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FUTURE STRUCTURAL CONSULTANTS
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f c = f c 1 − /
0.15 0.003 − εo
(εc − εo )
f c′
Stress
εc εc 2 f c = f c 2 − εo εo /
Strain
ε0
0.003
Figure 4.1: Theoretical Stress-Strain Plot of Concrete
f s = f y Es Stress f s = εsEs
εy
Strain
Figure 4.2: Theoretical Theoretical Stress-Strain Plot of Steel Reinforcement
44
b
ε ds
αf c
EC
f s
′
εs
c
γ c
Cc Cs
ds
dCFRP
εs ε
f s
CFRP
Section
Ts TCFRP
f CFRP Strain
Stress
Force
Figure 4.3: Strain, Stress, and Force Diagrams of CFRP Strengthened Concrete Beam
f c′ f c Stress dεc
εEC
Strain
0.003
Figure 4.4: Integration of Concrete Stress-Strain Curve
45
εc E
R
Neutral Axis
c
ϕ
M M ds
ε
S
Tension Steel Section
Strain Distribution
Figure 4.5: Rotation of Flexural Member
700 600 s 500 p i k n 400 i , t n e 300 m o M200
Control Two-Layer Three-Layer
100
Four-Layer
0 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Deflection, in.
Figure 4.6: Theoretical Moment-Deflection Plots of 0° /90° CFRP Strengthened Beams
46
700 600 s 500 p i k n 400 i , t n e 300 m o M200
Control Two-Layer Three-Layer Four-Layer
100 0 0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Deflection, in.
Figure 4.7: Theoretical Moment-Deflection Plots of ±45° CFRP Strengthened Beams
700 600
s 500 p i k n 400 i , t n e 300 m o M200
Two-Layer Three-Layer Four-Layer
100 0 0
0.002
0.004
0.006
0.008
0.01
CFRP Strain, in/in
Figure 4.8: Theoretical Moment-CFRP Strain Plots of 0 ° /90° CFRP Strengthened Beams
47
700 600
s 500 p i k - 400 n i , t n e 300 m o M200
Two-Layer Three-Layer Four-Layer
100 0 0
0.002
0.004
0.006
0.008
0.01
CFRP Strain, in/in
Figure 4.9: Theoretical Moment-CFRP Strain Plots of ±45° CFRP Strengthened Beams
48
Steel Reinforcement
FRP Reinforcement
Moment
Moment
Elastic Energy Released at Failure
Elastic Energy Released at Failure Deflection
Deflection
Figure 4.10: Comparison of Elastic Energy Released at Failure
EINEL, Inelastic Energy Consumed Prior to Failure MU M2 Moment
EEL, Elastic Energy Released at Failure
S2
* ETOT = EEL + EINEL
M1
S *S= S1
M1 S1 + (M2 - M1) S2 M2
* S, S1, S2: Slopes of Lines Deflection
Figure 4.11: Approximation Process for Energy Method
49
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