Dynamical systems approach for the evaluation of seismic structural collapse and its integration into PBEE framework
Introduction
The existing methods for predicting the collapse capacity of structures are based on subjective thresholds such as an acceptance criterion [1], limiting drift and/or the slope of the IDA (incremental dynamic analysis) curve (IM/DM rules) [2]. Typically, the IM/DM rules define collapse when the IDA curve approaches 20% of its initial slope or when the engineering demand parameter (EDP), such as interstorey drift ratio, approaches 10%. These thresholds, however, are not based on any phenomenological condition. Even though the IM/DM rules based conventional approach is popular in performance based earthquake engineering (PBEE) [1,3,4], it lacks the ability to accurately predict the dynamic instability leading to collapse, thus resulting in conservative designs [3,5,6]. Furthermore, the effects of force redistribution and local failures in the structure are not captured when the collapse capacity is predicted using these rules [5,7].
To address these shortcomings, Deniz et al. [5] and Zhou and Li [8] proposed energy based methods to predict collapse. In these methods, collapse is defined as the point where some form of the dissipated energy exceeds the supplied energy in the structural system. Since energy provides a holistic description of a structure's state, these methods are argued to be more capable of accurately predicting collapse. However, owing to their inherent nature of dependency on external work or gravitational energy, they have to rely on the occurrence of large deformations. Consequently, they act as lagging indicators of collapse because they are not formulated according to the principles of dynamic instability.
A number of studies have been performed to mathematically investigate the dynamic instability of structures [6,9]. In general, they are either based on the characteristics of the stiffness matrix [10], or the use of the direct Lyapunov method [11]. Instability is traditionally defined as the state when the determinant of the tangent stiffness matrix becomes negative. However, it has been shown that this does not represent a sufficient condition for instability [5,6,[9], [10], [11], [12]]. Using direct Lyapunov analysis, Bažant and Jiŕasek [11] found that instability is a consequence of exponential growth in the deviation of the response from the periodic solution. As a condition, if the eigenvalue of the tangent stiffness matrix past the static bifurcation state is negative, the system is rendered ‘‘temporarily’’ Lyapunov unstable. Yet, the study did not provide any criterion to assess global dynamic instability that can be used under the PBEE framework. On the other hand, Araki and Hjelmstad [10] proposed two dynamic collapse criteria for elastoplastic structures under earthquakes. The criteria are satisfied when the eigenvalues of the Hessian matrix of the total potential energy are negative and the direction of forcing is consistent with the response of the system. However, since the criteria are based on the system's response only near the initial equilibrium point, the unbounded response from the formulated equations may not be the same as the one obtained from the actual equations at the current equilibrium point. Additionally, the unloading response was only studied under free vibrations, which may not be accurate because force reversals can occur under earthquakes.
As far as the P-Delta instability is concerned, the most popular approach for its evaluation is by calculating the ‘‘stability coefficient’’ [[12], [13], [14], [15], [16], [17]]. Several variations of this coefficient have been proposed in the past [12,14]. In general, it is the ratio of the destabilising axial load to the lateral strength of the structure. Bernal [12] and Gupta and Krawinkler [14] in particular investigated the effects of modal, structural and material parameters on P-Delta instability that is defined using this coefficient. Additionally, design standards [13] base their recommendation for the consideration of P-Delta effect on critical values of this coefficient. Even though, the stability coefficient is not entirely related to the dynamics of the system, Bernal [12] demonstrated how the dynamic mode shapes control the failure mechanism and its effect on the stability coefficient. However, the formulation relied on empirical parameters and did not account for the system dynamics.
Using dynamical systems theory, Challamel and Gilles [18] studied the stability of a perfectly plastic oscillator under harmonic excitations by distinguishing its responses in the elastic and plastic regimes. The oscillator was considered as a non-smooth system comprising a set of smooth sub-systems. The stability of the non-smooth autonomous system was studied using the perturbation approach. A closed form solution of the boundary between the elastoplastic shakedown and the alternating plasticity was presented. Past this boundary, the oscillator was found to exhibit period doubling bifurcations. Nevertheless, simplification of the response in the elastic and plastic regimes may not be possible for hardening or softening oscillators [18]. Moreover, the dynamics was only studied under harmonic excitations. Hence, none of the above approaches can be employed for mathematically predicting collapse capacity of structures under earthquakes.
