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Slab – Column Frames
John WallaceUniversity of California, Los Angeles
with contributions from: Dr. Thomas H.-K. Kang Dr. Ian RobertsonUniversity of California, Los Angeles University of Hawaii, Manoa
2
Presentation OverviewCurrent Practice
Modeling & analysisConnection designProgressive collapseDeformation compatibility
Existing ConstructionPost-earthquake observationsModeling and Model AssessmentBackbone curves/Rehabilitation
3
Current PracticeNon-participating or “gravity” system
Post-tensioned slab-column frameSpan-to-depth ratios typically ~40+Use of shear reinforcement at slab-column connection to allow for thinner slabs or to eliminate drop panels
~1/3 scale shake table test specimen
4
Shear Reinforcement
Post-tensioning steel
~1/3 scale shake table tests: Kang & Walalce, ACI SJ, Sept-Oct. 2005
5
Gravity Load Analysis & DesignACI 318 Chapter 11, 13, & 21 Materials
Slab moments: Use direct design, Equivalent frame, or computer programConnection design – Chapter 11 & 13
EIcolumn
Effective slab width
wu = 1.2D + 1.6L
wu = 1.2D + 1.6L
EIslab = Ec(αβI2)
6
Gravity Load Analysis - MomentsGravity Analysis: 1.2D + 1.6L
Slab Moments
UnbalancedMoment
UnbalancedMoment
Design slab-column connection to transfer unbalanced moment to columnFEMA 356 refers to ACI 318 provisions
7
Unbalanced Moment TransferUnbalanced moment at the slab-column connection is transferred by two mechanisms:
Moment transfer (flexure) over a transfer width of c + 3h centered on the column
Eccentric shear on a critical section around the slab-column connection
Code provisions are covered in Chapter 13 (13.5) and Chapter 11 (11.12) of ACI 318
1 2
1 2
1where 1 (2 /3) /
, widths of critical section defined in 11.12.1.2
f f unbalanced
f
M M
b bb b
γ
γ
=
=+
=
1 2
(1 )
If b , then: 0.6 and 0.4
v f unbalanced v unbalanced
f v
M M M
b
γ γ
γ γ
= − =
== =
8
Unbalanced Moment TransferFlexural Transfer: c2 + 3h
γf Munb where γf is typically ~0.6 for square columnsRatio of top to bottom reinforcement of 2:1 recommended in ACI 318 (R13.5.3.3)
MRML
Munb = ML + MR
Unbalanced Moment(Interior connection)
c2+3h
h
c2
FEMA 356 6.5.4.3(2) allows use of c2+5h
9
Unbalanced Moment Transfer
( )u directgravity
o
Vv
b d=
Eccentric Shear transferCritical section is defined d/2 from column faceDirect shear stress
b0 = perimeter of critical section
Eccentric shear stress due to (1-γf)Munb = γvMunb
JzMv unb
vunb γ=
column
d/2d
c
a
b
c2+d
c1+d
zcentroid
z
c2+da
b c
d
c2+da
b c
d
10
Unbalanced Moment TransferCombined shear stressesCheck punching failure per 318
c2+da
b c
dDirect shearstress
z
c2+da
b c
dEccentric shearstress
+
Total shearstress
c2+da
b c
d
=
( )
'
' s
0
where =0.75
42
2 2
4
n c n u
cc
c c
'c
v v v v
f
dv Min f
b
f
φ φ φ φ
β
α
= ≥
⎧ ⎫⎛ ⎞+⎪ ⎪⎜ ⎟
⎪ ⎪⎝ ⎠⎪ ⎪⎛ ⎞+⎪ ⎪= +⎜ ⎟⎨ ⎬
⎝ ⎠⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭
ACI Eq. 11-33, -34, -35 ,maxu nv vφ≤
11
Laboratory Studies
Progressive collapse -continuous bottom steel (2 bars)ACI 318-05 7.13.2.5 (13.3.8.5)
Photo: Hwang and Moehle, ACI SJ, March-April 2000.
