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In this study, an adaptive finite element method (FEM) is developed for structural eigenproblems of cracked Euler-Bernoulli beams via the superconvergent patch recovery displacement technique. This research comprises the numerical algorithm and experimental results for free vibration problems (forward eigenproblems) and damage detection problems (inverse eigenproblems). Design/methodology/approach The weakened properties analogy is used to describe cracks in this model. The adaptive strategy proposed in this paper provides accurate, efficient, and reliable eigensolutions of frequency and mode (i.e. eigenpairs as eigenvalue and eigenfunction) for Euler-Bernoulli beams with multiple cracks. Based on the frequency measurement method for damage detection, utilizing the difference between the actual and computed frequencies of cracked beams, the inverse eigenproblems are solved iteratively for identifying the residuals of locations and sizes of the cracks by the Newton-Raphson iteration technique. In the crack detection, the estimated residuals are added to obtain reliable results, which is an iteration process that will be expedited by more accurate frequency solutions based on the proposed method for free vibration problems. Findings Numerical results are presented for free vibration problems and damage detection problems of representative non-uniform and geometrically stepped Euler-Bernoulli beams with multiple cracks to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method. Originality/value The proposed combination of methodologies described in the paper, leads to a very powerful approach for free vibration and damage detection of beams with cracks, introducing the mesh refinement, that can be extended to deal with the damage detection of frame structures.

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The development history and current state of studies on the characteristics and mechanisms of deformation and failure of rock materials were briefly reviewed from the viewpoint of energy. The main scope and the achievable objectives of the energy-based research system were expatiated. It was validated by experiments that the damage process of rocks can be well described by the rock damage evolution equation established based on energy dissipation. It was found from the uniaxial compression and biaxial compression tests that only a small proportion of the total input energy in hard rocks is dissipated before peak load and a large proportion in soft rocks is dissipated before peak load. For both hard and soft rocks, the energy dissipated after peak load accounts for a greater proportion. More energy would be required for rock failure under equal biaxial compression than under unequal biaxial compression. The total absorbed energy is different for rock failure under high-rate loading and low-rate loading. More fragmented failure pattern usually corresponds to higher energy absorption. The mesoscopic analysis on the damage and failure of bedded salt rocks showed that the energy dissipation is prominent and the total absorbed energy for rock failure is low when cracks propagate in the weak mud interlayer while it is contrary when cracks propagate in the salt rock. The energy accumulation, transfer, dissipation and release during the failure process of tunnel with impending failure under disturbance were analyzed theoretically based on the elastoplastic mechanics theory. Furthermore, the spatial distribution of energy dissipation and energy release of fractured rocks under unloading was simulated numerically. It was demonstrated that energy is likely to be released from the weakest surface under compression, which triggers the global failure of rocks.rock, deformation, failure, energy accumulation, energy dissipation, energy release Citation:Xie H P, Li L Y, Ju Y, et al. Energy analysis for damage and catastrophic failure of rocks.