Slender Wall Behavior & Modeling

John WallaceUniversity of California, Los Angeles

with contributions fromDr. Kutay OrakcalUniversity of California, Los Angeles

2

Presentation OverviewFEMA 356 Requirements

! General requirements! Modeling approaches

" Beam-column, fiber, general

! Stiffness, strength

Experimental Results! Model Assessment

" Rectangular, T-shaped cross sections

! FEMA backbone relations" Flexure dominant walls

3

FEMA 356 –Nonlinear Modeling for Buildings with Slender RC Walls

4

FEMA 356 – RC WallsGeneral Considerations – 6.8.2.1! Represent stiffness, strength, and

deformation capacity! Model all potential failure modes anywhere

along the wall (member) height! Interaction with other structural and

nonstructural elements shall be considered

! So, we must consider any and everything

5

Wall Modeling ApproachesEquivalent beam-column model ! hw/lw ! 3Modified equivalent beam-column! Rectangular walls (hw/lw " 2.5)! Flanged walls (hw/lw " 3.5)

Multiple-line-element and Fiber models! Concrete and rebar material models

General wall model

6

Equivalent Beam-Column Modelhw/lw ! 3:! Use of equivalent beam-

column permitted! Neutral axis migration not

considered! Interaction with in- and out-

of-plane elements not properly considered

! Axial load Impacts" Stiffness (EI)" Strength (P-M)

! L- or T-shaped walls" Where to locate the

element? " Elastic centroid?

3112

column w w

column cracking w w

A t l

I t l#

$

% &$ ' () *

Beams

Wall

Rigid end zones for beam

Column at wall centroid

Hinges

7

Modified Beam - Column ModelRectangular walls (hw/lw " 2.5)& Flanged walls (hw/lw " 3.5):

Use of modified beam-column elementwith added shear spring

Nonlinear flexure/shearare uncoupled using thisapproach

Beams

Wall

Shear spring

Column at wall centroid

Hinges

8

Modified Beam - Column ModelShear force – deformation properties

A

B

C

D

E

+/h

V Vn

1.0

0.2

CPLSIO

Deformation-controlled componenta b - a

c

, -

0.4

1 and 0.21 2

yy

c c

c c

Vh

G E A

G E ..

/ 0+ $ 1 21 2$3 4

/ 0$ 51 263 4

+y/h

9

Fiber Section Model

! Typically use a more refined mesh where yielding is anticipated;however,

! Nonlinear strains tend to concentrate in a single element, thus, typically use an element length that is approximately equal to the plastic hinge length (e.g., 0.5lw). Might need to calibrate them first (this is essential).

! Calibration of fiber model with test results, or at least a plastic hinge model, is needed to impose a “reality” check on the element size and integration points used.

Actual cross section

Concrete Fibers

Steel Fibers

10

MaterialsUnconfined Concrete

Maximum permissible compressive strain for unconfined concrete (FEMA 356 S6.4.3.1)

7 = 0.002 or 0.005

Limit state associated with crack

width

Str

ess

(ksi

)

Strain

, - , -

2' '

0 0

' '0 85

2

Linear descending branch defined by:

0.002; and 0.0038; 0.85

c cc c c

c c c

f f f

f f

7 77 7

7 7

% &/ 0' ($ 8 91 2' (3 4) *

$ $

In the absence of cylinder stress-strain tests, Saatcioglu & Razvi (ASCE, JSE, 1992) recommend relation based on work by Hognestad.

11

Materials

Confined Concrete (FEMA 356 6.4.3.1)! Use appropriate model, e.g.:

" Saatcioglu & Razvi (ASCE JSE, 1992, 1995)"Mander (ASCE JSE, 1988)"Modified Kent & Park (ASCE JSE, 1982)

! For reference

! FEMA 356 Qualifications: "Maximum usable compression strain based on

experimental evidence and consider limitations posed by hoop fracture and longitudinal bar buckling.