Interior connection
Exterior connection
12
ACI Committee 352.1R89Slab – Column Report
1 20.5 usm
y
w l lAfφ
=
Recommendations for the design of slab-column connections in monolithic RC Structures, ACI-ASCE Committee 352, Report 352.1R-89 (reapproved 1997)
spalling
kinkBottom bar at angle of30 degrees from horizontal
13
Deformation Compatibility
ImposedlateralDisplacements(new design)
EIcolumn
EIslab = Ec(αβI2)
wu = 1.2D + 0.5L
wu = 1.2D + 0.5L
Slab – column (gravity) frame assessmentIncluded in the model with the lateral system
Pushover Analysis(Assessmentof existing)
14
Deformation CompatibilityDetermine if the connection can resist the Vu & Munb without punching failure – Adequate strength. (ACI 318-05 21.11.5)
Flexural transfer, eccentric shear stress model
Limit analysis approach – for connections with a fuse
this does not consider the potential for shear strength degradation.
0 1 2 3 4 5 6 7Ductility (µ)
Shea
r Dem
and
(Vu)
Shea
r Cap
acity
(Vn)
Stress-inducedPunching
Shear CapacityShear Demand
Drift-induced Punching
Slab Moments
Munb , VuMunb , Vu
15
Alternative - Deformation CompatibilityVerify that punching failures do not occur for gravity shear combined with imposed interstory displacement for ∆M (new) or δtarget (Rehab). Adequate deformability. (ACI 318-05 21.11.5)
RC interior and exterior (limited data) connections
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Gravity Shear Ratio (Vg /φVc), where Vc = (1/3)f'c
1/2bod
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Drif
t Rat
io a
t Pun
chin
g
.
.
.
.
Isolated RC "Interior" Connections2,10,14,15,16,17
Subassemblies6
Nine-panel Frame18
Isolated RC "Edge" Connections9
Best-Fit Line for Interior Connections
without Shear Reinforcement
ACI 318-05 Limit
Relationshipfor RC with stud-rails
(Robertson et al.10)
16
Deformation CompatibilityPT Connections without shear reinforcement
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Gravity Shear Ratio (Vg /φVc), where Vc = (0.29f'c
1/2+0.3fpc)bod
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Drif
t Rat
io a
t Pun
chin
g
Qaisrani ('93) - Int.Trongtham et al. ('77) - Int.
Shatila ('87) - Ext.Trongtham ('77) - Ext.
Martinez ('93) - Ext.Martinez ('93) - Corner
.
.
.
.
.
.
.
.Best-Fit Line for
PT without Shear Reinforcement
ACI 318-05 Limit
Pimanmas ('04) - Int.
Foutch et al. ('90) - Ext.
Best-Fit Line forPT Subjected to Reversed Cyclic
Loading
17
Presentation OverviewCurrent Practice
Modeling & analysisConnection designProgressive collapseDeformation compatibility
Existing ConstructionBackground & observationsModeling and Model AssessmentBackbone curves/Rehabilitation
18
Older Construction Gravity design, or relatively low lateral forces used for design
No continuous bottom reinforcementthrough column cage
Bent reinforcementsometimes used
19
Post-Earthquake Observations
Bullock’s Department Store - Northridge Fashion Mall
20
Presentation OverviewCurrent Practice
Modeling & analysisConnection designProgressive collapseDeformation compatibility
Existing ConstructionBackground & observationsModeling and Model AssessmentBackbone curves/Rehabilitation
21
Modeling OverviewHow to model…
Lateral stiffness?Connection behavior?
How good are our models?Shake table studies
FEMA 356 backbone curvesBasis of existing curvesNew information?
22
Modeling Assumptions - Typical
EIeff = effectivecolumn stiffness
EIeff
Rigid end zones at joints
EIeff = αβEIg
Model slab with “an effective beam”
23
Analytical Model - Column Stiffness
P = PG
P = PG - PE
P = PG + PE
0.38 EIg
0.42 EIg0.39 EIg
M
φExterior Column
EIcr,col ≈ 0.4 EIg,col
Interior Column
EIcr ≈ 0.45 EIg
P = PGM
φ
My
Фy
PG = axial from gravity and PE = axial from earthquakeAnchorage slip – not likely as significant as noted for beam –column frames (see Elwood presentation)
24
Analytical Model – Slab Flexural StiffnessEffective Beam Width Model
l2
α: Effective Beam Width Factor
αl2
Applied lateral loads
l1
CL
CL
Allen & Darvall, ACI 74(7), 1977.Grossman, ACI 94(2), 1997. Hwang & Moehle, ACI 97(1), 2000.Kang & Wallace, ACI 102(5), 2005.
