Analytical Modeling of a

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Forced Vibration Testing &
Analytical Modeling of a


Four
-
story Reinforced Concrete
Frame Building

PI’s:

J. W. Wallace,
E. Taciroglu
, J.P. Stewart

Staff:

D.H. Whang, Y. Lei, S. Keown, S. Kang

Students:

E. Yu, D. Skolnik, W. Elmer


Forced Vibration Tests


Modal Identification


Finite Element Model Updating


Conclusions & Outlook

OUTLINE

Forced Vibration Tests


Goals of forced vibration tests/studies


Extract dynamic properties of the structure
experimentally


Validate the assumptions of analytical

model used
to predict structural response


Evaluate predictive capability of analytical

models



Until now, forced vibration tests have been performed
at
low
-
level response amplitudes



Two kinds of shakers were used as vibration sources


Eccentric mass shaker


Linear shaker

BACKGROUND


Eccentric Mass Shaker


Generate harmonic forces through rotation of mass


Steady state response
-
> frequency
-
response curve


Generally, larger maximum load capacity


Laborious tests; one frequency at a time



Linear Shaker


Arbitrary forcing function (Broadband excitation)


Transient response : reduce test time / more
computation


Effective in System Identification


Simulation of earthquake vibration

BACKGROUND


Produce a high
-
quality dataset


-

Low noise




155 dB accelerometer, 24
-
bit AD converter


-

High spatial density


Acceleration + Story Displacement + Strain


-

Low / high amplitude excitation




~max 200 kip force

Test Building : 4
-
story RC frame building


Damage survey


nees@UCLA equipment


Instrumentation scheme


Test procedure

OBJECTIVE &OVERVIEW


“Four Seasons Building”


4
-
Story RC Building with penthouse


Constructed in 1977


Damaged by the 1994 Northridge earthquake


Yellow Tagged (unoccupied, will be demolished)

Western Exterior of the Building

THE BUILDING


Located near the intersection of 101 & 405 Freeway, in
Sherman Oaks, California (16 km from UCLA)

UCLA

LOCATION


Lateral Load: Special Moment Frame (Beams+Columns)
around perimeter


Gravity Load: Post
-
tensioned flab slab with drop panels
+ Interior columns


Foundation: Belled Caissons + Grade beams


No shear walls

Typical Floor Plan

N
1
2
3
4
5
6
B
D
C
A
7
5@9.3 m
(5@30'-6")
3@9.3 m
(3@30'-6")
9.6 m
(31'-6")
1
2
3
4
5
6
7
2F EL 11.25' (3.43 m)
3F EL 22.75' (6.93 m)
GF : EL 0
4F EL 34.25' (10.44 m)
RF EL 47.75' (14.55 m)
PH. EL 59.58' (18.16 m)
3' (0.9 m)
N
Section along Lone B

STRUCTURAL SYSTEM


Beam : 24”x30” (Typical), 24”x36” (2nd Floor)


Column : 24”x24”



Slab : 8
-
1/2” with 7
-
1/2” drop panel (typical) ; Lightweight Concrete









(3000 psi)

Interior Columns


Slab
-
Column connection

Exterior Columns

Normal weight
concrete (4000 psi)

STRUCTURAL MEMBERS


Damage report (Sabol, 1994)


Previous analytical studies


Dovitch and Wight, 1994


Ascheim and Moehle, 1996


Hueste and Wight, 1998


Analytical results were not able to
identify the amount of damage observed
in the building


Effects of torsion / vertical response were
significant, or


Ground motions were more severe

PREVIOUS STUDIES


Interior frame


Punching shear failure at slab
-
column connections
around the perimeter of drop panel


Column B6 (3
rd

Floor)

Column B2 (2
nd

Floor)

Slab dropped 0.5 ~ 0.75 in. downwards

OBSERVED DAMAGE


Perimeter Frame


Beam
-
Column joint crack with concrete spalling


Spalling of cover concrete at beam end


Flexural cracks

Spalling at beam end

(Column A7 at 3F level)

Diagonal joint crack

(column A4 at 3F level)

Flexural cracks

(column B2 at 4
th

story)

OBSERVED DAMAGE


Non
-
structural Members


Separated from adjacent structural members


No structural contribution during the test was
expected, except possibly at the penthouse level

Masonry wall at ground floor

Partition wall at 2
nd

story

Penthouse drywall

OBSERVED DAMAGE

(B) Slight

(T) N.A.

