Modeling of Cement Based Composite Laminates Barzin Mobasher Department of Civil & Environmental Engineering Arizona State University Tempe, Arizona, 85287-5306
HPFRCC-4, Ann Arbor, Michigan, June 16-18, 2003.
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Scope of Presentation
Introduction Pultrusion Based Cement Composite Systems Theoretical Aspects of Formulation
Non-linear matrix response
Cracking Softening
Case Studies
Elastic response Traditional Composite laminate approach
Glass Fiber/ carbon fiber epoxy composites Cement based Glass fiber composite laminate Polypropylene fiber cement composites Fabric composites Comparison with Experimental results Repair and retrofit approach
Conclusions
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Areas of Application for high performance fiber reinforced Materials
high tensile-toughness characteristics superior impact, earthquake, and fatigue characteristics. prefabricated Structural elements, thin sheets, panels, cladding members. structural repair and retrofit. I-beams, structural members. Sound abatement walls.
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Scope of Presentation
Introduction Pultrusion Based Cement Composite Systems
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Filament Winding
Fiber Spool You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)
Fabric Reinforcing Methodology
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Cement composites with 50 MPa Tensile Strength and more than 1% strain Capacity 50 Unidirectional
Stress, MPa
40 30
0/90/0 20 10
GFRC Mortar
0 0.000
0.004 0.008 0.012 Strain, mm/mm
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0.016
Lamina stacking optimization for strength and toughness 60 50
Unidirectional
Stress, MPa
40
[0/-45/45/90]s
30 20 10 [45/-45]s 0 0.000
0.005
0.010 0.015 0.020 Strain, mm/mm
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0.025
Flexural response of unidirectional and angle ply composite laminates
Equivalent Flexural Stress, MPa
40
30 Unidirectional 20
0-90-0
10 Paste
4.8% Continuous AR Glass Fibers
0 0
4
8 12 16 Deflection, mm
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20
Polypropylene Fiber based Unidirectional Composites 20 PP #5
Polypropylene Based cementitious laminates with 7% volume fraction of continuous fibers.
Stress (MPa)
16 12 8 4 0 0.00 0.02 0.04 0.06 0.08 0.10 Strain (mm/mm) You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)
Microcrack Toughening Mechanisms
Cracked Laminate
Distributed microcracking in unidirectional composite
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Weak Interfaces Result in Strong & Tough Composites
Interfacial Microcracking in between plies is a toughening Mechanism You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)
Various stages of cracking
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Fabric Reinforced Cement composites 25
80
Stress, MPa
20
60
15 40 10 20
5
0
0
0.01
0.02
0.03
Crack Spacing, mm
BT-GNSP21
0 0.04
Strain, mm/mm
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Crack Spacing for AR-Glass and Polyethylene Fabrics 100
Crack Spacing, mm
80
