1 Background
2 Formulation for Reliability Analysis
2.1 Limit State Function
2.2 Algorithm for Reliability Analysis
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Nominal values of all random and deterministic variables;
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Statistical properties of all random variables (e.g. COV and bias factors) and select their probability distributions.
3 Data for Reliability Analysis
Constituent | Quantity |
---|---|
Cement | 520 kg/m3 |
Sand (fine) | 586 kg/m3 |
10 mm size coarse aggregate | 850 kg/m3 |
5 mm size coarse aggregate | 315 kg/m3 |
Water | 145 kg/m3 |
Super-plasticizer (Gli-110) | 3.0 L |
Retarder (LD10) | 1.5 L |
Fiber type | Length (mm) | Shape | Section dimensions (mm) | Aspect ratio | Specific gravity | Tensile strength (MPa) | Modulus of elasticity (GPa) | Bond factor, k |
---|---|---|---|---|---|---|---|---|
SF | 60 | Hooked ends | 0.75 \(\upphi\) (circular) | 80 | 7.85 | 1225 | 200 | 1 |
PP | 50 | Crimped | 1 × 0.6 (rectangular) | 57.2a | 0.9 | 550 | 4 | 1 |
KF | 45 | Plain | 0.50 \(\upphi\) (circular) | 90 | 1.45 | 3220 | 131 | 0.8 |
Concrete mix | Percentage of fiber by volume (by weight) | |||
---|---|---|---|---|
Polypropylene (PP) | Steel (SF) | Kevlar (KF) | Total | |
M0 | 0.0 (0.00) | 0.0 (0.00) | 0.0 (0.00) | 0.0 (0.00) |
M1 | 0.0 (0.00) | 1.2 (3.93) | 0.0 (0.00) | 1.2 (3.93) |
M2 | 0.2 (0.08) | 1.0 (3.27) | 0.0 (0.00) | 1.2 (3.35) |
M3 | 0.0 (0.00) | 1.4 (4.58) | 0.0 (0.00) | 1.4 (4.58) |
M4 | 0.2 (0.08) | 1.2 (3.93) | 0.0 (0.00) | 1.4 (3.98) |
M5 | 0.0 (0.00) | 0.9 (2.94) | 0.3 (0.11) | 1.2 (3.05) |
M6 | 0.0 (0.00) | 1.1 (3.60) | 0.3 (0.11) | 1.4 (3.71) |
M7 | 0.2 (0.08) | 0.9 (2.94) | 0.3 (0.11) | 1.4 (3.13) |
M8 | 0.2 (0.08) | 0.7 (2.29) | 0.3 (0.11) | 1.2 (2.48) |
Random variable | Nominal | Bias factor | Coefficient of variation (COV) | Distribution | References |
---|---|---|---|---|---|
HFRC slab | |||||
Concrete density, \(\rho_{c}\) (kg/m3) | 2500 | 1.05 | 0.10 | Lognormal | Choudhury et al. (2002) |
Concrete strength, \(f_{c}^{'}\) (MPa) | Variable | 1.10 | 0.10 | Lognormal | Siddiqui et al. (2014b) |
Thickness of slab, H (mm) | 90 | 1.00 | 0.05 | Normal | Choudhury et al. (2002) |
Reinforcement ratio, r (%) | 0.708 | 1.10 | 0.10 | Normal | Siddiqui et al. (2014b) |
Steel rebar spacing, cr (mm) | 100 | 0.90 | 0.05 | Lognormal | Assumed |
Reinforcing index, RIv | Estimateda | 1.10 | 0.12 | Lognormal | Assumed |
Projectile | |||||
Mass of the projectile, M (kg) | 0.8 | 1.10 | 0.05 | Lognormal | Penmetsa (2005) |
Diameter of the projectile, d (mm) | 40 | 1.05 | 0.05 | Normal | Penmetsa (2005) |
Impact velocity, V0 (m/s) | Variable | 0.90 | 0.10 | Extreme type I | Choudhury et al. (2002) |
4 Discussion of Results
Slab specimen |
V
0
|
V
BL
| V0/VBL | Failed/not failed |
P
f
| β |
---|---|---|---|---|---|---|
HFRC-M0S1 | 135.20 | 118.5 | 1.14 | F | 0.232 | 0.733 |
HFRC-M0S2 | 125.10 | 1.06 | N | 0.153 | 1.023 | |
HFRC-M0S3 | 108.10 | 0.91 | N | 0.051 | 1.635 | |
HFRC-M1S1 | 135.10 | 153.6 | 0.88 | N | 0.032 | 1.848 |
HFRC-M1S2 | 160.20 | 1.04 | F | 0.111 | 1.220 | |