(T) N.A.

(T) N.A.

(T) N.A.

(T) N.A.

(T) N.A.

(T) N.A.

(T) N.A.

(T) N.A.

1

2

3

4

1

2

3

4

5

6

B

D

C

A

7

(T) Slight

(T) Slight

(B) Slight

(B) Slight

(B) Slight

(B) Moderate

(B) Slight

(B) Slight

(B) Moderate

(T) Slight

(T) N.A.

(T) N.A.

(T) N.A.

(T) Severe

(B) Moderate

(B) Moderate

(T)

Slight

(T) N.E.

(B) Moderate

(B) Severe

(B) Severe

(B) Moderate

(B) Moderate

(B) Moderate

(B) Moderate

(T) N.A.

(T) Moderate

(T) Severe

N

T : Top face

B : Bottom face

N.A. : Not Accessible

(blank) : No Damage

(T) Severe

(T) Severe

(T) Severe

(T) N. A.

(B) Slight

(B) Slight

(T) N. A.

(T) N. A.

(B) Slight

(B) Slight

(B) Severe

(B) Slight

(B) Severe

(B) Moderate

(B) N. A.

(B) Slight

(T) N. A.

(B) Slight

Severe

: Big chunk crushed out, Floor level dropped or Reinforcements exposed

Moderate

: Large and developed cracks, small chunk crushed out, or aggregate exposed

Slight

: long crack around drop panel

Roof

4
th

Floor

2
nd

Floor

3
rd

Floor

INTERIOR DAMAGE

1

2

3

4

5

6

7

West Perimeter Frame (Line A)

East Perimeter Frame (Line D)

North Perimeter Frame (Line 7)

A

B

C

D

South Perimeter Frame (Line 1 & 2)

N

E

N

Diagonal joint crack

Diagonal joint crack with concrete spalling

Severe concrete crushing (at beam end) /Shear crack


Building experienced more deformation in
N
-
S direction than E
-
W direction

EXTERIOR DAMAGE


Two 100
-
kip capacity eccentric mass shakers


15
-
kip capacity linear shaker


Force
-
Balanced Accelerometers (FBA)


LVDTs (DC
-
DC Type)


Concrete strain gauges


24
-
bit AD converters


Wireless data
-
logging (Antelope) & Networking system


National Instrument signal conditioning units (LabView)


Mobile Command Center (MCC)


Power generators

TESTING EQUIPMENT


nees@UCLA


Two 100
-
kip capacity shakers


Generate harmonic forces through rotation of mass

nees@UCLA Eccentric Mass Shaker, MK
-
15



eccentricity
mass
of a basket mass eccentricity
e
m
 
mass
2
( ) 2 sin( )
e
P t m t
 

ECCENTRIC MASS SHAKER

69 Steel bricks

Empty basket

Half
-
full basket

Mass
-
eccentricity

(each basket)

16786 lb
-
in

56620 lb
-
in

Limiting

frequency

5.40 Hz

2.95 Hz

Pulse Marker



Basket configurations for this study



Adjustable basket

Hydrostone Leveling

ECCENTRIC MASS SHAKER


Produce force through linear motion of a moving mass


Moving mass (5 kip/g) + Dynamic Actuator (15 kip,
±
15”) + Hydraulic
system (90 gpm servo
-
valve, 30 gpm pump, 4 accumulators) +
Controller


Digital control : PD, LQG, adaptive ; displacement, acceleration


Broadband excitation ; white
-
noise, sine
-
sweep, earthquake
-
type

Linear Shaker

Example sine
-
sweep forcing function

LINEAR SHAKER

Linear

Shaker

Eccentric

Mass

Shaker

(South)

N

Eccentric
Mass
Shaker
(North)

Reference

Point

37.2 m (122 ft)

9.3 m (30.5 ft)

SHAKER LOCATIONS

Force
-
balance
Accelerometer

DCDT (DC
-
DC type LVDT)

High performance 24
-
bit Datalogger
(Kinemetrics, Q330)

National Instrument
Signal Conditioning
Module used for
concrete strain
gauges

(32 ch X 3 units)

Strain Gauge

Synchronization using GPS time

SENSORS & DATALOGGERS

Q330

WAP

Yagi Antenna

WAP

Antelope


server

Mobile

Command Center

Wireless

Communication

Wired

Sensors

Wireless

DC

: Data Concentration Point


WAP

:
Wireless Access Point

DC

Data
Concentration

Point (DC)

