Shi-Yu Xu, Ph.D. Student Jian Zhang, Assistant Professor

1Quake Summit 2010

10/08/2010

Coupled Axial-Shear-Flexure Interaction Hysteretic Model for Seismic Response

Assessment of Bridges

Shi-Yu Xu, Ph.D. StudentJian Zhang, Assistant Professor

Department of Civil & Environmental EngineeringUniversity of California, Los Angeles

2Quake Summit 2010

Outline

Introduction Motivation & Objectives Shear-Flexure Interaction Under Constant Axial Load

Proposed Axial-Shear-Flexure Interaction (ASFI) Scheme Primary Curves and Hysteretic Models Considering Combined Actions Generation of Primary Curve Family Stress Level Index & Two-stage Loading Approach

Model Verification Static Cyclic Tests Comparison with Fiber Section Model under Seismic Loadings Limitations and Known Issues

Factors Affecting ASFI & Effects on Bridge Responses Arrival Time of Vertical Ground Motion Vertical-to-Horizontal PGA Ratio

Summary

3Quake Summit 2010

Introduction

Motivation Bridge columns are subjected to combined actions of

axial, shear and flexure forces due to structural and geometrical constraints (skewed, curved etc.) and the multi-directional earthquake input motions.

Axial load variation can directly impact the ultimate capacity, stiffness and hysteretic behavior of shear and flexure responses.

Accurate seismic demand assessment of bridges needs to realistically account for combined actions.

Objectives An efficient analytical scheme considering axial-shear-

flexural interaction Shear and flexural hysteretic models reflecting the

effects of axial load variation and accumulated material damage (e.g. strength deterioration, stiffness degrading, and pinching behavior)

4Quake Summit 2010

Axial-Shear-Flexural Interaction

• Significance of Non-linear Shear-Flexural Interaction (Ozcebe and Saatcioglu 1989) Shear displacement can be significant -- even if a RC member is not

governed by shear failure (as is the case in most of RC columns). Inelastic shear behavior -- RC members with higher shear strength

than flexural strength do not guarantee an elastic behavior in shear deformation.

• Coupling of Axial-Shear-Flexural Responses (ElMandooh and Ghobarah 2003) Dynamic variation of axial force -- will cause significant change in

the lateral hysteretic moment-curvature relationship and consequently the overall structural behavior in RC columns.

5Quake Summit 2010

Axial-Shear-Flexure Interaction at Material Level

MCFT

fsx

fsy

fcx

fcy

fx

fy

vxy

vcxy

x1 y

fc1

fc2

c 2

11

2

2 22 2,max ' '

,

,

1 200

2

crc

c cc c

sx s x y x

sy s y y y

ff

f f

f E ff E f

Equilibrium Strain Compatibility Constitutive Law

(Vecchio and Collins 1986)Modified Compression Field Theory

γ

τ

φ

M

+

6Quake Summit 2010

Derivation of Flexural and Shear Primary Curves

Discretize RC member into small pieces. For each piece of RC element, estimate M-φ and τ-γ relationship by Modified Compression Field Theory (MCFT, Vecchio and Collins 1986).

M

M=V*h

dy

VN

yi

V

MCFTγ

τ

γ

τ

…φ

M

φ

M

…+

+

F-UEL

S-UEL

SSI springFNDN

DECK

S-UEL

F-UEL

Rigid Column

Input the V-Δs and M-θ curve to Shear-UEL & Flexural-UEL.

Δs

VS-UEL

Δm

M

θ

MF-UEL

Integrate curvature and shear strain to get displacement.

δ=Σ { φi*dy*yi + γi*dy } Flexural deformation Shear deformation

= h *θ + Δs

7Quake Summit 2010

Shear-Flexure Interaction (SFI) under Constant Axial Load

0 2 4 6 8 10 120

50

100

150

200

250

300

Shear Strain (mm/m)

Shea

r (kN

)

V-1

V-2

V-3

V-4

V-5

V-6

V-7

V-8

V-9

0 5 10 15 20 25 300

30

60

90

Curvature (rad/km)

Mom

ent (

kN-m

)

M-1

M-2

M-3

M-4

M-5

M-6

M-7

M-8

M-9

0 5 10 150

10

20

30

40

50

60

70

Total Displacement (mm)

Shea

r (kN

)

total displ.shear displ.f lexural displ.