The aim of this study, therefore, is to propose a mathematical criterion that not only predicts structural instability but can also be used in conjunction with IDA under the PBEE framework. Indeed, in this era of sustainability awareness, it is essential to develop more accurate methods for predicting collapse that can result in less conservative designs. The proposed criterion detects the onset of collapse using a dynamical systems theory based algorithm. This is accomplished by assessing the instantaneous P-Delta stability of an idealised SDOF (single-degree-of-freedom) structure [11]. This condition is then extended to formulate a global instability criterion for first mode governed structures. For illustration, a generic case of a bi-linear hardening/softening SDOF system is considered, which, under seismic excitation, acts as a non-smooth non-autonomous system.
Section snippets
Mathematical formulation
The SDOF oscillator shown in Fig. 1a is considered. A flexible system is chosen in this study because it has been previously found that such systems are highly vulnerable to P-Delta effects [19]. It is a reduced model of the shake table experiment structure tested by Ref. [16] as shown in Fig. 2. It has similar dimensions and non-linear material properties as the experimental structure. However, the column section (height, ) of the oscillator is assumed to be rigid with a lumped mass m
Identifying instability in the oscillator under harmonic excitations
The dynamics of the described oscillator system is initially studied under a sinusoidal acceleration time history. The oscillator system is lumped with a mass equivalent to 356 kN. Following the modelling parameters (Table 1) proposed by Kanvinde [16], the initial stiffness, , of the spring is taken as 903.87 N-m/rad (8 kip-in/rad) with a hardening coefficient (). The modal analysis results in the fundamental circular frequency, rad/sec. The sinusoidal time history has
General
The non-smooth nature of the oscillator system becomes more pronounced when actual seismic ground motion time histories are used as forcing functions. Owing to the nature of seismic time histories, the solution can only be obtained numerically. Consequently, the stiffness of the rotational spring, , may also change rapidly. Therefore, similar to the approach discussed in Section 3.1, the system is considered as a set of subsystems.
Stability of the oscillator under seismic excitation
To illustrate the dynamics and predict collapse under a
Impact on collapse performance
The performance evaluation of the oscillator is carried out using collapse fragility curves [3,29], which describe the probability of collapse in the demand-capacity format [30]. Both IM based and EDP based collapse fragility curves [3] are developed using the IDA results derived from the three approaches. These fragility curves are then compared to evaluate the effect on collapse performance using the proposed approach.
The collapse fragility curves are developed for the intensity measure (IM)
Applicability of the proposed approach to MDOF structures
As discussed previously, the dynamic stability based approach is developed for SDOF systems. This makes it applicable only to MDOF structures that are predominantly first-mode governed or structures that are susceptible to fail in first-story mechanism [31]. To illustrate the applicability of the proposed method to MDOF structures, a code-conforming archetype 2-storey reinforced concrete building is considered. Moreover, contrary to the previously discussed steel SDOF oscillator, this structure
Summary and conclusions
This paper presents a dynamical systems theory based approach for mathematically evaluating instability in single-degree-of-freedom (SDOF) oscillators under harmonic and seismic excitations. The proposed approach is intended for structures that can be idealised as SDOF, or those exhibiting predominant first mode governed behaviour. The collapse performance of a code-conforming first-mode governed RC framed structure is demonstrated using the proposed approach by idealizing it to an equivalent
CRediT authorship contribution statement
Shivang Pathak: Formal analysis, Investigation, Conceptualization, Validation, Visualization, Writing - original draft. Simon Watt: Methodology, Supervision, Writing - review & editing. Amar Khennane: Writing - review & editing, Funding acquisition. Safat Al-Deen: Resources, Funding acquisition.
Declaration of competing interest
The authors whose names are listed below certify that there are no conflicts of interest about the manuscript titled “Dynamical systems approach for the evaluation of seismic structural collapse and its integration into PBEE framework”.
Acknowledgements
The authors are grateful for the financial support provided by University of New South Wales (UNSW), Australia in conducting this research.
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