Wireless Access
Point (WAP)

WIRELESS DATA ACQUISITION

Power for the shakers

Battery box/portable power

Power for DAQ

POWER GENERATORS



Acceleration

Force
-
balance type
Accelerometer




Strain

Strain gauges placed at
top and bottom of floor
slabs and 3 faces of
columns




Interstory


Displacement

DCDTs measure
displacement from
bottom of one column to
top of the consecutive
column



197 Total channels



16 tri
-
axial + 27 uniaxial accelerometers



26 DCDT’s



96 Strain gauges

INSTRUMENTATION

Vertical Accelerometer

NS Accelerometer

EW Accelerometer

Column with strain gauge

LVDT

3u1

3v1

3u2

3v2

3u3

3v3

3u4

3v4

1

2

3

4

5

6

7

1u1

1v1

1w5

1w1

1w4

1v4

1w3

1v3

1w8

1w7

1w6

1u2

1v2

1w2

LVDT
-
NS1

LVDT
-
NS2

LVDT
-
EW

Rv1

Ru1

Ru4

Rv4

Rv3

Ru3

Rv2

Ru2

Pu1

N

LVDT
-
NS2

LVDT
-
NS1

LVDT
-
EW

Pv1

Pv2

N

5

2

1

4

3

7

6

Rw1

Rw4

Rw3

Rw2

3w1

3w4

3w3

3w2

Roof / Penthouse

3
rd

floor level

Ground floor

Elevation (A
-
A)

A

A

Roof Level

3F Level

Ground

INSTRUMENTATION PLAN

8"

8"

8"

4"

12"

24"

12"

Curtain Wall



Column Strain Gauges


3 faces for curvature
calculation in both
directions


Along A2 & B2 column
from ground floor to roof
floor


Below and above the
floor slab level

S1

S2

S3

S4

S5

S6

S8

S10

S7

S9

0.25
L

L

=30'
-
6"

60"

60"

42"

1

2

3

B

A

0.25
L

0.25
L

0.25
L



Floor Slab Strain Gauges


Top and bottom faces of
3
rd

& 4
th

floor slab

INSTRUMENTATION PLAN

Date

Test

6/22/04


E
-
W translational excitation with
empty

basket


Run1

7/2/04


Ambient vibration measurement


Run1

7/13/04


E
-
W translational excitation with
empty

basket


Run2


Torsional excitation with
empty

basket


Run1

7/14/04


E
-
W translational excitation with
half
-
full

basket


Run1


Torsional excitation with
half
-
full

basket


Run1

7/19/04


E
-
W translational excitation with
half
-
full

basket


Run2


Torsional excitation with
half
-
full

basket


Run2


Ambient vibration measurement


Run2


Linear shaker sinesweep / whitenoise


Run1

7/22/04


N
-
S translational excitation with
half
-
full

basket


Run1

7/28/04


N
-
S translational excitation with
empty

basket


Run1


Linear shaker seismic simulation test

8/2/04


N
-
S translational excitation with
empty

basket


Run2


Linear shaker sinesweep / whitenoise


Run2

8/3/04


Ambient vibration measurement


Run3


E
-
W translational excitation with
empty

basket


Run3

TESTING SEQUENCE

Eccentric Mass Shaker Test

VIDEO CLIPS

Modal Identification

TESTING & DATA ACQUISITION


Identification and updating
performed with data from the linear
shaker white noise excitation


Data recorded with four tri
-
axial
accelerometers used derive three
story responses

SYSTEM IDENTIFICATION

N
4
SID

(
N
umerical

Algorithm

for

S
ubspace

S
tate

S
pace

S
ystem

Id
entification)


Discrete

time

domain

method

uses

measured

data

directly


Makes

projections

of

certain

subspaces

generated

from

the

input/output

observations

to

estimate

state

sequence

using

linear

algebra

tools

such

as

QRD

and

SVD
.



