Slender Wall Behavior & Modeling -...
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Slender Wall Behavior & Modeling
John WallaceUniversity of California, Los Angeles
with contributions fromDr. Kutay OrakcalUniversity of California, Los Angeles
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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
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FEMA 356 –Nonlinear Modeling for Buildings with Slender RC Walls
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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
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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
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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
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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
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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
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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
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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.
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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.
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MaterialsSteel Material:
Str
ess
(ksi
)
Strain
Maximum usable strain limits perFEMA 356 S6.4.3.1
7 = 0.02 7 = 0.05
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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
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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.
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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)
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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
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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
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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
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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
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Experimental ResultsRW2 & TW1: ~ ¼ scale tests
Thomsen & Wallace, ASCE JSE, April 2004.
Uncoupled designDisplacement-based design
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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
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Experimental ResultsRW2 & TW2: ~ ¼ scale tests
Thomsen & Wallace, ASCE JSE, April 2004.
Displacement-based design of T-shape
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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
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Model Assessment –Comparison of Analytical and Experimental results
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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.
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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, >
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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
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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.
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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
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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 . . . . . . .
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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
+
-
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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)
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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
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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
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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
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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
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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
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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
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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).
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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).
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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
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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
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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%
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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
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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
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46
FEMA 356 Table 6-18
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47
FEMA 356 Table 6-18
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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 $
$ $
!
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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 $
$ $
!
!
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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
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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
$ $
% &' ($' () *
$ $
$ $$ $
$ $
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52
FEMA Backbone Relations – TW2
, -, -
, -
4
3
y
3
4 48
40.2 kips
3 0.5
40.2 (150")3(4400 )(40,700 )0.25"
2.2 =34.5"
0.015(144") 2.16"0.020(144") 2.88"
0.75(40.2 ) 30.2 kips
nlateral
w
lateral load
c g
k
ksi in
g g x
a
bk
residual
MPh
P h
E I
I I y
P
A
AA
$ $
% &' ($' () *
$
$
$
$ $$ $
$ $
, -, -
, -
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
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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
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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
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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
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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
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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
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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
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Slender Wall Behavior & Modeling
John WallaceUniversity of California, Los Angeles
With contributions fromDr. Kutay OrakcalUniversity of California, Los Angeles