Design, Scaling, Similitude, and Modeling of ShakeModeling ...nheri.ucsd.edu/eot/docs/2010-user-perspective-seismic-testing/UCSD... · Design, Scaling, Similitude, and Modeling of

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


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


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


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


Dynamic EffectsDynamic Effects

• Quasi-static test • Shake-table testQuasi static test


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


Common SolutionsCommon Solutions

• Testing portions of structures g p(i.e. substructures)

• Building scaled specimens

• Expanding the platen area

• Redundancy in the instrumentation scheme


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


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


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


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


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


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


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


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


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


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


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


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


Drift, %

Example of Adequate/Distorted (?) ModelExample of Adequate/Distorted (?) ModelLarge-scale specimenSmall-scale specimen

δ = 1 %


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


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


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


Infill ExampleInfill Example

• 2/3-scale, three-,story, masonry-infilled, non-ductile RC fRC frame

• tested in Fall 2008 @ UCSD


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”


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


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


Gravity LoadsGravity Loads

`

`

`

`

`

`

`

`

3#5 bars3#5 bars27.9

Transverse Beam Transverse BeamSlab

28.374.775.1 61.8 75.1

15.4


Transverse Beam Transverse BeamSlab

Mismatch Between Gravity and Inertia Mismatch Between Gravity and Inertia MM Gravity Mass MassesMasses

agravity

Inertia Mass


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:


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:


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:


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


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


Failure PatternsFailure Patterns


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


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


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


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


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


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


Potential Cracking PatternsPotential Cracking PatternsFlexural Shear


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


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)


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


Thank youThank you


Documents

Design, Scaling, Similitude, and Modeling of ShakeModeling ...nheri.ucsd.edu/eot/docs/2010-user-perspective-seismic-testing/UCSD... · Design, Scaling, Similitude, and Modeling of