1 Introduction
2 Fire Resistance Column Curves Method
2.1 Concept of Fire Resistance Column Curves
2.2 Construction of Column Curves
Column | L m (in.) | \(\lambda\) \(\left( \frac{L}{r} \right)\) | Series 1 | Series 2 | Series 3 | Series 4 | Series 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
e0 = 5 | e0 = 15 | e0 = 25 | e0 = 15 | e0 = 15 | ||||||||
δ0 = 10 | δ0 = 10 | δ0 = 10 | δ0 = 0 | δ0 = 0 | ||||||||
(mm) | (mm) | (mm) | (mm) | (mm) | ||||||||
\(R_{f}\)(min) | \(R_{f}\)(min) | \(R_{f}\)(min) | \(R_{f}\)(min) | \(R_{f}\)(min) | ||||||||
F | H | F | H | F | H | F | H | F | H | |||
1 | 0.380 (15) | 4.3 | 224 | 224 | 214 | 214 | 199 | 199 | 188 | 188 | 190 | 190 |
2 | 0.635 (25) | 7.2 | 222 | 221 | 211 | 208 | 197 | 196 | 186 | 185 | 188 | 185 |
3 | 1.270 (50) | 14.5 | 218 | 214 | 205 | 195 | 189 | 178 | 182 | 178 | 185 | 178 |
4 | 1.910 (75) | 21.7 | 212 | 204 | 198 | 180 | 178 | 157 | 175 | 170 | 177 | 165 |
5 | 2.54 (100) | 28.9 | 202 | 190 | 188 | 159 | 167 | 138 | 167 | 158 | 170 | 138 |
6 | 3.18 (125) | 36.2 | 192 | 169 | 178 | 136 | 158 | 117 | 160 | 138 | 160 | 120 |
7 | 3.81 (150) | 43.4 | 183 | 148 | 168 | 115 | 148 | 98 | 147 | 90 | 150 | 102 |
8 | 4.45 (175) | 50.6 | 173 | 128 | 158 | 96 | 138 | 79 | 141 | 78 | 138 | 87 |
9 | 5.08 (200) | 57.9 | 163 | 105 | 148 | 75 | 125 | 59 | 133 | 64 | 132 | 68 |
10 | 5.72 (225) | 65.1 | 152 | 86 | 138 | 60 | 117 | 47 | 125 | 50 | 118 | 55 |
11 | 6.35 (250) | 72.3 | 140 | 68 | 127 | 47 | 107 | 37 | 109 | 35 | 112 | 41 |
12 | 6.99 (275) | 79.6 | 129 | 53 | 117 | 37 | 99 | 28 | 102 | 25 | 107 | 33 |
13 | 7.62 (300) | 86.8 | 119 | 40 | 109 | 28 | 89 | 19 | 95 | 18 | 102 | 29 |
14 | 8.26 (325) | 94.0 | 111 | 27 | 104 | 19 | 83 | 11 | 88 | 9 | 95 | 22 |
Load (kN) | 1200 | 1200 | 1200 | 1200 | 1200 | 1200 | 1710 | 1710 | 1200 | 1200 | ||
Load ratio | 35 | 35 | 35 | 35 | 35 | 35 | 50 | 50 | 35 | 35 | ||
Cover to main bars (mm) | 48 | 48 | 48 | 48 | 48 | 48 | 48 | 48 | 25 | 25 |
Column | L m (in.) | \(\lambda\) \(\left( \frac{L}{r} \right)\) | Series 6 | Series 7 | Series 8 | |||
---|---|---|---|---|---|---|---|---|
e0 = 15 | e0 = 25 | e0 = 50 | ||||||
δ0 = 0 | δ0 = 0 | δ0 = 0 | ||||||
(mm) | (mm) | (mm) | ||||||
\(R_{f}\)(min) | \(R_{f}\)(min) | \(R_{f}\)(min) | ||||||
F | H | F | H | F | H | |||
1 | 0.380 (15) | 4.3 | 222 | 222 | 222 | 222 | 170 | 170 |
2 | 0.635 (25) | 7.2 | 221 | 218 | 220 | 218 | 167 | 167 |
3 | 1.270 (50) | 14.5 | 217 | 211 | 215 | 205 | 160 | 152 |
4 | 1.910 (75) | 21.7 | 209 | 194 | 207 | 190 | 152 | 133 |
5 | 2.54 (100) | 28.9 | 200 | 162 | 199 | 168 | 141 | 117 |
6 | 3.18 (125) | 36.2 | 189 | 141 | 179 | 145 | 133 | 98 |
7 | 3.81 (150) | 43.4 | 178 | 125 | 177 | 124 | 124 | 81 |
