5.1 Introduction

The primary purpose of performance-based seismic engineering design is to produce a structure that has a definable and predictable performance during future earthquakes. This design concept was described in detail, as it applies to building structures, in Chapter 2. The performance-based seismic engineering philosophy can be applied to both new and existing buildings and includes a performance evaluation stage whereby the reliability, or alternately the probability of failure, of the building is calculated for multiple performance limit states. Following the conceptual design phase for new buildings, the building design is evaluated to determine whether it is on target to meet the desired performance objectives. This evaluation is repeated throughout the remainder of the building design following the preliminary and final design stages. For existing buildings, a performance evaluation of the existing building design can be performed to determine whether satisfactory performance will be achieved during future earthquakes and to aid in the conceptual development of an appropriate retrofit scheme.

There are a number of ways to quantify the seismic performance of a building, the form of which should be dictated by both structural engineers and building owners.

Information regarding the reliability of different building designs can be used by the structural engineer to compare the relative performance of structural systems or configurations for new buildings and alternate retrofit strategies for existing buildings. Building owners desire that the performance of buildings be explicitly defined and quantified such that the economic and social implications of the building's future performance can be considered. For example, assume that it was determined that a new building housing a manufacturing business would remain operational during an earthquake that has a return period of 72 years. A stronger earthquake would result in a loss of function for the business and therefore, have an immediate economic impact An alternative structural system incorporating supplemental energy dissipation devices, such as viscous fluid dampers, can be developed that would keep the business operational during a 475 year earthquake, but, at the added initial cost of the supplemental energy dissipation system. With this information, the building owner can weigh the costs of the initial investment with the future costs of repairs, business losses, content damage, and death or injury in order to make the appropriate financial and social decisions. The optimal building design will be one in which the present and future costs, also known as life-cycle costs (Ang and De Leon, 1996), are minimized. The concept of life-cycle cost optimization is qualitatively illustrated in Figure 5.1 (Lee, 1996). This figure shows that in general, as the initial investment increases, the future costs resulting from damage decreases, hence, the optimal design strategy is that which minimizes the total overall cost. The total life-cycle cost is a function of the building reliability, or probability of failure, therefore, it is necessary to have available the tools to quantify the reliability of a building design. Alternately, if a specific level of building performance is desired, then a performance design and evaluation procedure is necessary to quantify the reliability of meeting the specified level of performance. This research will focus purely on methods for quantifying building performance in terms of its probability of failing to meet specific performance objectives.

As discussed in Chapter 2, performance-based seismic design requires the selection of performance objectives that consist of a level of performance coupled with a level of earthquake ground motion. The performance objective for a building can be quantified in terms of a performance Junction, where x = (x;,x2,---,xn) is a vector of building demand and capacity related variables that are used to quantify building performance. If the performance function is greater than zero, i.e. g(x) > 0, then the building design meets the intended performance objective. This is described as a "safe" building design. On the other hand, if the value of the performance function is less than zero, i.e. g(x) < 0, then the building design does not meet the desired performance objective. This is described as a "failed" building design. Consider, for example, a building where performance is quantified in terms of the maximum interstory drift In this case, the performance function is expressed as, g(x) = g(x;,x2,-,xn)

capacity of the building for the desired level of performance. Similarly, 6d is a random variable that quantifies the maximum interstory drift demand on the building at the specified level of earthquake ground motion. The failure and safe regions defined by the performance function given in Eq. (5.2) are illustrated in Figure 5.2. Consider in this example the occurrence of a Rare (475-year) earthquake and the Vision 2000 performance objective called Life Safety, see Figure 2.1, Chapter 2. In this case, 5d in Eq. (5.2) corresponds to the maximum interstory drift demand from a 475-year earthquake ground motion, and 5C is the maximum interstory drift angle capacity corresponding to the Life Safety performance objective. Recall that the demand and capacity are random variables because their values cannot be estimated with certainty. If the random variables, 5d and 5C are correlated random variables, then they have a joint probability density function (PDF). Define the joint PDF of 8d and Sc to be p(5c,8d). Since g(5d,5c) is a function of the random variables 5d and 5C, the performance function itself is a random variable. For a given design, the probability that g(5d,8e) is less than zero is called the probability of failure. The probability of failure, Pf, is expressed mathematically as,

If it is assumed that the maximum interstory drift demand and capacity are independent random variables, then Eq. (5.3) becomes,

The probability of failure calculated using Eq. (5.4) is qualitatively illustrated in Figure 5.3. In this figure, the probability of failure is represented by the overlapping area of the PDF's. The probability of failure depends on the shape of the PDF's, the degree of dispersion in these curves, and the relative position of these curves.

In the general case where the performance function defined in Eq. (5.1) is a function of many random variables x;,x2,---,xn, the joint PDF of the variables is p(xi,xJ,-",xn). Then the equation for the probability of failure can be written as,

In a structural engineering context, performance functions can be thought of as representing different building failure modes. For the performance function expressed in Eq. (5.2), the failure of a building design to meet the specified performance objective, or failure mode, is defined to occur when the maximum interstory drift demand exceeds the interstory drift capacity. In general, a building can have more than one mode of failure. Each failure mode can be defined by performance functions expressed in terms of multiple building response quantities. In Chapter 2, Section 2.5, it was shown that the FEMA 273 document (ATC, 1997) provides guidelines for acceptable building

performance in terms of maximum inters to ry drift, maximum beam plastic rotation, and maximum panel zone plastic rotation. Other measures of performance can include maximum column plastic rotation, story drift ductility, and damage indicators such as the Park and Ang (1985) damage index.

In general, for failure mode /, failure is defined by the event,

Consider a building with k failure modes corresponding to k performance functions. Building failure is defined to occur when the design fails in any one of the k failure modes. This is written as,

where Et i^Ej denotes the union of failure events / and j. Equation (5.5) can now be generalized to represent the probability of building failure, where building performance is quantified in terms of k failure modes and k performance functions, g,(x/,x2,---,xn), that are a function of n random demand and capacity variables. This generalization can be written as,

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