Prediction of Concrete Fracture Mechanics Behavior and Size Effect using Cohesive Zone Modeling

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26 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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Prediction of Concrete Fracture Mechanics Behavior
and Size Effect using Cohesive Zone Modeling

Department of Civil and Environmental Engineering

University of Illinois at Urbana
-
Champaign

Kyoungsoo Park
, Glaucio H. Paulino, Jeffery R. Roesler

Federal Aviation Administration

Center of Excellence for Airport Technology, UIUC

2

Stage I

Elastic behavior

Stage II

Crack initiation

Tensile strength

Stage III

Non
-
linear cohesive law

Bi
-
linear softening curve for concrete

Stage IV

Traction
-
free macro
-
crack

Concept of Cohesive Zone Model


w
f
w
t
f

I

II

III

IV

Penalty stiffness

F f
G G

f
G
t
f


w
f
w
1
w
t
f


3

Determination of the Cohesive Law

Cohesive strength : f
t


Splitting test

Initial fracture energy : G
f

Size effect method (SEM)

Two
-
parameter fracture model (TPFM)

Total fracture energy: G
F

Hillerborg’s work
-
of
-
fracture method

The stress ratio of the kink point :
ψ

Peterson : 1/3

Wittmann : 0.25

Bazant : 0.15~0.33


1
2
f
t
G
w
f


2
(1 )
f F f
t
w G G
f


 
  
 

Bi
-
linear softening curve

F f
G G

f
G
t
f


w
f
w
1
w
t
f


Penalty stiffness

Kink point

4

FEA Implementation

Principle of Virtual Work

Virtual Internal Work = External Virtual Work



FEA Formulation

c
T T T
c
d d d
  
  
    
  
ε
σ
w T u F
c
T T
c
d d d
  

 
    
 

 
  
T
B E B N N u P
w
t t
t n
n n
t n
T T
w w
T T
w w
 
 
 
 

 

 
  
 
 
 
T
w
t cr
f w



1
t cr
f w w

 
n
n
T
w












t f k
f w w


 
0


0
n cr
w w
 


cr n k
w w w
 


k n f
w w w
 


n f
w w

1
u
1
v
2
v
2
u
3
u
3
v
4
u
4
v
t
w
n
w
5

ABAQUS User Element (UEL)

Nodal coordinates

Transformation Matrix ( )

Crack opening width ( )

Bi
-
linear softening curve ( )


Global coordinate system


Element stiffness matrix


Load vector

Element stiffness matrix ( )

Load vector ( )

UEL Properties

(


,thickness)

,,,,
F f t cr
G G f w


,
n t
w w
,


T
T
w
Numerical Integration

Local coordinate system

T
N T
T


T
N N
w
R
R
6

Three
-
Point Bending Test

Obtain fracture parameters

Compare load
-
CMOD curves

Size effect

P
0
a
D
S
L
[mm]

7

Experimental Results

Fresh and Hardened Properties of the Concrete




Fracture Parameters

Hillerborg

TPFM

SEM

G
F

(N/m)

K
I

(MPa m
1/2
)

CTOD
c

(mm)

G
f

(N/m)

c
f

(mm)

B250
-
80a

B250
-
80b

B250
-
80c

193

139

169

1.261

1.203

1.497

0.0167

0.0181

0.0319

52.1

24.36

B150
-
80a

B150
-
80b

B150
-
80c

N/A

170

159

N/A

1.086

0.983

N/A

0.0255

0.0115

B63
-
80a

B63
-
80b

B63
-
80c

CB63
-
80a

CB63
-
80b

CB63
-
80c

N/A

106

N/A

123

124

123

N/A

1.012

0.834

1.130

1.002

1.293

N/A

0.0159

0.0115

0.0142

0.0075

0.0184

Fresh Concrete

Hardened Concrete

Density

2403 kg/m
3

Compressive strength

58.3 MPa

Slump

100 mm

Split strength

4.15 MPa

Air content

2.8 %

Modulus of elasticity

32.0 GPa

8

Specimen Geometry and FE Mesh

Cohesive elements

9

Numerical Simulation of TPB


10

Concrete Fracture Simulation (TPB)


11

Numerical Validation


Small Beam

0
a


D = 63 (mm)


f
t
’ = 4.15 (MPa)


G
f

= 56.6 & 52.1 (N/m)


G
F

= 119 (N/m)


Ψ

= 0.25

12

Numerical Validation


Intermediate Beam

0
a


D = 150 (mm)


f
t
’ = 4.15 (MPa)


G
f

= 56.6 & 52.1 (N/m)


G
F

= 164 (N/m)


Ψ

= 0.25

13

Numerical Validation


Large Beam

0
a


D = 250 (mm)


f
t
’ = 4.15 (MPa)


G
f

= 56.6 & 52.1 (N/m)


G
F

= 167 (N/m)


Ψ

= 0.25

14

Model Sensitivity



Initial fracture energy



Tensile strength



Total fracture energy



Stress ratio of the kink point

CMOD

Load

Tensile strength

Stress ratio of the kink point

Total fracture energy

15

1.00E+05
1.00E+06
10
100
1000
10000
Size Effect





log
D mm




log
Nu
Pa

0
1
t
N
Bf
D D




Experimental data

Numerical Result


TPFM

SEM

*

D = 63mm

D = 150mm

D = 250mm

16

Summary

Predict Load
-
CMOD Curve

Bi
-
linear softening cohesive zone model


Without calibration of the fracture parameters.


Investigate Size Effect

Cohesive Zone Model with bi
-
linear softening


Experiment results from Three
-
point bending tests


Size effect expression:


Good agreement between the results from the three methods.


0
1
t
N
Bf
D D