8 | 4.45 (175) | 50.6 | 162 | 106 | 169 | 105 | 115 | 64 |
9 | 5.08 (200) | 57.9 | 148 | 92 | 159 | 85 | 105 | 48 |
10 | 5.72 (225) | 65.1 | 138 | 73 | 149 | 69 | 97 | 38 |
11 | 6.35 (250) | 72.3 | 130 | 57 | 138 | 57 | 89 | 29 |
12 | 6.99 (275) | 79.6 | 125 | 46 | 127 | 46 | 81 | 21 |
13 | 7.62 (300) | 86.8 | 121 | 39 | 119 | 37 | 73 | 14 |
14 | 8.26 (325) | 94.0 | 118 | 35 | 115 | 28 | 67 | 6 |
Load (kN) | 1200 | 1200 | 1000 | 1000 | 1200 | 1200 | ||
Load ratio % | 35 | 35 | 30 | 30 | 35 | 35 |
Column | L m (in.) | \(\lambda\) (\(\frac{L}{r}\)) | Series 9 | Series 10 | ||
---|---|---|---|---|---|---|
e0 = 5 | e0 = 15 | |||||
δ0 = 10 | δ0 = 10 | |||||
(mm) | (mm) | |||||
\(R_{f}\)(min) | \(R_{f}\)(min) | |||||
F | H | F | H | |||
1 | 0.380 (15) | 4.3 | 228 | 228 | 220 | 220 |
2 | 0.635 (25) | 7.2 | 225 | 225 | 217 | 216 |
3 | 1.270 (50) | 14.5 | 221 | 218 | 211 | 205 |
4 | 1.910 (75) | 21.7 | 216 | 209 | 205 | 191 |
5 | 2.54 (100) | 28.9 | 210 | 197 | 198 | 170 |
6 | 3.18 (125) | 36.2 | 204 | 177 | 190 | 147 |
7 | 3.81 (150) | 43.4 | 196 | 158 | 179 | 125 |
8 | 4.45 (175) | 50.6 | 187 | 138 | 170 | 105 |
9 | 5.08 (200) | 57.9 | 176 | 116 | 160 | 85 |
10 | 5.72 (225) | 65.1 | 166 | 98 | 152 | 70 |
11 | 6.35 (250) | 72.3 | 157 | 78 | 141 | 57 |
12 | 6.99 (275) | 79.6 | 147 | 62 | 132 | 48 |
13 | 7.62 (300) | 86.8 | 136 | 48 | 123 | 38 |
14 | 8.26 (325) | 94.0 | 128 | 36 | 115 | 28 |
Column | L m (in.) | \(\lambda\) (\(\frac{L}{r}\)) | Series 11 | Series 12 | ||
---|---|---|---|---|---|---|
e0 = 5 | e0 = 15 | |||||
δ0 = 10 | δ0 = 10 | |||||
(mm) | (mm) | |||||
\(R_{f}\)(min) | \(R_{f}\)(min) | |||||
F | H | F | H | |||
1 | 0.380 (15) | 3.3 | 252 | 252 | 248 | 248 |
2 | 0.635 (25) | 5.4 | 251 | 250 | 247 | 247 |
3 | 1.270 (50) | 10.9 | 249 | 246 | 245 | 240 |
4 | 1.910 (75) | 16.3 | 246 | 239 | 241 | 227 |
5 | 2.54 (100) | 21.7 | 242 | 231 | 236 | 215 |
6 | 3.18 (125) | 27.1 | 236 | 220 | 227 | 201 |
7 | 3.81 (150) | 32.6 | 229 | 207 | 220 | 176 |
8 | 4.45 (175) | 38.0 | 221 | 188 | 210 | 162 |
9 | 5.08 (200) | 43.4 | 214 | 171 | 202 | 141 |
10 | 5.72 (225) | 48.8 | 206 | 155 | 193 | 124 |
11 | 6.35 (250) | 54.3 | 196 | 135 | 183 | 105 |
12 | 6.99 (275) | 59.7 | 189 | 115 | 175 | 86 |
13 | 7.62 (300) | 65.1 | 179 | 100 | 166 | 74 |
14 | 8.26 (325) | 70.5 | 169 | 85 | 156 | 62 |
15 | 8.89 (350) | 76.0 | 158 | 69 | 145 | 50 |
16 | 10.16 (400) | 86.8 | 143 | 51 | 133 | 37 |
17 | 11.43 (450) | 97.7 | 132 | 31 | 124 | 23 |
Column | L m (in.) | \(\lambda\) (\(\frac{L}{r}\)) | Series 13 | Series 14 | ||
---|---|---|---|---|---|---|
e0 = 30 | e0 = 50 | |||||
δ0 = 0 | δ0 = 0 | |||||
(mm) | (mm) | |||||
\(R_{f}\)(min) | \(R_{f}\)(min) | |||||
F | H | F | H | |||
1 | 0.760 (30) | 4.3 | 320 | 320 | 308 | 308 |