12

MaterialsSteel Material:

Str

ess

(ksi

)

Strain

Maximum usable strain limits perFEMA 356 S6.4.3.1

7 = 0.02 7 = 0.05

13

General Wall Models/FE Modelse.g., RAM-PERFORM:! Flexure - fiber model (2-directions)! Shear - Trilinear backbone relation! Flexibility to model complex wall

geometry! Mesh refinement issues

Flexure/Axial Shear

Concentration of nonlinear Deformations in one element

14

Stiffness ModelingFEMA 356 Section 6.8.2.2 – Use Table 6.5! Uncracked: EIeffective = 0.8EIg

! Cracked: EIeffective = 0.5EIg

30 x 2 ft Wall Section16 - #14 Boundary#6@12" Web

CURVATURE

MO

MEN

T

P=0.30Agf'cP=0.20Agf'cP=0.10Agf'c1.0, 0.75, 0.5, 0.4EcIg

0.75EcIg 0.5EcIg

Wallace, et al., 4NCEE, Vol. 2, pp 359-368, 1990.

15

Response Correlation Studies! Ten Story Building in San Jose, California! Instrumented: Base, 6th Floor, and Roof! Moderate Intensity Ground Motions – Loma Prieta

4.53 m (14.88 ft)

1.68 m(5.5 ft)

PLAN VIEW: CSMIP BUILDING 57356

8.84 m (29 ft)

8.84 m (29 ft)

5 @ 10.97 m (36 ft)

16

Response Correlation Studies! Ten Story Building in San Jose, California! Instrumented: Base, 6th Floor, and Roof! Moderate Intensity Ground Motions – Loma Prieta

0 10 20 30Time (sec)

-1.5

0

1.5

Dis

plac

emen

t (in

.)

Analysis - 0.5Ig

Measured

17

Strength RequirementsACI 318 Provisions! Pn- Mn

" For extreme fiber compression strain of 7c =0.003.

! Vn" ACI 318-99,02,05 Equation 21-7

'

3.0 for / 1.52.0 for / 2.0

n cv c c t y

c w w

c w w

V A f f

h lh l

# :

##

% &$ 6) *$ "$ !

Linear interpolationallowed for intermediatevalues

18

Definition of Wall Cross Section

Flexural strength! Consider all vertical reinforcement within web

and within the effective flange width

Consider the influence of openings on the strength and detailing requirements ! ACI 318-02, 05 Appendix A – Strut & Tie Approach

Cross-Section Definition

beff

0.25hw

' ', ,

', ,

s bound s flange s

s bound s flange s

A A A

A A A

6

6

19

Behavior of Flanged WallsFlange Compression versus Tension

7t

7c

sAbeff

Flange CompressionLow compressive strainLarge curvature capacityMn & Vu similar rectangle

beff

Flange TensionLarge compressive strainLess curvature capacityMn ; Vu ;

7t

7c

, ,s bound s flangeA A6

20

Experimental ResultsRW2 & TW1: ~ ¼ scale tests

Thomsen & Wallace, ASCE JSE, April 2004.

Uncoupled designDisplacement-based design

21

Experimental Results

P = 0.09Agf'cvu,max = 4.85<f'c

-4.0 -2.0 0.0 2.0 4.0Top Displacement (in.)

-80

-40

0

40

80

Lat

eral

Loa

d (k

ips)

-2.8 -1.4 0.0 1.4 2.8Lateral Drift (%)

TW1RW2

P = 0.07Agf'cvu,max = 2.32<f'c

Abrupt Lateral Strength lossDue to buckling; Axial load Maintained

RW2

TW1

22

Experimental ResultsRW2 & TW2: ~ ¼ scale tests

Thomsen & Wallace, ASCE JSE, April 2004.

Displacement-based design of T-shape

23

Experimental Results

P = 0.075Agf'cvu,max = 5.5<f'c

-4.0 -2.0 0.0 2.0 4.0Top Displacement (in.)

-80

-40

0

40

80

Lat

eral

Loa

d (k

ips)

-2.8 -1.4 0.0 1.4 2.8Lateral Drift (%)

TW2RW2

P = 0.07Agf'cvu,max = 2.32<f'c

TW2

RW2Lateral strength loss due to lateralInstability due to spalling; Axial load maintained

24

Model Assessment –Comparison of Analytical and Experimental results

25

MVLE (Fiber) Model

h

(1-c)h

ch

12

3

45

6 Rigid Beam

Rigid Beam

k 1 k 2 knkH. . . . . . .

m

RC WALL WALL MODEL

1

2

. . . . .

