reinforcement
Shear steel reinforcement
Nodal location
Smeared crack
Interface concrete element
Smeared crack concrete element
Shake Table Training Workshop 2010 – San Diego, CA47
Potential Cracking PatternsPotential Cracking PatternsFlexural Shear
Shake Table Training Workshop 2010 – San Diego, CA48
Finite Element ModelFinite Element Model
200
300
400
Shake-Table Tests
-100
0
100
200
Base
She
ar, k
ips FEAP-Prediction
-400
-300
-200
-2 -1 0 1 2
B
Shake Table Training Workshop 2010 – San Diego, CA49
1st Story Drift, %
Finite Element ModelFinite Element Model(by(by KoutromanosKoutromanos et al)et al)(by (by KoutromanosKoutromanos et al)et al)
• Gilroy 67% (design level earthquake)Gilroy 67% (design level earthquake)
Shake Table Training Workshop 2010 – San Diego, CA
Laws in Experimental StudiesLaws in Experimental Studies• Murphy’s law
– If something can go wrong, it will!
• O’Toole’s law– Murphy is wildly optimistic
• Dan’s law– Things are never as bad as they turn out to be
• Conte’s law– No model is as good as the prototype
• Seible’s law– The most important aspect of a test are the pictures and videos
Shake Table Training Workshop 2010 – San Diego, CA