2 | 1.270 (50) | 7.2 | 318 | 314 | 304 | 301 |
3 | 2.54 (100) | 14.5 | 312 | 304 | 296 | 283 |
4 | 3.81 (150) | 21.7 | 300 | 281 | 286 | 261 |
5 | 5.08 (200) | 28.9 | 288 | 235 | 272 | 230 |
6 | 6.35 (250) | 36.2 | 272 | 206 | 257 | 197 |
7 | 7.62 (300) | 43.4 | 249 | 171 | 242 | 168 |
8 | 8.86 (350) | 50.6 | 232 | 155 | 229 | 141 |
9 | 10.16 (400) | 57.9 | 214 | 135 | 214 | 110 |
10 | 11.43 (450) | 65.1 | 199 | 108 | 200 | 88 |
11 | 12.70 (500) | 72.3 | 188 | 85 | 184 | 69 |
12 | 13.97 (550) | 79.6 | 182 | 67 | 169 | 54 |
13 | 15.24 (600) | 86.8 | 176 | 57 | 158 | 40 |
14 | 16.51 (650) | 94.0 | 172 | 50 | 152 | 26 |
Column | Load | Load | L | \(\lambda\) | e | \(R_{f}\) |
---|---|---|---|---|---|---|
(kN) | Level | (m) | (\(\frac{L}{r}\)) | (mm) | (min) | |
(%) | (in.) | |||||
CE1 | 1200 | 0.29 | 0.38 (15) | 4.3 | 0 | 224 |
CE2 | 1200 | 0.43 | 0.38 (15) | 4.3 | 15 | 224 |
CE3 | 1200 | 0.37 | 0.38 (15) | 4.3 | 25 | 215 |
CE4 | 1200 | 0.41 | 0.38 (15) | 4.3 | 35 | 199 |
CE5 | 1200 | 0.48 | 0.38 (15) | 4.3 | 50 | 170 |
C1 (305 × 305) | e (mm) | 0.0 | 5.0 | 10.0 | 20.0 | 25.0 | 50.0 | 100.0 |
\(e/b\) | 0.0 | 0.01639 | 0.03278 | 0.06557 | 0.08196 | 0.16393 | 0.32786 | |
\(\frac{{N_{{Rd,\lambda^{\min } }} }}{{N_{{Rd,\lambda^{\min } }}^{0} }}\) | 1.0 | 0.949 | 0.902 | 0.823 | 0.789 | 0.641 | 0.424 | |
Notes | b = 305 mm, \(A_{s}\) = 4No25, \(f^{\prime}_{c}\) = 40 MPa, \(f_{y}\) = 444 MPa, \(\lambda\) = 4.3 | |||||||
C2 (610 × 610) | e (mm) | 0.0 | 10.0 | 20.0 | 40.0 | 50.0 | 100.0 | 200.0 |
\(e/b\) | 0.0 | 0.01639 | 0.03278 | 0.06557 | 0.08196 | 0.16393 | 0.32786 | |
\(\frac{{N_{{Rd,\lambda^{\min } }} }}{{N_{{Rd,\lambda^{\min } }}^{0} }}\) | 1.0 | 0.955 | 0.911 | 0.835 | 0.800 | 0.657 | 0.437 | |
Notes | b = 610 mm, \(A_{s}\) = 8No25, \(f^{\prime}_{c}\) = 40 MPa, \(f_{y}\) = 444 MPa, \(\lambda\) = 4.3 | |||||||
C3 (305 × 305) | e (mm) | 0.0 | 5.0 | 10.0 | 20.0 | 25.0 | 50.0 | 100.0 |
\(e/b\) | 0 | 0.01639 | 0.03278 | 0.06557 | 0.08196 | 0.16393 | 0.32786 | |
\(\frac{{N_{{Rd,\lambda^{\min } }} }}{{N_{{Rd,\lambda^{\min } }}^{0} }}\) | 1.0 | 0.951 | 0.904 | 0.824 | 0.788 | 0.645 | 0.445 | |
Notes | b = 305 mm, \(A_{s}\) = 8No25, \(f^{\prime}_{c}\) = 40 MPa, \(f_{y}\) = 444 MPa, \(\lambda\) = 4.3 | |||||||
C4 (305 × 305) | e (mm) | 0.0 | 5.0 | 10.0 | 20.0 | 25.0 | 50.0 | 100.0 |
\(e/b\) | 0.0 | 0.01639 | 0.03278 | 0.06557 | 0.08196 | 0.16393 | 0.32786 | |
\(\frac{{N_{{Rd,\lambda^{\min } }} }}{{N_{{Rd,\lambda^{\min } }}^{0} }}\) | 1.0 | 0.950 | 0.903 | 0.825 | 0.791 | 0.641 | 0.417 | |
Notes | b = 305 mm, \(A_{s}\) = 4No25, \(f^{\prime}_{c}\) = 50 MPa, \(f_{y}\) = 444 MPa, \(\lambda\) = 4.3 | |||||||
