Equation 1
| nisee |
National Information Service for Earthquake Engineering
University of California, Berkeley |
Note: this paper was presented at the EERC-CUREe Symposium in Honor of Vitelmo V. Bertero, January 31 - February 1, 1997, Berkeley, California.
One concept that can help in formulating a rational and transparent earthquake-resistant design approach is that of performance-based design. In performance-based design, the desired behavior of the building during ground motions of different intensities (design objectives), should be established in a qualitative manner, before the numerical phase of the overall design process starts. Then, the qualitative definition of these design objectives should be quantified, and used as instrumental design information during the numerical phase.
One way in which the design objectives can be introduced into the overall design process is to establish, during the numerical phase, a design methodology that incorporates energy concepts and damage indices. Although currently it is possible to formulate performance-based numerical design methodologies, it is not possible to implement them in practice. It is still necessary to carry out ambitious research programs that can help us understand some of the many issues that currently prevent us, from achieving such implementation.
Conceptual Phase. This phase should focus on the conception of an efficient solution to the design problem at hand. To do so, it is first necessary to define the design objectives, which is the set of the desired behavior(s) of the building, generally formulated in terms of acceptable structural and nonstructural damage, for all relevant levels of design ground motion. Next, it is necessary to establish, according to the local seismicity and the design objectives, if the construction site is suitable. If the site is suitable, the designer proceeds to the conceptual design, in which the designer conceptually establishes the global and structural configuration of the building, the structural systems and materials, and the foundation and nonstructural systems. It is suggested that the postelastic behavior of the building is contemplated at this stage of the design; that is, the conceptual design should demand from the designer a clear understanding of the desired nonlinear response of the building. The intuition, experience and good judgement of the designer are essential to a successful formulation of the Conceptual Phase of the overall design process. The Numerical Phase should not be started unless the Conceptual Phase has been concluded satisfactorily.
Numerical Phase. Once the Conceptual Phase is finished, the overall design process proceeds to the Numerical Phase, which is constituted by two steps: Preliminary Design and Final Design. These two steps involve the sizing and detailing of the structural and nonstructural systems. Any numerical methodology used to carry out the Numerical Phase should be: transparent, so that the designer can lay out numerically a solution to the conceptual formulation of the design problem; flexible, so that the designer can solve the Numerical Phase for different design objectives and structural systems; simple, to make possible its practical application; and concise, so that the designer can consider improving or reformulating its initial conceptual design during the preliminary stages of the Numerical Phase. One of the largest challenges of performance-based earthquake engineering is to formulate and develop a numerical methodology that has the above mentioned characteristics.
Implementation phase. During this phase, the quality of the design should be guaranteed by means of a detailed and independent revision of the same. Also, this phase should contemplate adequate quality control during the fabrication (construction) of the building, and of its maintenance, occupation and function.
All relevant aspects of Equation (1) should be satisfied efficiently, from technical and economical point of views, under the restrictions imposed by the design objectives. Ideally, Equation (1) should be formulated explicitly for all the levels of design ground motion under consideration.
Damage indices, that establish analytical relationships between the maximum and/or cumulative response of structural and nonstructural components and the level of damage they exhibit, have been developed and quantified in recent years. A performance-based numerical methodology is possible if, through the use of damage indices, limits can be established to the maximum and cumulative response of the structure, as a function of the desired behavior(s) of the building for the different levels of design ground motion. Once the response limits have been established, it is then possible to estimate the mechanical characteristics that need to be supplied to the building so that its response is likely to remain within these limits.
For instance, the level of damage in typical nonstructural members strongly depends on the interstory drift index (IDI = interstory drift normalized by story height) to which they are subjected to. Through the use of a response index that relates the IDI demands in these members to their level of damage, it is possible to set, according to the acceptable levels of nonstructural damage established by the different limit states, maximum acceptable values for the IDI demands (IDI subscript max) during the different levels of design ground motion. Only then it becomes possible to estimate the mechanical characteristics (usually stiffness and strength), that need to be supplied to the building, so that its maximum global displacement demand, during all design ground motions, is limited to values that are consistent with limiting its maximum IDI demands within the values of [IDI subscript max] established for the different limit states.
The response index most often used to estimate damage in RC ductile members is that developed by Park y Ang (Park et al. 1987). If some regularity conditions are met, a global value of this index can be used to characterize damage in the ductile members of RC frames (Teran-Gilmore 1996). In this case, the Park and Ang damage index, [DMI subscript PA], for a framed structure can be estimated using the following expression:
where [delta] is the maximum lateral displacement demand during the design ground motion, [delta subscript u] the ultimate displacement under monotonically increasing lateral deformation, [E subscript H mu] the plastic energy dissipation demand during the design ground motion, [F subscript y] the yield strength and [beta] a parameter determined experimentally. A value of [DMI subscript PA] less or equal than 0.4 can be interpreted as repairable damage; from 0.4 to less than 1.0, as nonrepairable damage; and larger than 1.0 as failure; while 0.15 corresponds to the median of the values of [beta] obtained experimentally. According to the levels of structural damage implied by the limit states established for all levels of design ground motion, it is possible to determine a maximum value of [DMI subscript PA] for each limit state. Then it becomes possible to estimate the mechanical characteristics (usually stiffness,strength and maximum and cumulative deformation capacities), that need to be supplied to the building, so that its maximum and cumulative deformation demands, during the design ground motions, are limited to values that are consistent with limiting the structural damage to the levels implied by the selected maximum values of [DMI subscript PA].
If the average plastic energy dissipation can be used to characterize the expected cumulative nonlinear response of the ductile members of a structure, its use in earthquake-resistant design is possible through the use of simple damage indices, like the one summarized in Equation (2). That is, the designer would be able to establish the mechanical characteristics that need to be supplied to the structure, so that the damage produced by the plastic energy dissipation is consistent with the acceptable level of structural damage. Thus, it becomes necessary to determine whether the way in which the plastic energy is dissipated in the member is instrumental or not in determining its level of damage.
