Application of rigid-perfectly plastic spectra in improved seismic response assessment by Endurance Time method
Introduction
Among various methods of analysis that have been adopted by seismic codes for the analysis of structures subjected to earthquake loadings, the time-history analysis provides the most realistic prediction of structural behavior. However, the complexity and huge computational effort involved in this type of analysis has limited its widespread practical application. This drawback has motivated researchers during the past years to develop alternative analysis methods. These new methods are much less computationally intensive as compared to the time-history analysis; additionally, they can estimate the seismic demands with an acceptable degree of accuracy.
Within these methods, Endurance Time (ET) is one of the promising approaches, introduced by Estekanchi et al. in 2004 [1]. This method is a time-history-based analysis procedure, in which the structure is subjected to a set of predesigned accelerograms that increase their intensity in time—referred to as the ET excitation functions. The excitation functions are generated in such a way that their response spectra increase in time; hence, the response of the structure under this kind of accelerogram gradually increases with time. The performance of the structure is estimated based on the time interval during which it can sustain the imposed dynamic excitation. By using a properly designed excitation function, this endurance can be correlated to the intensity level of ground motions that the structure can carry on. More details about the concept of the ET method, as well as the characteristics of the ET excitation functions, can be found in the literature [2].
The main advantage of the ET method over the regular time-history method is that it requires a small number of analyses. In the ET method, the structural responses at different excitation levels are obtained in a single time-history analysis, thereby significantly reducing the computational demand. Therefore, by using the ET method and adopting the concepts of performance-based design [3], [4], the performance of a structure at various seismic hazard levels can be predicted using a single time-history analysis. The application of the ET method in the seismic performance assessment of steel frames has been studied by Hariri-Ardebili et al. [2] as well as Mirzaee and Estekanchi [5].
The results of the ET analysis are usually presented by increasing ET response curves. The ordinate at each time t corresponds to the maximum absolute value of the required engineering demand parameter in the time interval [0, t] as given in Eq. (1):In this equation, Ω is the Max_Abs operator as was defined above, and P(t) is the desired response history of an engineering demand parameter such as interstory drift ratio, base shear, or other parameters of interest. The abscissa of an ET response curve is time, which is an indicator of the intensity in the ET analysis. Fig. 1(a) shows a typical ET response curve in which the maximum interstory drift is used as the demand parameter. ET curves are usually serrated, due to the statistical characteristics and dispersion of the results of the ET analysis in the nonlinear range. Sometimes the response value does not pass the maximum value experienced before in a time interval and, therefore, the resulting ET curve has a constant value in that interval. In order to get more accurate and consistent ET curves, Estekanchi et al. [6] recommended using the average of the results from three ET excitation functions.
In a study accomplished by Mirzaee et al. [7], the correlation between time—as an indicator of the intensity in ET analysis—and seismic hazard return period was investigated. Substituting the time with the return period increases the readability and efficiency of response curves and can considerably improve the presentation of the results of the ET analysis. They used the elastic response spectrum defined in ASCE41-06 as an intermediate criterion [8]. The response spectrum (Sa) for any hazard level can be expressed as a function of the return period (R) and the period of free vibration (T). Acquiring the inverse of this function with respect to variable R, the return period can be expressed as a function of T and Sa (Eq. (2)):where R is the return period, and f is a function that relates the return period to Sa and T. Apart from this, the response spectrum for the ET excitation function is defined as:where T is the period of free vibration, t is time, and a is acceleration. In view of Eqs. (2), (3), it turns out that the seismic hazard return period can be expressed as a function of T and t (Eq. (4)):where h is a function that relates the return period to T and t. Since the establishment of an explicit formulation for this function is not straightforward, they developed a matrix form for the return period where for each period of vibration (T) and each ET time (t), a particular return period was specified.
Applying Eq. (4), the abscissa of response curves, originally expressed by time (see Fig. 1(a)), can be replaced with the return period (R). A sample response curve obtained in such a manner is illustrated in Fig. 1(b), where the return period axis is displayed in a logarithmic scale. The ET time at which Eq. (4) holds is referred to as the equivalent time corresponding to return period R and period of free vibration T.
