1 Introduction
2 Equivalent Curvature Blocks at Ultimate State
2.1 Assumptions
2.2 Equivalent Rectangular Curvature Blocks and Unbonded Tendon Strain
Sections |
L
s
|
ω
s
|
κ
s
| |
---|---|---|---|---|
Locations | Symbols (s) | |||
Beam end | es |
\( - \frac{{M_{es} }}{{M_{us} }}L_{sp} \)
|
L
es
|
ω
es
(3L
sp
- L
es
) |
Cracked section | cs |
\( \frac{{M_{es} + M_{cs} }}{{M_{us} }}L_{sp} \)
| − (L
cs
+ L
us
) |
ω
cr
(L
cs
+ 2L
ys
+ 3L
us
) |
Yield section | ys |
\( \frac{{M_{ys} - M_{cs} }}{{M_{us} }}L_{sp} \)
| − (L
ys
+ L
us
) |
ω
ys
(L
ys
+ 2L
us
) |
Ultimate section | us |
\( \left( {1 - \frac{{M_{ys} }}{{M_{us} }}} \right) \cdot L_{sp} \)
| − L
us
|
ω
us
· L
us
|
2.3 Moment and Curvature at Characteristic Sections
3 Iterative Algorithm for the Prediction of ɛ pfu at Ultimate State
3.1 Modified Algorithm to Include Frictional Losses
3.2 Validity of the Model
Types | Diameter (mm) | Effective area (mm2) | CFRP | Steel | |||
---|---|---|---|---|---|---|---|
f
pfm
or f
bfm
(MPa) |
E
pf
or E
bf
(MPa) |
ɛ
pfm
or ɛ
bfm
(%) |
f
y
(MPa) | ||||
Tendons | CFCC | 7.5 | 30.4 | 1880 | 144,000 | 1.30 | – |
10.5 | 55.7 | 1880 | 144,000 | 1.30 | – | ||
15.2 | 113.6 | 1750 | 135,000 | 1.30 | – | ||
DWC | 9.5 | 70.9 | 2500 | 135,000 | 1.85 | – | |
Auxiliary bars | DWC | 6.0 | 28.3 | 2200 | 139,000 | 1.58 | – |
Steel | 6.0 | 31.6 | – | – | – | 420 | |
Stirrups | DWC | 6.0 | 28.3 | 2200 | 139,000 | 1.58 | – |
Steel | 6.0 | 31.6 | – | – | – | 420 |
Specimens | Prestressing reinfocement ratio
\( \rho_{pf}^{U} /\rho_{pfb}^{B} \)
| Initial prestrssing
f
pfi
/f
pfm
| Loading type, L
sp
(mm) | Reinforcement | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Concrete compressive strength, \( f_{c}^{'} \)
(MPa) | Unbonded FRP tendons | Bonded reinforcing bar | ||||||||
Types | Diameter (mm) |
A
pf
(mm2) | Types | Diameter (mm) |
A
bf
(mm2) | |||||
RU50 | 0.53 | 0.5 | 4-point (900) | 35 | CFCC | 10.5 | 55.7 | Steel | 6.0 | 63.2 |
RU70 | 0.37 | 0.7 | 4-point (900) | 35 | CFCC | 10.5 | 55.7 | Steel | 6.0 | 63.2 |
RO50 | 1.51 | 0.5 | 4-point (900) | 35 | CFCC | 7.5 | 30.4 | Steel | 6.0 | 63.2 |
CFCC | 15.2 | 113.6 | ||||||||
RO55 | 1.85 | 0.55 | 4-point (900) | 40 | DWC | 9.5 | 70.9 | CFRP | 6.0 | 84.9 |
TB45 | 0.84 | 0.45 | 4-point (1200) | 40 | DWC | 9.5 | 141.8 | CFRP | 6.0 | 56.6 |
TO45 | 1.36 | 0.45 | 4-point (1200) | 40 | DWC | 9.5 | 70.9 | CFRP | 6.0 | 56.6 |
DWC | 9.5 | 141.8 |
Specimens | Prestressing reinforcement ratios | Max. load | Tendon strains | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
