reinforced concrete beam - column joint: macroscopic super...
TRANSCRIPT
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Reinforced Concrete Beam-Column Joint: Macroscopic
Super-element models
-Nilanjan Mitra
(work performed as a PhD student while at University of Washington between 2001-2006)
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Need for the study
Reinforced concrete beam column jointssubjected to earthquake loading
Experimental Investigation@ UW
I-280 Freeway, San Francisco, CAfollowing Loma Prieta Earthquake in 1989
Courtesy: NISEE, Univ. of California, Berkeley.
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Loading in a joint region
Earthquake Loading of Beam-Column Joint
compression resultant (concrete and steel)
shear resultant (concrete)
Earthquake Loading of Beam-Column Joint
compression resultant (concrete and steel)
compression resultant (concrete and steel)
shear resultant (concrete)
shear resultant (concrete) tension resultant (steel)
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anchorage bond stress acting on joint core concrete
compression force carried by joint core concrete
Internal load distribution in a joint
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Macroscopic beam-column joint element models
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Macroscopic beam-column joint element models
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Macroscopic beam-column joint element models
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shear panel
external node
internal node
rigid externalinterface plane
shown with finite widthto facilitate discussion
beam element
zero-width region
interface-shear spring
bar-slip spring
zero-length
zero-length
elem
ent
colu
mn
Proposed Beam-column super-element model• 4-noded 12-dof element• 8 bar-slip springs to simulate
anchorage failure• 4 interface-shear springs to simulate
shear transfer failure at joint interface• 1 shear-panel to simulate inelastic
action of shear within joint core
Note: The location of the bar-slipsprings is at the centroid of thetension-compression couple at nominalstrength of the beams.
[Mitra & Lowes; J. Structural Eng. ASCE, 2007: 133 (1): 105-20 ]
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Joint element formulation: Kinematics
External, Internal and Component deformation
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Joint element formulation: Equilibrium
External, Internal and Component forces
Solution of element state achieved by an iterative procedure and requires solving for zero reaction in the 4 internal degrees of freedom
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Characterized by Response envelope Unload reload path Damage rules
Hysteretic one dimensional material model
deformation
load
(ePd1,ePf1)
(ePd4,ePf4)
(ePd3,ePf3)(ePd2,ePf2)
(eNd3,eNf3)(eNd2,eNf2)
(*,uForceP.ePf3)
(dmin,f(dmin))
(dmax,f(dmax))
(rDispP.dmax,rForceP.f(dmax))
(rDispN.dmin,rForceN.f(dmin))
(*,uForceN.eNf3)
(eNd1,eNf1)
(eNd4,eNf4)
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Damage simulation in material model
-0.015 -0.01 -0.005 0 0.005 0.01 0.015-8
-6
-4
-2
0
2
4
6
8
deformation
load
without damagewith unloading stiffness damage
-0.015 -0.01 -0.005 0 0.005 0.01 0.015-8
-6
-4
-2
0
2
4
6
8
deformationlo
ad
without damagewith reloading stiffness damage
-0.015 -0.01 -0.005 0 0.005 0.01 0.015-8
-6
-4
-2
0
2
4
6
8
deformation
load
without damagewith strength damage
( ) ( )( )3 4
1 max 2i dα αδ α α χ= +
max minmax
max min
max. ,i id dd
def def
=
( ).f No of load cyclesχ =
( )f Accumulated Energyχ =
( )0 1 ki ik k δ= −
( ) ( ) ( )max max 01 f
iif f δ= −
( ) ( ) ( )max max 01 d
iid d δ= +
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Damage simulation in material model
load historyi
monotonic
monotonic load history
dEE
EgE dE
χ = =
∫
∫
-0.015 -0.01 -0.005 0 0.005 0.01 0.015-8
-6
-4
-2
0
2
4
6
8
deformation
load
without damagewith all 3 damage rules
-0.015 -0.01 -0.005 0 0.005 0.01 0.015-5
-4
-3
-2
-1
0
1
2
3
4
5
deformation
load
with all 3 damages (Energy)with all 3 damages (Cyclic)
max4duu
χ =∑
Energy criterion
No. of load cycle criterion: rain-flow-counting algorithm
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Shear-panel calibration
column
shear panel
• Shear panel envelope calibration• MCFT• Diagonal compression strut
• Compression envelope reduction
• Determination of hysteretic model parameters
Typical response envelope
Observed Simulated
Spec
imen
SE8
(S
teve
ns e
t al.
1987
)
-0.012 -0.008 -0.004 0 0.004 0.008 0.012-10
-8
-6
-4
-2
0
2
4
6
8
10
Shear strain
Shea
r st
ress
(MPa
)
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Shear panel envelope calibration using proposedDiagonal compression strut mechanism
_ cosc strut strut strutstrut
jnt
f ww
ατ
⋅ ⋅=
• Mander et al. (1988) concrete
• Column longitudinal and joint hoop
steel confine the strut.
• Reduction in concrete to account for
perpendicular tensile stress to the strut
cyclic loading.