(Average) | \(\frac{{N_{{Rd,\lambda^{\min } }} }}{{N_{{Rd,\lambda^{\min } }}^{0} }}\) | 1.0 | 0.951 | 0.905 | 0.827 | 0.792 | 0.646 | 0.431 |
3 Evaluation
3.1 Evaluation of the Proposed Method by Comparison with Experimental Data
Lab | \(A_{s}\) cm2 | \(a\) mm | \(b_{1}\) mm | \(b_{2}\) mm | \(L\) m | \(f_{cm}\) MPa | \(f_{ym}\) MPa | \(e\) mm | \(P\) kN | \(R_{f}\) min | \(\lambda\) (L/r) | \(k_{e}\) | \(\overline{R}_{f}^{0}\) min | \(\overline{R}_{f}\) min | \(\frac{{R_{f} }}{{\overline{R}_{f} }}\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
TUBr | 9.2 | 30 | 200 | 200 | 5.71 | 42 | 480 | 100 | 140 | 31 | 99.13 | 0.392 | 128 | 50 | 0.62 |
TUBr | 9.2 | 30 | 200 | 200 | 5.71 | 42 | 480 | 50 | 172 | 35 | 99.13 | 0.750 | 138 | 103 | 0.34 |
TUBr | 12.6 | 38 | 200 | 200 | 4.76 | 31 | 462 | 20 | 240 | 36 | 82.64 | 0.964 | 160 | 154 | 0.23 |
TUBr | 18.9 | 38 | 300 | 300 | 4.70 | 35 | 505 | 5 | 1548 | 38 | 54.40 | 1.0 | 117 | 117 | 0.32 |
TUBr | 12.6 | 38 | 200 | 200 | 5.76 | 32 | 443 | 10 | 208 | 40 | 100.00 | 1.0 | 168 | 168 | 0.24 |
TUBr | 9.2 | 30 | 200 | 200 | 5.71 | 42 | 477 | 10 | 245 | 40 | 99.13 | 1.0 | 141 | 141 | 0.28 |
TUBr | 12.6 | 38 | 200 | 200 | 4.76 | 24 | 487 | 0 | 340 | 48 | 82.64 | 1.0 | 153 | 153 | 0.31 |
TUBr | 18.9 | 38 | 300 | 300 | 4.76 | 38 | 404 | 5 | 1224 | 48 | 55.09 | 1.0 | 132 | 132 | 0.36 |
TUBr | 12.6 | 38 | 200 | 200 | 4.76 | 31 | 462 | 10 | 280 | 49 | 82.64 | 1.0 | 159 | 159 | 0.31 |
TUBr | 12.6 | 38 | 200 | 200 | 4.76 | 31 | 462 | 60 | 170 | 49 | 82.64 | 0.678 | 156 | 106 | 0.46 |
TUBr | 18.9 | 38 | 300 | 300 | 4.70 | 32 | 503 | 150 | 280 | 49 | 54.40 | 0.357 | 152 | 54 | 0.90 |
TUBr | 9.2 | 30 | 200 | 200 | 5.71 | 42 | 482 | 10 | 175 | 49 | 99.13 | 1.000 | 149 | 149 | 0.33 |
TUBr | 18.9 | 38 | 300 | 300 | 4.70 | 32 | 526 | 150 | 465 | 50 | 54.40 | 0.357 | 128 | 46 | 1.10 |
TUBr | 9.2 | 30 | 200 | 200 | 5.71 | 42 | 485 | 50 | 122 | 52 | 99.13 | 0.750 | 146 | 109 | 0.48 |
TUBr | 12.6 | 38 | 200 | 200 | 4.76 | 31 | 462 | 100 | 130 | 53 | 82.64 | 0.392 | 153 | 60 | 0.88 |
TUBr | 18.9 | 38 | 300 | 300 | 4.70 | 32 | 503 | 10 | 970 | 55 | 54.40 | 1.0 | 139 | 139 | 0.40 |
TUBr | 18.9 | 38 | 300 | 300 | 3.76 | 42 | 452 | 5 | 1695 | 57 | 43.52 | 1.0 | 118 | 118 | 0.48 |
TUBr | 18.9 | 38 | 300 | 300 | 4.70 | 32 | 526 | 10 | 1308 | 57 | 54.40 | 1.0 | 122 | 122 | 0.47 |
TUBr | 18.9 | 38 | 300 | 300 | 5.76 | 24 | 487 | 0 | 800 | 58 | 66.67 | 1.0 | 143 | 143 | 0.41 |
TUBr | 12.6 | 38 | 200 | 200 | 3.76 | 24 | 487 | 0 | 420 | 58 | 65.28 | 1.0 | 144 | 144 | 0.40 |
RUG | 6.8 | 31 | 300 | 200 | 3.90 | 31 | 493 | 20 | 300 | 60 | 45.14 | 0.976 | 145 | 142 | 0.42 |
RUG | 6.8 | 41 | 300 | 200 | 3.90 | 33 | 493 | 20 | 283 | 60 | 45.14 | 0.976 | 181 | 177 | 0.34 |
TUBr | 8 | 33 | 300 | 300 | 3.90 | 34 | 576 | 0 | 950 | 61 | 45.14 | 1.0 | 119 | 128 | 0.48 |
