Design, Scaling, Similitude, and
Modeling of ShakeShake-Table Test
Structures
Andreas Stavridis, Benson Shing, and Joel Conte
University of California, San Diego
NEES@UC San Diego
NEES@UBuffalo
NEES@UNevada‐Reno
Shake Table Training Workshop 2010 – San Diego, CA
Topics Covered
• Overview of shake-table test considerations
• Dimensional analysis
• Similitude law
• Scaling and design of test structures
• Modeling of test structures
• Case Study:
y Shake Table tests of an infilled
frame
Shake Table Training Workshop 2010 – San Diego, CA
Needs for Shake
Shake--Table Tests
• Study the seismic performance of (non-)
structural
t t l components
t and
d complex
l systems
t
• Provide data to validate/calibrate analytical
models
• Validate design/construction concepts and
details
Shake Table Training Workshop 2010 – San Diego, CA
Specimens Tested on Shake Tables
• Non-Structural Components
– e.g. anchors,
h
racks
k
• Structural Components
p
– e.g. columns, dampers
• Substructures
S
– e.g. frames, joints, walls
• Complete Structures
– e.g.
g buildings,
g , bridges,
g , wind turbines
Shake Table Training Workshop 2010 – San Diego, CA
Advantages of Shake Table Tests Over
Other Testing Methods
• More realistic consideration of dynamic effects
– inertia forces
– damping forces
– no need
d tto attach
tt h loading
l di devices
d i
th
thatt may iinfluence
fl
the structural performance
• Best / more direct way to simulate earthquake
ground motion effects
Shake Table Training Workshop 2010 – San Diego, CA
Dynamic Effects
• Quasi
Quasi-static
static test
• Shake-table test
Shake Table Training Workshop 2010 – San Diego, CA
Constraints of ShakeShake-Table Tests
•
•
•
•
•
•
Cost
Shake table availability
qu p e t capac
capacity
ty
Equipment
Accuracy of certain measurements
Boundary conditions
Limited time to react if things go wrong
Shake Table Training Workshop 2010 – San Diego, CA
Common Solutions
• Testing
gp
portions of structures
(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 Chart
Step 1
• define need for
research
Step 2
• facility/cost
constraints
• similitude law
Step 3
• data
processing
Identify structural
system,
system concept
etc. to be tested
Design
Test
Structure
Analyze
Test Data
Design
Prototype
Structure
Design
Instrumentation
Plan
Validate
Analytical
Models
Shake Table Training Workshop 2010 – San Diego, CA
Design
Testing
Program
Evaluate
Concept,
System
Extraction of Test Substructures
• Special
p
considerations to be
paid on
– Boundary conditions
– Kinematic constraints existing
i prototype
in
t t
structure
t t
– Gravity loading conditions
– Seismic loading conditions
Shake Table Training Workshop 2010 – San Diego, CA
Mismatch Between Gravity and Inertia
Masses
Masses
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
strain rate effects may become important
Shake Table Training Workshop 2010 – San Diego, CA
Background
• Scale models
– should satisfy similitude requirements so that
they can be used to study the response of fullscale structures
• Similitude
Si ilit d requirements
i
t
– based on dimensional analysis
Shake Table Training Workshop 2010 – San Diego, CA
Background
• Dimensional analysis
y
– a mathematical technique to deduce the theoretical
relation of variables describing a physical
phenomenon
• Dimensionally homogeneous relations
– relations valid regardless of the units used for the
physical
p
y
variables
Shake Table Training Workshop 2010 – San Diego, CA
Fundamental Dimensions in Physical
P bl
Problems
•
•
•
•
•
•
Length (L)
Force (F) or Mass (M)
Time ((T))
Temperature (θ)
Electrical charge
…
Most important for
problems in structural
engineering
Any equation describing a physical phenomenon
should be in dimensionally homogeneous form
Shake Table Training Workshop 2010 – San Diego, CA
Example
w(x)
Deflection of a beam
Governing Differential Equation
d 4u
EI 4  w x 
dx
 L  L L L   F L
F
4
2
4
Shake Table Training Workshop 2010 – San Diego, CA
Buckingham’s π Theorem
• Ag
general approach
pp
for dimensional analysis
y
• Any 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 π Theorem
• Initial equation
f  X 1 , X 2 , X 3 ,..., X n 
• Equivalent
q
equation
q
of
dimensional parameters
g  1 ,  2 ,,...,,  m 
with
m  nr
in which:
Xi
 i  X X ... X
a
k
r
b
l
physical variable
c
m dimensionless product of the physical variables
number of fundamental dimensions
Shake Table Training Workshop 2010 – San Diego, CA
Properties of πi’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 Object
initial assumption
S  kg a t b
F g , t   0
or
in dimensional terms