AR Glass Fabric Polyethylene
60 AR-Glass fabric
40 20 0
0
0.01 0.02 0.03 Strain, mm/mm
0.04 Polyethylene Fabric
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Stiffness degradation and Crack Spacing Relationship 1000
Tangent Stiffness, MPa
Glass Fabrics 100
10
1 80
Polyethylene Fabric
60 40 20 Crack Spacing, mm
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0
Scope of Presentation
Introduction Pultrusion Based Cement Composite Systems Theoretical Aspects of Formulation
Elastic response Traditional Composite laminate approach
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Composite Elastic Properties
Rule of Mixtures for longitudinal stiffness:
E1( ) E f V f Em ( )( 1 V f )
Halpin-Tsai equations for transverse modulus E2 , G12, and n12
E2 ( )
Em ( )( 1 V f ) 1 V f
E f Em ( ) E f Em ( )
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Incremental Approach
Unidirectional approach
ij S jki k i k
S
i 1 jk
or in matrix form:
1 S11 k 2 S 21 12 i 0 S11
ij ik1
1 E1( )
S12
S12 S 22 0 12 E1( )
1
0 1 1 0 2 2 S 66 12 i 12 i 1 S 22
1 E2 ( )
S 66
1 G12 ( )
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Q11 Q12 0 Sij1 Q21 Q22 0 0 0 Q66
Orientation Effects and Stress Transformation Cos 2 1 x T Sin 2 2 y ij Sin Cos 12 xy
1 R 0 0
0 0 1 0 0 2
Sin 2 Cos 2 Sin Cos
2Sin Cos Cos 2 Sin 2
2Sin Cos
x y xy
m=n qn
Q T R Q R T 1 ij ij ij ij hm-1
n A Q m( h h ) ij m m1 ij m1 1 n 2 h2 ) Bij Qijm( hm m1 2 m1 1 n D Q m( h3 h3 ) ij m m1 ij 3 m1
hm
m=1 q1
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Incremental Approach to Laminate Formulation Piecewise linear approach: ΔN ΔM
A B
B
D
N or M
0 Δε Δκ
n A Q m( h h ) ij m m1 ij m1 1 n B Q m( h 2 h 2 ) ij 2 ij m m1 m1 1 n D Q m( h3 h3 ) ij m m1 ij 3 m1
For uniaxial loads in a symmetric lamina (B=0) : For bending only in a symmetric lamina (B=0) :
or d
ΔN A Δε 0
ΔM D Δ
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Modes of failure 1t
2t
1c
12
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2c
23
Initial Failure Criteria
F ( 1 , 2 , 12 ) 1
General Yield surface Strength criteria 1 1fu Tsai-Wu Criteria
2 2fu
2
2
12 12fu
2
F ( 1 , 2 , 12 ) F111 2 F111 2 F22 2 F66 12 F11 F2 2 1 F1
R :=
1 1 1t 1c
F2
t1 := 50
c2 := 15
t2 := 10
12 := 5
c1 := 15
13 := 5
7
1
1 1 2t 2c
F11
1 σ 1t σ 1c
F22
1 σ 2t σ 2c
23 := 8
1
1 1 2 2 150 1 30 2 750 1 150 2 112500 1 2
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F66
1 σ 6u2
Scope of Presentation
Introduction Pultrusion Based Cement Composite Systems Theoretical Aspects of Formulation
Elastic response Traditional Composite laminate approach
Non-linear matrix response
Cracking Softening
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Three zones of stress-strain response for the matrix
Elastic matrix
Cracking matrix
failure by means of the yield surface criterion, ultimate strength, Tsai-Wu reduced stiffness, model the stiffness degradation use a scalar damage variable stress-strain response obtained using Nemat-Nasser and Hori’s approach.
2 1
3
Softening matrix, distributed cracking
strain-softening response, Karihaloo, and Hori’s softening Model
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Cracking Matrix- Zone 2 Stress