Lab | \(A_{s}\) cm2 | \(a\) mm | \(b_{1}\) mm | \(b_{2}\) mm | \(L\) m | \(f_{cm}\) MPa | \(f_{ym}\) MPa | \(e\) mm | \(P\) kN | \(R_{f}\) min | \(\lambda\) (L/r) | \(k_{e}\) | \(\overline{R}_{f}^{0}\) min | \(\overline{R}_{f}\) min | \(\frac{{R_{f} }}{{\overline{R}_{f} }}\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
TUBr | 18.9 | 38 | 300 | 300 | 5.76 | 24 | 487 | 30 | 600 | 61 | 66.67 | 0.929 | 145 | 135 | 0.45 |
TUBr | 18.9 | 38 | 300 | 300 | 4.76 | 24 | 487 | 30 | 650 | 63 | 55.09 | 0.929 | 141 | 131 | 0.48 |
TUBr | 12.6 | 38 | 200 | 200 | 3.76 | 24 | 487 | 0 | 420 | 66 | 65.28 | 1.0 | 144 | 144 | 0.46 |
TUBr | 18.9 | 38 | 300 | 300 | 4.76 | 31 | 462 | 30 | 650 | 69 | 55.09 | 0.929 | 148 | 137 | 0.50 |
TUBr | 9.2 | 30 | 200 | 200 | 5.71 | 42 | 478 | 10 | 128 | 72 | 99.13 | 1.0 | 154 | 154 | 0.47 |
TUBr | 18.9 | 38 | 300 | 300 | 4.76 | 31 | 462 | 90 | 460 | 75 | 55.09 | 0.643 | 144 | 93 | 0.81 |
TUBr | 18.9 | 38 | 300 | 300 | 4.76 | 31 | 462 | 30 | 650 | 80 | 55.09 | 0.929 | 148 | 137 | 0.58 |
TUBr | 18.9 | 38 | 300 | 300 | 3.76 | 24 | 487 | 0 | 930 | 84 | 43.52 | 1.0 | 135 | 135 | 0.62 |
TUBr | 18.9 | 38 | 300 | 300 | 4.76 | 31 | 462 | 15 | 740 | 85 | 55.09 | 1.0 | 148 | 148 | 0.57 |
TUBr | 18.9 | 38 | 300 | 300 | 3.76 | 24 | 487 | 30 | 740 | 86 | 43.52 | 0.929 | 135 | 125 | 0.69 |
RUG | 16.1 | 33 | 400 | 400 | 3.90 | 30 | 576 | 20 | 1650 | 93 | 33.85 | 0.982 | 146 | 143 | 0.65 |
TUBr | 18.9 | 38 | 300 | 300 | 4.76 | 24 | 487 | 0 | 880 | 108 | 55.09 | 1.0 | 138 | 138 | 0.78 |
TUBr | 18.9 | 38 | 300 | 300 | 2.66 | 33 | 458 | 30 | 845 | 111 | 30.79 | 0.929 | 137 | 127 | 0.87 |
RUG | 8 | 33 | 300 | 300 | 3.90 | 29 | 576 | 0 | 422 | 116 | 45.14 | 1.0 | 150 | 150 | 0.77 |
RUG | 8 | 33 | 300 | 300 | 3.90 | 35 | 576 | 622 | 120 | 45.14 | 1.0 | 142 | 142 | 0.85 | |
RUG | 6.8 | 31 | 300 | 200 | 3.90 | 30 | 493 | 20 | 178 | 120 | 45.14 | 0.976 | 167 | 163 | 0.74 |
RUG | 6.8 | 41 | 300 | 200 | 3.90 | 32 | 493 | 20 | 334 | 120 | 45.14 | 0.976 | 174 | 170 | 0.71 |
RUG | 8 | 48 | 300 | 300 | 3.90 | 37 | 576 | 20 | 349 | 123 | 45.14 | 0.976 | 155 | 151 | 0.81 |
RUG | 8 | 33 | 300 | 300 | 3.90 | 37 | 576 | 20 | 220 | 125 | 45.14 | 0.976 | 164 | 160 | 0.78 |
TUBr | 18.9 | 38 | 300 | 300 | 2.66 | 33 | 418 | 50 | 780 | 125 | 30.79 | 0.833 | 140 | 117 | 1.07 |
RUG | 16.1 | 33 | 300 | 300 | 3.90 | 36 | 576 | 20 | 370 | 126 | 45.14 | 0.976 | 157 | 153 | 0.82 |
TUBr | 18.9 | 38 | 300 | 300 | 3.76 | 24 | 487 | 0 | 930 | 138 | 43.52 | 1.0 | 135 | 135 | 1.02 |
TUBr | 18.9 | 38 | 300 | 300 | 3.33 | 31 | 433 | 15 | 735 | 160 | 38.54 | 1.0 | 153 | 153 | 1.05 |
