A case-study in performance-based design using yield frequency spectra

An analytical formulation is offered to allow performance-based seismic design to be achieved following a direct code-compatible procedure. The approach builds upon the use of the yield displacement as a robust system characteristic. A new format for displaying seismic demands known as Yield Frequency Spectra is introduced to quantitatively link performance objectives with the base shear seismic coefficient for a fixed value of yield displacement. Analytical expressions allow estimating the design base shear strength to satisfy any number of performance requirements, foregoing the need for a behaviour factor. The effect of uncertainties is naturally introduced to inject the proper conservatism for, e.g., the natural randomness in the ground motion or lack of knowledge in modelling and analysis. Finally, an 8-story reinforced concrete frame is designed, showing that EN1998 may not achieve the stated performance targets, while the proposed approach can match them with a single iteration. Introduction As a result of economic damage in the 1994 Northridge and 1995 Hyogo-Ken Nambu (Kobe) earthquakes, significant attention has been directed at augmenting the life safety performance objective, characteristic of traditional codes, with additional criteria to limit economic losses in more frequent earthquakes. Basic notions of performance-based design, elaborated in the Vision 2000 report (SEAOC 1995), are widely accepted and are now incorporated in mainstream documents such as ASCE/SEI 41/06 (ASCE 2007) and EN1998 (CEN 2004). Several approaches have been suggested, mainly conforming to the displacement-based design paradigm, as presented by Moehle (1992), Priestley (2000) and Aschheim (2002). They invariably incorporate some form of an equivalent single-degree-offreedom (or first-mode) representation for use in preliminary design. More importantly, they use as starting point an estimate of the yield displacement, rather the fundamental period of the structure, the former being a more stable parameter for a given structural configuration (Aschheim 2002). On the other hand, though, they are deterministic in focusing design on a specific intensity of shaking represented by a design response spectrum associated with a specified hazard level. The hazard level is typically set at a 10% probability of exceedance in 50 years, equivalent to a 475-year mean return period, or a Po = –ln(1–0.10)/50 = 0.0021 mean annual frequency (MAF) of exceedance. When facing the significant uncertainty associated with ground motions, modeling and structural response, deterministic methods are inherently limited. Cornell et al (2002) showed that in the presence of variability due to either aleatory or epistemic sources, the determination of performance at a single level of “design” intensity is unconservative: The more frequent appearance of significant damage at lower levels of intensity will always bias the results. Thus, the use of the “design” intensity results in buildings that may be subject to damage with a higher mean annual frequency of occurrence (greater than the typically desired Po = 0.0021). To achieve uniform levels of safety in the presence of uncertainties, additional hazard and structural response data are needed, in order to consider site and structure characteristics. At present, modern seismic codes use blanket safety factors, typically embodied into the definition of the strength reduction factor R (or behavior factor q) 1 Lecturer, National Technical University of Athens, Athens, Greece, divamva@central.ntua.gr 2 Research Engineer, KANTIA AE, Herakleion, Greece, katsanosvagelis@gmail.com 3 Professor, Santa Clara University, Santa Clara, CA, maschheim@scu.edu D VAMVATSIKOS, EI KATSANOS and MA ASCHHEIM 2 and other design requirements that provide inconsistent levels of safety, even for different buildings within the same class and site. Still, despite the apparent advantages, a fully probabilistic performance-based seismic design approach is difficult to achieve in practice. Design is an inverse problem that, in the case of earthquakes, is based on the non-invertible nonlinear relationships between seismic intensity and structural demands. Thus, iterations are needed, in which each cycle involves the re-design of the structure and its full performance-based assessment via nonlinear static or dynamic procedures (e.g., Krawinkler et al 2006). Figure 1. YFS contours at Cy = 0.1,0.2,...,1.0 for an elastoplastic system (δy = 0.06m) for a site in Los Angeles, CA, overlaid by the design points of three performance objectives for μ = 1, 2, 4 at 50%, 10% and 2% in 50yrs rates, respectively. The oscillator must have sufficient strength to satisfy each performance objective; the third objective governs with base shear coefficient of Cy ≈ 0.93 and a period of T ≈ 0.51s As a partial solution, Vamvatsikos et al (2013) proposed “Yield Frequency Spectra” (YFS) as a rapid means to establish the strength required for a preliminary design to provide a desired level of confidence in satisfying one or more performance objectives related to system drift and ductility demands. YFS provide a visual representation of a system’s performance that quantitatively links the MAF of exceeding any displacement value (or ductility μ) with the system yield strength (or seismic coefficient at yield, Cy). As with other methods, an “equivalent” single-degree-of-freedom model is utilized to establish the preliminary design, which may be based on current code criteria. Fig.1 presents an example of YFS developed for an elastic-perfectly-plastic oscillator. In this case, three performance objectives are specified (the red “x” symbols) while curves representing the site hazard convolved with the system fragility are plotted for fixed values of Cy. Thus, the minimum acceptable Cy that fulfils the set of performance objectives for the site hazard can be readily determined. Code-compatible YFS design formulation YFS application necessitates the use of a full set of hazard curves: One for each period of interest for the specific site. For cases where such complete information is not available, it would be desirable to have at least an approximate solution that can be based on the basic tools of the seismic code: Smoothed, uniform hazard design spectra with the addition of an estimate of the slope of the hazard curve in the region of interest. Following the work of Vamvatsikos and Aschheim (2014), we shall outline the derivation of a set of closed-form expressions that can achieve YFS-like results with a minimum of computations. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 −4 10 −3 10 −2 10 −1 10 0