Damage B
1
A
O
0
0
Matrix parallel cracking You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)
Strain
Damage Evolution Law-Zone 2 0 ( ) 1 um 0 k i
f ( 1 , 2 , 12 ) 1
, 0 i 1 f ( 1 , 2 , 12 ) 1 1.00
Em
Em 1
b = 0.8
0.80 E( ) / E0
Karihaloo and Fu, 1990 a= 0.16, b= 2.3 um= ultimate strain at failure for uniaxial conditions
b = 0.6 b = 0.4
0.60
16 i (1 - m2 ) 3 0.40
0.000 Nemat Nasser and Hori, Micromechanics: overall properties of Heterogeneous Materials, 1993
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0.002 0.004 Axial Strain, mm/mm
0.006
Computations Within a load step Compute overall stiffness Calculate the strains, stresses, curvature, and elongation Check for the failure of the entire lamina using ultimate strength update the structure
Check for the failure in the components of the lamina using the failure criteria (Tsai-Wu or ultimate strength Theory) update the lamina properties
Initialize all the geometrical and historical parameters for each lamina. Impose the strain distribution, update the stresses in this increment
N A ( Q ( )) M B ( Q ( ))
B ( Q ( )) D ( Q ( ))
0
N xj ,i N xj ,i 1N xj ,i N xj ,i 1 A( Q( )) i 0
M xj ,i M xj ,i 1M xj ,i M xj ,i 1 D (Q( )) i You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)
Ultimate Failure & Strain Softening Response 1.0
fu is ultimate strength of fiber •Softening Response Post peak as a fraction of ultimate matrix stress, Model by Karihaloo and Fu, 1990
/ft
•Ultimate FailureFailure Criteria for each lamina: un = max(VffuCos2qm, t2)
H = 0.25 gage length (in meters) E0= 30000 MPa B= 200e-6 A= 0.3* ft/ E0 0= B * H 2.3 Cd0=.16*(1-A/ B ) w0=16*Cd0 / 9
0.8
0.6
0.4
0.2 0.0000
0.0001
0.0002 0.0003 Crack Opening,
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0.0004
Scope of Presentation
Introduction Pultrusion Based Cement Composite Systems Theoretical Aspects of Formulation
Non-linear matrix response
Elastic response Traditional Composite laminate approach Cracking Softening
Case Studies
Glass Fiber/ carbon fiber epoxy composites Cement based Glass fiber composite laminate Polypropylene fiber cement composites Fabric composites Comparison with Experimental results Repair and retrofit approach
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Glass-epoxy Composites
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Unidirectional Glass-Epoxy and Woven Carbon-Epoxy Composites Vf (%)
Strength (MPa)
Elastic Modulus (GPa)
Glass Fiber epoxy, 0º, 90º
45
fu = 1654 mu = 70
Ef =72, f = 0.2 Em= 4.0, m=.18
Woven Carbon Textile
50
t1=35-70
Ef = 100, m=.25 Em= 6.0, m=.18
Glass-Epoxy composites
Woven Carbon
Experiment
Simulation
60 Theory
40 Load, KN
Load, KN
0 degree
40
Sample 1
Vf =.5 =0.2 Em= 6000. MPa Ef= 100000. MPa m=.18 f =0.25
20
20 90 degree Theory Experiment
0 0.00
2.00 4.00 6.00 Elongation, mm
8.00
0 0.00
Sample 2
0.20
0.40
0.60
0.80
Clip Gage Elongation, mm
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1.00
Unidirectional and 0/90/0 fiber compositeseffect of fiber volume fraction 600
Unidirectional Glass Fiber Composites
500
= 10 MPa t1
= 5 MPa 400
= 40 MPa Vf = 6%
= 40 MPa c2
400
= 5 MPa 12
= 5 MPa 23
Vf = 0%
