13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- Table 2. Comparison of natural frequencies computed by analytical method and finite-element method for Timoshenko portal frame with crack located in horizontal segment. Natural frequencies (Hz) Analytical method Finite element method % Difference End cond. Crack loc. δ a/h ω1 ω2 ω3 ω1 ω2 ω3 ω1 ω2 ω3 nocrack 278.97 1247.63 1874.68 284.72 1253.30 1915.70 2.02 0.45 2.14 0.2 276.55 1246.04 1865.88 282.13 1252.10 1905.40 1.98 0.48 2.07 0.4 269.46 1241.25 1839.67 275.00 1248.90 1874.90 2.01 0.61 1.88 0.2 0.6 258.46 1233.46 1798.69 262.03 1242.80 1816.40 1.36 0.75 0.98 0.2 278.33 1233.51 1855.02 284.05 1240.20 1893.60 2.01 0.54 2.04 0.4 276.38 1190.17 1804.91 282.21 1201.00 1834.20 2.07 0.90 1.60 0.4 0.6 273.01 1119.64 1746.49 278.46 1119.40 1739.40 1.96 -0.02 -0.41 0.2 278.82 1231.95 1874.48 284.52 1238.60 1915.60 2.00 0.54 2.15 0.4 278.37 1186.61 1873.94 283.86 1197.80 1913.80 1.93 0.93 2.08 F-H 0.7 0.6 277.61 1118.15 1873.15 282.40 1123.20 1908.10 1.70 0.45 1.83 nocrack 171.26 1127.17 1699.66 175.89 1127.60 1725.90 2.63 0.04 1.52 0.2 170.06 1121.71 1689.28 174.53 1123.00 1714.70 2.56 0.11 1.48 0.4 166.48 1105.35 1660.88 170.61 1110.50 1685.60 2.42 0.46 1.47 0.2 0.6 160.72 1079.11 1622.47 162.97 1087.80 1638.10 1.38 0.80 0.95 0.2 171.12 1112.64 1698.17 175.71 1114.20 1724.10 2.61 0.14 1.50 0.4 170.71 1069.01 1693.32 175.24 1075.70 1718.60 2.59 0.62 1.47 0.4 0.6 169.97 999.73 1684.19 174.22 999.78 1701.00 2.44 0.01 0.99 0.2 170.72 1116.71 1694.33 175.29 1118.20 1720.20 2.61 0.13 1.50 0.4 169.08 1085.34 1678.86 173.48 1091.70 1703.30 2.54 0.58 1.43 H-H 0.7 0.6 166.31 1035.35 1655.99 169.74 1040.60 1669.40 2.02 0.50 0.80 nocrack 371.92 1412.93 2299.39 379.70 1426.70 2333.60 2.05 0.97 1.47 0.2 370.06 1408.26 2284.20 377.59 1422.60 2317.40 1.99 1.01 1.43 0.4 364.63 1394.43 2240.90 371.73 1410.90 2274.90 1.91 1.17 1.49 0.2 0.6 356.26 1372.69 2179.08 361.01 1388.60 2202.70 1.32 1.15 1.07 0.2 371.71 1387.86 2296.75 379.42 1402.20 2330.50 2.03 1.02 1.45 0.4 371.06 1316.61 2286.83 378.70 1333.30 2318.10 2.02 1.25 1.35 0.4 0.6 369.92 1213.84 2266.57 377.13 1207.20 2265.20 1.91 -0.55 -0.06 0.2 371.08 1397.73 2290.78 378.77 1412.20 2324.10 2.03 1.02 1.43 0.4 368.54 1353.49 2265.65 376.00 1371.00 2295.40 1.98 1.28 1.30 F-F 0.7 0.6 364.36 1286.74 2229.64 370.48 1292.60 2234.50 1.65 0.45 0.22 Note: F-H - Fixed-hinged, H-H – Hinged-hinged and F-F – fixed-fixed end conditions. References [1] R.Y. Liang, J. Hu, F.K. Choy, Detection of crack in beam structures using measurement of natural frequencies, Journal of Franklin Institute, 328 (1991) 505-518. [2] W.M. Ostachowicz, M. Krawczuk, Analysis of the effect of cracks on the natural frequencies of a cantilever beam, Journal of Sound and vibration, 150 (1991) 191-201. [3] O.S. Salawu, Detection of structural damage through changes in frequency: a review, Engineering structures, 19 (1997) 718-723.
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