13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- dz z EI K MSE j j b b i j j i T i ij 2 2 2 ) ( ) ( 2 1 1 ∫ + ∂ ∂ = = φ φ φ （15） dz z EI K MSE j j b b d i d j d j i d T i j d ij ∫ + ∂ ∂ = = − 1 2 2 2 ) ( ) ( 2 1 )( ) (1 φ φ φ α (16) j EI ( ) and d j EI ( ) represent the structural flexural rigidity of th j element before and after damage respectively. jb and 1+jb represent z coordinates of node j and node 1+j respectively. The bending stiffness d j EI ( ) of structure with damage is unknown for experimental modal analysis, it can be substituted by bending stiffness before injury ( j EI ( ) )in the case of small injuries. Consider the selected elements is relatively small, the bending stiffness j EI ( ) of the th j element can be approximated as a constant, it can be mentioned the integral sign. Eq. (15) and (16) can be rewritten as: dz z EI MSE j j b b i j ij 2 2 2 ) ( ( ) 2 1 1 ∫ + ∂ ∂ = φ （17） dz z EI MSE j j b b d i j d ij ∫ + ∂ ∂ = 1 2 2 2 ) ( ( ) 2 1 φ （18） 2 2 2 ) ( z i ∂ ∂ φ in Eq. (17) can be instead by the mean value of '' 2 ( ) ijφ and 2 '' ( 1) ) ( + i j φ , so Eq.(17) can be expressed as: ) ) ]( ( ) [( ) ( 4 1 1 2 '' ( 1) '' 2 + + + + = j j i j ij j ij b b EI MSE φ φ (19) Similarly, the equation (18) can be rewritten as: ) ) ] }( ) ] [( ( ) {[( 4 1 1 2 ( 1) 2 + + ′′ + ′′ + = j j d i j d ij j d ij b b EI MSE φ φ (20) Where, '' 2 ( ) ijφ 、 2 '' ( 1) ) ( + i j φ 、 2 [( ) ]′′ d ijφ and 2 ( 1) ) ] [( ′′ + d i j φ can be got by measuring displacement mode shapes. 2.4.Damage location determination The original signal of injury can be got by the difference of element modal strain energy of the before and after damage: ij d ij j f z MSE MSE − = ( ) （21） Fitting signal ( ) f z j using cubic spline interpolation, the signal are transformed by wavelet function, we get the transform coefficients which can be used to detect the damage location. i j ij DI C a b , ( , ) = （22） i j C a b , ( , ) represents the th j element transform coefficient, ij DI represent element injury positioning indicators. So index ij DI can be used to determine the damage element, if the index ij DI is a large value, we can judge the th j element occur injury.
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