0 5 10 150.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

Total Displ. (mm)

Shea

r-to

-Tot

al D

ispl

. Rat

io

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4 M/V =0.076(m)1

M/V =0.229(m)2

M/V =0.381(m)3

M/V =0.534(m)4

M/V =0.686(m)5

M/V =0.838(m)6

M/V =0.991(m)7

M/V =1.143(m)8

M/V =1.296(m)9

M/V ratio

Col

umn

Hei

ght (

m)

0 2 4 6 8 10 120

50

100

150

200

250

300

Shear Strain (mm/m)

Shea

r (kN

)

V-1

V-2

V-3

V-4

V-5

V-6

V-7

V-8

V-9

0 5 10 15 20 25 300

30

60

90

Curvature (rad/km)

Mom

ent (

kN-m

)

M-1

M-2

M-3

M-4

M-5

M-6

M-7

M-8

M-9

0 5 10 150

10

20

30

40

50

60

70


Shea

r (kN

)

total displ.shear displ.f lexural displ.

0 5 10 150.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

Total Displ. (mm)

Shea

r-to

-Tot

al D

ispl

. Rat

io

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4 M/V =0.076(m)1

M/V =0.229(m)2

M/V =0.381(m)3

M/V =0.534(m)4

M/V =0.686(m)5

M/V =0.838(m)6

M/V =0.991(m)7

M/V =1.143(m)8

M/V =1.296(m)9

M/V ratio

Col

umn

Hei

ght (

m)

dy

VN

yi

M

M=V*h

Sections with different M/V ratio (level of shear-flexural interaction) demonstrate different mechanical properties and behaviors

Section with higher M/V ratio: Larger moment capacity Smaller shear capacity

Maximum moment capacity is bounded by pure bending case

8Quake Summit 2010

Improved Hysteretic Rules for Shear & Flexural Springs

Unloading & reloading stiffness depend on: Primary curve (Kelastic, Crack, & Yield) Cracked? Yielded? Shear force level Max ductility experienced Loading cycles at max ductility level Axial load ratio

-80 -60 -40 -20 0 20 40 60 80-25

-20

-15

-10

-5

0

5

10

15

20

25

Shear Displacement

She

ar F

orce

Hysteretic Loop

B

E

FA

C

D

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

Shear DisplacementSh

ear F

orce

Vcr

Vy

maximum peak (Δm,Vm)

hardening reference point(Δm,V’m)

previous peak (Δp,Vp)

pinching reference point (Δp,V’p)

G

H

Structural characteristics Damage in the column

Loading history

Varying during earthquake !!

(Ozcebe and Saatcioglu,1989)

Xu and Zhang (2010 ), EESD

9Quake Summit 2010

Outline





Summary

10Quake Summit 2010

Effects of Axial Load Variation on Total Primary Curves

• Ultimate capacity and stiffness increase with compressive axial load level.• Yielding displacement is almost fixed, regardless of applied axial load.• Cracking point is getting smaller as axial force decreasing, implying the

column being relatively easy to be cracked.

0 10 20 30 400

20

40

60

80

100

Column Tip Drift (mm)

She

ar (k

N)

PEER-93

P/P0=-5%(T)P/P0=-2%(T)P/P0= 0 (-)P/P0= 5%(C)P/P0=10%(C)P/P0=20%(C)

0 5 10 15 20 250

100

200

300

400

500

600

700

Column Tip Drift (mm)S

hear

(kN

)

PEER-121


0 50 100 1500

50

100

150

200

250

Column Tip Drift (mm)

She

ar (k

N)

PEER-122


Kunnath et al.H/D=4.5

Calderone-828 H/D=8.0

Calderone-328 H/D=3.0

11Quake Summit 2010

Normalization of Primary Curves

-10 0 10 20 30 400

0.5

1

1.5

2

P/P0 (%), Compression is "+".