• Strut force is converted to panel shear
stress as
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2_
_3.62 2.82 1 0.39
0.45 0.39
c strut t t t
c Mander cc cc cc
t
cc
ff
ε ε εε ε ε
εε
= − + ∀ <
= ∀ ≥
Proposed concrete compression envelope reduction
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
εt / εcc
f c_ob
s / f c_
Man
der
Data with ρj > 0
Data with ρj = 0
Vecchio 1986Stevens 1991Hsu 1995Noguchi 1992Proposed eq. for ρ
j > 0
Proposed eq. for ρj = 0
2_
_0.36 0.60 1 0.83
0.75 0.83
c strut t t t
c Mander cc cc cc
t
cc
ff
ε ε εε ε ε
εε
= − + ∀ <
= ∀ ≥
0jρ =0jρ >Eq. for Eq. for
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Comparison of MCFT and Diagonal Compression Strut model in shear-panel envelope calibration
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
φ
τ mcf
t_cy
clic
/ τ m
ax
0.55JFBYJFBY
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
φ
τ diag
onal
_stru
t / τ
max
JFBYJFBY
[Lowes, Altoontash and Mitra, J. Structural Eng. ASCE, 2005: 131 (6) ]
Transverse steel contribution to shear stress
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Bar slip material model calibration
column
Bar-slip spring• Mechanistic model :- envelope
• Hysteretic model calibration
• Strength deterioration model
-2 0 2 4 6 8 10 12 14 16-1000
-500
0
500
1000
slip (mm)
bar-
spri
ng fo
rce
(kN
)
Typical response envelope
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Bar slip mechanistic model
Assumptions for anchorage response of bond within the joint region:
• Bond stress uniform for elastic reinforcement, piecewise uniform for reinforcement
loaded beyond yield
• Slip is the relative movement of reinforcement bar with respect to the joint perimeter
• Slip is a function of strain distribution in the joint
• Bar exhibits zero slip at zero bar stress
2
0
2fsl
fsE b Eslip s y
b b
ldd x dx f fA E E d
τ π τ⋅= = ∀ <
⋅∫
( )0
e ye
e
l llyE b Y b
slip eb b hl
fd dd x dx x l dxA E E A E
τ π τ π+ ⋅ ⋅
= + + − ⋅ ⋅ ∫ ∫
22
2 2y y yeE Ys y
b b
f l ll f fE d E E dτ τ
= + + ∀ ≥Mechanistic model
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Strength deterioration– Is activated once slip exceeds the slip level corresponding to ultimate stress
in the reinforcing bars.– Is observed upon reloading, with the result that bar-slip springs always
exhibit positive tangent stiffness.
0 5 10 15 200
1
2
3
4
5
specimen number
max
imum
slip
/
slip
with
anc
hora
ge le
ngth
equ
al to
join
t wid
th
BYJFBY
0 5 10 15 200
5
10
15
20
specimen number
Sim
ulat
ed m
axim
um b
ar-s
lip
BYJFBY
( )max, lim max,f f
i i ult i ultd d d dδ α δ= − ≤ ∀ ≥
Strength deterioration calibration for bar-slip spring
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Steps for calibrating the joint model• Calculate moment curvature of beams and columns• From moment curvature analysis determine
• moment associated with first yield of the reinforcing bar• tension-compression couple distance at nominal yield strength• neutral axis depth at nominal yield strength
• Define joint elements parameters using joint geometry and tension-compression couple distance
• Determine concrete compression strut response• Mander model for concrete• Concrete strength reduction eq. proposed to account for perpendicular cracks
and cyclic loading• Hysteretic parameters defined for shear panel
• Determine bar-slip response• Mechanistic model for bond• Hysteretic parameters defined for bar-slip model
• Interface slip-springs are defined to be stiff and elastic
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Concrete Stress-Strain(Compressive only, no tensile strength)
Reinforcing Steel Stress-Strain
Beam-Column Elements:Force based lumped plasticity element
Plastic Hinge region
Elastic region
Fiber discretisation
joint element
plastic hinge length
column axial load applied under load control
beam-column element
lateral load applied under displacement
control
Model simulationL
ab te
st
OpenSees Model
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Validation study
-6 -4 -2 0 2 4 6-300
-200
-100
0
100
200
300
Drift (%)
Col
umn
shea
r (kN
)
-6 -4 -2 0 2 4 6-300
-200
-100
0
100
200
300
Drift (%)
Col
umn
shea
r (kN
)
-6 -4 -2 0 2 4 6-300
-200
-100
0
100
200
300
Drift (%)
Col
umn
shea
r (kN
)Specimen OSJ10:
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Validation study discussion & conclusion• Failure mechanism
– For joints exhibiting JF (joint failure prior to beam yielding), 82% accurate.
– For joints exhibiting BYJF (beam yielding followed by joint failure), 89% accurate.
– For joints exhibiting BY (beam yielding), 94% accurate.• Initial and unloading stiffness
– For all joints, mean of simulated to observed ranges from 1.03 to 1.06 with an average C.O.V. = 0.15.
• Post-yield tangent stiffness– For joints that exhibit BYJF, mean ratio of simulated to observed is 1.0
with a C.O.V. = 0.22.• Maximum strength
– For all joints, mean of simulated to observed is 1.03 with a C.O.V. = 0.17.
• Drift at maximum strength– For all joints, mean of simulated to observed is 1.12 with a C.O.V. =
0.27.• Strength at final drift level
– For all joints, strength for final drift cycle is 1.04 with a C.O.V = 0.2.• Pinching ratio (ratio of strength at zero drift to maximum strength)
– For all joints, pinching ratio is 1.04 with a C.O.V = 0.12.