MULTIDISCIPLINARY CENTER FOR EARTHQUAKE ENGINEERING RESEARCH

Weakening and damping structure using rocking columns and energy dissipating devices

Hwasung Roh: Ph. D. CandidateAndrei M. Reinhorn: Advisor

Department of Civil, Structural and Environmental Engineering

ABSTRACTIn yielding structures acceleration response is proportional to structural strength after yielding. Structural strength needs to be controlled in order to protect structural and nonstructural components. The current retrofit methods, such as bracing systems, lead to increase in the strength. Correspondingly, story accelerations are increased, while the displacement demands are decreased. Also the ductility demand is decreased. Damping and strength reduction can control both accelerations and displacement response. Reducing structural strength, defined here as ”weakening”, can be implemented by allowing the columns or beams to rotate freely at connections. Rocking columns, the subject of this work, are alternative techniques for strength reduction. In the conventional structures, damage may occur at all stories due to the story drifts. When the rocking columns are installed to support only gravity loads, only small damage is developed at the toes of such columns. Therefore, with rocking column systems, the structures can control more easily the damage. Globally however, the control of deformations should be done by other parts of the structural system. This study is focused to the evaluation of rocking columns. Comparing with the conventional columns, the boundary conditions of rocking column are varied continuously. Furthermore, the rocking column has two stress distribution types, linear and nonlinear. The nonlinear stress distribution is developed around column base because of the lack of tension resisting capacity due to the boundary condition. The range of nonlinear stress distribution near the contact face, defined as “transitional height”, should be determined. In this study, this height contributing to the lateral and vertical displacement significantly is derived from the mechanical models. Analytical results of force-displacements are derived and compared with FEM results showing nonlinear response even in elastic range. Macroscopic models are developed. Hysteretic behavior of rocking column is determined from the permanent deformation of each toe under cyclic load. However, this energy dissipating quantity is not enough to provide the required capacity. Therefore, devices to provide energy dissipating are needed to make damping structures. Such devices can be replaced after earthquakes.

.

Comparison of structural characteristics b/w conventional and rocking column

OBJECTIVESDetermine:§Height of nonlinear stress distribution§Analytical tools for the lateral and axial stiffness of rocking column§Comparison the analytical results with FEM analysis results§Macroscopic models of the rocking column element for IDARC2D (Inelastic analysis program)

METHODS & RESULTS§Height of nonlinear stress distribution (called “Transitional height” in this research)- The lack of tensile resisting capacity in the rocking column- Separating tensile and compressive resisting components in “Vertex up” and “Vertex down”

§Lateral displacement and lateral stiffness of rocking column

- Moment Area Method <For unconfined condition>

§Vertical displacement and axial stiffness of rocking column-Mechanical deformation +Geometric uplifting displacement

<For unconfined condition>§Macroscopic rocking column element for IDARC2D-Structural element (Conventional : Rocking)

-Spread plasticity model (Conventional : Rocking) -Yield penetration model

-Structural combination (Conventional : Rocking)

ACKNOWLEDGEMENTSThis work is supported by MCEER Program area Thrust 2: Seismic Retrofit of Acute Care Facilities, Principal investigator and/or faculty advisor: Prof. A. M. Reinhorn, University at Buffalo (SUNY).

<BMD>

Cracking: Yielding: UltimateLE: Linear elasticNE: Nonlinear elasticP : PlasticNK: Negative stiffnessLS: Linear stressNS: Nonlinear stress

-Controlled global stiffness-Small local plastic damage at toes-Small lateral resisting capacity(when without confinement effects) -Axial displacement (large) is not equal to the axial deformation-Self-centering-Non Linear behavior in the elastic range

-Limited global stiffness-Plastic hinge-High lateral resisting capacity

-Axial deformation is equal to the axial displacement-Permanent displacement-Linear behavior in elastic range

Rocking columnConventional column

PLELateral displacement

Lat

eral

forc

e 1

23

ITF

Inve

rted

trian

gula

rdi

strib

utio

n

0=ITH

0h

cbcbb −

ITh

NS

LS

( )( )( ) 344716

1742

0

+−

−−=

Ω=∴

bb

bbh

rr

rrhh

r

<Effective section>

both0

toph0

2h

2h

.botbM

0h

h

bM

0Mx

.0botM

topM 0

topbM

.botx

topx

cb

b

LS

NS

.botcb

LS

NS

topcb

NS

NSMLLSMLML fff ,,,,, += NSSLLSSLSL fff ,,,,, +=SLML

L ffK

,,

1+

=

cW

cWN +

H( )