V y(P

/P0=n

%) /

Vy(P

/P0=5

%C

) Y =-2.15*(X-0.60)2+1.65

-10 0 10 20 30 400

0.5

1

1.5

2

2.5


V u(P/P

0=n%

) / V

y(P

/P0=5

%C

) Y =-3.20*(X-0.60)2+2.32

(c) yield load (d) ultimate capacity

20

0 0

( / %) 0.68*( 0.25) 0.01( / 5%)

cr

y

P P n PP P P

20

0 0

( / %) 1.47*( 0.25) 0.02( / 5%)

cr

y

V P P n PV P P P

0 2

0 0

( / %)2.15*( 0.6) 1.65

( / 5%)y

y

V P P n PV P P P

20

0 0

( / %) 3.20*( 0.6) 2.32( / 5%)

u

y

V P P n PV P P P

-10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5


cr(P

/P0=n

%) /

y(P

/P0=5

%C

) Y = 0.68*(X+0.25)2+0.01

-10 0 10 20 30 400

0.2

0.4

0.6

0.8

1


V cr(P

/P0=n

%) /

Vy(P

/P0=5

%C

) Y = 1.47*(X+0.25)2+0.02

12Quake Summit 2010

Generation of Primary Curve Family

(i) 0crack: straight line

(ii) crackyield: interpolation

(iii) yieldultimate: interpolation

% %( )* %

% %( )

% %

% * % %( )

% % % %( )

.

*( )

*( )

I Iii cr

Iy cr

I Iii crI I

y cr

n n nii y cr cr

n n n nii y cr cr

DL def level

V VSL stress level

V V

DL

V SL V V V

(iv) ultimatefailure: constant residual strength ratio% %

( ) ( )

% %( )

% %

% % % %( )

*( )

n Iiii iii

I Iiii yI I

u y

n n n niii u y y

ductility unchanged

V VSL stress level

V V

V SL V V V

% %( ) ( )

%( )

%

% %( )

*

n Iiv iv

IivI

u

n niv u

ductility unchanged

VRSR residual strength ratio

V

V RSR V

n% primary curve (predicted)

I% initial primary curve (given)

n% critical points, predicted from equations

loading

deflection

I% critical points, on initial primary curve

a a

a

b bb

i ii iii iv

%( ) % %

( )%

%( ) % %

( )%

. *

. *

Ii n n

i crIcr

Ii n n

i crIcr

DL def level DL

VSL stress level V SL V

V

Objective: Generating the primary curves related to various axial load levels from a given primary curve subject to an initial axial load

13Quake Summit 2010

Stress Level Index & Two-stage Loading Approach

Equivalentstress level Equivalent

stress level

-5%

Δy

5%yV

Δ1

dc

Δmax

5%effV

5%mV

0%

Δy

0%yV

Δ1

dc

Δmax

0%effV

0%mV

10%

dc

Δy

10%yV

Δ1 Δmax

10%effV

10%mV

Keep Δ, change N: 10% -5% Keep N, change Δ : Δ1 Δ2

10%

-5%

10%effV

5%effV

Δ1Δ2

10%

c d-5%

10%effV

5%effV

Δ1

c d

Δmax

Assumption:Effective stress level of a loaded column at fixed ductility is independent of axial load.max

Effective Lateral Load, Stress Level Index

Lateral Capacity at , eff

m

V cV d

14Quake Summit 2010

Outline





Summary

15Quake Summit 2010

Cyclic Test: Experimental Program – TP031 ~ TP034

TP-033 TP-034

HeightDiameter

=

16Quake Summit 2010

Verification of Primary Curve Prediction

-80 -60 -40 -20 0 20 40 60 80-200

-150

-100

-50

0

50

100

150

200

Displacement (mm)

She

ar F

orce

(kN

)

Hysteretic Loop

AnalyticalExperimental

-80 -60 -40 -20 0 20 40 60 80-200

-150

-100

-50

0

50

100

150

200

Displacement (mm)

She

ar F

orce

(kN

)Hysteretic Loop


TP-032Sakai and Kawashima H/D=3.375


TP-031

TP-032Given the primary curve of TP-031, predicts the response of TP-

032.