2 a
L  K MT
Tb
from dimensional homogeneity
M :1  a
S  kg t 2 or
T : 0  2a  b
K can be determined experimentally
 S 
G 2   0
 gt 


Shake Table Training Workshop 2010 – San Diego, CA
Application of Similitude Theory
• The π terms are general, non-dimensional,
non dimensional,
and independent; hence they apply to any
system. In tthis
syste
s case tthe
ep
prototype
ototype
structure (p) and the scaled model (m).
• If   
we have complete similarity
between the prototype and the model
p
i
m
i
– true model
Shake Table Training Workshop 2010 – San Diego, CA
If   
p
i
m
i
• In case πi‘ss are not important
– the model maintains ‘first-order’ similarity
– adequate model
• In case πi‘ss are important
– the model does not maintain ‘first-order’
similarity
– distorted model
Shake Table Training Workshop 2010 – San Diego, CA
Example of Adequate/Distorted (?) Model
Small-scale specimen
Large-scale specimen
350
Late
eral force, kips
300
250
200
150
1/5-scale specimen
p
100
2/3-scale specimen
50
0
0
0.5
1
1.5
2
Drift, %
Shake Table Training Workshop 2010 – San Diego, CA
Example of Adequate/Distorted (?) Model
Small-scale specimen
Large-scale specimen
δ=1%
Shake Table Training Workshop 2010 – San Diego, CA
Application of Similitude Theory
• Rewriting
g the equations
q
for the p
prototype
yp and
model structures
m
m
m
m
p
p
p
p




,

,...,

 i    k ,  l ,,...,,  n and i
k
l
n


• Scale factors:
Si


i quantity
tit in
i scaled
l d mod
d ell

i quantity in prototype
• Obtained by equating the π-terms
solving
g for the S ratio
 
p
i
i
Shake Table Training Workshop 2010 – San Diego, CA
m
i
and
Example of Scale Factor Derivation
S 


m
F

A

 




3


V
aV
aL
aL


m


m
F

A


 



3


V
aV
aL
aL


p
S

S S l
Shake Table Training Workshop 2010 – San Diego, CA
Similitude 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
– one off the remaining variables
– the dimensionally independent variables
Shake Table Training Workshop 2010 – San Diego, CA
Calculating the Scale Factors
• Select scale factors for 3 dimensionallyy
independent 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 Example
• 2/3-scale,, threestory, masonryinfilled, non-ductile
RC fframe
• tested in Fall 2008
@ UCSD
Shake Table Training Workshop 2010 – San Diego, CA
Prototype Structure
• Represents structures built in California 1920’s
• Earliest
E li t b
building
ildi code
d we ffound:
d 1936
• Design
g considerations
–
–
–
–
–
Currently available materials used
Only gravity loads considered
Allowable stress design procedure
Contribution of infills ignored
No shear reinforcement in beams
• Three-wythe
Th
th masonry walls
ll on th
the perimeter
i t
Shake Table Training Workshop 2010 – San Diego, CA
29
Design of Prototype Structure
.30*L = 5’ 5’’
90o
bend
0.20*L = 3’ 8’’
0.25*L = 4’ 6’’
Story
level
Design of beams
Width
Depth
Bent
bars
Straigh
t 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
Design of columns
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
16
2%
8#7
#3@16”
#3@16
Shake Table Training Workshop 2010 – San Diego, CA
30
Considerations
– Amount of gravity
mass to be added
– Scaling issues
– Attachment of mass
– Out-of-plane stability
– M
Measurementt off floor
fl
displacements
– Loading protocol
Shake Table Training Workshop 2010 – San Diego, CA
31
Layout of Prototype Structure
3
5.50
2
5.50
1
Exterior
frame
6.70
A
6.70
6.70
B
C
D
T ib t
Tributary
area for
f seismic
i i mass
Tributary area for gravity mass
Masonry-infilled bays
Shake Table Training Workshop 2010 – San Diego, CA
Gravity Loads
`
`
`
`
3#5 bars
27.9
3#5 bars
15.4
75.1
Transverse Beam
61.8
74.7
75.1
Slab
Transverse Beam
Shake Table Training Workshop 2010 – San Diego, CA
33
28.3

