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 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= +

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

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

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

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

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

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

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

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

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

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

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:

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.