Basic assumptions: • Plane sections (rigid rotation of top/bottom beams• Uniaxial material relations (vertical spring elements)

MVLE Model versus Fiber Model:• Similar to a fiber model except with constant curvature

over the element height (vs linear for fiber model)

Orakcal, Wallace, Conte; ACI SJ, Sept-Oct 2004.

26

Strain, 7

O

TensionNot to scale

Compression

( =7c ' , = f c

' )

(70, 0)

(70+ 7t , ft)

Material (Uni-axial) Models

Strain, 7

7y

E0

E1= bE0>y

OR

Concrete :• Chang and Mander (1994)

# Generalized (can be updated)# Allows refined calibration# Gap and tension stiffening

Reinforcing Steel :• Menegotto and Pinto (1973)• Filippou et al. (1984)

# Simple but effective# Degradation of

cyclic curvature

r

Stre

ss, >

27

Model Assessment$ Approximately 1/4 scale$ Aspect ratio = 3$ Displacement – based

evaluation for detailing provided at the wall boundaries

$ 12 ft tall, 4 ft long, 4 inches thick

$ #3 vertical steel, 3/16” hoops/ties

$ #2 deformed web steel$ Constant axial load$ Cyclic lateral

displacements applied at the top of the walls

28

Instrumentation

Wire Potentiometers(horizontal displacement)

Wire Potentiometers (X configuration)

Steel Strain Gage Levels

Wire Potentiometers(vertical displacement)

LVDT's

Concrete Strain Gages

Linear Potentiometers (Pedestal Movement)

Rigid Reference Frame

RW2

• Extensive instrumentation provided to measure wall response at various locations

Massone & Wallace; ACI SJ, Jan-Feb 2004.

29

Applied Lateral Displacement

-80

-40

0

40

80

-2

-1

0

1

2RW2

0 100 200 300 400 500 600 700 800Data Point Number

-80

-40

0

40

80

Top

Dis

plac

emen

t (m

m)

-2

-1

0

1

2

Drif

t Rat

io (%

)

Applied displacementPedestal movement excludedPedestal movement and shear deformations excluded

TW2

30

Model Details – RW2 1219 mm

19 mm 19 mm3 @ 51 mm 153 mm 3 @ 191 mm 153 mm 3 @ 51 mm

64 mm

19 mm

19 mm

102 mm

#2 bars (db=6.35 mm) Hoops (db=4.76 mm)8 - #3 bars

1 2 3 4 5 6 7 8uniaxial element # :

(db=9.53 mm) @ 191 mm @ 76 mm

m=16

1

2

. . . . .h

(1-c)h

ch

k 1 k2 knkH . . . . . . .

31

Model Details – TW2

19 mm 19 mm

3 @ 51 mm153 mm 3 @ 191 mm 153 mm

3 @ 51 mm

64 mm

19 mm

19 mm

1219 mm

3 @ 140 mm

102 mm

4 @ 102 mm

19 mm

102 mm

19 mm

3 @ 51 mm

102 mm

1219 mm

uniaxial element # : 1

2

345

6

7

8

9

10

12-19

118 - #3 bars(db=9.53 mm)

#2 bars (db=6.35 mm) @ 191 mm

Hoops (db=4.76 mm)@ 76 mm

#2 bars (db=6.35 mm) @ 140 mm

2 - #2 bars (db=6.35 mm)

Hoops and cross-ties (db=4.76 mm)@ 38 mm

8 - #3 bars(db=9.53 mm)

Hoops (db=4.76 mm)@ 32 mm

+

-

32

Concrete Model - Unconfined

0 0.001 0.002 0.003 0.004

Strain

0

10

20

30

40

50

Stre

ss (M

Pa)

Test Results1st Story2nd Story3rd Story4th Story

Analytical (Unconfined)

33

Concrete Model - Confined

0 0.005 0.01 0.015 0.02 0.025

Strain

0

10

20

30

40

50

60

70

Stre

ss (M

Pa)

Unconfined ModelMander et al. (1988)Saatcioglu and Razvi (1992)

RW2

TW2 Flange

TW2 Web

34

Concrete Model - Tension

0 0.0005 0.001 0.0015 0.002 0.0025

Strain

0

0.5

1

1.5

2

2.5

Stre

ss (M

Pa)

Chang and Mander (1994)Belarbi and Hsu (1994)

