1 Introduction
2 Experimental Program
2.1 Materials and Specimen
Fibre type | σ
u
a
(MPa) | σ
y
b
(MPa) | lf (mm) | df (mm) | Hook length (mm) | Hook angles (°) | Hook height (mm) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
L1
|
L2
|
L3
|
L4
| θ1 | θ2 | β | H1 | H2 | |||||
3D 65/60 BG | 1150 | 775–985 | 60 | 0.90 | 2.12 | 2.95 | – | – | 45.7 | 45.5 | 67.5 | 1.85 | – |
4D 65/60 BG | 1500 | 1020–1165 | 60 | 0.90 | 2.98 | 2.62 | 3.05 | – | 30.1 | 30.8 | 75.0 | 4.37 | 2.20 |
5D 65/60 BG | 2300 | 1177–1455 | 60 | 0.90 | 2.57 | 2.38 | 2.57 | 2.56 | 27.9 | 28.2 | 76.0 | 2.96 | 1.57 |
Concrete type | Mix proportions (kg/m3) | W/B (−) | Compressive strength | |||||
---|---|---|---|---|---|---|---|---|
Cement (type) | Fly ash | Aggregates | Sand | Superplasticizer | Water | (MPa) | ||
8–10 mm | 0-mm | |||||||
NSC | 364 (32.5R) | – | 979 | 812 | – | 200 | 0.55 | 33 |
MSC | 350 (52.5N) | 107 | 660 | 1073 | – | 205 | 0.45 | 54 |
HSC | 480 (52.5N) | 45 | 850 | 886 | 6 | 210 | 0.40 | 72 |
2.2 Test Setup
2.3 Experimental Results
3 Elastic–Plastic Moment Expression
3.1 Plastic Penetration Section Geometry
3.2 Curvature–Plastic Penetration Relationship
3.3 Account of Springback
4 Method
x | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
---|---|---|---|---|---|---|---|---|---|---|
1 − x | 0.9 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 | 0.0 |
2 − x | 1.9 | 1.8 | 1.7 | 1.6 | 1.5 | 1.4 | 1.3 | 1.2 | 1.1 | 1.0 |
Equation (16) | 0.7826 | 0.7029 | 0.673 | 0.6509 | 0.6136 | 0.5482 | 0.450 | 0.3208 | 0.1673 | 0.0 |
y | 4.14 | 2.693 | 2.14 | 1.719 | 1.294 | 0.8854 | 0.5455 | 0.2945 | 0.1201 | 0 |
Method | 3DH | 4DH | 5DH |
---|---|---|---|
Step 1 (\( \rho_{R} \)) | 39.2 | 57.95 | 62 |
Step 2 (\( {\text{y}} \)) | 0.40 | 0.74 | 0.98 |
Step 3 (\( x \)) | 0.74 | 0.60 | 0.56 |
Step 4 (\( \theta_{e} \)) | 15.07 | 23.57 | 26.10 |
Step 5 (\( {\text{R}} \)) | 0.67 | 0.50 | 0.45 |
Step 6 (\( h \)) | 0.33 | 0.27 | 0.25 |
Step 7 (\( M_{ep} \)) | 108 | 117 | 140 |
5 Incorporating the Elastic–Plastic Moment Expression Into Frictional Pulley Model
5.1 For 3DH fibre
5.2 For 4DH fibre
5.3 For 5DH fibre
6 Model Validation
Fibre type | Matrix type |
P
1
|
P
2
|
P
3
|
P
4
|
P
5
|
---|---|---|---|---|---|---|
3DH | NSC | 55 | 404 | 264 | – | – |
MSC | 76 | 593 | 285 | – | – | |
HSC | 108 | 625 | 317 | – | – | |
4DH | NSC | 55 | 545 | 454 | 229 | – |
MSC | 76 | 766 | 475 | 250 | – | |
HSC | 108 | 798 | 507 | 282 | – | |
5DH | NSC | 55 | 746 | 834 | 510 | 255 |
MSC | 76 | 1162 | 855 | 531 | 276 | |
HSC | 108 | 1194 | 887 | 563 | 308 |