13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- junctions of two segments and continuity conditions at the crack location. These characteristic equations are used to compute the natural frequencies of Timoshenko portal frame by numerically. Finally, these computed natural frequencies are compared with that of natural frequencies obtained from finite element method for fixed-hinged, hinged-hinged and fixed-fixed end conditions. 2. Theoretical formulation For in-plane free vibration analysis of Portal frame without and with crack, initially a beam with crack has been studied and natural frequencies have been compared with available literature. For the In-plane free vibration analysis of portal frame, transverse and longitudinal motions of each member are taken into consideration. In analytical modelling of frame, Timoshenko beam theory approach is used for analysis of transverse vibration, while axial vibration of rod is considered for analysis of longitudinal vibration of each member. A portal frame containing a part through-the-thickness edge crack undergoing free transverse vibration, gives rise to a deformation pattern corresponding to natural frequencies. This in turn will change the slope-mode shape, curvature mode shape, etc. The forward problem of determination of natural frequencies knowing the crack details for fixed-hinged, hinged-hinged and fixed-fixed end conditions have been examined. Accuracy obtainable in connection with natural frequencies is compared numerically using commercially available FE tool for different boundary conditions. 2.1. Formulation for portal frame without crack For a free, in-plane, symmetric transverse and axial motions of each segments in portal frame without any crack is modelled by using Timoshenko beam theory i.e., taking the effects of shear deformation and rotational inertia. Neglecting damping effect, the mode shape equations of each segments are governed by (Fig. 1)[11] Transverse motion: (1) Slope due to bending: (2) Longitudinal motion: (3) where vi is transverse displacement, ui is axial displacement, φi is the rotation due to bending of the segments, these are function of non-dimensional position, ηi along the length of segment in a particular mode for the segment i, a prime indicates differentiation with respect to ηi and additional parameters given by ( ) for 1,2,3 ( ) 0, 0 ) ( ) ( ) ( = < < = ′′ − − ′′′′ + + i v v v i i i i i i i i β η η στ α η τ σ η for 1,2,3 ( ) 0, 0 ( ) 2 = < < = ′′ + i v u i i i i i i β η η γ η ( ) for 1,2,3 ( ) 0, 0 ) ( ) ( ) ( = < < = ′′ − − ′′′′ + + i i i i i i i i i β η ηφστ α ηφτσ ηφ Figure 1. Schematic of Timoshenko portal frame: (i) front view and (ii) side view. h (i) (ii) η2 v3 η3 η1 v2 L1 L2 L3 v1 b
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