Given the primary curve of TP-032, predicts the response of TP-

031.

17Quake Summit 2010

-80 -60 -40 -20 0 20 40 60 80-200

-150

-100

-50

0

50

100

150

200

Displacement (mm)

She

ar F

orce

(kN

)

Hysteretic Loop


-80 -60 -40 -20 0 20 40 60 80-200

-150

-100

-50

0

50

100

150

200

Displacement (mm)

She

ar F

orce

(kN

)Hysteretic Loop


Verification of Mapping between Different Axial Load Level



TP-031

TP-032

TP-033

TP-034

Axial load decreasing

Axial load decreasing

Axial load increasing

Axial load increasing

18Quake Summit 2010

Dynamic Validation with Fiber Section Model

0 2 4 6 8 10-500

0

500

1000

Axi

al F

orce

(kN

)

0 2 4 6 8 10-200

0

200

Time (s)

She

ar (k

N)

0 2 4 6 8 10

-20

0

20

Time (s)

Tip

Dis

pl. (

mm

)

-20 -10 0 10 20 30-200

-150

-100

-50

0

50

100

150

200

Tip Displ. (mm)

She

ar (k

N)

OpenSees w/ V-EQOpenSees w/o V-EQABAQUS w/ V-EQABAQUS w/o V-EQ

• Proposed ASFI model in general produces larger displacement demand than the fiber section model.

• Vibration frequencies of the two models agree with each other indicating reasonable prediction on the tangent stiffness of the proposed ASFI model.

• Considering only the SFI can yield good prediction on the displacement demand.

ABAQUS ASFI Model

OpenSees Fiber Model

19Quake Summit 2010

V

Δs

M

θ

Limitations and Known Issues

• Estimation on post-peak stiffness of primary curve family may not be adequate.

• May converge at an incorrect solution for systems with yielding platform.

• May converge at an inconsistent deformed configuration for softening systems.

• Use of full stiffness matrix can somehow improve the above-mentioned convergence issues, however, it is an asymmetric matrix which offsets most of the advantages.

20Quake Summit 2010

Outline





Summary

21Quake Summit 2010

Factors Affecting ASFI: Arrival Time of Vertical Ground Motion

-0.4s -0.3s -0.2s -0.1s 0.0 0.1s 0.2s 0.3s 0.4s0

1

2

3x 10

6

Max

Bas

e S

hear

(N)

tpeakV

- tpeakH

w /o V-EQno shift on V-EQ

-0.4s -0.3s -0.2s -0.1s 0.0 0.1s 0.2s 0.3s 0.4s0

1

2

3x 10

6

Max

Bas

e S

hear

(N)

tpeakV

- tpeakH


-0.4s -0.3s -0.2s -0.1s 0.0 0.1s 0.2s 0.3s 0.4s0

2

4

6

8

10x 10

6

Max

Bas

e M

omen

t (N

-m)

tpeakV

- tpeakH


-0.4s -0.3s -0.2s -0.1s 0.0 0.1s 0.2s 0.3s 0.4s0

2

4

6

8

10x 10

6

Max

Bas

e M

omen

t (N

-m)

tpeakV

- tpeakH


-0.4s -0.3s -0.2s -0.1s 0.0 0.1s 0.2s 0.3s 0.4s0

0.02

0.04

0.06

0.08

Max

Col

umn

Drif

t (m

)

tpeakV

- tpeakH


-0.4s -0.3s -0.2s -0.1s 0.0 0.1s 0.2s 0.3s 0.4s0

0.02

0.04

0.06

0.08

Max

Col

umn

Drif

t (m

)

tpeakV

- tpeakH


(a) H: WN22; V: WN22 (b) H: WN22; V: NO4

0 2 4 6 8 10-0.5

0

0.5 0.4521(g)

