Static pushover analysis

The static pushover analysis has no rigorous theoretical foundation. It is based on the assumption that the response of the structure (MDOF system) can be estimated using the results of the analysis of an equivalent SDOF oscillator (see Fig. 1). This means that it is assumed that the response is governed by one invariant mode of vibration. In general this is incorrect. However, the assumption is approximately fulfilled in many (regular) structures, where the influence of the higher modes is negligible and the deflection shape is almost invariable. Thus the seismic response of these MDOF systems is quite accurately estimated based on the analysis of an equivalent SDOF model. It is not the intention of this article... 

The basics of aU these methods are presented in a series of articles in the encyclopedia and will not be repeated herein. Instead, some aspects of analysis (other than member modeling that was presented in section Modeling of Bridge Components ) specific to concrete bridges will be briefly described, and the current trends in nonlinear static (pushover) analysis of bridges (not covered elsewhere here) will also be... 

The ductility factor fi for the ultimate limit state is calculated in accordance with the appropriate material standard, where such data are provided. Contrary to NZS 1170.5 2004, where reference is made to material standards only and a special study is required otherwise, maximum ductility factors for typical structural systems are provided directly in AS 1170.4 2007 (Table 5). Alternatively, for a specific structure, /x and Sp can be determined via a nonlinear static pushover analysis. Note also that AS 1170.4 2007 applies only to structures with ductility factor jU < 3, and when a higher ductility factor is considered, the design must be performed in compliance with NZS 1170.5 2004 provisions (Table 5). 

The response sensitivity, probabilistic response and reliability analysis methods presented are based on nonlinear FE quasi-static pushover and time-history analyses, which are used extensively in earthquake engineering and referred to by structural design codes. 

Fragility analysis Fragility curve Limit state Nonlinear static procedure Pushover analysis SDOF model Seismic demand Target displacement Uncertainty... 

From the historical point of view, pushover analysis has been typically used as a convenient tool to estimate the nonlinear properties and capacity of individual structural components or the whole structure. It typically represents the first step of different static inelastic methods. More specifically, the static nonlinear analysis is performed using the multi-degree-of-freedom (MDOF) model of the analyzed structure. The main purpose of this analysis is to define the properties of the equivalent single-degree-of-freedom (SDOF) model, which is then further... 

The seismic input is represented by the acceleration response spectrum included in the codes and the displacement spectrum or with a set of accelerograms. The type of analysis is static, dynamic, or response history. The stiffness that is used to calculate the maximum displacement demand is the equivalent pre-yielding stiffness or the equivalent secant stiffness, usually obtained based on the pushover analysis. This variety of solutions can result in a different estimation of the response. In section Different Applications of the Inelastic Pushover-Based Analysis, Adopted in the (Pre)Standards and Guidelines, those solutions (methods), which are adopted in Eurocode 8/3 (CEN 2005), ATC-40 (ATC 1996), and EEMA-356 (ASCE 2000), the recommended modifications in EEMA-440 (ATC FEMA 2005), and the displacement-based method, included in the NZSEE (2006), are presented. They were selected since they illustrate many of the basic concepts that are currently in use. 

Pushover analysis is the static nonlinear analysis which is typically performed as the first step of the majority of the inelastic static methods for the seismic assessment of the (existing) structures. The structure is subjected to the lateral load (representing the inertial forces), the intensity of which is gradually increased. The corresponding lateral displacement at a certain location (the reference point or control point) in the structure is recorded. Then the pushover curve, representing the relationship between the base shear of the structure and the registered displacements, is constructed (see Fig. Ic). 

The static nonlinear analysis of the MDOF system is repeated up to the estimated seismic displacement in order to be able to analyze different aspects of the response (e.g., story drifts, shear forces, bending moments). The same loading pattern as in the first pushover analysis is employed. 

The 2003 NEHRP Recommended Provisions (BSSC 2004) specify procedures for ncmtinear static analysis and nonlinear dynamic (response-history) analysis. The nonlinear static analysis procedure is similar to the ELF procedure, in that the pushover capacity curve is used to define the nonlinear behavior of the structure. However, in the nonlinear static analysis procedure, the actual nonlinear force-displacement relation is used, rather than an idealized elastoplastic curve. In addition, since actual pushover strength is known from the nonlinear pushover analysis, the force reduction for design of the seismic-force-resisting system is based on overstrength alone with no additional reduction (Symans et al. 2008). 

Nonlinear static analysis, which is commonly referred to as pushover analysis in earthquake engineering, enables the simple evaluation of the bearing of structural loads and the deformation capacities, as well as the evaluation of... 

Based on the knowledge of the FRS, one of the following methods can be adopted (a) static equivalent forces (including nonlinear pushover analysis), (b) modal analysis, and (c) time history (linear or nonlinear) analyses based on accelerograms compatible with the FRS. 

Pushover analysis was originally developed for first-mode dominated strucmres and later refined to account also for higher mode effects (e.g., Chopra and Goel 2002). The basic assumption of these procedures is that the nonlinear static response can be related to the nonlinear dynamic response. This is, however, not the case for many structures because the dynamic response is strongly path dependent, and effects, such as cyclic deterioration, damping, and duration of an earthquake, among others, cannot be captured. Nonetheless, nonlinear static analysis... 

Nonlinear static monotonic (pushover) analysis shows that the conventional MRF and the SC-MRF have comparable base shear strength and initial stiffness. The conventional MRF experiences significant damage in beams at the DBE drift. On the other hand, the SC-MRF has damage-free beams for drifts even higher than the MCE drift. 

An alternative nonlinear static procedure was proposed by Aydinoglu (2003). The procedure, called Incremental Response Spectrum Analysis (IRSA) does not properly belong to the pushover methods. The procedure is displacement-based, it uses the equal displacement rule and the structure nonlinear behavior is modeled as piece-wise linear. The initial modes are computed and the elastic response spectrum for the initial structure is carried out. In the first stop the spectral ordinates are scaled to the formation of the first plastic hinge, which corresponds in the piece-wise linear capacity curve, to the first change of stiffness. The updated modal quantities are computed and additional spectral ordinates are computed up to the formation of the second plastic hinge. The procedure continues until the entire spectral ordinates are applied. The method does not require any transformation to an equivalent SDOE system. 

Nonlinear static analysis Pushover-based analysis... 

Assessment of Existing Structures Using Inelastic Static Analysis, Fig. 1 Version of the inelastic pushover-based analysis, included in EC8/3... 

The accuracy of different lateral load patterns in structures where the higher modes have an important influence on the response is discussed in sections Some Issues that Influence the Accuracy of the Inelastic (Single-Mode) Pushover-Based Methods and Application of the Inelastic Static Methods to the Analysis of Bridges . 

Since the methods included in different codes are conceptually different, they are presented in more detail in the next subsections. The FEMA-356 and ATC-40 methods were evaluated in the document FEMA-440, where their modifications are proposed. The main observations and the proposed modifications are presented in section ATC-40. The matters of accuracy and different issues that can influence the pushover-based analysis of buildings are discussed in section Some Issues that Influence the Accuracy of the Inelastic (Single-Mode) Pushover-Based Methods. The estimation of the response of bridges using the inelastic pushover-based analysis is presented in section Application of the Inelastic Static Methods to the Analysis of Bridges. ... 

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