Identifies

system

matrices

from

estimated

states

based

on

a

linear

least

squares

solution



Can

be

applied

to

systems

subjected

to

known

or

unknown

excitation


Well

implemented

in

MATLAB’s

System

Identification

Toolbox

1
k k k
k k k
X X u
y X u

 
 
A B
C D




2
Re 2
sign Re
i i
i i i
i i i
f
f
C C
 
  
  


 
 
 
u:

input force applied
with linear shaker

y
:

output measured
floor responses

SYSTEM IDENTIFICATION

Stability Tolerances



D
f
≤ 1.5%



D

≤ 5%



MAC
≥ 98%

Stability Plot

EW

NS
Tor







2
(,)
T
A B
T T
A A B B
MAC A B
 
   


 
SYSTEM IDENTIFICATION

EW

NS
Tor

Frequencies and Damping Ratios

For
Amb

Mode

Forced

f

(Hz)


⠥

A浢楥湴

f

(Hz)


⠥

A浢楥湴m/

䙯牣敤

1 EW

0.88

5.6

1.09

3.4

1.24

0.61

2 NS

0.94

6.9

1.25

3.1

1.33

0.45

3 Tor

1.26

6.0

1.55

2.1

1.23

0.35

4 EW

2.73


5.6

3.23

3.0

1.18

0.54

5 NS

2.94


7.7

3.63

3.1

1.23

0.40

6 Tor

3.44


6.1

4.16

2.1

1.21

0.34

7 Mix

4.54


13.5

-

-

-

-

K
s
1
K
s
2

Ambient vibration > linear shaker test > EMS test


=> Stiffness degradation of structural member


(contribution of nonstructural elements is negligible ; damage survey)



3 ~ 4% frequency drop in ambient vibration after EMS test


due to the high amplitude vibrations during Half
-
full basket testing


=> degradation of (cladding / Foundation & soil / structural member) ??


Larger frequency drop in N
-
S
direction => effect of damage

DISCUSSION

Finite Element Model
Updating

FINITE ELEMENT MODELING

Modeling Assumptions



Lumped Mass



Rigid Diaphragms



Classical Damping

From Core Tests



r
n

=140pcf,
r
l

= 115pcf



E
cn

= 4028ksi, E
cl

= 2517ksi

Effective Stiffness (
FEMA 356 ,
Paulay&Priestley, “Effective
Beam Method”
)



Columns: 0.5E
cn
I
g



Beams: 0.42E
cn
I
g



Slabs: 0.4E
cl
I
g

N

FINITE ELEMENT MODELING

Mode

FE

SID

FE / SID

1 EW

0.92

0.88

1.05

2 NS

1.12

0.94

1.19

3 Tor

1.35

1.26

1.07

4 EW

2.6

2.73

0.95

5 NS

2.94

2.94

1.00

6 Tor

3.53

3.44

1.03

Natural Frequencies (Hz)

EW

NS
Tor

FINITE ELEMENT MODELING

FRF
-

NS direction

2
( )
H( ) x( )/( )
i
f
  
  
 
   
 

B M C K
MODEL UPDATING

x( ) x( ) x( ) ( )
t t t L f t
  
M C K
2
x( ) ( )
i L f
   
 
   
 
M C K
( )H( )
L
 

B
2
 
K M
Sensitivity
-
Based Updating Procedure using Frequency
Response Function (FRF) and Modal Frequencies

MODEL UPDATING

1 2
p [,,,]
T
k
p p p





0
0
0
p p
0
p p
(p,)
H( ) L (p,)H( )
(p)
Ω (p )
F
M
p
p
p
p

   



D
D
 

  
 

 
 
 

  
 

 
 
B
B
Error residuals

d
p
d
F F F
M M M


D
     
 
   
 
     
C
C
ε L (p,)H( )
ε Ω (p)
F
M
 
 
 
B
Parameter Vector

Non
-
linear functions of p

Linearize with a first
-
order Taylor series expansion

MODEL UPDATING

0
p p p p
lb ub
D
  
2
p
Min p d
D

WC W
lim
1 cor(C,C ) , if cor(C,C )
i j i j i j
p p c
   
such that

and

Objective Function

MODEL UPDATING

Parameter(s) associated with

Bounds

Initial Values

Mass of 2F

85 %
-

115 %

65.0 (kips sec
2
/ft)

Mass of 3F & 4F

64.7 (kips sec
2
/ft)

Mass of RF

62.1 (kips sec
2
/ft)

Mass of PH

50 %
-

150 %

7.6 (kips sec
2
/ft)

Radius of gyration of 2F & 3F

75 %
-

135 %

64.2 (ft)

Radius of gyration of 4F

64.0 (ft)

Radius of gyration of RF

57.5 (ft)

Radius of gyration of PH

26.7 (ft)