0 0.005 0.01 0.015 0.02 0.025 0.03

0

0.5

1

1.5

2

2.5(7t ,ft )

r

35

Reinforcement Material Model

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Strain

-600-500-400-300-200-100

0100200300400500600

Stre

ss (M

Pa)

#3 (RW2 & TW2 Flange)#3 (TW2 Web)#2 (TW2 Web)#2 (RW2 & TW2 Flange)

#3#2

0 0.02 0.04 0.06 0.08 0.1

0100200300400500600700

#3 rebar#2 rebar4.76 mm wire

Tension

Compression Test Results

36

Model Assessment – RW2

-80 -60 -40 -20 0 20 40 60 80

Top Flexural Displacement, +top (mm)

-200

-150

-100

-50

0

50

100

150

200

Late

ral L

oad,

Pla

t (k

N)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Lateral Flexural Drift (%)

TestAnalysis

5Pax 0.07Ag=f c'

Plat , +top

0100200300400500

P ax

(kN

)

RW2

37

Model Assessment – RW2

-80 -60 -40 -20 0 20 40 60 80

Lateral Flexural Displacement (mm)

0

1

2

3

4

5

Stor

y N

umbe

r

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Lateral Flexural Drift (%)

TestAnalysis

1.5%2.0%2.5%

0.75%1.0 %

RW2

Applied LateralDrift Levels:

Top

38

Model Assessment – RW2

-0.01

0

0.01

0.02

Rot

atio

n

( r

ad)

0 100 200 300 400 500 600 700-15-10

-505

1015

Dis

plac

emen

t

(

mm

)

TestAnalysis

RW2 (First Story)

Results based on recommended values for material parameters; however, results could vary, maybe significantly, for different element lengths and material parameters (particularly if no strain hardening)

1.5%2.0%

Data Point

0.008 FEMA 356 CP limit

39

Model Assessment – RW2

RW2Boundary Zone

100 150 200 250 300 350 400 450 500 550 600

Data Point

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Con

cret

e St

rain

Concrete Strain GageLVDTAnalysis

0.25% 0.5%0.75%

1.0%

1.5%

1.0%

2.0%

1.5%

Orakcal & Wallace; ACI SJ, in-press for publication in 2006 (see 13WCEE).

40

Model Assessment – RW2

RW2Boundary Zone

100 150 200 250 300 350 400 450 500 550 600

Data Point

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Con

cret

e St

rain

Concrete Strain GageLVDTAnalysis

0.25% 0.5%0.75%

1.0%

1.5%

1.0%

2.0%

1.5%

Orakcal & Wallace; ACI SJ, in-press for publication in 2006 (see 13WCEE).

41

Model Assessment – TW2

-80 -60 -40 -20 0 20 40 60 80

Top Flexural Displacement, +top (mm)

-400

-300

-200

-100

0

100

200

300

400

Late

ral L

oad,

Pla

t (k

N)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Lateral Flexural Drift (%)

TestAnalysis

5Pax 0.075Ag=f c'

Plat , +top

0250500750

P ax

(kN

)

TW2

C

T

T

C

42

Model Assessment – TW2

-80 -60 -40 -20 0 20 40 60 80

Lateral Flexural Displacement (mm)

0

1

2

3

4

5

Stor

y N

umbe

r

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Lateral Flexural Drift (%)

TestAnalysis

1.5%2.0%2.5%

0.75%1.0 %

TW2

Applied LateralDrift Levels:

Top

C

T

T

C

43

Model Assessment – TW2

-600 -400 -200 0 200 400 600

Distance along Flange from Web (mm)

-0.005

0

0.005

0.01

0.015

0.02

0.025

Flan

ge C

oncr

ete

Stra

in (

LVD

Ts)

TestAnalysis

0.5%1.0%2.0%2.5%

TW2

C

T

T

C

y7

2.0%

2.5%

2.5%

2.0%

44

Model Assessment – Stability

P = 0.09Agf'cvu,max = 4.85<f'c

-4.0 -2.0 0.0 2.0 4.0Top Displacement (in.)

