Nonlinear Analysis of Reinforced Concrete

Xuehui AN

Today’s Agenda

• Review of last week’s lecture• Constitutive Models for RC (3)• Today’s Topic

1 Method of Modeling

2 Concrete Model Prior to Generation of Cracks (1) Elasto-Plastic and Fracture Model (2) Cracking Criteria

3 Modeling of Cracked Concrete (1) Modeling of Concrete under Tensile Stress (2) Modeling of Concrete under Compressive stress (3) Modeling of Concrete under Shear Stress (4) Inter-relation of Each Model

4 Modeling of Reinforcing Bar in Concrete

5 Verification

(1) Modeling of Concrete under Tensile Stress

(1) Modeling of Concrete under Tensile Stress

(1) Modeling of Concrete under Tensile Stress

This area is tension stiffening

Cracking Yeilding

Tensile strain

Tot

al t

ensi

le s

tres

s/A

c

xEs

ft=xFy

steelThis area is tension stiffening

Cracking Yeilding

Tensile strain

Tot

al t

ensi

le s

tres

s/A

c

xEs

ft=xFy

steel

lr

ftt Gdu

t

Discrete crack

t

t

Un-cracked concrete

t

t

r

ftt l

Gd

Area = fracture energy

ft

cu r

tc

l

u

cu

:total strain c

u :strain of un-cracked concrete:strain due to crack

Area = fracture energy

ut

ut

ut

Fracture energy based stress-strain model for concrete

l ru t

r

ftt l

Gd

A re a = f ra c tu re e n e rg y

t

ftt u

Gd

A re a = f ra c tu re e n e rg y

t

t

t

t

Fracture energy based stress-strain model for concreteFrom real crack band to finite element

0 1000 2000 3000 40000.0

0.2

0.4

0.6

0.8

1.0

1.2

Tensile strain (micro)

Normalized tensile stress

t t tu tcf ( / ) t

t t

f

r

dG

l lr

Plastic zone

No. 1 2 3 5 6lr (mm) 70 140 170 230 500

c 0.6 0.9 1.6 2.5 5.5

Gf = 113 N/m

Cracking

c = 0.6

0.9

1.6 2.5

5.5

tft

tt f/

)f,( ttu

t

Fracture energy based stress-strain model for concreteApplied in FEM Analysis

Concrete

Stress

Strain

Steel

Stress

Strain

y

c

2000-3000 micro

100-200 micro

cE

tf

yf

sE

Reinforced ConcreteTotal Force

Average Strain

Steel Force

Tension Stiffening

Concrete

Steel

Stress

Strain

tf

Tension Softening

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Average strain ( x10 )

fc1/fcr

c

1

cr

cr

1c

f

f

-3

c = 0.2 (Welded mesh)

c = 0.4 (Deformed bar)

Cracking

Deformed Bar

Concrete

Stress

Strain

Steel

Stress

Strain

y

c

2000-3000 micro

100-200 micro

cE

tf

yf

sE

As

Ac

+

30

040

050

0m

m

Casting direction

Bolt Reaction plate

Center-hole load cell

Support shoe

Center-hole jack

Support plate

Steel bar

Guide for the meter

Free-end-slipDisplacement meter

Support ring

Steel bar

10D Unbonded region

Concretespecimen

Displacement transducer

Specimen

Roller

Load cell

Center – hole jack

Position along bar (m)

Stress distribution

Strain distributionSpecimen no.4St

ress

(G

Pa)

Stra

in (

10-2

)

sh

y

fy

0.5 1.0 1.5 2.0 2.50.4

0.5

0.6

0

1

Concreteprimary crack

secondary conical bond crack

Control volume contains several primary cracks

Control volume contains several secondary bond cracks

Control volume between two adjacent secondary bond cracks(fully microscopic)

Experimentally based macroscopic bond momel

Spatially averaged macro model based on bond slip

Local bond slip relation

Bar lug-surrounding concrete bearing action

Control volume = the whole member (fully macroscopic)Macroscopic modeling

Microscopic modeling

Deformed bar

Col

umn

Beam

Footing

10.00

20 .00

40 .00

60 .00

80 .00

100.00

120.00

10 .00

20 .00

40 .00

60 .00

80 .00

100 .00

120 .00

ConcreteSteel

CrackingFEM Model

One Element

Fine Elements

A x i a l t e n s i o n t e s t

d x

dc

0 x 00S 0

x

0dxS

Zero at crack plane

Stresses in concrete

Stresses in steel

average

average

Total Force Total Force

Total Force = Steel Force (x) + Concrete Force (x)