Column Stiffness at 2F
-

RF

35 %
-

150 %

0.5E
cn
I
g

Column Stiffness at PH

0.75E
cn
I
g,
(NS)

2.5E
cn
I
g
(EW)

Slab Stiffness at 2F
-

RF

0.4E
cl
I
g,

Slab Stiffness at PH

0.6E
cl
I
g
(NS)


2.0E
cl
I
g
(EW)

Beam Stiffness at 2F
-

RF

0.42E
cn
I
g

Damping ratios

2.5 %
-

20 %

5 %

Dimensionless

Parameters


10

Mass


52

Stiffness



9

Damping

MODEL UPDATING

Ratios of Initial Mass

2F

3F

4F

RF

PH

Translational Mass

94%

97%

104%

105%

97%

Radius of gyration

102%

104%

97%

102%

104%

Stiffness Factors

2F

3F

4F

RF

PH

NS Interior, North & South Frame Columns

0.40

0.48

0.32

0.45

0.73

NS of East Frame Columns

0.36

0.41

0.22

0.49

-

NS of West Frame Columns

0.45

0.39

0.26

0.46

-

EW of Interior, East & West Frame Columns

0.46

0.62

0.49

0.42

2.20

EW of North Frame Columns

0.49

0.52

0.59

0.49

-

EW of South Frame Columns

0.52

0.56

0.34

0.46

-

East Frame Girders

0.45

0.23

0.38

0.41

-

West Frame Girders

0.42

0.17

0.39

0.40

-

South Frame Girders

0.49

0.32

0.36

0.39

-

North Frame Girders

0.43

0.57

0.43

0.42

-

Slab NS

0.43

0.19

0.36

0.40

0.58

Slab EW

0.44

0.36

0.35

0.37

1.86

Damping
Ratios

7th

8th

9th

10th

11th

12th

13th

14th

15th

9.6%

15.9%

7.3%

15.5%

2.5%

8.8%

8.8%

5.4%

13.5%

MODEL UPDATING

Ratios of Initial Mass

2F

3F

4F

RF

PH

Translational Mass

94%

97%

104%

105%

97%

Radius of gyration

102%

104%

97%

102%

104%

Stiffness Factors

2F

3F

4F

RF

PH

NS Interior, North & South Frame Columns

0.40

0.48

0.32

0.45

0.73

NS of East Frame Columns

0.36

0.41

0.22

0.49

-

NS of West Frame Columns

0.45

0.39

0.26

0.46

-

EW of Interior, East & West Frame Columns

0.46

0.62

0.49

0.42

2.20

EW of North Frame Columns

0.49

0.52

0.59

0.49

-

EW of South Frame Columns

0.52

0.56

0.34

0.46

-

East Frame Girders

0.45

0.23

0.38

0.41

-

West Frame Girders

0.42

0.17

0.39

0.40

-

South Frame Girders

0.49

0.32

0.36

0.39

-

North Frame Girders

0.43

0.57

0.43

0.42

-

Slab NS

0.43

0.19

0.36

0.40

0.58

Slab EW

0.44

0.36

0.35

0.37

1.86

Damping
Ratios

7th

8th

9th

10th

11th

12th

13th

14th

15th

9.6%

15.9%

7.3%

15.5%

2.5%

8.8%

8.8%

5.4%

13.5%

MODEL UPDATING

Mode

Initial

Updated

SID

1 EW

0.92

0.90

0.88

2 NS

1.12

0.97

0.94

3 Tor

1.35

1.25

1.26

4 EW

2.6

2.72

2.73

5 NS

2.94

2.93

2.94

6 Tor

3.53

3.44

3.44

Natural Frequencies (Hz)

EW

NS
Tor

MODEL UPDATING

FRF
-

NS direction

MODEL UPDATING

2
nd

Floor

3
rd

Floor

4
th

Floor

Roof

Penthouse

Predicted

and
Measured

NS response to 0.5
-

5 Hz linear shaker sine sweep

Conclusions & Outlook

CONCLUSIONS


Identified modal properties of the first seven modes
using N4SID


Frequencies identified from ambient vibrations
represent a stiffer structure than that identified from
white noise excitation


FE model is updated using a modal
-

FRF
-
sensitivity based method


Frequencies, mode shapes, and FRF of the
updated model compare well with those identified


Predicted acceleration response of the updated
model compares quite well with the measured data