13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- 4. Results and Discussions In this study, non-destructive direct solutions for the estimation of the natural frequency of a portal frame with and without crack have been presented. Changes in the natural frequencies of a portal frame due to the presence of a crack may provide additional information for damage detection of these structures. The presence of crack has been theoretically considered by an equivalent torsional spring. To take into account the effects of rotational inertia and shear deformation, Timoshenko beam theory has been employed. The three set of end conditions, fixed-hinged, hinged-hinged and fixed-fixed, have been considered. Crack locations ranging from 20% to 70% of length of segment and crack sizes ranging from 20% to 60% of depth are considered. By means of these boundary conditions and applying suitable compatibility conditions at the cracked section, the characteristic equations have been derived explicitly, whose solution provides the natural frequencies of the portal frame. A MATLAB code has been written to compute the frequencies numerically. The computed natural frequencies have been compared with those obtained by the finite element tool. The geometry of the portal frame with following cross-sectional dimensions and material properties are considered: Length of each no-crack segments (Li)=0.225m, width (b)=0.0125m and depth (h)=0.025m. The material data employed are: mass density ( ρ)= 7800kg/m3, modulus of elasticity E=210GPa, Poisson’s ratio μ=0.3 and Timoshenko shear coefficient κ =5/6. The first three natural frequencies calculated by forward analysis are presented in Tables 1 and 2. The percentage difference in the frequencies taking finite element results as the reference is shown in the Tables 1 and 2. The maximum difference among all results is 7.09% which proves usefulness of proposed method. As expected, the trend of natural frequencies of portal frame with crack, decreases as the crack size increases in comparison with natural frequencies of portal frame without crack (Fig. 5). 5. Conclusion Solution to forward problems i.e. determination of natural frequencies knowing the crack details in Timoshenko portal frame has been studied. The presence of crack has been modelled by an equivalent torsional spring. It is found that the maximum percentage differences between the natural frequencies computed by analytical approach are less than 7.09% as compared to the finite element result. Changes in the natural frequencies of the portal frame due to the presence of a crack may provide additional information for damage detection of these structures. Figure 5. Plot of percentage change in natural frequencies vs. crack size: (i) Fixed-hinged ends of frame with crack located in vertical segment at δ=0.4 and (ii) Fixed-fixed ends of frame with crack located in horizontal segment at δ=0.4.
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