13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- where ξ is the non-dimensional flexibility of the torsional spring representing the crack and the relation through crack size can be written in the following form[6]: where r is crack size a to segment depth h ratio and f is crack geometry parameter defined by (40) By inserting general solutions of type (5-7) into Eqs. (35-38) results in following 24 homogeneous equations. (41) (42) (43) (44) (45) (46) (47) (48) (49) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) Eqs. (41-64) can be expressed conveniently in the following matrix equation. (65) where {C}={A1,B1…F4} T are unknown arbitrary constants. For non-trivial solution (66) Evaluation of Eq. (66) numerical method yields natural frequencies of portal frame with crack. 2.3. Formulation for Portal frame with crack located in horizontal segment The modelling of portal frame with crack located in horizontal segment (Fig.3) is done in the similar way as explained in the preceding section. The characteristic equations of type (66) can be obtained by incorporation of the following compatibility conditions at two junctions and crack location with associated boundary conditions. 0 sin cos sinh cosh 2 4 4 2 4 4 1 4 4 1 4 4 = + + + β λ β λ βλ βλ D C B A 0 sin cos sinh cosh 2 4 4 2 2 2 4 4 2 2 1 4 4 1 1 1 4 4 1 1 = + + + β λ λ β λ λ βλ λ βλ λ D q C q B q A q 0 cos sin 4 4 4 4 = + γβ γβ F E 0 sin cos sinh cosh 2 2 2 1 1 2 1 1 1 1 1 1 1 1 − − = + + + A C D C B A β λ β λ βλ βλ 0 sin cos sinh cosh 2 2 2 2 1 1 1 2 1 2 2 2 1 1 2 2 1 1 1 1 1 1 1 1 1 1 = − − + + + q A qC D q C q B q A q λ λ β λ λ β λ λ βλ λ βλ λ ( ) ( ) ( ) ( ) ( ) ( ) 0 cos sin cosh sinh 2 2 2 2 1 2 1 1 2 1 1 2 2 1 1 1 2 2 1 1 1 1 1 1 1 1 − − − + = + + − + + − − B q D q D q C q B q A q λ λ βλ λ βλ λ βλ λ βλ λ 0 cos sin 2 1 1 1 1 − = + F F E γβ γβ 0 sin cos 2 1 1 1 1 − = − E F E γβ γβ 0 cos sin 3 3 2 2 2 2 − − = + A C F E γβ γβ 0 sin cos sinh cosh 3 2 2 2 2 2 2 1 2 2 1 2 2 + = + + + F D C B A β λ β λ βλ βλ 0 cos sin cosh sinh 2 3 1 3 2 2 2 2 2 2 2 2 1 2 2 1 1 2 2 1 − − = + − + B D D C B A λλβλλ βλλ βλ λ βλ λ 0 sin cos sinh cosh 2 2 1 1 3 2 2 3 2 2 2 2 2 2 2 2 1 2 2 1 1 1 2 2 1 1 = − − + + + q A qC D q C q B q A q λ λβλ λβλ λβλ λ βλ λ ( ) ( ) ( ) [ ] 0 sin cos 2 3 2 3 1 1 2 2 2 2 − + + = − − GB q D q F E E λ λ κ γβ γβ γ ( ) ( ) ( ) ( ) [ ] 0 cos sin cosh sinh 3 2 2 2 2 2 2 2 2 2 2 1 2 2 1 1 1 2 2 1 1 + = + + − + + − − E E D q C q B q GA q γ βλ λ βλ λ βλ λ βλ λ κ 0 cos sin 4 4 3 3 3 3 − − = + A C F E γβ γβ 0 sin cos sinh cosh 4 2 3 3 2 3 3 1 3 3 1 3 3 + = + + + F D C B A β λ β λ βλ βλ 0 cos sin cosh sinh 2 4 2 3 1 4 3 2 2 3 3 2 1 3 3 1 1 3 3 1 − − = + − + B D D C B A λ λβλλ βλλ βλ λ βλ λ 0 sin cos sinh cosh 2 2 4 2 3 1 1 4 3 2 2 2 3 3 2 2 1 3 3 1 1 1 3 3 1 1 = − − + + + q A q C D q C q B q A q λ λ β λ λ β λ λ βλ λ βλ λ ( ) ( ) ( ) [ ] 0 sin cos 2 4 2 4 1 1 3 3 3 3 − + + = − − GB q D q F E E λ λ κ γβ γβ γ ( ) ( ) ( ) ( ) [ ] 0 cos sin cosh sinh 4 2 3 3 2 2 2 3 3 2 2 1 3 3 1 1 1 3 3 1 1 + = + + − + + − − E E D q C q B q GA q γ βλ λ βλ λ βλ λ βλ λ κ 0 1 1 + = A C 0 1 1 2 1 − = qB q D 0 1 = F ( ) ( ) ( ) ( ) (50) 0 sin cos sin cos sinh cosh cosh sinh 2 2 2 1 2 1 2 2 2 1 1 2 2 1 2 1 2 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎭ ⎬ ⎫ − − = + + − + + + + λ λ β λ ξλ β λ λ β λ λ β λ ξλ βλ ξλ βλ λ βλ ξλ βλ λ B D q D C q q B q A ( , ) 0 Δ = ξ ω 0; , at 2 1 1 = = η β η ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) (67) 0 , 0 0 ( ) (0) ( ), (0) (0), ( ) ( ), (0) 2 1 1 1 1 1 1 2 2 1 1 2 1 1 2 2 1 1 1 1 2 ⎭ ⎬ ⎫ =− ′ ′ − ′ ′ − = ′ = ′ ′ = ′ − = = Eu Eu Gv Gv v v u v u v β φ β κ β φ κ β φ φ β β β [ ] { } { }24 1 24 1 24 24 0 ( , ) × × × = Δ C ξ ω (39) 6 2 f L h r π ξ= ( ) 6 5 4 3 2 2.4909 7.332 7.533 5.1773 0.6384 1.035 3.7201 r r r r r r f r + − + − + − =
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