13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- According to the principle of energy balance, cracks release of elastic strain energy into plastic strain energy and surface energy when the pile cracks, thus contributing to the crack propagation, eventually lead to structural strain can be reduced. Based on this, the Eq. (4) can be expressed in the form of the strain energy. For piles structures, define the th i modal th j element strain energy as: j i T i ij K MSE φ φ = （5） d j i d T i j d i d j d T i d ij K K MSE φ φ α φ φ )( ) (1 ( ) = − = （6） Where, the superscript "d" represents the damaged structure. According to the above formulas we can get: i n j ij j i n j T i i Ti MSE MSE K K = = =∑ ∑ = = 1 1 φ φ φ φ （7） ∑ ∑ ∑ = = = =− Δ = Δ =− n j ij j j i n j T j i j i n j T i i T i MSE K K K 1 1 1 α φ φ α φ φ φ φ （8） Ignore second-order or more high-end items, we can get: ) ( ] )( ) [(1 ] ( ) [( ) 1 1 1 1 ij n j d ij j i T i d j i d T i n j j d j i d T i j i T i n j d j i d T i i j n j T i i Ti MSE MSE K K K K K K K − = − = − + Δ − Δ = Δ = ∑ ∑ ∑ ∑ = = = = φ φ φ φ α φ φ φ φ φ φ φ φ φ φ （9） ) ( 1 1 1 j n j d ij j i j n j T j i i j n j T i i Ti MSE MSE K K K − Δ =− Δ Δ = Δ Δ =− ∑ ∑ ∑ = = = α φ φ α φ φ φ φ （10） Eq.(7), Eq. (8), Eq. (9) and Eq. (10) substituted into Eq. (4), Eq. (4) can be expressed as: ∑ = ⋅ = ⋅ n j d ij j d i i MSE MSE 1 ) ( α δ （11） Where, ∑ = = n j d ij d i MSE MSE 1 , the total strain energy of the th i modal of Injury structure. The formula (11) is structural damage identification equation, which is composed by modal damage sensitivity factor, modal strain energy of injury structure and the element damage factor.The strain energy is more sensitive to the small damage of the structure[16], Measuring modal the damage sensitivity factor is relatively easy, and it has a high measuring accuracy[17]. So the Eq.(11) can be used to identify small damage of the pile structure with higher accuracy. iδis the indicator used to determine the modal sensitivity of structural damage, the value of iδ is the larger, the modal damage identification of structural damage is the more sensitive, or vice versa. Therefore, according to the modal damage sensitivity factor iδ, we can select the mode used for structural damage identification. Practical application is setting a threshold based on the environment and damage of the structure, it is the efficient mode when the threshold value greater than the set threshold value. 2.3. Element modal strain energy calculation For the pile structure, in the case of element is sufficiently small, element modal strain energy can be represented by the formula as follow:
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