RC PTαβ
0.75 0.650.33 0.5
Kang & Wallace (2005)
αβl2
β: Coefficient accounting for Cracking
25
Analytical Modeling - Connections
Munbalanced @connectionFlexure c2+3h (5h)Eccentric shear Mn = Mf/0.6
M-n / M+
n @ column strip
Punching before or after yieldingKang, 13WCEE, Aug. 2004, Paper 1119
Slab
Connection( rigid plastic spring )Column
( fiber element )
Column strip spring
This model satisfies FEMA 356 6.5.4.2.2, which states that the connection must be modeled separately from slab and column elements.
M
θ, δMn
Vg / Vo0.75
0.0375
Limit State Model: Mean – 1 σ
θ, δ
Limit
26
Connection Modeling - Punching
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Gravity Shear Ratio (Vg /φVc), where Vc = (1/3)f'c
1/2bod
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Drif
t Rat
io a
t Pun
chin
g
.
.
.
.
Isolated RC "Interior" Connections2,10,14,15,16,17
Subassemblies6
Nine-panel Frame18
Isolated RC "Edge" Connections9
Best-Fit Line for Interior Connections
without Shear Reinforcement
ACI 318-05 Limit
Relationshipfor RC with stud-rails
(Robertson et al.10)
27
Shake Table Studies
Two stories, 2 × 2 baysApproximately 1/3 scale
W C E
S
N
CL
CL
1.82 m
Kang and Wallace, ACI SJ, Sept-Oct, 2005, another paper in-press.
28
RC Specimen
Six 200 x 200 mm columns90 mm thick slab
9.5 mm rebar fy = 414 MPaf’c = 28 MPa
4.3 m
4.1 m
29
RC Specimen - Reinforcement
Interior Connection Shear Reinforcement
Expected connection behavior: Flexural yielding, followed by punching
30
5.7 m
5.7 m
8 mm 7-wire strand
6.35 mm deformed rebar
PT Specimen
31
PT Specimen – Interior ACI318-05 Requires only bottom (integrity) reinforcement
32
PT Video – Run 5
33
Model Assessment - NSP-4 -3 -2 -1 0 1 2 3 4
Top Drift Ratio [%]
-1.5
-1
-0.5
0
0.5
1
1.5
Bas
e S
hear
[W]
-80 -60 -40 -20 0 20 40 60 80
Top Displacement Relative to Footing [mm]
-250
-200
-150
-100
-50
0
50
100
150
200
250B
ase
Shea
r [kN
]RC-RUN4-ExpPush-over (2:1)Push-over (1:2)
θu = 2.5%
RC-RUN4
H
2HH
2H(1:2 Ratio) (2:1 Ratio)
Top displacement relative to footing [mm]
Base
She
ar [
kN]
Base
She
ar [
W]
Top Drift [%]
34
Model Assessment - PT
10 12 14 16 18 20 22
Time (sec)
-0.04
-0.02
0
0.02
0.04
-0.04
-0.02
0
0.02
0.04
Top
Drif
t Rat
io
(a) PT-RUN4 : ym - σres caseExperimental Response HistoriesAnalytical Response Histories
Measured Top Drift at Peak Base Shear (2.78%)
Measured Top Drift at Peak Base Shear (2.78%)
See Kang et al., 13WCEE, August 2004, paper 1119Direct measurement of footing rotations
35
Presentation OverviewCurrent Practice
Modeling & analysisConnection designProgressive collapseDeformation compatibility
Existing ConstructionBackground & observationsModeling and Model AssessmentBackbone curves/Rehabilitation
Shear reinforcement
36
Deformation – Backbone CurvesModel Parameters, Radians Slabs Controlled by
Flexure
0
gravityVV
Continuity Reinforcement
Plastic Hinge
a
Plastic Hinge
b
Residual Strength
c
≤ 0.2 Yes 0.02 0.05 0.2 > 0.4 Yes 0.0 0.04 0.2 ≤ 0.2 No 0.02 0.02 --
≥ 0.25 No 0.0 0.0 --