\( \rho_{pf}^{U} \)
|
\( \rho_{pfb}^{B} \)
|
\( \rho_{pfb}^{U} \)
|
\( \rho_{pf}^{U} /\rho_{pfb}^{U} \)
| Test (kN) | Thy/Test |
ɛ
pfu
| Δɛ
pfu
| |||||
Test (mm/mm) | Model/Test | Test (mm/mm) | Model/Test | Equation (14)/Test | Equation (15)/Test | |||||||
RU50 | 0.0019 | 0.0036 | 0.0014 | 1.36 | 47.2 | 1.03 | 0.0088 | 1.10 | 0.0032 | 1.45 | 1.16 | 1.62 |
RU70 | 0.0019 | 0.0052 | 0.0025 | 0.76 | 58.4 | 0.94 | N.A | N.A | N.A | N.A | N.A | N.A |
RO50 | 0.0053 | 0.0035 | 0.0013 | 4.08 | 80.2 | 0.95 | 0.0082 | 1.06 | 0.0020 | 1.34 | 1.14 | 0.94 |
RO55 | 0.0024 | 0.0030 | 0 | – | 116.1 | 0.81 | 0.0142 | 0.99 | 0.0040 | 0.92 | 0.74 | 0.11 |
TB45 | 0.0021 | 0.0025 | 0.0013 | 1.62 | 118.5 | 1.06 | N.A | N.A | N.A | N.A | N.A | N.A |
TO45 | 0.0034 | 0.0025 | 0.0012 | 2.83 | 150.3 | 0.94 | 0.0112 | 1.13 | 0.0043 | 1.39 | 1.11 | 0.64 |
Average Standard deviation | 0.96 0.09 | 1.07 0.06 | 1.27 0.24 | 1.04 0.20 | 0.83 0.63 |
3.3 Practical Equation of Δf ps
4 Iterative Algorithm for the Determination of ρ pfb U
4.1 Development of a Balanced Ratio
4.2 Estimation of \( \rho_{pfb}^{U} \) and Failure Modes for Tested Beams
5 Parametric Studies for \( \rho_{pfb}^{U} \)
No. | Effects | Initial prestress | Compressive concrete strength | Bonded FRP tendon | Effective depth | Distance between loading points | Balanced ratio | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\( f_{pfi} \)
(MPa) |
\( f_{pfi} /f_{pfm} \)
|
\( f_{c}^{'} \)
(MPa) |
\( f_{c}^{'} /f_{c,std}^{'} \)
|
A
bf
(mm2) |
A
bf
/A
bf,min
|
d
pf
(mm) |
L/d
pf
|
L
p
(mm) |
L
p
/L
|
\( \rho_{pfb}^{U} \)
|
\( \rho_{pfb}^{B} \)
|
\( \rho_{pfb}^{U} /\rho_{pfb}^{B} \)
| ||
1 | Standard | 845 | 0.5 | 40 | 1.0 | 745 | 1.0 | 560 | 26.8 | 4667 | 0.33 | 0.0027 | 0.0039 | 0.68 |
2 |
\( f_{pfi} \)
| 676 | 0.4 | 40 | 1.0 | 745 | 1.0 | 560 | 26.8 | 4667 | 0.33 | 0.0021 | 0.0033 | 0.63 |
3 | 1014 | 0.6 | 0.0037 | 0.0051 | 0.73 | |||||||||
4 |
f
c
’
| 845 | 0.5 | 30 | 0.75 | 745 | 1.0 | 560 | 26.8 | 4667 | 0.33 | 0.0021 | 0.0032 | 0.65 |
5 | 50 | 1.25 | 0.0032 | 0.0046 | 0.70 | |||||||||
8 |
d
pf
| 845 | 0.5 | 40 | 1.0 | 745 | 1.0 | 600 | 25 | 4667 | 0.33 | 0.0029 | 0.0042 | 0.71 |
9 | 520 | 28.8 | 0.0025 | 0.0039 | 0.65 | |||||||||
10 |
A
bf
| 845 | 0.5 | 40 | 1.0 | 372.5 | 0.5 | 560 | 26.8 | 4667 | 0.33 | 0.0030 | 0.0043 | 0.70 |
11 | 1118 | 1.4 | 0.0024 | 0.0037 | 0.65 | |||||||||
12 |
L
p
| 845 | 0.5 | 40 | 1.0 | 745 | 1.0 | 560 | 26.8 | 0 | 0 | 0.0015 | 0.0035 | 0.43 |
13 | 2333 | 0.17 | 0.0020 | 0.0036 | 0.55 | |||||||||
14 | 7000 | 0.50 | 0.0032 | 0.0039 | 0.83 |