221 c

c

bbWNHh

−+−

HN

CV ,δ

opV ,δ rotθ

( ) mVrotc

rotCV hbb

,, cos12

δθθδ −−−

−

=

−=

2,c

rotopV

bbθδ

rotθ : Rotational angle at thecenter of contact face

(Cumulative unbalanced curvature + Strain of contact face after cracking)

aθ'

aθ

'bθ

bθ

'LLaλ Lbλ

aM'

aM

'bM

bM

L

'aM

'bM

'aθ

'bθ

"L'Laλ 'Lbλ

',0 bhM

',0 ahM

'L

[ ]

=

'

'

'

'

'b

ac

b

a KMM

θθ

[ ]

=

b

as

b

a KMM

θθ

[ ] [ ][ ] [ ]LKLK cs~

'~

=

[ ]

=

'

'

'

'

'b

ar

b

a KMM

θθ

0== ba λλ

[ ] [ ] rrs KK '=

MM ='θθ ='

[ ] cK ' from spread plasticity model

[ ]ba

ab

ab

Lλλ

λλλλ

−−

−

−

=1

11

~

'LAα 'LBα

AA EI

f1

=

AcrM

BcrM

00

1EI

f = BB EI

f1

=

( ) '1 LBA αα −−

'L

AhM ,0

BhM ,0

00

1EI

f =

'LAα

'L

AA EI

f1

=B

B EIf

1=

'LBα( ) '1 LBA αα −−∫∫ +=

L

Z

jiL jiij dx

GA

xvxvdx

xEI

xmxmf

00

)()(

)(

)()(

=

B

A

BBBA

ABAA

B

A

MM

ffff

θθ

=

−

BBBA

ABAA

BBBA

ABAA

kkkk

ffff 1

Laλ'LAα

Lbλ'LBα

()

'1

LB

Aα

α−

−

L

EI1

0

0

M

Con

vent

iona

l col

umn

'aM

'bM

aM Beam

BeambM

Aα Bα

00

000

2

BA

BA

EIEIEIEI

EI+

=

AhA r ,=α BhB r ,=α

00 EIEI =

Rocking columnConventional column

Laλ'LAα

Lbλ'LBα

()

'1

LB

Aα

α−

−

L

EI1

0

0

M

Roc

king

col

umn

'aM

'bM

aM

bM

Beam

Beam

L

'L

: Height of conventional column: Height of rocking column

Vertex up model

Vertex down model

(in IDARC2D Report)

Unconfined case Confined case Unconfined / confined case

mV ,δ : Mechanical nonlinearaxial deformation

K: Stiffness: f: Flexibility, L: Lateral, M: Moment, S: Shear

=Ωcondition confinedfor 2

condition unconfinedfor 1

( ) εθθθ +

−

−= −

chMcbb

bsinsin

2sin

0,1

0

10

20

30

40

50

60

70

80

-0.5 0.0 0.5 1.0 1.5 2.0 2.5Vertical top displacement at the center of column (mm)

Lat

eral

forc

e (k

N)

Analytical (N=0.0kN)

Analytical (N=100.0kN)

Analytical (N=500.0kN)

FEM (N=0.0kN)

FEM (N=100.0kN)

FEM (N=500.0kN)

b/h=1/4, h=4m, Wc=94.08kN

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10Lateral top displacement (mm)

Lat

eral

forc

e (k

N)

Analytical (N=0.0kN)

Analytical (N=100.0kN)

Analytical (N=500.0kN)

FEM (N=0.0kN)

FEM (N=100.0kN)

FEM (N=500.0kN)

b/h=1/4, h=4m, Wc=94.08kN

+≤ 1

298.0

cc WN

hb

WH(Pre-rocking)

(Pre-rocking)

+≤ 1

298.0

cc WN

hb

WH

:Vertical displacementat the opening tip

0

10

20

30

40

50

60

70

80

0.0 0.5 1.0 1.5 2.0 2.5Vertical displacement at the tip of opening (mm)

Lat

eral

forc

e (k

N)

Analytical (N=0.0kN)

Analytical (N=100.0kN)

Analytical (N=500.0kN)

FEM (N=0.0kN)

FEM (N=100.0kN)

FEM (N=500.0kN)

b/h=1/4, h=4m, Wc=94.08kN

TF

0h

TH

cbcbb −

Th LS

NS

Triangular distribution

topCV ,δ

botCV ,δ

2h

2h

N

cWN +

H

HH

H

221 top

cbbNHh

−−

cW21

cW21

( )22

1 .botc

cbb

WNHh−

+−

(Even elastic)

Ω

−==cT

ccb WFhb

WHbb

r/1

//1

21

3

Lateral displacementLE NE P

Lat

eral

forc

e

1

2 3

Pre-rocking(Physical)

Post-rocking(Non-physical)

NK

Weakening structure Weakening and Damping structure Capacity-demand curve

StrengtheningWeakening

DampingLateral displacement

Bas

e sh

ear Demand Capacity

cσtσ

0=tσ

NS

LS

LS

cσ