non-linear modeling of reinforced concrete … · non-linear static analysis (pushover)...

Non-Linear Modeling of Reinforced Concrete Structures for Seismic Applications

02/18/2010

Luis A. MontejoAssistant Professor

Department of Engineering Science and MaterialsUniversity of Puerto Rico at Mayaguez

OutlineJustificationExtended moment curvature analysisFinite Element Modeling:

Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity)

Non-linear static analysis (pushover)Non-linear time history analysis:

Seismic InputDamping

Application ExampleConclusion

When do we need non-linear modeling?Design verification of very important and/or unusual structures.

When do we need non-linear modeling?

Have you ever use a force-reduction factor (R) in your design?

WTn

0 2

0.3

0.4

0.5

0.6

0.7

cele

ratio

n [g

]

A [g]

Fel=A [g]*W

0 1 2 3 4 50

0.1

0.2

period [s]ac

cTn

Fel/R

Δ1

Δ1*Sd <=Δlimit

When do we need non-linear modeling?Have you ever use a force-reduction factor (R) in your design?

20

25

e

Fn (Mn) μ1

0 1 2 3 4 5 6 7 80

5

10

15

displacement

late

ral f

orce

first yield

φεc

εy

ΔFΔ1*Sd ~Δlimit


Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity




Moment-Curvature Analysis

φεc

εy

Stre

ss

Strain0 0.01 0.02 0.03

0

10

20

30

40

50

60

: Confined Concrete: Unconfined Concrete

Stre

ss [M

Pa]

Strain-0.1 -0.05 0 0.05 0.1

-600

-400

-200

0

200

400

600concrete steel

Moment-Curvature Analysis

4000

5000

6000kN

-m)

0 0.05 0.1 0.150

1000

2000

3000

Curvature(1/m)

Mom

ent (

k

From M-Φ to F-Δ

(M) L

Δy ΔpP

actualidealized

φp φy

LpLsp

θp

structure and moment distribution curvature profile displacements

yppy

y

LL

L

φφφ

φφφ

>+Δ=Δ

≤=Δ 3/2

(Park and Priestley, 1988)

From M-Φ to F-Δ

2500

3000

3500

4000kN

)

0 0.05 0.1 0.15 0.20

500

1000

1500

2000

Displacement(m)

Forc

e (k

Shear CapacityV = Vp + Vs + Vc

revised UCSD shear model (Kowalsky and Priestley, 2000), illustration by Pablo Robalino

02

>−

= PL

cDPVp


picture by Pablo Robalino

( )θcots

cdclbHfAV hyhsxs

−+−=


ecs AfV 'αβγ=

α : aspect ratioβ : longitudinal reinforcementγ : aggregate interlock

Shear Capacity

2500

3000

3500

4000

kN)

2500

3000

3500

4000

kN)

2500

3000

3500

4000

kN)

0 0.05 0.1 0.150

500

1000

1500

2000

Displacement(m)

Forc

e (k

0 0.05 0.1 0.150

500

1000

1500

2000

Displacement(m)

Forc

e (k

0 0.05 0.1 0.150

500

1000

1500

2000

Displacement(m)

Forc

e (k

shear failure

Shear Capacity

45

90

-2.4 -1.6 -0.8 0 0.8 1.6 2.4[in]

200

400

e [k

N]

μ8

μ6μ4

-90

-45

0

[kip

s]

-60 -40 -20 0 20 40 60

-400

-200

0

displacement [mm]

late

ral f

orce

μ8

μ6 μ4

Onset of bucklingst

rain

characteristic capacity

flexural tension strain b kli

curvature ductility

stee

l ten

sion

growth straincolumn strain-ductilit

y

behavior

n buckling

Moyer and Kowalsky 2003

Onset of buckling

1020

[kip

s]

-5.9 -3.9 -2.0 0 2.0 3.9 5.9displacement [in]

50

100

e [k

N]

μ8μ6

μ5μ4

-20-10

0

late

ral f

orce

-150 -100 -50 0 50 100 150

-100

-50

0

displacement [mm]

late

ral f

orce

: bar buckl.: theor. buckl.