Lab | \(A_{s}\) cm2 | \(a\) mm | \(b_{1}\) cm | \(b_{2}\) cm | \(L\) m | \(f_{cm}\) MPa | \(f_{ym}\) MPa | \(e\) mm | \(P\) kN | \(R_{f}\) min | \(\lambda\) (L/r) | \(k_{e}\) | \(\overline{R}_{f}^{0}\) min | \(\overline{R}_{f}\) min | \(\frac{{R_{f} }}{{\overline{R}_{f} }}\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 3.5 | 44.4 | 0 | 1778 | 146 | 43.37 | 1.0 | 198 | 198 | 0.74 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 3.7 | 44.4 | 0 | 1333 | 170 | 43.37 | 1.0 | 227 | 227 | 0.75 |
NRC | 12.6 | 58 | 20 | 20 | 381 | 4.2 | 44.2 | 0 | 169 | 180 | 66.15 | 1.0 | 254 | 254 | 0.71 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 3.8 | 44.4 | 0 | 1333 | 187 | 43.37 | 1.0 | 229 | 229 | 0.82 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 4.4 | 44.4 | 0 | 1044 | 201 | 43.37 | 1.0 | 252 | 252 | 0.80 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 3.6 | 44.4 | 0 | 1067 | 208 | 43.37 | 1.0 | 242 | 242 | 0.86 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 3.5 | 44.4 | 0 | 916 | 210 | 43.37 | 1.0 | 250 | 250 | 0.84 |
NRC | 65.5 | 80 | 40.6 | 40.6 | 381 | 4.6 | 41.4 | 0 | 2978 | 213 | 32.58 | 1.0 | 336 | 336 | 0.63 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 3.4 | 44.4 | 0 | 800 | 218 | 43.37 | 1.0 | 256 | 256 | 0.85 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 3.5 | 44.4 | 0 | 711 | 220 | 43.37 | 1.0 | 263 | 263 | 0.84 |
NRC | 40.9 | 61 | 30.5 | 30.5 | 381 | 3.7 | 44.4 | 0 | 1333 | 225 | 43.37 | 1.0 | 244 | 244 | 0.92 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 5.3 | 44.4 | 0 | 1178 | 227 | 43.37 | 1.0 | 254 | 254 | 0.89 |
NRC | 20.4 | 61 | 30.5 | 30.5 | 381 | 5 | 44.4 | 0 | 1067 | 234 | 43.37 | 1.0 | 257 | 257 | 0.91 |
NRC | 40.9 | 61 | 40.6 | 40.6 | 381 | 3.9 | 44.4 | 0 | 2418 | 262 | 32.58 | 1.0 | 254 | 254 | 1.03 |
NRC | 65.5 | 64 | 40.6 | 40.6 | 381 | 3.8 | 41.4 | 0 | 2795 | 285 | 32.58 | 1.0 | 263 | 263 | 1.08 |
3.2 Comparison with the Results of EC2 Method
4 Numerical Examples
4.1 Fire-Resistance-Column-Curves for Design
Term | Expression | CI-7 (fixed–fixed) | CIII-3 (pinned–pinned) |
---|---|---|---|
\(f^{\prime}_{c}\)(MPa) | Given | 35 | 35 |
\(f_{ym}\)(MPa) | Given | 420 | 420 |
\(A_{s}\) cm2 | Given | 12.56 | 12.56 |
\(a\) mm | Given | 58 | 58 |
\(b_{1}\)(mm) | Given | 250 | 250 |
\(L\)(m) | Given | 4.0 | 4.0 |
\(L^{\min }\)(m) | \(r \times 4.3\) | 0.31 | 0.31 |
\(e\)(mm) | Given | 0.0 | 20 |
\(N_{Ed,\,fi}\)(kN) | Given | 1000 | 1000 |
\(\lambda\) | \(L/r\) | 55.55 | 55.55 |
\(\lambda_{e}\) | \(\lambda - 4.3\) | 51.25 | 51.25 |
\(k_{e}\) | \(\begin{gathered} k_{e} = \,1.0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,e\,/b\, \le \,\,0.05 \hfill \\ k_{e} \, = \,1\, - \,1.43\left( {\frac{e}{b}\, - 0.05} \right)\,\,\,\,\,\,\,\,0.05\,\, < \,\,e\,/b\, \le \,\,0.17 \hfill \\ \end{gathered}\) | 1.0 | 0.96 |