Steel Force= Steel Stress (x) * As

Concrete Force= Concrete Stress (x) * AcA=As+Ac

x

(x)

(x)

Zero at crack plane

Stresses in concrete

Stresses in steel

average

average

c

c

dvvv )(1

Total Force Total Force

Total Force = Steel Force (x) + Concrete Force (x)

Steel Force= Steel Stress (x) * As

Concrete Force= Concrete Stress (x) * AcA=As+Ac

x

(x)

Stress

Strain

tfStress

Strain

tf

?

Concrete

Steel

Stress

Strain

y

2000-3000 micro

100-200 micro

cE

Stress

Strain

c

tf

yf

sE

Total Force - Steel Force (x) = Concrete Force (x)

Concrete SteelTotal Force

Average Strain

Steel Force

Total Force – Average Steel Force = Average Concrete Force

Average Concrete Force

Average Steel Force

ss AE

Reinforced ConcreteTotal Force

Average Strain

Steel Force

Total Force – Average Steel Force = Average Concrete Force

Average Concrete Force

Average Steel Force

ss AE

Total Force

Reinforced ConcreteTotal Force

Average Strain

Steel Force

Total Force – Average Steel Force = Average Concrete Force

Average Concrete Force

Average Steel Force

ss AE

Total Force

Str

ess

dis

trib

utio

n

Strain distribution

Average

Local

Steel strain at cracking location

Total Force

Average Strain

Steel Force

Average Concrete

Force

Average Steel Force

Average yielding

Steel strain at cracking locationAverage Strain

Average yieldingstrength

Stress-strain for steel in RC

P P

f y

c r

s ycr f

f y

ycr f)( y

S t r e s s d i s t r i b u t i o n i n r e i n f o r c e m e n t

C r a c k s

s

Bare bar

c

s

ss

0 y

fyS

tres

s at

the

crac

ks

Average strain

Yield plateau jump!

V

dvyxV

),(1

dvy)(x,V

1

v

Str

ess

dis

trib

utio

n

stress

strain

dvy)(x,V

1

v

Plastic zone

dvy)(x,V

1

v

stressstress

strainstrainS

train distribution

Strain distribution

Str

ess

dis

trib

utio

n

Str

ess

dis

trib

utio

n

Strain distribution

Plastic zone

V

dvyxV

),(1

V

dvyxV

),(1

Str

ess

dis

trib

utio

nstrain

stress

dvy)(x,V

1

v

V

dvyxV

),(1

Strain distribution

Average

Local1

43

2

?

Izumo &Shima(1987)

Hsu(1991)

= ??

Crack Crack

ConcreteReinforced ConcreteSteel

1. Experimental 2. Simplified 3. Microscopic Bond

Morita and Kaku (1964)

Gilbert and Warner (1978)

Rizkalla and Hwang (1984)

Yoshikawa and Tanabe (1986)Linear Bond-Slip RelationshipChan et al (1992)Assumed Bond Distribution

0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

Average strain (x10 )

fc1/fcr

1

cr1c

2001

ff

-3

Stress-Strain Relationship of Concrete in Tension by Vecchio and Collins

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

Average strain ( x10 )

fc1/fcr

c

1

cr

cr

1c

f

f

-3

c = 0.2 (Welded mesh)

c = 0.4 (Deformed bar)

Stress-strain relationship of concrete in tension by Okamura et al.

<input data> : strain {path-dependent parameters} { }

Selection of active crack coordinate

Compute stress according toquasi-orthogonal two-way crackmodel

New crack formed?

Switch of coordinate orintroduction of new

coordinate

Already cracked?

Update of path-dependent parameters for both twocoordinates

<output data> : stress{ }c

Un-crackedconcreteroutine

crack formed?

Introduce thefirst coordinate

No

Yes

No

No

Yes

Yes

Depending on change of active crackThe coordinate is re-adjusted or switched

x1

y1

y1

x1

x2

y2

y2

x2

The first coordinate x1-y1

(READJUST)

The second coordinate x2-y2

(SWITCH)

(READJUST)

Today’s Topic

• English Ability