The time-history displacement response, *u*(*t*), of a linear-elastic SDOF system, for a fixed value of damping, subjected to a given ground motion, \(\ddot{u}\)g*(t)*, is characterized by only one variable, which is usually period *T*. From this period, *ω* = 2π/*T* and *k* = *ω*2*m* can be computed, with *k* as the initial stiffness, *m* the mass and *ω* the natural circular frequency of the system (see Fig. 1). The maximum displacement of a linear-elastic SDOF for a given ground motion is called elastic spectral displacement *S**d* = max[*u*(*t*)], for which the SDOF develops an elastic force *F**e* = *k S**d*. Typically, this elastic force is presented in the literature as being divided by mass *m*, and then it is called *spectral pseudo-acceleration*: *S**a*=*F**e*/*m*. From the simple relationships above, it can be demonstrated that *S**a* = *ω*2*S**d*. Since *T* characterizes the SDOF response for a given ground motion, all the variables defined above can be expressed as functions of *T*, i.e.: *S**a*(*T*), *S**d*(*T*) and *F**e*(*T*). For example, the (pseudo-acceleration) response spectrum of a ground motion can be represented by *S**a*(*T*) versus *T*, as shown in Fig. 2.

In structural engineering design, the seismic action is not unique (note that Fig. 2 corresponds only to one seismic accelerogram, \(\ddot{u}\)g*(t)*). Furthermore, a probabilistic treatment of the seismic action is required. Simple envelope approximations of uniform-hazard response spectra derived from probabilistic seismic hazard analysis (PSHA, Cornell 1968) are used in professional standards for structural design purposes: the *design spectra*. For a SDOF system of period *T* and mass *m*, the design spectrum gives the maximum elastic force that can be developed by the SDOF system, *S**a*(*T*)*m*, for a certain probability of occurrence, e.g. 10% in 50 years.

Likewise, a non-linear SDOF system modeled as elasto-plastic can be characterized by only two variables, usually *C**y* and *u**y*, with *u**y* as the yield displacement and *C**y* as the yield strength coefficient defined as *C**y* = *F**y*/*mg* where *F**y* is the yield strength (see Fig. 3). Note the absence of a period in this characterization, which is defined via *C**y* and *u**y* as

$$T=2\pi \sqrt{\frac{{u}_{y}}{{C}_{y}g}}$$

1

If the maximum displacement of this non-linear system for a given ground motion is *u*max so that if *u*max > *u**y*, then the maximum displacement is in the plastic range. It is important to note that once the system is in the yield plateau, the main recovery factor of the system in terms of displacement is the cyclical nature of the seismic action itself, whereby the system alternates between unloading/reloading at elastic stiffness (or near-elastic for non-kinematic-hardening hysteresis rules), and it sustains further damage along the yield plateau. In other words, the importance of a period as a performance-prediction variable is limited when the system enters the post-yield range. The reader should be aware that the authors are looking for a simple preliminary design tool, not a complete and comprehensive description of the response.

In the time-history analysis, ductility (*µ*) is defined as *u*max/uy (i.e., ductility demand). Given that demand should not overcome capacity (D < C), it is necessary to ensure that the ductility demanded by the earthquake is at least equal to the ductility capacity in the structure. In other words, when designing for a certain level of ductility, the designer needs to ensure that the structure is going to allow reliable plastic behavior in the areas of (pre-designated) structural detailing where plasticity is going to take place (i.e., ductility capacity). Although it is a little unconventional, for the sake of mathematical completeness, values of *µ* smaller than 1 are going to be considered (even if they are not proper ductility) in order to also include the cases where *u*max is smaller than *u**y* in the study. Even though the SDOF is characterized by one or two variables depending on whether it is linear or elasto-plastic, the (maximum or worst-case) response of the system can be described by only one variable: the maximum displacement, *S**d* or *u*max. The inelastic displacement ratio *C*1 is defined to relate *S**d*(*T*) and *u*max, so that *u*max *= C*1*S**d*(*T*), FEMA 440(FEMA 2005).

A simple way to describe the intensity of a ground motion is *S**d*, because it shows, as no other parameter does, the effect of a specific ground motion on an elastic SDOF of period *T*. Notice that, as mentioned, *S**a* = *ω*2*S**d*, lending the same properties of *S**d* to *S**a*. As discussed, the strength of this relationship is weakened when entering the non-linear range. This is quantified by the use of inelastic displacement ratios *C*1(*C**y*, *Τ*), also known as *R*-*µ*-*T* strength ratio/ductility/period relationships (where *R* is essentially *C**y*, see Fig. 3). Since the connection is no longer deterministic, these relationships convey the statistics of inelastic displacement *u*max for systems of a given *C**y* and *T*, typically offering the mean estimate, \(\stackrel{-}{{u}_{\text{m}\text{a}\text{x}}}\), for use in tandem with the static pushover (e.g., (FEMA 2005), (EN1998-3 2005)):

$$\stackrel{-}{{u}_{\text{m}\text{a}\text{x}}}={C}_{1m}\left({C}_{y},T\right){S}_{d}\left(T\right)$$

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where *C*1*m* is the mean inelastic displacement ratio function for a given *C**y* and *T*.

The predictive ability of *S**d* and *S**a*, the intensity measures (IMs) within the framework of Performance-Based Earthquake Engineering (Cornell and Krawinkler 2000) can be characterized by implementing the associated dispersion of the distribution of *C*1(*C**y*,*Τ*), as offered by e.g. (Vamvatsikos and Cornell 2006) or (Ruiz-García and Miranda 2007). This is the so-called IM efficiency (Luco and Cornell 2007) and the lower the dispersion, the higher the efficiency and associated predictive capability. The issue of IM sufficiency also arises (Nicolas Luco and Cornell 2007), which characterizes the capability of *S**d* and *S**a* to render the distribution of *C*1(*C**y*,*Τ*), or equivalently of *u*max conditioned on *S**d* or *S**a*, independent of other seismological characteristics.

Sufficiency is a particularly useful property, as it negates (or generally reduces) bias when assessing performance (see Section 4). *S**a*(*T*) is moderately efficient and sufficient, but clearly imperfect whenever large excursions into non-linearity or significantly higher-mode effects are involved (e.g. (Luco and Bazzurro 2007). At the same time, *S**a*(*T*) is clearly better than peak ground acceleration *PGA* (e.g., (Kazantzi and Vamvatsikos 2015) ) for everything but the shortest periods, in which the two parameters are practically identical. Nevertheless, in the Yield Displacement Charts (YDC) presented, the authors are going to use *PGA* as the intensity measure, emphasizing the fact that it is not period dependent, which makes it more versatile than *S**a*(*T*) at the cost of reducing efficiency and sufficiency. The authors believe this tradeoff is acceptable for practical design applications, where simplicity is important, and conservativeness can be added as needed to make up for the probabilistic deficiencies of *PGA*.