Seismic demand

Californian earthquakes, such as Whittier 1987, Loma Prieta 1989 and Northridge 1994, demonstrated the vulnerability of older reinforced concrete bridge columns to failure under seismic demands [230]. 

One of the methods developed for the seismic risk evaluation of structures is the SAC-FEMA method, which enables probability assessment in closed form (Cornell et al. 2002), and represents a part of a broader PEER probabilistic framework (Deierlein 2004). Within the framework of SAC-FEMA method, the relationship between the seismic intensity measure and the engineering demand parameter is usually determined by Incremental Dynamic Analysis (IDA) developed by Vamvatsikos Cornell (2002). IDA is a powerful tool for estimation of seismic demand and capacity for multiple levels of intensity. However, it requires a large number of inelastic time-history analyses (and corresponding detailed data on ground motion time-histories and hysteretic behavior of structural elements) and is thus very time-consuming. Often it is possible to create summarized IDA curves with less input data, with less effort, but with still acceptable accuracy. One possible approach is to determine seismic demand for multiple levels... 

Figure 6 Capacity diagrams, IN2 curves, and elastic and inelastic spectra defining seismic demand at the NC limit state for the bare and infilled frame presented in AD format.

Chopra, A.K. 6c Goel, R.K. 2002. A modal pushover analysis procedure for estimating seismic demands for buildings, Earthquake Engineering and Structural Dynamics, 31, 561-582. 

All seismically compact W14 and W18 column sections were subsequently studied parametrically Fig. 8 illustrates the relationship developed between column web stiffness and seismic demand for W14 and W18 column sections. An equation for determining the need for continuity plates was formulated by applying a regression analysis to the column data and solving for the web thickness, tew (Adan 2006). Continuity plates are required in W14 columns if the web thickness is less than the value given by ... 

The main objective of the assessment is to compare the seismic demand (input) with the capacity of the plant and to decide on the need for upgrades. 

Luco N, Cornell CA (2007) Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions. Earthq Spectra 23(2) 357-392 Shome N, Cornell CA (1999) Probabilistic seismic demand analysis of nonlinear structures. RMS program, report no. RMS35, PhD thesis, Stanford University, CA... 

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

The main objective of this derivation is to introduce the theoretical background for the determination of the parameters of the SDOF model, which is one of the key components of the pushover-based methods. The advantage of the SDOF model is its simplicity. Thus, it can be used to determine the seismic demand based on nonlinear response history analyses. However, it is important to understand the theoretical limitations of such approach. 

The pushover analysis becomes an approximate analysis method if it is used for the prediction of expected seismic performance of a structure (e.g., Saiidi and Sozen 1981 Fajfar and Fischinger 1988 Fajfar 2000). In this case, it is important that the structural model implicitly accounts for the cyclic deterioration. However, the results of pushover analysis are used for the definition of the SDOF model, which represents a link between the seismic action and the seismic demand. 

The seismic demand at the level of the SDOF model can be obtained by means of numerous procedures. Once the displacement at the level of the SDOF model has been obtained, it can be transformed to the selected EDP at the structural level. Since several phenomena associated with the response of a structure during an earthquake are neglected in pushover-based methods, it is clear that the results of such simplified procedure are approximate also from the theoretical point of view. Due to the approximate nature of pushover-based procedures, many types of pushover analysis have been developed in recent decades. Some of them are briefly described below. 

Definition of limit states. In general, limit states can be defined at the component or structural level. As briefly discussed above, there is no critical restriction regarding the definition of limit states if the seismic demand is based on a pushover analysis procedure. 

Different types of pushover analysis can be performed. Conventional pushover analysis, which is based on the invariant lateral load, is the simplest and is the most often applied. The results of the pushover analysis are pushover curves and the seismic demand on the structure given the top displacement. In general, more than one pushover analysis per structural model is required. 

Assessment of seismic demand at the level of structural model. The displacement obtained from the response history analysis of the SDOF model should be transformed to the top displacement using Eqs. 8 and 9. Any other engineering demand parameter at the level of structure can be obtained from the results of the pushover analysis on the basis of the known top displacement However, if more than one pushover analysis is performed per structural model, in order to account for the effect of higher failure modes, then the total demand can be obtained by enveloping the results of all pushover... 

Vamvatsikos D, Cornell CA (2006) Direct estimation of the seismic demand and capacity of oscillators with multi-linear static pushovers through IDA. Earthq Eng Stmct Dyn 35(9) 1097-1117... 

In most of the modem (pre)standards and guidelines for the assessment of existing stmc-tures, the simplest form of the pushover-based analysis is included. All methods that are included in these documents are based on the same basic concept presented above. However, these methods are not the same. The procedures which are used to define the properties of the equivalent SDOF model and those which are used to define the seismic demand of this model are different. In general two different approaches are used. The properties of the equivalent SDOF system are defined either based on the equivalent pre-yielding stiffness or based on the equivalent secant stiffness (see Fig. 4a, b). 

The application of the capacity spectrum technique means that both the structural capacity curves and the demand response spectra are plotted in the spectral acceleration versus the spectral displacement domain and compared. Therefore in the next step, the pushover curve is converted to the capacity spectrum curve using the modal shape vectors, participation factors, and modal masses obtained from a modal analysis of the structure. The capacity spectrum curve represents the relationship between accelerations Sa and displacements Sd of the equivalent SDOF oscillator. Then the standard elastic acceleration spectrum (corresponding to 5 % damping) is converted to the ADRS format, where the spectral accelerations are presented as a function of the corresponding spectral displacements (see Fig 4d). In this way, the capacity curve and the seismic demand can be plotted on the same axes and compared. 

The equivalent period T q and effective viscous damping figq (which is used to reduce the seismic demand) are defined as... 

Based on the comparison of the capacity curve and seismic demand, the target displacement is defined. This displacement is then converted to roof displacement and other aspects of the response are defined. 

The ATC-40 method is also well accepted due to the clear and useful visualization of the procedure. Note, however, that the procedure, defined in EC8/3, can also be presented in a similar manner nevertheless, it is essentially different. The capacity curve as well as the seismic demand, defined in EC8/3, can also be converted to the ADRS format, plotted on the same axes and compared, as is presented in Fajfar (1999) and illustrated in Eig. 4c. 

Miranda E, Ruiz-Garcia J (2002) Influence of stiffness degradation on strength demands of structures built on soft soil sites. Eng Stmct 24(10) 1271-1281 Nassar AA, Krawinkler H (1991) Seismic demands for SDOE and MDOE systems, TR 95, The J. Blume Earthquake Engineering Center, Stanford University, Palo Alto... 

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