E n
200
m=
30000
m=
0.18
t1
= 5 MPa t2
= 40 MPa c1
c1
Vf = 6% Nominal Load, N/mm
Nominal Force, N/mm
t2
0/90/0 Glass Fiber Composites
= 10 MPa
= 40 MPa c2
= 5 MPa 12
= 5 MPa
300
23
E Vf = 4%
n
m=
30000
m=
0.18
200 Vf = 2%
Vf = 2%
100 Vf =0%
Vf = 0%
0 0.000
0.001 0.002 0.003 0.004 Axial Strain, mm/mm
0
0.005
0.000
0.001 0.002 0.003 Axial Strain, mm/mm
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0.004
Response of a 6 stack 0/90/0 lamina Strain Distribution
150 Nominal Stress, MPa
10 8 6
z,mm
4 2 0 -2
100
50
-4 -6 -8 -10 0
0.5
1
1.5
2
2.5
3
0
1.5
2
2.5
3
3.5 x 10
10
Transverse Stress
6
8 6 4
z,mm
4 2
z,mm
1
-3
x 10
8
0
2 0
-2
-2 -4
-4
-6
-6
-8
-8
-10 0
0.5
-3
mm/mm x
Stress 10 Distribution
0
3.5
2
4
6
8
10
12
-10 0
0.5
1
1.5
MPa You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)
2
MPa
2.5
3
3.5
4
Response of a 6 stack 0/90/0 lamina
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Response of an 8 stack [0/45/-45/90/90]s lamina Strain Distribution 10 8 6
140
2 0
120
-2 -4 -6 -8 -10 0
0.5
1
1.5
2
mm/mm x Stress Distribution 10
2.5
3
3.5 x 10
-3
8 6
Nominal stress, MPa
z,mm
4
100
80
60
40
z,mm
4
20
2 0
0
-2
0
0.5
1
1.5
2
2.5
-4
Nominal strain, mm/mm
-6 -8 -10 0
5
10
15
20
25
MPa You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)
3
3.5 -3
x 10
Response of an 8 stack [0/45/-45/90/90]s lamina
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[0/45/-45/90/90]s Composites - Effect of Vf 200
= 10 MPa
[0/45/-45/90/90]s
t1
= 5 MPa t2
Vf =6%
Nominal Load, N/mm
160
= 40 MPa c1
= 40 MPa c2
= 5 MPa 12
= 5 MPa
120
Vf = 4%
23
E n
Vf =2%
80
m=
30000
m=
0.18
Vf = 0%
40
0 0.000
0.001 0.002 0.003 Axial Strain, mm/mm
0.004
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Polypropylene Fiber Composites Effect of Vf
Effect of Lamina Orientation
3000
40 t1= 5 MPa
0= 3.5e-4
Unidrectional Polypropylene Fiber Composites
t2= 5 MPa c1= 40 MPa 12= 5 MPa
2000
Nominal Stress, MPa
Nominal Load, N/mm
c2= 40 MPa 23= 5 MPa
Vf = 20%
Vf = 15%
1000 0= 3.5e-4
Vf = 10%
0 0.00
30
Em = 30000 MPa Ef = 8000 MPa nm = 0.18 nf = 0.25 t1 = 5 MPa
unidirectional
20 0/90/0
10
Em = 30000 MPa Ef = 12000 MPa nm = 0.18 nf = 0.25
0.02 0.04 Axial Strain, mm/mm
Vf = 6%
0.06
90/0/90
0 0.00
0.02 0.04 Axial Strain, mm/mm
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0.06
Comparison With Experimental Results of unidirectional and 0/90/0 composites 60 = 10 MPa t1
50
Unidirectional Experiment
= 5 MPa t2
= 40 MPa c1
= 40 MPa c2
40 Stress, 30 MPa 20
= 5 MPa 12
Theory
= 5 MPa 23
Theory
[0/90]s
Experiment Em = 30000 Ef = 70000 Vf = 5%
10
n = 0.18 m
0 0.000
0.005 Strain, mm/mm
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0.010
Comparison of PPFRC with Experiments
Model Simulation
Stress, MPa
15
Damage Evolution Law
ik 0 (1 um ) 0 0., 5., 0.8 10
Experiments, Pivacek, Haupt, and Mobasher, 1998
5
Vf = 6% Em = 30000 MPa Polypropylene Ef = 8000 MPa Fiber Composites nm = 0.18 nf = 0.25 t1 = 5 MPa w0= 3.5e-4 Softening Coefficient
0.000
0.005
0.010
0.015
Strain, mm/mm You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)
Flexural Response of Glass Fiber Composite Laminates 1600 = 5 MPa t1
Nominal Moment, N-mm/mm
= 5 MPa t2