-0.4432(g)

Time (s)

Acce

lera

tion

(g)

HV

0 2 4 6 8 10-0.5

0

0.5

1

0.4521(g) 0.5352(g)

Time (s)

Acce

lera

tion

(g)

HV

(a) Horizontal: WN22 (Tp=0.488s); Vertical: WN22 (Tp=0.138s)

(b) Horizontal: WN22 (Tp=0.488s); Vertical: NO4 (Tp=0.322s)

No significant correlation is found.

22Quake Summit 2010

Factors Affecting ASFI: Vertical-to-Horizontal PGA Ratio

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3x 10

6

Max

Bas

e Sh

ear

(N)

PGAV / PGA

H

w /o V-EQ

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3x 10

6

Max

Bas

e S

hear

(N)

PGAV / PGA

H

w /o V-EQ

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10x 10

6

Max

Bas

e M

omen

t (N

-m)

PGAV / PGA

H

w /o V-EQ

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10x 10

6

Max

Bas

e M

omen

t (N

-m)

PGAV / PGA

H

w /o V-EQ

0.0 0.2 0.4 0.6 0.8 1.00

0.02

0.04

0.06

0.08

Max

Col

umn

Drif

t (m

)

PGAV / PGA

H

w /o V-EQ

0.0 0.2 0.4 0.6 0.8 1.00

0.02

0.04

0.06

0.08

Max

Col

umn

Drif

t (m

)

PGAV / PGA

H

w /o V-EQ

(a) H: WN22; V: WN22 (b) H: WN22; V: NO4

0.0 0.2 0.4 0.6 0.8 1.0-1

0

1

2x 10

7

PGAV / PGA

H

Axia

l For

ce (N

), co

mp.

is "+

"

Column of Bridge#4 (H/D=2.5, P/P0=15%)

subject to WN22 (T&V)

Maxmin

0.0 0.2 0.4 0.6 0.8 1.0-1

0

1

2x 10

7

PGAV / PGA

H

Axia

l For

ce (N

), co

mp.

is "+

"

Column of Bridge#4 (H/D=2.5, P/P0=15%)

subject to WN22(T) & NO4(V)

Maxmin

(a) Horizontal: WN22 (Tp=0.488s); Vertical: WN22 (Tp=0.138s)

(b) Horizontal: WN22 (Tp=0.488s); Vertical: NO4 (Tp=0.322s)

tVpeak – tH

peak = -0.1s

• Larger PGAV/PGAH ratio tends to have larger influence on force demand.

• No significant correlation exists with drift demand.

23Quake Summit 2010

Bridge Responses Considering ASFI

1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

Max

Col

umn

Drif

t Rat

io

Bridge #4, H/D=5.0

V+HH only

1 2 3 4 5 6 7 8 9 100.5

1

1.5

2

2.5x 10

6

Max

Sec

tion

Forc

e (N

)

1 2 3 4 5 6 7 8 9 100

5

10

15x 10

6

Max

Sec

tion

Mom

ent (

N-m

)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

Max

Dec

k Ac

c. (g

)

Earthquake Index Number

1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

Max

Col

umn

Drif

t Rat

io

Bridge #4, H/D=2.5

V+HH only

1 2 3 4 5 6 7 8 9 101.5

2

2.5

3

3.5x 10

6

Max

Sec

tion

Forc

e (N

)

1 2 3 4 5 6 7 8 9 104

6

8

10x 10

6

Max

Sec

tion

Mom

ent (

N-m

)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

Max

Dec

k Ac

c. (g

)

Earthquake Index Number

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06-3

-2

-1

0

1

2

3x 10

6 Force-Displacement //Longi.

Column Drift (m)

Shea

r Fo

rce

(N)