-80

-40

0

40

80

Lat

eral

Loa

d (k

ips)

-2.8 -1.4 0.0 1.4 2.8Lateral Drift (%)

TW1TW2

P = 0.075Agf'cvu,max = 5.5<f'c

TW1 – Abrupt failure due to bucklingTW2 – Lateral instability due to spalling

and large compression

45

Model Assessment - Stability

Rebar Buckling at Wall Boundary Rebar Fracture Following Buckling at Wall Boundary

Instabilities, such as rebar buckling and lateral web buckling, and rebar fractureare typically not considered in models; therefore, engineering judgment is required. Loss of lateral-load capacity does not necessarily mean loss of axial load capacity

46

FEMA 356 Table 6-18

47

FEMA 356 Table 6-18

48

FEMA 356 – Modeling Parameters

' '

2

's

& 0.07 & Hoops @ 2" o.c.

2(0.027 in ) 0.09( )( 6" 3/ 8" 3 /16")(5 ksi / 63 ksi) 1.2" Non-confo

WALL RW2:

WALL TW2: Flange Compre

rming

8 - #3

ssio

A 10 - #

n

s s g c

c

s

A A P A f

s hs

A

$ $

$ $ 6 69

$ $

, - , -' 2

'

'

3 and 4 - #2 63 ksi & Hoops/Ties @ s=4"

No special detailing required: Conforming

0.42 in 63 ksi0.075(2) 0.127

4"(48")( 6 ksi)40 kips 2.7

4"(48") 6000 /1000

y

s s y

w w c

u

w w c

f

A A f Pt l f

Vt l f

5

% &8 6 8) *$ 6 $

$ $

!

49

FEMA 356 – Modeling Parameters

's

2

8 - #3 & 2 - #2 A 24 - #3 and 8 - #2 & 63 ksi

Hoops/Ties @ s=1.25" (5 legs and 2 legs)5(0.027 in ) 0.09( )( 16" 3/

WALL TW2: Flange

8" 3/16")(6 ksi / 63 ksi) 1.

Tension

"

(

0

2 0

s y

c

A f

s h s

$ $ 5

$ $ 6 6 9

, - ? @, -

2

'

'

'

.027 in ) 0.09( )( 2.5" 3/ 8" 3/16")(6 ksi / 63 ksi) 2.1" Conforming

16(0.11) 6(0.049) 63 ksi0.075(2) 0.26

4"(48")( 6 ksi)80 kips 5.4

4"(48") 6000 /1000

c

s s y

w w c

u

w w c

s h s

A A f Pt l f

Vt l f

$ $ 6 6 9

8 6 6$ 6 $

$ $

!

!

50

FEMA 356 – Modeling Parameters

Tables 6-18 (partial):

Model Parameters, Radians Walls Controlled by Flexure

'

')(

cww

yss

fltPfAA 68

Conf.

Bound. 'cww flt

V Plastic Hinge

a

Plastic Hinge

b

Residual Strength

c

" 0.1 Yes " 3 0.015 0.02 0.75

" 0.1 No " 3 0.008 0.015 0.60 ! 0.25 Yes ! 6 0.005 0.010 0.30

! 0.25 No ! 6 0.002 0.004 0.20

RW2TW2Flange Tension

TW2Flange Comp

51

FEMA Backbone Relation – RW2

, -, -

4

3

y

3

29.4 kips

3 0.5

29.4 (150") 0.41"3(4000 )(18,432 )0.008(144") 1.15"0.015(144") 2.16"

0.6(29.4 ) 17.6 kips

nlateral

w

lateral load

c g

k

ksi in

a

bk

residual

MPh

P h

E I

P

A

AA

$ $

% &' ($' () *

$ $

$ $$ $

$ $

52

FEMA Backbone Relations – TW2

, -, -

, -

4

3

y

3

4 48

40.2 kips

3 0.5

40.2 (150")3(4400 )(40,700 )0.25"

2.2 =34.5"

0.015(144") 2.16"0.020(144") 2.88"

0.75(40.2 ) 30.2 kips

nlateral

w

lateral load

c g

k

ksi in

g g x

a

bk

residual

MPh

P h

E I

I I y

P

A

AA

$ $

% &' ($' () *

$

$

$

$ $$ $

$ $

, -, -

, -

4

3

y

3

4 48

77.0 kips

3 0.5

77.0 (150")3(4400 )(40,700 )0.48"

2.2 =34.5"

0.005(144") 0.72"0.010(144") 1.44"

0.30(77.0 ) 23.1 kips

nlateral

w

lateral load

c g

k

ksi in

g g x

a

bk

residual

MPh

P h

E I

I I y

P

A

AA

$ $

% &' ($' () *

$

$

$

$ $$ $

$ $

Flange Compression Flange Tension

53

Backbone Curve – RW2

, -, -3/3n w w

yc cr

M h hE I

A $

P = 0.07Agf'cvu,max = 2.2<f'c psi

-4.0 -2.0 0.0 2.0 4.0Top Displacement (in.)