Vf = 6%
= 10 MPa
1200
c1
= 10 MPa c2
= 5 MPa 12
= 5 MPa 23
Vf = 4%
800
Vf = 2%
400
0 0.0000
Vf =0%
Em = 30000 Ef = 70000 nm = 0.18
0.0001 0.0002 0.0003 Curvature, 1/mm
0.0004
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Response of an 8 stack unidirectional lamina to Flexure 10
Strain Distribution
a b
4 2 0
8
c
-2
c
-4 -6 -8 -10-3
b
4 2 0
c
xx
-2
a
b
c
a
6 z,mm
z,mm
10 8 6
-4
a
b
-6 -8
-2
-1
0 1 mm/mm
-10 -10
2 3 x 10-3
-5
0
10 5 MPa
15
20
25
10000 8
b
a
6
c
4 z,mm
Moment/unit width, N mm /mm
10
2 0 -2 -4
c
yy
b
-6
a
-8 -10
-4
-3
-2
-1
0
MPa
1
2
3
4
5
9000 8000
c
7000
b
6000 5000 4000 3000 2000 1000 0
a 0
0.5 1 1.5 Curvature, 1/mm
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2
2.5
3
3.5 x 10-4
Flexural Load-Deflection Strain Distribution
sigma x
0.25
0.25
0.2
0.15
0.1
0.1
0.05
0.05 z,mm
0.15
z,mm
Stress Distribution 8 layers 0 degree
0.2
0
0 -0.05
-0.05
Strain Distribution
-0.1 -0.15
-0.1 -0.15 -0.2
-0.2 -0.25 -1
-0.5
0
0.5 1 mm/mm
1.5
2
-0.25 -4000
2.5 x 10
-3000
-2000
-1000
0
1000
2000
3000
psi
-3
sigmax 0.25
250
0.2 0.15
200
0 degree
0.1 0.05
force, lbs
z,mm
150
0
Stress Distribution 8 layers [02/902]s
-0.05
100 -0.1
[02/902]s
50
-0.15 -0.2
0
-0.25 -4000
0
1
2
3
4 5 deflection, in
6
7
8
-3000
-2000
9 x 10
-3
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-1000
0 psi
1000
2000
3000
Composite Retrofit - Reinforced Concrete beamGlass Epoxy Bonded Laminate Concrete Steel-concrete layer Concrete Glass-epoxy material
Young’s Modulus psi
Posisson’s ratio
Compressive Strength, psi
Tensile strength, Psi
Coefficient of Thermal expansion
epoxy
6.e5
0.28
20000
6000
2.6e-6
glass fibers
11.5e6
0.25
200000
200000
1.e-6
paste
3.e6
0.17
8000.
500.
1.e-6
steel
29.e6
0.3
36000
36000
6.e-6
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Stress Distribution in Flexural Sample
Vs= 0.12 tgl-ep
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Moment Curvature computations of a Retrofitted RC beam 60000
Composite Retrofit
Vg= 0.4
Moment, Lb-in
Vs= 0.12
tgl-ep= 0.2"
12”
40000
tgl-ep= 0.1"
10” tgl-ep= 0.0"
20000
tgl-ep 0
0
0.0001
0.0002
0.0003
0.0004
Curvature, 1/in
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Modeling of Fabrics 20
12
200
Theoretical Modeling
8
160
max = 5.0 MPa f = 3.0 MPa d = 1.3 MPa X = 30 mm (assumed crack spacing)
4
0
0.01
Debonding Length
Crack Spacing
Load, N
Stress, MPa
16
0
BGNS200_1
Em=15000 MPa Ef=5000 MPa tu = 5 MPa fu= 3700 MPa nm=0.17 nf= 0.25
0.02
0.03
120
80
40
0.04
Strain, mm/mm
0
0
0.4
0.8
1.2
1.6
Slip, mm
Pullout Slip response
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2
Conclusions
Theoretical models based on composite laminate Theory can be used to predict the mechanical response of various cement based composite systems. Combination of damage mechanics based methods and composite laminate theory can result in a useful method to evaluate the response of various fiber matrix, geometry, and loading combinations. The formulation is applicable to a wide range of materials studied including unidirectional, angle ply, sandwich, fabric reinforced, and also retrofit composite systems.
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