μ8μ6

μ5 μ4

Finite element modeling of RC structures

Spring moment - rotationnon-linear spring

Lumped Plasticity Model

Lumped plasticity modelsDisadvantages:

-Axial force-moment interaction and axial-force stiffness interaction are separate from the element behavior

-Need to use M-C analysis to find:Elastic and post-yield stiffnessp yNon-linear axial force/moment interaction envelope

force-moment interaction and axial-force stiffness interaction

-5%

0%+5%

+10%+20%

+40%

T C C T

Finite element modeling of RC structures

Longitudinal steel fibers

Unconfined concreteA A

Section A-A

Unconfined concrete cover fibers

Confined concrete core fibers

Material stress - strain

Distributed Plasticity Model

Material constitutive relationships

1.5

2.9

4.4

[ksi

]

10

20

30

Stre

ss [M

Pa]

Mander monotonic envelope

Concrete02 Unconfined concrete

0

0 0.005 0.01 0.015 0.02

0

Strain

0

1.5

2.9

4.4

[ksi

]

0 0.005 0.01 0.015 0.02

0

10

20

30

Strain

Stre

ss [M

Pa]

Mandermonotonicenvelope

Concrete02

Confined concrete

Material constitutive relationshipsReinforcingSteel material (Mohle and Kunnath, 2006), account for degrading strength and stiffness due to cyclic reversals.

73

109

500

750 Raynor monotonic envelope

-109

-73

-36

0

36

73

[ksi

]

-0.02 0 0.02 0.04 0.06-750

-500

-250

0

250

500

Strain

Stre

ss [M

Pa]

ReinforcingSteelmaterial

Distributed plasticity modelsAdvantages:

-No prior M-C analysis required-No need to define hysteretic response (it’s defined by the material models)-The influence of axial load is directly modeledPost peak strength reduction factor resulting from material-Post-peak strength reduction factor resulting from material

strain-softening or failure can be directly modeled.

Disadvantages:

-Shear strength and shear deformations still under development-Time consuming-Strain localization

Strain localization problemas

e sh

ear

3 IP

5 IP 3 E

5 E20 E

Force based Displacement based

Ba

Bas

e cu

rvat

ure

Displacement at top Displacement at top

8 IP

5 IP

3 IP

8 IP

3 E

5 E

20 E

Fiber based lumped plasticity model

Force-based-fiber sectionsE, A, I

Linear Elastic

Lpi Lpj

L

node i node j

BeamWithHinges element (Scott and Fenves, 2006)

Fiber based lumped plasticity model

-1

-0.5

0

0.5

1

Nor

mal

ized

For

ce

0

0.5

1

Nor

mal

ized

For

ce

-0.05 0 0.05Drift

N 0 0.02 0.04 0.06 0.080

Drift

N

0 10 20 300

10

20

30

40

cycle #

AB ξ

[%]

0 0.02 0.04 0.06 0.080

10

20

30

Drift

Cur

vatu

re d

uctil

ity: Experimental: Simulation

Non-linear Static Analysis (Pushover)

1348

1798

6000

8000

kN]

449

899 [kip

s]

0 0.02 0.04 0.06 0.08 0.10

2000

4000

Lateral drift

Forc

e [k

: +20°C: -40°C : serviceability : dam. control

Non-linear Static Analysis (Pushover)Disadvantages:

•Higher mode effects are missed

•With an unidirectional push the hysteretic characteristics f th t t t b l t dof the structure can not be evaluated

•If force controlled: tends to become unstable after the peak force is reached

•If displacement controlled: how do you specified the displacement vector in a multistory building… potential soft-storey building mechanisms can be inhibited

Non-linear Time History Analysis: Seismic inputSeismic input:

0.4

0.6

0.8

1

PSA

[%g]

•Real records (Historic)•Artificial records:

•Full artificial (Simqke – Gasparini and Vanmarke, 1976)•Modified historic records:

•Using Fourier: Wes Rascal (Silva and Lee, 1987)•Using CWT: ArtifQuakeLet (Suarez and Montejo, 2003)•Using Wavelets: rspmatch (Hancock et al, 2005)

0 1 2 3 40

0.2

T (s)

Non-linear Time History Analysis: Seismic Input

0.2

0.4

0.6

acce

lera

tion

[g]

T1 T2

0 0.5 1 1.5 2 2.5 30

Period [s]

a

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

Period [s]

disp

lace

men

t/g

period shift

error

Non-linear Time History Analysis: Damping

Damping = hysteretic + elastic (viscous) elhyst ξξξ +=

Hysteretic damping:

Non-linear Time History Analysis: DampingElastic damping: represents damping not captured by the hysteretic model:

•Hysteretic damping on the elastic range•Foundation compliance and non-linearity•Radiation dampingI t ti b t t t l t t l•Interaction between structural an non-structural

members

Non-linear Time History Analysis: DampingElastic damping:

mkmc

xmkxxcxm

ξωξ 22 ==

−=++ &&&&&

What values of k and ξ are appropriate?

•Traditional lumped plasticity model: 5% concrete, 2% steel •Fiber model: very low 0-2%

•Initial stiffness based viscous damping may result in inelastic damping forces that are unrealistically high. Use tangent-stiffness viscous damping.