HFRC-M1S3 | 147.25 | 0.96 | N | 0.064 | 1.522 | |
HFRC-M2S1 | 135.10 | 142.9 | 0.95 | N | 0.062 | 1.540 |
HFRC-M2S2 | 160.20 | 1.12 | F | 0.184 | 0.902 | |
HFRC-M2S3 | 147.25 | 1.03 | N | 0.109 | 1.233 | |
HFRC-M3S1 | 135.10 | 162.6 | 0.83 | N | 0.018 | 2.087 |
HFRC-M3S2 | 178.50 | 1.10 | F | 0.153 | 1.025 | |
HFRC-M3S3 | 147.25 | 0.91 | N | 0.040 | 1.748 | |
HFRC-M4S1 | 135.10 | 151.8 | 0.89 | N | 0.035 | 1.809 |
HFRC-M4S2 | 168.20 | 1.11 | F | 0.168 | 0.962 | |
HFRC-M4S3 | 147.25 | 0.97 | N | 0.069 | 1.486 | |
HFRC-M5S1 | 135.15 | 139.7 | 0.97 | N | 0.070 | 1.475 |
HFRC-M5S2 | 125.00 | 0.89 | N | 0.036 | 1.801 | |
HFRC-M5S3 | 147.10 | 1.05 | F | 0.121 | 1.169 | |
HFRC-M6S1 | 135.00 | 147.1 | 0.92 | N | 0.044 | 1.703 |
HFRC-M6S2 | 160.20 | 1.09 | F | 0.149 | 1.041 | |
HFRC-M6S3 | 147.25 | 1.00 | N | 0.083 | 1.384 | |
HFRC-M7S1 | 135.20 | 141.4 | 0.96 | N | 0.0653 × 10−1 | 1.512 |
HFRC-M7S2 | 158.10 | 1.12 | F | 0.181 × 10−1 | 0.911 | |
HFRC-M7S3 | 147.30 | 1.04 | N | 0.114 × 10−1 | 1.206 | |
HFRC-M8S1 | 135.34 | 134.3 | 1.01 | N | 0.983 × 10−1 | 1.291 |
HFRC-M8S2 | 153.67 | 1.14 | F | 0.207 × 10−1 | 0.815 | |
HFRC-M8S3 | 147.30 | 1.10 | N | 0.170 × 10−1 | 0.954 |
4.1 Effect of Impact Velocity
4.2 Effect of Steel Fiber Proportion
4.3 Effect of Slab Thickness
5 Conclusions
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As the impact velocity of projectile increases, reliability of HFRC slabs decreases sharply. Reliability is around 4 when impact velocity is about half of the ballistic limit and decreases to nearly 0, when it is around 1.4 times the ballistic limit of the slab. The reliability is 3 and above when the ratio V0/VBL is 0.7 or less.
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The ballistic limit of the HFRC slabs should be kept around 1.4 V0 in order to achieve desired reliability index of the slab (i.e. 3.0).
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The addition of fibers improves the slab reliability substantially. The maximum increase was observed for that HFRC slab which contains highest amount of steel fibers (i.e. 1.4%). The second highest reliability is obtained for the slab that contains second highest percentage of steel fibers (i.e. 1.2%).
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Those HFRC slabs which have a relatively higher percentage of steel fibers have higher tolerance for projectile impact than HFRC slabs with lower percentage of steel fibers.
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The reliability index of present HFRC slabs will achieve desired value of reliability index (i.e. 3.0) if the steel fiber percentage is increased to 1.8%.
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As the slab thickness increases, the ballistic limit of the slab increases which consequently increases the overall reliability of the HFRC slab.
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To achieve the reliability index = 3.0 for HFRC-M5 and HFRC-M8 slabs, thickness of slab should be at least 100 mm. However, for HFRC-M3 slabs, this thickness requirement reduces to about 95 mm.