-40

-20

0

20

40

Late

ral L

oad

(kip

s)-2.8 -1.4 0.0 1.4 2.8

Lateral Drift (%)

Plat@Mn(7c=0.003)=29.4k-100

0

100

Late

ral L

oad

(kN

)

FEMA 356 NC/C

NC C

54

Backbone Curve – TW2

, -, -3/3n w w

yc cr

M h hE I

A $

P = 0.075Agf'c

-4.0 -2.0 0.0 2.0 4.0Top Displacement (in.)

-120

-80

-40

0

40

80

Late

ral L

oad

(kip

s)

-2.8 -1.4 0.0 1.4 2.8Lateral Drift (%)

Plat@Mn(7c=0.003)=77.0k

Plat@Mn(7c=0.003)=40.2k

-400

-200

0

200

Late

ral L

oad

(kN

)

FEMA 356 Conformingvu,max = 5.4<f'c psi

vu,max = 2.7<f'c psi

55

Cantilever Wall TestsPaulay, EERI, 2(4), 1986 [Goodsir, PhD 1985 NZ]

h = 3.3 m= 10.83 ft

(3.94”)

' 'g

3 3

y 3 '

& 0.163 A & Assume conforming

(70 )(130") 700.4" (10.0 ) 4.63 0.5 3(~

WALL Goodsir

3750 )(0.5)(4")(59") /12 (4")(59") 3750

0.01(33

, 1985:

00 ) 33

s s c

u

c g w w c

a

A A P f

VPL k kmmE I ksi psit l f

mm m

A

A

$ $

$ $ $ $ $

5 $ 0.015(3300 ) 50bm mm mmA 5 $

(59”)

ConformingP=10%, V=3

ConformingP=10%, V=6

56

Cantilever Wall TestsPaulay, EERI, 2(4), 1986 [Goodsir, PhD 1985 NZ]

h = 3.3 m= 10.83 ft

' 'g

3 3

y 3 '

& 0.12 A & Assume conforming

(70 )(130") 700.4" (10.0 ) 4.63 0.5 3(~ 3

WALL Goodsir,

750 )(0.5)(4")(59") /12 (4")(59") 3750

0.01(330

1

0

8

)

5

3

:

3

9

s s c

u

c g w w c

a

A A P f

VPL k kmmE I ksi psit l f

mm mm

A

A

$ $

$ $ $ $ $

5 $ 0.015(3300 ) 50b mm mmA 5 $

ConformingP=10%, V=3

ConformingP=10%, V=6

57

SummaryFEMA 356 Backbone Curves! In general, quite conservative! This appears to be especially true for cases where

moderate detailing is provided around boundary bars! Possible reformat

" Compute neutral axis depth" If s <12db over c/2, then modest ductility " If s < 8db and transverse steel ratio is ~1/2 of ACI 318-05,

then moderate ductility " If s < 8db and transverse steel ratio is > 3/4 of ACI 318-05,

then high ductility " Do not reduce deformation capacity for shear stress below 5

roots f’c

58

Shear DesignWall shear studies! Aktan & Bertero, ASCE, JSE, Aug. 1985! Paulay, EERI 1996; Wallace, ASCE, JSE, 1994.! Eberhard & Sozen, ASCE JSE, Feb. 1993

Design Recommendations! Based on Mpr at hinge region! Uniform lateral force distribution

, -, -, -lim

0.9 /10

0.3

prwall v u v

u

wall it m e

MV V n

M

V V D W weight A EPA

B B/ 0

$ $ 61 23 4

$ 6 $ $ $

Paulay, 1986

Eberhard, 1993

Slender Wall Behavior & Modeling

John WallaceUniversity of California, Los Angeles

With contributions fromDr. Kutay OrakcalUniversity of California, Los Angeles