13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- For other end conditions; hinged-hinged and fixed-fixed, of portal frame with crack are as follows. For hinged-hinged ends: For fixed-fixed ends: 3. Finite element computation for natural frequencies The natural frequencies of portal frame with and without crack are computed for a numerical verification of the solution to forward problem by a standard finite element software (i.e., ANSYS-11[22]). A frame is discretized by Eight-node quadrilateral shell elements and quarter-point singular elements employed around the crack-tip is as shown in Fig. 4. 0; , at 4 3 3 = = η β η ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) (68) 0 , 0 0 ( ) (0) ( ), (0) (0), ( ) ( ), (0) 4 2 3 3 3 3 3 4 4 3 3 4 3 3 4 4 3 3 3 3 4 ⎭ ⎬ ⎫ =− ′ ′ − ′ ′ − = ′ = ′ ′ = ′ = − = Eu Eu Gv Gv v v u v u v β φ β κ β φ κ β φ φ β β β ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (69) , 0 , 0 0 , , 0 0 , 0 0 2 2 2 2 3 2 2 3 2 2 3 2 2 2 2 3 2 2 3 2 2 3 3 ⎭ ⎬ ⎫ ′ = ′ = ′ ′ − ′ = ′ − = ′ − ′ = ′ = β β β φξ β β φ β φ β φ φ β u u u u v v v v v v 0; , at 3 2 2 = = η β η (70) ( ) 0, ( ) 0 ( ) 0, (0) 0, (0) 0, (0) 0 4 4 4 4 4 4 1 1 1 ⎭ ⎬ ⎫ = ′ = = = ′ = = β β φ β φ u v u v ; at 0; at 4 4 1 β η η = = (71) ( ) 0, ( ) 0 ( ) 0, (0) 0, (0) 0, (0) 0 4 4 4 4 4 4 1 1 1 ⎭ ⎬ ⎫ = = = = = = β β φ β φ u v u v ; at 0; at 4 4 1 β η η = = Figure 3. Schematic of Timoshenko portal frame with crack located in horizontal segment represented by torsional spring. v3 η4 v4 v2 v1 η1 η3 L1 L2 L4 L3 η2 ξ Enlarged portion of crack region Figure 4. Schematic of finite element modelling of Timoshenko portal frame with crack located in horizontal segment. Crack region Crack tip Crack faces
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