\(N_{{Rd,\lambda^{\min } }}^{0}\)(kN) | \(A_{s} \frac{{f_{y} }}{{\gamma_{s} }}\, + \,A_{c} \frac{{0.85f^{\prime}_{c} }}{{\gamma_{c} }}\),\(\gamma_{s} = 1.15,\,\,\gamma_{c} \, = \,1.5\) | 1673 | 1673 |
\(k_{R}\) | \(- 8.49\left( \frac{e}{b} \right)^{3} \, + \,6.91\left( \frac{e}{b} \right)^{2} \, - \,\,3.049\frac{e}{b}\, + \,1.0\) | 1.0 | 0.8 |
\(N_{{Rd,\lambda^{\min } }}\)(kN) | \(k_{R} .N_{{Rd,\lambda^{\min } }}^{0}\) | 1673 | 1332 |
\(\mu_{{fi,\lambda^{\min } }}\) | \(\mu_{{fi,\lambda^{\min } }} = \frac{{N_{Ed,fi} }}{{N_{{Rd,\lambda^{\min } }} }}\) | 0.6 | 0.75 |
\(\omega\) | \(\omega = \frac{{A_{s} .f_{yd} }}{{A_{c} .f_{cd} }}\), \(f_{yd} = \frac{{f_{ym} }}{{\gamma_{s} }}\),\(f_{cd} = \frac{{f_{cm} }}{{\gamma_{c} }}\) | 0.32 | 0.32 |
\(R_{{\eta fi,\lambda^{\min } }}\) | \(R_{{\eta fi,\lambda^{\min } }} \, = \,83\left( {1.0 - \mu_{{fi,\lambda^{\min } }} \frac{{\left( {1 + \omega } \right)}}{{\left( {0.85/\alpha_{cc} } \right) + \omega }}} \right)\) | 27.04 | 12.7 |
\(R_{a}\) | \(R_{a} \, = \,1.6\left( {a - 30} \right)\) | 48.8 | 48.8 |
\(R_{{L^{\min } }}\) | \(R_{{L^{\min } }} \, = \,9.6\left( {5 - L^{\min } } \right)\) | 45.027 | 45.027 |
\(R_{b}\) | \(R_{b} \, = \,0.09b^{\prime}\) | 22.047 | 22.047 |
\(R_{n}\) | \(\begin{gathered} R_{n} \, = \,0\,\,\,\,\,\,\,\,\,\,\,\,\,for\,n \ge \,\,4 \hfill \\ R_{\,n} \, = \,12\,\,\,\,\,\,\,\,\,\,\,\,\,for\,n < \,4 \hfill \\ \end{gathered}\) | 0.0 | 0.0 |
\(\overline{R}_{f}^{0}\) | \(\overline{R}_{f}^{0} \, = \,\,120.\left( {\,\frac{{R_{{\eta fi,\lambda^{\min } }} \, + \,R_{a} \, + \,R_{{L^{\min } }} \, + \,R_{b} \, + \,R_{n} }}{120}} \right)^{1.8}\) | 156 | 128 |
\(\overline{R}_{f}\)(min) | \(\overline{R}_{f} \, = \,k_{e} .\overline{R}_{f}^{0}\) | 156 | 123 |
\(k_{\lambda }\) | 0.75 | 0.50 | |
\(R_{f}\)(min) | \(R_{f} \, = \,k_{\lambda } .\overline{R}_{f}\) | 117 | 61 |
5 Limitations
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Columns subjected to ASTM E119 standard fire, ISO 834 standard fire, or any design fire of similar profile as that of standard fire.
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Dimension of the column cross-section b (square or rectangular): up to 600 mm.
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Concrete cover: 25–80 mm.
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Percentage of longitudinal bars: 1%–4%.
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Concrete compressive strength: 24 MPa–53 MPa.
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Eccentricity ratio e/b: 0–0.4.
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Slenderness ratio \(\lambda\): 0–100.
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Load ratio: 0.15–0.7.
