reinforced concrete beam-column joint macroscopic super-element models
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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 joints
subjected to earthquake loading
Experimental
Investigation
@ UW
I-280 Freeway, San Francisco, CA
following 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 concretecompression 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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shearpanel
external node
internal node
rigid externalinterface plane
shown with finite widthto facilitate discussion
beamelement
zero-width region
interface-shear spring
bar-slip spring
zero-length
zero-length
element
column
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-slip
springs is at the centroid of the
tension-compression couple at nominal
strength 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 damage
with 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
deformation
load
without damage
with 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 damage
with strength damage
3 41 max 2i d
max min
max
max min
max. ,i id d
ddef def
.f Noofloadcycle
fAccumulatedEner
01 ki ik k max max 01
f
iif f
max max0
1d
ii
d d
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Damage simulation in material model
loadhistoryi
monotonic
monotonicloadhistory
dEE
EgE d
-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 damage
with 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)
max4
du
u
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
SpecimenSE
8
(Stevensetal.1
987)
-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
Shearstress(MPa)
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Shear panel envelope calibration using proposed
Diagonal compression strut mechanism
_ coscstrut strut strstrut
jnt
f w
w
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.
0.45 0.
cstrut t t t
cMander cc cc cc
t
cc
f
Proposed concrete compression envelope reduction
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
t
/ cc
f
c_obs
/fc
_Mander
Data with j
> 0
Data with j
= 0
Vecchio 1986
Stevens 1991
Hsu 1995
Noguchi 1992
Proposed eq. forj
> 0
Proposed eq. forj
= 0
2
_
_
0.36 0.60 1 0.
0.75 0.
cstrut t t t
cMander cc cc cc
t
cc
f
0j 0j 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
mcft_cyclic/max
0.55
JF
BYJF
BY
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
diagonal_strut
/max
JF
BYJF
BY
[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)
ba
r-springforce(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
2
fsl
fsE b Eslip s
b b
ldd xdx f
AE Ed
0
eye
e
lll
yEb Y bslip e
b bhl
fd dd xdx xlAE EAE
22
2 2yy yeE Y
s
b b
fl llf
EdEEd
Mechanistic model
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Strength deterioration
Is activated once slip exceeds the slip level corresponding to ultimate stressin the reinforcing bars.
Is observed upon reloading, with the result that bar-slip springs alwaysexhibit positive tangent stiffness.
0 5 10 15 200
1
2
3
4
5
specimen number
maximumslip/
slipwithanchoragelengtheq
ualtojointwidth
BYJF
BY
0 5 10 15 200
5
10
15
20
specimen number
Simulatedmaximum
bar-slip
BYJF
BY
max, lim max,f fi i ult i l d d d
Strength deterioration calibration for bar-slip spring
S f lib i h j i d l
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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
M d l i l i
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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 simulation
Labt
est
OpenSees Model
V lid ti t d
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Validation study
-6 -4 -2 0 2 4 6-300
-200
-100
0
100
200
300
Drift (%)
Columnshear(kN)
-6 -4 -2 0 2 4 6-300
-200
-100
0
100
200
300
Drift (%)
Columnshear(kN)
-6 -4 -2 0 2 4 6-300
-200
-100
0
100
200
300
Drift (%)
Columnshear(kN)
Specimen OSJ10:
V lid ti t d di i & l i
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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 1.06with an average C.O.V. = 0.15.
Post-yield tangent stiffness
For joints that exhibit BYJF, mean ratio of simulated to observed is 1.0with 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.
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