Mismatch Between Gravity and Inertia
M
Masses
Gravity Mass
agravity
 Inertia Mass
aseismic
Shake Table Training Workshop 2010 – San Diego, CA
34
Derivation of Scale Factors: Gravityy
SL  2
• Length:
g
3
• Stress:
S  1
cce e at o S a  1
• Acceleration:
•Strain:
•Curvature:
•Area:
•Volume:
•Moment of inertia:
S  1
S  1
SL
•Force:
S F  S A S  4
2
•Moment:
SM  SF SL  8
27
9
•Mass:
Sm  S F
9
•Time:
St 
3
S A  SL SL  4
SV  S L S L S L  8
27
S I  S L S L S L S L  16
81
SL
Sa
4
Sa
•Frequency: S f  1
9
St
 2
3
 0.816
 1.224
Shake Table Training Workshop 2010 – San Diego, CA
Derivation of Scale Factors: Inertia
Mismatch of gravity
and inertia masses
M 
Scaling of the inertia mass
The force scale factor
needs to be preserved
•Seismic acceleration:
•Time
•Frequency:
q
y
seis
M prot
M
S
seis
m

grav
prot
S mgrav

M
seis
M spec
grav
M spec
 0.20
grav
S seis

S
 Sf  4
f
f
9
S aseis   M S agrav  2 . 273
S
seis
t

S Lseis
S seis
 M 1
f
M S
S
grav
t
grav
a

1
M
S tgrav  0.542
 1.846
Shake Table Training Workshop 2010 – San Diego, CA
Alternative Derivation
S
•Seismic mass:
S
•Seismic acceleration:
•Time:
•Frequency:
F
S
seis
a
seis
t
seis
i
m