6 Conclusions
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The results illustrated that by using an individual column curve for each end condition case, the deviation between the actual and the assessed fire resistance of the columns will be reduced significantly. Influential parameters such as slenderness ratio, load ratio, load eccentricity, cross-section size, concrete cover thickness, and reinforcement ratio imply a realistic basis for the computation. By explicitly taking into consideration the initial out-of-straightness, the solution appears to assume even further closeness to reality.
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Two equations were suggested to describe the proposed column curves. These equations are applicable for slenderness ratio in the range between 4.3 and 100, which cover most RC column lengths in practice.
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The characteristic fire resistance, which includes almost all the parameters that affect fire resistance, implies a realistic basis for the computation. This characteristic fire resistance is determined using EC2 equation at slenderness ratio of 4.3. The equation was modified by adding a parameter that accounts for the effect of load eccentricity.
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An equation was derived to calculate the column capacity at normal temperature conditions for very short eccentric columns. The proposed equation is simple, accurate, and helpful in determining the characteristic fire resistance.
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It was found that the resulted column curves represent a median for many experimental data. Therefore, these fire-resistance-column-curves can be safely used for design purpose when appropriate safety factors are adopted.
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Results revealed that for columns with smaller eccentricity ratio, explicitly taking the effect of load eccentricity in FRCC method leads to only slight improvement when compared to EC2 method. For columns with high eccentricity ratio, FRCC method is safe but conservative. EC2 method, however, is unsafe in many column cases.
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By using FRCC method, the difficulty in determining exact solution of column capacity at normal temperature conditions that encountered in EC2 method has been eliminated.
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The proposed FRCC method can be treated as a modification for EC2 method. The method is practical, simple, and can be conveniently used by engineers for every day design; thus, it can be easily incorporated in design codes.