13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- (4) E is modulus of elasticity, I is second moment of inertia, A is cross sectional area, G is shear modulus, ρ is density of material, κ is Timoshenko’s shear coefficient and its value is 5/6 for rectangular cross-section, L is total length of portal frame. The solutions of Eq. (1-3) are given by (5) (6) (7) Ai, Bi , Ci, Di , Ei and Fi are arbitrary constants evaluated from the boundary conditions. The boundary and compatibility conditions for a fixed-hinged frame (Fig. 1) are as follows. (8) (9) (10) By substituting Eqs. (5-7) in to Eqs. (8-10) results in following 18 homogeneous equations. (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) These can be expressed conveniently in the following form. (29) where {C}={A1,B1…F3} T are unknown arbitrary constants. 1 2 3 2 2 2 4 2 2 2 2 2 , , , , , L L L L L L E L EI L A G L E L i i = = + + = = = = β ω ρ γ ω ρ α κ ω ρ τ ω ρ σ for 1,2,3 sin , 0 cos sinh ( ) cosh 2 2 1 1 = < < + + + = i D C B v A i i i i i i i i i i i i β η ηλ ηλ ηλ ηλ η for 1,2,3 , 0 cos sin cosh sinh ( ) 2 2 2 2 1 1 1 1 = < < − + + = i q D q C qB q A i i i i i i i i i i i i β η ηλ ηλ ηλ ηλ η φ 0; , at 2 1 1 = = η β η 0 1 1 + = A C 0 1 1 2 1 − = qB q D 0 1 = F 0 sin cos sinh cosh 2 3 3 2 3 3 1 3 3 1 3 3 = + + + β λ β λ βλ βλ D C B A 0 sin cos sinh cosh 2 3 3 2 2 2 3 3 2 2 1 3 3 1 1 1 3 3 1 1 = + + + β λ λ β λ λ βλ λ βλ λ D q C q B q A q 0 cos sin 3 3 3 3 = + γβ γβ F E 0 cos sin 2 2 1 1 1 1 − − = + A C F E γβ γβ 0 sin cos sinh cosh 2 2 1 1 2 1 1 1 1 1 1 1 1 + = + + + F D C B A β λ β λ βλ βλ 0 cos sin cosh sinh 2 2 2 1 1 2 1 2 2 1 1 2 1 1 1 1 1 1 1 1 − − = + − + B D D C B A λλβλλ βλλβλ λβλ λ 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 λβλ λβλ λβλ λ βλ λ ( ) ( ) ( ) [ ] 0 sin cos 2 2 2 2 1 1 1 1 1 1 − + + = − − GB q D q F E E λ λ κ γβ γβ γ ( ) ( ) ( ) ( ) [ ] 0 cos sin cosh sinh 2 2 1 1 2 2 2 1 1 1 1 2 2 1 1 1 1 1 1 1 1 + = + + − + + − − E E D q C q B q G A q γ β λ λ β λ λ βλ λ βλ λ κ 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 2 2 1 3 2 2 2 2 2 2 1 2 2 1 1 2 2 1 − − = + − + B D D C B A λ λβλλ βλλ βλ λ βλ λ 0 sin cos sinh cosh 2 2 3 2 2 1 1 3 2 2 2 2 2 2 2 2 1 2 2 1 1 1 2 2 1 1 = − − + + + q A q C 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 1 2 2 2 2 2 1 1 1 2 2 1 1 + = + + − + + − − E E D q C q B q G A q γ β λ λ β λ λ βλ λ βλ λ κ for 1,2,3 , 0 cos ( ) sin = < < + = i F u E i i i i i i i i β η γη γη η ⎭ ⎬ ⎫ = ′ = = = = = ( ) 0, ( ) 0 ( ) 0, (0) 0, (0) 0, (0) 0 3 3 3 3 3 3 1 1 1 β β φ β φ u v u v 0; , at 3 2 2 = = η β η ; at 0; at 3 3 1 β η η = = ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ⎭ ⎬ ⎫ =− ′ ′ − ′ ′ − = ′ = ′ ′ = ′ − = = 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 ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ⎭ ⎬ ⎫ =− ′ ′ − ′ ′ − = ′ = ′ ′ = ′ = − = 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 [ ] { } { }18 1 18 1 18 18 0 ( ) × × × = Δ C ω , 2 2 wher e 2 1 τ σ α τ σ λ + ⎟ + − ⎠ ⎞ ⎜ ⎝ ⎛ − = , 2 2 3 G L κ ω ρ λ= , 1 2 3 2 1 1 λ λ λ + = q , 2 2 2 2 3 2 λ λ λ − = q , 2 2 2 2 τ σ α τ σ λ + ⎟ + + ⎠ ⎞ ⎜ ⎝ ⎛ − =
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