Continuity reinforcement defined as at least one bottom bar or pt bar continuous through the column cage in each direction
a b - a
c
Vu = 1.2D + 0.5L
37
Slab – Column TestsTypical test setup
Strong Floor
Reactionblock
Cyclic lateral loadLC
column
Gravity load
Load cell
Load cell
slabδ
columnδ
Axial load
38
New Data – Test Results #1Slab reinforcing details
Robertson & Johnson, 13WCEE, August 2004, Paper 143
10ft x 10 ft x 4.5” 2 Continuous bottom bars
39
Test Results #1 – Control Specimen
Robertson & Johnson, 13WCEE, August 2004, Paper 143
0.23g
c
VV
=
Yield: 1.5% (assumed)2
n
p
(12)(71 mm )(414 MPa)(95mm)=33,500 kN-mmP = 33,500 kN-mm/1524mm = 22 kN
0.015; 0.02(17 / 20) 0.017
n
e
M
θ θ
=
= = =
continuity
40
Test Results #1
0.28g
c
VV
=
Robertson & Johnson, 13WCEE, August 2004, Paper 143
2
n
p
(12)(71 mm )(414 MPa)(95mm)=33,500 kN-mmP = 33,500 kN-mm/1524mm = 22 kN
0.015; 0.02(12 / 20) 0.012
n
e
M
θ θ
=
= = =
continuity
41
Test Results #1
0.48g
c
VV
=
Robertson & Johnson, 13WCEE, August 2004, Paper 143
2
n
p
(12)(71 mm )(414 MPa)(95mm)=33,500 kN-mmP = 33,500 kN-mm/1524mm = 22 kN
0.015; 0
n
e
M
θ θ
=
= =
42
Test Results #1 - SummaryFEMA 356 – Overall comparison
Robertson & Johnson, 13WCEE, August 2004, Paper 143
43
Test Results - #2Slab reinforcing details – less reinforcement
Robertson & Johnson, 13WCEE, August 2004, Paper 143
10ft x 10 ft x 4.5” 2 Continuous bottom bars
44
Test Results
Robertson & Johnson, 13WCEE, August 2004, Paper 143
0.36g
c
VV
=
2
n
p
(7)(71 mm )(414 MPa)(95mm)=19,500 kN-mmP = 19,500 kN-mm/1524mm = 13 kN
0.015; 0.02(4 / 20) 0.004
n
e
M
θ θ
=
= = =
45
Test Results - Summary
Durrani, Du, Luo, ACI SJ, July-Aug. 1995
0.30g
c
VV
≈ p0.01 0.02eθ θ≈ =
Backbone relation:P = 10 kip (arbitrary)
spandrel
Straight bars vsBent up bars- no difference - except for collapse
46
Presentation OverviewCurrent Practice
Modeling & analysisConnection designProgressive collapseDeformation compatibility
Existing ConstructionBackground & observationsModeling and Model AssessmentBackbone curves/Rehabilitation
Shear reinforcement
47
SummaryModeling
Effective beam width modelConnection behavior – Limit state model
Backbone curves - RCConservative – In generalReview allowable plastic rotation for low gravity stress ratios < 0.2, mean - σPotential to increase plastic rotation for low reinforcement ratiosRemove residual capacity for RC connections
48
SummaryBackbone curves - PT
ConservativeIncrease plastic rotation from 0.02 (RC) to 0.03 at gravity shear ratio of 0.2Review higher gravity shear ratios – allowable plastic rotation of 0.01 at a gravity shear ratio of 0.5Allow residual capacity of 0.2 up to drifts of about 5% where one strand pass within the column cage in both directions.
Slab – Column Frames
John WallaceUniversity of California, Los Angeles
with contributions from: Dr. Thomas H.-K. Kang Dr. Ian RobertsonUniversity of California, Los Angeles University of Hawaii, Manoa
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