Estekanchi et al. [9] studied the results of the ET analysis in the estimation of the maximum interstory drift of the frames with elastic-perfectly plastic (EPP) material model and compared them with the results of time-history analysis, using ground motions. It was shown that in the frames which experience a linear behavior, the results of ground motions match the results of the ET analysis more closely. However, in the frames which experience large nonlinear deformations, the difference between the results increases, and the ET method gives unreliable estimations. It is believed that this inconsistency can be alleviated by using a more appropriate intensity measure [10], [11] in correlating the ET time and the return period.
The basic philosophy of the modern earthquake resistant design procedures is to achieve a reliable structural performance while maintaining an economical design. This goal is achieved by ensuring energy dissipation capabilities under strong ground motions [12], [13]. Such energy dissipation is usually attained through high levels of plastic deformation in certain specially designed elements, namely displacement-controlled elements, while elastic behavior is ensured in other structural members, namely force-controlled members. Regarding this design philosophy, the improvement of the ET method in the prediction of the maximum response of structures that experience large nonlinear deformations is necessary, which is the main purpose of this paper.
The key idea is that, if elastic deformations become relatively insignificant (e.g. at large deformation levels), rigid-perfectly plastic (RPP) models rather than EPP models can be adopted to assess the structural response. Paglietti and Porcu [14] concluded that RPP single-degree-of-freedom (SDOF) systems can be used to evaluate the maximum plastic displacement of any EPP-SDOF system under strong earthquakes. Their study also introduced the idea of the so-called RPP spectrum defined as the relationship between a structural response parameter (such as peak structural displacement) and the plastic capacity of the RPP model. Such a spectrum is much easier to be developed than the EPP one and depends only on one parameter, namely the yield strength of the system over its own weight. Fan and Ji [15] remarked the same conclusions by applying a finite element program. Makris and Black [16] used a rigid-plastic model to study the response of ductile structures as well.
Domingues Costa et al. [17] presented a simplified seismic design method, based on the assumption that the dynamic response of the structures with large ductility capacity may be derived by neglecting the contribution of their elastic properties. Málaga-Chuquitaype et al. [18] demonstrated the applicability of response history analysis based on RPP models for the seismic assessment and design of steel buildings. They also indicated that such RPP models are able to predict global seismic demands with reasonable accuracy, especially when the inelastic deformations are large. Hibino et al. [19] used the rigid-plastic model to predict the story displacement where the plastic drift dominates in two-story buildings under strong ground motion. Moreover, the idea of RPP systems has been adapted to model sliding-type isolation system [20].
In the present study, an effective method is investigated in order to correlate the seismic hazard return period and the intensity in ET analysis. Motivated by the studies cited above, the nonlinear RPP spectra are used instead of linear elastic spectra to define this correlation. The application of the proposed method in estimating the maximum displacement response of several EPP-SDOF systems is demonstrated. Additionally, the proposed method is applied to estimate the response curves of two three-story frames, namely a moment-resisting frame (MRF) and a frame equipped with friction dampers. All ET analyses are carried out using ETA20inx01-3 excitation functions available at the ET method website [21].
It is shown that the application of this method improves the accuracy and reliability of the response curves in the nonlinear range in comparison to the procedures that are based on linear elastic spectra. Further, in cases where the structure experiences large nonlinear displacements, the response curves resulted from this method match well the ones obtained by performing a conventional nonlinear time-history method. The modeling and nonlinear analyses have been performed using PEER’s OpenSees platform [22].
Section snippets
Generating rigid-perfectly plastic spectra
A rigid-perfectly plastic (RPP) system is a system characterized by a force–displacement relationship indicated in Fig. 2(a). No deformation occurs until F reaches the yield force, Fy, and the force cannot exceed the yield force, i.e., |F| ⩽ Fy. The RPP model can be simulated by a Coulomb friction block, as is illustrated in Fig. 2(b).
