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)
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.
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)
anchorage bond stress acting on joint core concrete
compression force carried by joint core concrete
Internal load distribution in a joint
Macroscopic beam-column joint element models
Macroscopic beam-column joint element models
Macroscopic beam-column joint element models
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 ]
Joint element formulation: Kinematics
External, Internal and Component deformation
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
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)
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 41 max 2i d = +
max minmax
max min
max. ,i id d
ddef def
=
( ).f No of load cycles =
( )f Accumulated Energy =
( )0 1 ki ik k = ( ) ( ) ( )max max 0 1
fii
f f = ( ) ( ) ( )max max 0 1
dii
d d = +
Damage simulation in material model
load historyi
monotonic
monotonic load history
dEE
EgE dE
= =
-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
=
Energy criterion
No. of load cycle criterion: rain-flow-counting algorithm
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
)
Shear panel envelope calibration using proposedDiagonal compression strut mechanism
_ cosc strut strut strutstrut
jnt
f ww
=
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
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
= + 0
Data with j = 0
Vecchio 1986Stevens 1991Hsu 1995Noguchi 1992Proposed eq. for
j > 0
Proposed eq. for j = 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
= + Eq. for Eq. for
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
mcf
t_cy
clic
/ m
ax
0.55JFBYJFBY
0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
dia
gona
l_st
rut /
m
ax
JFBYJFBY
[Lowes, Altoontash and Mitra, J. Structural Eng. ASCE, 2005: 131 (6) ]
Transverse steel contribution to shear stress
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
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
= =