C1@B1C2@B1C1@B2C2@B2

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-3

-2

-1

0

1

2

3x 10

6 Force-Displacement //Trans.

Column Drift (m)

Shea

r Fo

rce

(N)

-0.01 -0.005 0 0.005 0.01-8

-6

-4

-2

0

2

4

6

8x 10

6 Moment-Rotation @Longi.

Rotation Angle @ F-UEL (rad)

Mom

ent (

N-m

)

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015-8

-6

-4

-2

0

2

4

6

8x 10

6 Moment-Rotation @Trans.

Rotation Angle @ F-UEL (rad)

Mom

ent (

N-m

)

Force v.s. total column drift (H/D=2.5)

Considering axial variation does not change overall bridge responses

much.

24Quake Summit 2010

Summary

• Axial load considerably affects the lateral responses of RC columns.

• Primary curves of the same column under different axial loads can be predicted very well by applying the normalized primary curve and parameterized critical points.

• Mapping between loading branches corresponding to different axial load levels is made possible by breaking the step into two stages: constant deformation stage and constant loading stage.

• Model verification shows that the proposed method is able to capture the effects of axial load variation on the lateral responses of RC columns.

• Transient time analysis on individual bridge column and on prototype bridge system shows that considering axial load variation during earthquake events does not change the drift demand significantly.

25Quake Summit 2010

ACKNOWLEDGEMENT

Thanks for your attention !

The research presented here was funded by National Science Foundation through the Network for Earthquake Engineering Simulation Research Program, grant CMMI-0530737, Joy Pauschke, program manager.

Thank You!

26Quake Summit 2010

Analytical Models for RC Columns

Plastic Hinge Models Using equivalent springs to simulate shear

and flexural responses of columns at the element level

Empirical and approximate Difficult to couple together the axial, shear,

and flexural responses Numerical instability in the adopted

hysteretic models may induce convergence problem

Fiber Section Formulation Controlling the element responses directly at

the material level Coupling the axial-flexural interaction Rotation of principal axes in concrete (as

large as ～ 30°) due to the existence of shear stress is not considered

Elastic or rigid beam

Linear or Nonlinear spring elements

x

y

z

fiber

y

z

27Quake Summit 2010

Deficiencies of Current Numerical ModelsDeficiencies of Current Models

Non-linearity in shear deformation is not accounted for. Material damage (strength deterioration and pinching) due to cyclic loading is not considered. Axial-Shear-Flexural interaction is not captured.

-60 -40 -20 0 20 40 60-150

-100

-50

0

50

100

150

Displacement (mm)

Shea

r (kN

)

(a) Nonlinear Timoshenko Beam Element

Test TP-021nonLinear M-

-60 -40 -20 0 20 40 60-150

-100

-50

0

50

100

150

Displacement(mm)

Shea

r (kN

)

(b) OpenSees Fiber Element

Test TP-021OpenSees Fiber

28Quake Summit 2010

-60 -40 -20 0 20 40 60

-200

-100

0

100

200

-10.0%

25.0%


Late

ral L

oad

(kN

)

TP033: Axial Load= -10(-0.3%) ~ +310(+8.5%) kN predicted by equations

0 5 10 15 200

50

100

150

200


Late

ral L

oad

(kN

)

EXPP/Po= 12.80%proposed Eq's

0 10 20 30 400

50

100

150

200

-10.0%

25.0%


Late

ral L

oad

(kN

)

Primary Curve Family of TP-033

0 5 10 150

50

100

150

200


Late

ral L

oad

(kN

)

P/P0= 12.80%

OpenSees

Comparison of Primary Curve Family with Fiber Model

• Similar trends are observed except post-yield response.• Fiber Section Model overestimates initial stiffness.• Fiber Section Model underestimates axial load effects.

0%

10%

Shi-Yu Xu, Ph.D. Student Jian Zhang, Assistant Professor

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Transcript of Shi-Yu Xu, Ph.D. Student Jian Zhang, Assistant Professor