13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- For non-trivial solution, (30) which gives the characteristic equations. Solving this equation numerically, the natural frequencies of portal frame without crack are obtained. 2.2. Formulation for portal frame with crack located in left vertical segment One of the convenient methods of modelling the vibration of a beam segment with a crack is to split the segment into two around the crack section and connect them by massless spring element, whose flexibility is given by a matrix of size 6×6 [20,21]. When the various modes of vibration become uncoupled, the size of the flexibility matrix reduces. Particularly, for a pure transverse vibration the matrix is of size 1×1. That is, there is only one spring element, which is a torsional spring. A typical representation of a portal frame with a crack located in left vertical segment is shown in Fig. 2. The governing mode shape equations of each segment are of the form: (31) (32) (33) where values of βi are as follows. (34) The general solutions of Eq. (31-33) are of the form Eqs. (5-7). Boundary and compatibility conditions at the junctions are as follows. (35) (36) (37) The compatibility conditions at crack location are given by (38) ( ) 0 Δ = ω ( ) for 1,2,3,4 ( ) 0, 0 ) ( ) ( ) ( = < < = ′′ − − ′′′′ + + i v v v i i i i i i i i β η η στ α η τ σ η for 1,2,3,4 ( ) 0, 0 ( ) 2 = < < = ′′ + i v u i i i i i i β η η γ η ( ) for 1,2,3,4 ( ) 0, 0 ) ( ) ( ) ( = < < = ′′ − − ′′′′ + + i i i i i i i i i β η ηφστ α ηφτσ ηφ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎭ ⎬ ⎫ ′ = ′ = ′ ′ − ′ = ′ − = ′ − ′ = ′ = 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 2 1 1 2 1 1 2 2 , 0 , 0 0 , , 0 0 , 0 0 β β β φξ β β φ β φ β φ φ β u u u u v v v v v v ⎭ ⎬ ⎫ = ′ = = = = = ( ) 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 = = 0; , at 3 2 2 = = η β η ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ⎭ ⎬ ⎫ =− ′ ′ − ′ ′ − = ′ = ′ ′ = ′ = − = 0 , 0 0 ( ) (0) ( ), (0) (0), ( ) ( ), (0) 3 2 2 2 2 2 2 3 3 2 2 3 2 2 3 3 2 2 2 2 3 Eu Gv Eu Gv v v u v u v β φ β κ β φ κ β φ φ β β β 0; , at 4 3 3 = = η β η ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ⎭ ⎬ ⎫ =− ′ ′ − ′ ′ − = ′ = ′ ′ = ′ = − = 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 0; , at 2 1 1 = = η β η 0 1 location, and crack , , (1 ) , 4 4 3 3 2 2 1 1 ≤ ≤ = = = − = δ β β δ β δ β L L L L L L L L v3 (i) Figure 2. (i) Schematic of Timoshenko portal frame with crack in left vertical segment and (ii) Representation by rotational spring. (ii) L2 L4 L3 L1 a η1 η2 η4 η3 v4 v2 v1 L3 L2 L4 L1 ξ
RkJQdWJsaXNoZXIy MjM0NDE=