According to the related theory about structural dynamics, the fundamental period is the one of the three major dynamic characteristics of structures, and the value of the structural vibration period is inversely proportional to the square root of the structural stiffness. The structural stiffness degradation is the one of the specific performance characteristics of structural seismic damage. Therefore, during the assessing process of the structural seismic damage, it can be directly considering the change of fundamental period to indirectly considering the change in structural stiffness due to seismic actions.

In the process of establishing structural seismic damage assessment method based on dynamic characteristics, it is necessary to focus on two problems. a) It should be established a fundamental period estimation formula which is simpleness and clear physical significance as well as have relatively accurate estimation results. b) It should be constructed a functional relationship between fundamental period and structural seismic damage. For the above, the relevant researchers have conducted a lot of research and achieved certain research results.

Gilles et al. (2011) analyzed the theoretical basis of the seismic action calculation in the equivalent static method of NBCC standard, and researched the differences of base shear calculation results for different height buildings between using experience fundamental period formula and standard methodology. The results show that when the fundamental period is calculated by empirical formula for estimating results, the base shear of the structure is lowered by 3.5 times.

Sofi et al. (2015) focused on analyzing the mechanical principles and main characteristics of various fundamental period computational formulas, and analyzed the impact of masonry infilled wall, concrete or cement block partition wall on the fundamental period.

Sangamnerkar and Dubey (2015) analyzed 36 reinforced concrete frame structures with different underlying dimensions and shaft-spacing structures. And influencing factors on the fundamental period are researched such as the underlying width, the column cross-section size and the stiffness of the structural basis. The results show that the growth ratio of structural fundamental period is proportional to the underlying width growth.

Young and Adeli (2016) designed 12 eccentric cantilever steel frame structures with different heights, spans and spatial stiffness distributions, and used ETABS to analyze the fundamental periods of different structures. Comparing to ASCE7-10 formula, Raylei formula and ETABS analysis results, the recommendation fundamental period calculation formula for different types of eccentric cantilever steel frame structures are given.

Wang et al. (2018) analyzed 414 high-rise, super-high-rise reinforced concrete structures and mixed structures. The main influencing factors of the fundamental period are comprehensively analyzed, and the fundamental period calculation formula for high-rise building structure is fitted. And 15 shake table test data, 27 pulsation tests, wind test data and Chinese standardized calculation formula calculate results are used to verify the fitted formula. After the correction, the fundamental period calculation formula and the first three-order cycle ratio relationship for high-rise and supper-high-rise building are given.

Based on 90 fundamental period data of steel plate shear wall structures collected in literatures, Jiang et al. (2020) determined a new calculation formula according to the multi-freedom structure dynamic characteristics calculation theory. And the research formula is verified by the shake table tests.

In response to the shortcomings of the fundamental period of shear wall structures in the Indian seismic code, Mandanka et al. (2020) selected 23 irregular shear walls considering the stiffness regular with different planar dimension, structural height, shear wall size, etc., and analyzed the fundamental period of these buildings by using ETABS software. Based on the numerical simulation analysis data, a new fundamental period estimation formula for the stiffness irregular reinforced concrete shear wall structure is fitted considering influencing factors such as total structural height, structural width, and inertial moments.

Elfath and Elhout (2020) applied the Egyptian code to design 36 steel frame structures with different structural height, seismic intensity and elastic story-drift angle, and the change pattern of fundamental period was analyzed focusing on the changes of structural stiffness and height distribution. It is believed that the fundamental period of the bending frame structure is closely related to the seismic intensity and story-drift angle.

For the coupling relationship between the fundamental period and the structural seismic damage, the relevant researchers have been researched.

Eleftheriadou and Karabinis (2013) statistically analyzed the damage data of 164,135 buildings in the Parnitha 5.9 earthquake in 1999. The relationships between the range of different fundamental periods and the damage ratio of buildings corresponding to different damage levels have been focused on.

Based on 300,000 nonlinear seismic response time history analysis data, Katsanos and Sextos (2015) used the theory of elastoplastic response spectrum to study the calculation method of the period elongation of the building structure under the seismic damage state. Research results shows that the periodic elongation rate of damaged structures is significantly affected by the period of structural elasticity and the rate of structural stiffness degradation.

Sarno and Amiri (2019) established a nonlinear single-degree-of-freedom structural system for reinforced concrete structures, taking into account structural factors such as structural ductility coefficient and stiffness degradation rate, and used OpenSees software input to consider the ground motion of the main aftershock sequence for nonlinear time history analysis. The relationship between the period extension rate after structural failure and the epicenter distance, the main aftershock PGA ratio, site type, duration, elastic basic period, ductility coefficient, stiffness degradation rate, cumulative damage and other factors has been emphatically studied, and the final period extension rate estimation formula is given.

The structural failure factor is established according to the structural dynamic equation by Gunawan (2019). The structural failure factor iterative calculation method is constructed using the high-order Runge-Kutta method, and the sensitivity assessment of the structural failure factor was verified using single-degree-of-freedom and double-degree-of-freedom system.

Gunawan et al. (2021) applied Euler-Bernoulli beam theory to construct a structural damage assessment formula that taking structural natural vibration period as the dominant factor.

According to the above research literature, it is known that necessary research has been carried out in the calculation of structural fundamental period and structural damage assessment based on periodic changes. However, most of the existing period calculation formulas are based on empirical formulas, and most of the formulas use the height of the structure or the number of floors to directly estimate the fundamental period of the structure. Although a fewer independent variables can increase the convenience of formula application, it also sacrifices the accuracy of the formula for calculating the fundamental period of complex and diverse structures, which is unfavorable for structural damage assessment based on periodic changes. At the same time, the existing research results mostly focus on the structural period extension ratio after the structure is completely destroyed. While the relative research results on the period change interval corresponding to different damage levels including slight damage, moderate damage, severe damage, and destroy are few. In this paper, aiming at the shortcomings of the existing research work, the mechanical analysis of the generalized single degree of freedom system is carried out by using the structural dynamics theory, and the basic period estimation formula of the structure based on displacement is obtained. Using the direct coupling relationship of structural damage, structural displacement response, structural stiffness degradation and structural periodic change, the estimation interval of structural periodic change factor corresponding to different failure levels is established. Finally, the seismic damage example is used to verify the research method.