Non-linear Time History Analysis: Damping

Petrini et al. 2008

Application: Alaska DOT bridges

Cap Beam

Super Structure

Columnn/Pile


8#7 or8#9

#3@60mm

linear potentiometers

457mm OD steel tubeplane stub

post-tensioning

base plate

supportblock

string potentiometer

(2) hydr.jacks

Crossbeam

Crossbeam

1651 mm

(F, Δ)

(2) loadcells

strong floor

pin pin

pin

specimen

pinfixed

dL

API-5L x52OD: 24 inThick.: 12 in

12#7 ASTM A706string pot.

Δ

actuator 1 actuator 2

TOP HINGE

BOTTOM HINGE


TOP HINGE

BOTTOM HINGE

picture by Lennie Gonzales


44.9

56.20.8 1.6 2.4 3.1 3.9 4.7

[in]

200

250

N] 202.2

0.0 2.0 3.9 5.9 7.9 9.8 11.813.815.717.719.7[in]

900

1200

N]

MOMENT-CURVATURE ANALYSIS

TOP HINGE

11.2

22.5

33.7

[kip

s]

20 40 60 80 100 1200

50

100

150

displacement [mm]

late

ral f

orce

[kN

: measured: calculated: εs = εy

: εs = 0.015

: εs = 0.06

67.4

134.8

[kip

s]

0 50 100 150 200 250 300 350 400 450 5000

300

600

displacement [mm]

late

ral f

orce

[kN

: measured: calculated: εst = εy

: εst = 0.008

: εst = 0.028

BOTTOM HINGE


0

1

mal

ized

forc

e

0.5

1

mal

ized

forc

e

TOP HINGE

FIBER MODEL CALIBRATION

-0.05 0 0.05-1

drift

norm

0 0.02 0.04 0.060

drift

norm

0 10 20 300

20

40

cycle #

AB ξ

[%]

0 0.02 0.04 0.060

20

40

drift

curv

atur

e du

ctili

ty: Experimental: Simulation


BOTTOM HINGE

0

1

mal

ized

forc

e

0.5

1

mal

ized

forc

e

Experimental

FIBER MODEL CALIBRATION

-0.05 0 0.05-1

drift

norm

0 0.01 0.02 0.03 0.04 0.050

drift

norm

0 10 20 300

20

40

cycle #

AB

ξ [%

]

0 0.01 0.02 0.03 0.04 0.050

5

10

drift

curv

atur

e du

ctili

ty

pe e aSimulation


BeamWithHingesLinearelastic

FINITE ELEMENT MODEL

p-ysprings Distributed

plasticity


8847

11796

12000

16000

-m]

: top hinge: bottom hinge: first yied: serviceability: damage control

M-C ANALYSIS

2949

5898 [kip

s-ft]

0 0.02 0.04 0.06 0.08 0.10

4000

8000

φD

Mom

ent [

kN- : damage control


1798

2247

8000

10000

12000

kN]

fi t i ld b tt hi

damage control top hingeserviceabilitybottom hinge

PUSHOVER RESULTS

449

899

1348

1798

[kip

s]

0 0.05 0.1 0.150

2000

4000

6000

8000

late

ral f

orce

[

drift

first yield top hinge

serviceability top hinge

first yield bottom hinge


1

1.2

atio

n [g

]

0 02

0.03

nt/g

SEISMIC INPUT

0 0.5 1 1.5 2

0.2

0.4

0.6

0.8

Period [s]

Pse

udo

Acc

eler

a

0 0.5 1 1.5 20

0.01

0.02

Period [s]

Dis

plac

emen

Application: Alaska DOT bridgesINCREMENTAL DYNAMIC ANALYSIS

0.08

0.1

0.12t

: Average: Eq. records damage control

0 0.5 1 1.50

0.02

0.04

0.06

peak ground acceleration [g]

late

ral d

rift

0.20g

0.76g

serviceability

first yield

Conclusion

It was shown that non-linear analyses provide us with valuable information regarding the seismic behavior of RC structures otherwise impossible to obtain through conventional linear analyses.

It is expected that, with the available computational tools, non-linear analyses become more popular in the design office environment.

Available (FREE) toolsMoment-curvature analysis:

http://www.ecf.utoronto.ca/~bentz/home.shtml http://blogs.uprm.edu/montejo/

Nonlinear FEM (lumped and distributed plasticity):Nonlinear FEM (lumped and distributed plasticity):

http://opensees.berkeley.edu/index.php http://www.seismosoft.com

Generation of spectrum compatible earthquake records:

contact the author http://blogs.uprm.edu/montejo/

non-linear modeling of reinforced concrete … · non-linear static analysis (pushover)...

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