Structural health monitoring is crucial to maintain the structural performance safely. Moreover, the Kullback-Leibler divergence (KLD) is applied usually to asset the similarity between different probability density functions in the pattern recognition. In this study, the KLD is employed to detect the damage. However the asymmetry of the KLD is a shortcoming for the damage detection, to overcoming this shortcoming, two other divergences and one statistic distribution are proposed. Then the damage identification by the KLD and its three descriptions from the symmetric point of view is investigated. In order to improve the reliability and accuracy of the four divergences, the gapped smoothing method (GSM) is adopted. On the basis of the damage index approach, the new damage index (DI) for detect damage more accurately based on the four divergences is developed. In the last, the grey relational coefficient and hypothesis test (GRCHT) is utilized to obtain the more precise damage identification results. Finally, a clear remarkable improvement can be observed. To demonstrate the feasibility and accuracy of the proposed method, examples of an isotropic beam with different damage scenarios are employed so as to check the present approaches numerically. The final results show that the developed approach successfully located the damaged region in all cases effect and accurately.