An RPP-SDOF system of yield force Fy and mass m is assumed to be subjected to an earthquake excitation. This SDOF system can be modeled with a friction block
Correlating time and return period by RPP spectra
In this section, the RPP spectra obtained in the foregoing discussion are used to correlate the return period and the ET time. In order to detect this correlation, the ETA20inx01-3 excitation functions are imposed to RPP-SDOF systems of several Ay/g’s. The resulting displacement responses can be calculated by using Eq. (5). The RPP spectrum for an ET excitation function is defined as is indicated in Eq. (11):where t is time, and Δ (τ) is the displacement of the
Analysis of SDOF systems
In this section, the application of the ET analysis method to estimate the response curves of systems with elastic-perfectly plastic (EPP) behavior is investigated. Therefore, various EPP-SDOF systems with free vibration periods (T) of 0.1, 0.2, 0.3, 0.4, 0.8, 1.0, 2.0, and 4.0 s and with Ay/g’s equal to 0.05, 0.1, 0.2, 0.3, and 0.4 are considered. The viscous damping is assumed to be equal to zero. These systems are once analyzed by applying ETA20inx01-3 excitation functions, and once again by
Study of a three-story moment-resisting frame
In this section, the application of the RPP spectra in producing the ET response curves for multi-story structures is demonstrated (Eq. (12)). The case study is a three-story single bay MRF, which is adopted from the models developed by Estekanchi et al. [6] (see Fig. 10). This frame is designed according to the Iranian National Building Code (INBC), using codified base shear for a high seismicity area [26], which is almost identical to the AISC-ASD design recommendations [27]. Some properties
Summary and conclusions
In the structures that experience large nonlinear deformations, the difference between the results obtained by the conventional ET analysis and the nonlinear time-history analysis, which uses recorded ground motions as the seismic inputs, can be significant. This is mainly due to the fact that elastic spectra are not good representatives of intensity in the largely plastic response range. Significant improvement in the prediction of the maximum response of the structures that experience large
References (29)
- et al.
Performance-based seismic assessment of steel frames using endurance time analysis
Eng Struct
(2014) Performance-based design in earthquake engineering: state of development
Eng Struct
(2001)- et al.
Performance-based design in earthquake engineering: a multi-disciplinary review
Eng Struct
(2001) - et al.
Application of endurance time method in linear seismic analysis
Eng Struct
(2007) - et al.
Improved methodology for endurance time analysis: from time to seismic hazard return period
Sci Iran
(2012) - et al.
Application of endurance time method in seismic assessment of steel frames
Eng Struct
(2011) A broad-range power-law form scalar-based seismic intensity measure
Eng Struct
(2009)- et al.
Intensity measures for the seismic response prediction of mid-rise buildings with hysteretic dampers
Eng Struct
(2015) A direct displacement-based seismic design procedure of inelastic structures
Eng Struct
(2001)- et al.
Performance-based seismic design of structures: a direct displacement-based approach
Eng Struct
(2003)
Endurance time method for seismic analysis and design of structures
Sci Iran
Performance-based seismic retrofitting of steel frames by endurance time method
Earthq Spectra
Seismic rehabilitation of existing buildings
Rigid-plastic approximation to predict plastic motion under strong earthquakes
Earthq Eng Struct Dynam
Cited by (22)
An innovative variable target time method for probabilistic-based seismic performance assessment of multi-storey buildings
2022, Journal of Building EngineeringSeismic damage and life cycle cost assessment of unanchored brick masonry veneers
2022, Engineering StructuresCitation Excerpt :Low computational demand is the major advantage of this method over conventional time history analyses, which makes it suitable for computationally-costly structures such as masonry infill-façade systems. Numerous studies have demonstrated the accuracy of the ET method in examining the seismic performance of structures, including steel and concrete buildings [35–38], multi-span bridges [39], offshore structures [40], controlled systems with frictional [41] and viscous [42] dampers, and power substation equipment [43]. Valuable features of endurance time excitation functions (ETEFs) provide the efficiency for this method.
Experimental study of deconstructable bolt shear connectors subjected to cyclic loading
2021, Journal of Constructional Steel ResearchCitation Excerpt :Similar results are also reported in the literature [36–38]. The cyclic behaviour of the shear connectors can be modelled using zero-length elements to simulate the entire structural response of buildings to earthquake motions [39–42]. Hence, an theoretical model replicating the realistic cyclic behaviour of the shear connectors would be of interest.
Generating high spectral consistent endurance time excitations by a modified time-domain spectral matching method
2021, Soil Dynamics and Earthquake EngineeringCitation Excerpt :Compared to the traditional incremental dynamic analysis (IDA) method [14–16], the endurance time method substantially improves the computational efficiency of nonlinear dynamic time history analysis in engineering practice, which benefits from the ETAF being used as an alternative to seismic records [13]. By now, it has been widely used in different engineering fields, such as in the seismic assessment [17–25], structural design [26–31] and probabilistic based earthquake engineering [32,33]. Because of the predominance of the use of ETAFs in the endurance time method, efficiently generating an accurate ETAF is critical to the success of the subsequent performance evaluation [13].