S Fseisi
seis
i
S
L

grav
M spec
seis
M prot
S mseis
S
seis
a

SF
S Mseis
 SL
S seis
 1
f
S aseis
S tseis
Shake Table Training Workshop 2010 – San Diego, CA
Instrumentation
• Instrumentation
– 135 strain-gauges
– 66 accelerometers
– 79 displacement
transducers
• Story displacements
– Mass-less poles
– Deformation of
triangles attached on
the RC frame
• 8 GB of raw data
Shake Table Training Workshop 2010 – San Diego, CA
38
Seismic Loading
 Elastic range
2.5
 6 low-level earthquakes
 10%-40%
2
Sa, g
 Mild nonlinearity
 67% of Gilroy
 67% of Gilroy
 83% of Gilroy
Structural
Period
1.5
DBE
MCE
67% of Gilroyy
100% of Gilroy
1
0
0.5
0
 Significant nonlinearity
 91% of Gilroy
 100% of Gilroy
 “Collapse”
Collapse of structure
 120% of Gilroy
 250% El Centro 1940
0
0.5
1
Period, sec
1.5
 Before and after each
earthquake test
 Ambient vibration was recorded
 White noise tests were
performed
Shake Table Training Workshop 2010 – San Diego, CA
39
2
Failure Patterns
Shake Table Training Workshop 2010 – San Diego, CA
40
Test Summary
Frequency
S aMCE
Hz
18
0.64
16.7
0.69
15.9
0.77
14.8
0 96
0.96
13 5
13.5
1.43
8.5
1.55
5.3
1.04
recorded
S ai
–
–
–
–
–
–
–
–
Initial Structure
Gilroy 67%
Gilroy 67%
Gilroy 83%
Gilroy 91%
Gilroy 100%
Gilroy 120%
El Centro 250%
Damage
minor
minor
some
some
significant
severe
collapse
Max Drift
%
0.01
0.10
0.17
0.28
0
0.40
40
0.55
1.06
V1 / W
V1 / W
Specimen Prototype
0.97
1.41
1.75
1.77
1 76
1.76
1.68
1.68
Shake Table Training Workshop 2010 – San Diego, CA
41
0.43
0.62
0.77
0.78
0 78
0.78
0.74
9.74
Analytical Methods for Infilled Frames
• Limit analysis methods
– Predefined failure modes
– Limited information on the behavior Smeared Crack
Only
Smeared +
Discrete Crack
• Strut models
– Not all failure modes captured
– Empirical
E i i l fformulas
l b
based
d on casespecific experimental data
– A variety of proposed implementation
schemes
• Finite element analysis
–
–
–
–
–
Frame elements
Shear panel element
Smeared crack elements
Interface elements
Bond slip elements
Shi and
Shing
d Spencer
S
(1999)
Shake Table Training Workshop 2010 – San Diego, CA
Simplified Modeling
Lateral fo
orce, kN
 Consider single-bay w/
diagonal struts
 Obtain response of frame
w/ solid infill
600
OpenSEES model
500
Simplified curve
Bare Frame
120
400
80
300
200
40
100
0
 Obtain response of bare
frame
0
0
02
0.2
04
0.4
11,34,35
10,33
26
3
9
6
24
27
8,26,27,31
7,25,29
12
9,28,32
13
21
22
2
 Assemble multi-bay, multistory
t
model
d l
12,36
15
25
 Calibrate struts to simulate
failure of the RC columns
08
0.8
Drift ratio, %
14
 M
Modify
dif for
f panels
l with
ith
openings
06
0.6
8
5
20
23
5 18 19 23
5,18,19,23
4,17,21,
10
17
1
6,20,24
11
18
4
16
1,13
19
2,14,15
i,j
Nodes (with bold letters the
master nodes for the RC frame)
k
Strut Elements
k
RC elements
Shake Table Training Workshop 2010 – San Diego, CA
43
7
3,16
1
12
1.2
Lateral forrce, kips
700
Simplified Model
400
Ba
ase shear, kN
1350
900
Shake-Table Tests
300
Strut
St
ut model
ode
200
450
100
0
0
-450
50
-100
00
-900
-200
-1350
-300
-1800
-1.5
-400
-1
-0.5
0
0.5
1
st
1 Story drift, %
Shake Table Training Workshop 2010 – San Diego, CA
44
Bas
se shear, kips
s
1800
Behavior of Physical Specimen
Concrete
Shear
Crack
Tensile failure
of head
jjoint
Brick
Crushing
Concrete
Flexural
Crack
Sliding of
bed joint
Tensile Splitting
of a Brick
Shake Table Training Workshop 2010 – San Diego, CA
45
•
Modeling Scheme for Masonry
El
Elements
t
Brick units
– Split into two smeared-crack
elements
Half Brick
½ Brick to ½ Brick
joints
– Interface element allows tensile
splitting
Mortar Joint
•
Mortar joints
– Interface elements
Interface element for brick interface
Interface elements for mortar joints
Smeared crack brick element
Shake Table Training Workshop 2010 – San Diego,
CA
46
Modeling Scheme for Concrete
•
Concrete members
– Smeared crack elements
– Interface elements allow for
diagonal cracks
 Longitudinal reinforcement
 Shear Reinforcement
 Distributed in 8 bars
 Distributed in 2 bars p
per x-section
 Zig-zag pattern
Flexural steel
reinforcement
Shear steel
reinforcement
Nodal location
Smeared crack
concrete element
Interface
concrete element
47
Shake Table Training Workshop 2010 – San Diego,
CA
Potential Cracking Patterns
Flexural
Shear
Shake Table Training Workshop 2010 – San Diego, CA
48
Finite Element Model
Base Shear, kips
B
400
300
Shake-Table Tests
200
FEAP-Prediction
100
0
-100
-200
-300
-400
-2
-1
0
49
1
1st Story Drift, %
Shake Table Training Workshop 2010 – San Diego, CA
2
Finite Element Model
(by Koutromanos et al)
• Gilroy 67% (design level earthquake)
Shake Table Training Workshop 2010 – San Diego, CA
Laws 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
Thank you
Shake Table Training Workshop 2010 – San Diego, CA