Figure I shows constant damage strength spectra obtained from elasto-perfectly-plastic single--degree-of-freedom (SDOF) systems, having equivalent damping coefficients (xi) of 0.05 and 0.20, and subjected to El Centro N-S ground motion. The strengths summarized in Figure I represent those required by the SDOF systems so that their level of damage, after subjected to El Centro N-S motion, is representative of incipient collapse. Two sets of constant damage spectra are shown: one obtained using a response index that neglects the way in which the plastic energy is dissipated (DMI subscript PA), and the other obtained using a response index that heuristically accounts for this energy dissipation (DMI subscript MH). The details on how the spectra shown have been estimated can be found in Teran-Gilmore (1996).
As shown in Figure 1, both damage indices yield similar constant damage strength spectra for [xi] of 0.05 and 0.20. This fact implies that the way in which the plastic energy is dissipated has a small influence in the final state of the SDOF systems or, in other words, that the plastic energy dissipated by the SDOF systems is enough to characterize, on average, their cumulative nonlinear behavior. Figure 2 shows similar constant damage strength spectra obtained for the SCT E-W ground motion. The latter motion, recorded during the 1985 Mexico Earthquakes, has a very narrow frequency content around a fundamental period of excitation of 2 sec. As shown in Figure 2, the strength spectra derived from [DMI subscript MH] has larger ordinates than those derived from [DMI subscript PA] for SDOF systems having a period around the fundamental period of excitation.
The results shown in Figure 1, illustrate the conclusion derived from an extensive statistical study of the response of SDOF systems to ground motions having different durations and frequency content: due to the random nature of ground motion, the response of the SDOF systems does not tend to concentrate in load cycles of a given amplitude, but rather the response will oscillate between cycles of different amplitude. Thus, the plastic energy is not dissipated in a specific manner, and the total plastic energy demand during the ground motion is, on average, a good way of characterizing the cumulative hysteretic response of the earthquake-resisting system. Furthermore, the total plastic energy demand may be used, in a reasonable manner, in lieu of the time-history of the plastic energy dissipation to estimate failure due to low-cycle fatigue. As shown in Figure 2, an exception to the above is the response of SDOF systems subjected to ground motions with a very narrow frequency content, particularly when the period of these systems is close to that of the excitation.
The above conclusions can not be extended to other levels of damage significantly different to incipient collapse. The statistical studies from which the above conclusions were derived only considered incipient collapse. This issue needs to be clarified through further research.
For many years, methods of elastic analysis have provided the designer with a reasonable tool to estimate the local strength "demands" in earthquake-resistant structures. Thus, it seems convenient to keep the use of elastic methods of analysis for this purpose. Nevertheless, the sizing of the structural elements and the determination of other seismic supplies should not be based on the use of such methods. Previous to carrying out an elastic analysis, the mechanical characteristics of the structure, other than strength, should be determined using a numerical procedure that involves the use of damage indices and energy concepts.
The first thing the designer needs to accomplish during the Numerical Phase, is the quantification, through the use of damage indices, of the qualitative definition of the desired behavior(s) of the building for the different levels of design ground motion. This quantification implies establishing limits to the maximum or cumulative demands of all relevant response parameters. Then, the designer needs to establish the mechanical characteristics, other than strength, that should be supplied to the structure, so that its response is controlled within the specified limits. These mechanical characteristics will be denoted as design parameters and, for simplicity sake, it is proposed that they be estimated using a SDOF model of the structure and the maximum demands (through the use of response design spectra) of all relevant response parameters. That is, the designer should establish the mechanical characteristics that should be supplied to the SDOF model so that its response satisfies the response limits. The mechanical properties estimated in the SDOF model constitute the design parameters for which the structure should be designed. The author has discussed in detail the definition and use of SDOF models to estimate the response of multi-degree-of-freedom structures (Teran-Gilmore 1996).
Some of the most common design parameters are: the maximum fundamental period of translation, which defines the minimum stiffness supply in the structure; the viscous damping coefficient, which can be used for the design of passive energy dissipation systems; the global maximum and cumulative deformation capacities of the structure, which set requirements for the detailing of the structural members and their supports and connections; and the maximum global displacement ductility demand that the structure may undergo during the design ground motion, which sets requirements for the lateral strength that need to be supplied to the structure. It is important to emphasize that it is necessary to estimate design parameters for all relevant levels of ground motion and their corresponding limit states.
Once the critical set of design parameters have been established for all relevant limit states, the designer can size the structural members (stiffness design), estimate their longitudinal reinforcement (strength design), and estimate their transverse reinforcement and detailing (deformability design). It should be emphasized that, by using a methodology as the one discussed above, the designer can obtain a reasonable idea of which limit state controls the design before the structural members are sized and the elastic analysis carried out. In this way, appropriate decisions can be taken in the preliminary stages of the design procedure.
To make all the above possible, the designer should handle simultaneously, during the Numerical Phase, the maximum demands of all relevant response parameters (e.g., displacement, velocity, acceleration, and plastic and viscous damping energy dissipation). The explicit definition of design spectra for all the above mentioned response parameters, may result too cumbersome for practical application. Recently, the author has studied the relationships that exist between the energy inputted by the ground motion to the structure (input energy, E subscript I) and the plastic energy (E subscript H mu) and viscous damping energy (E subscript H xi) dissipated by the same; and between all these energy demands and other relevant seismic demands. From this experience, the author has learned that these relationships are stable and can be quantified through simple expressions.
To illustrate the nature of these expressions, Figure 3a shows the average [E subscript H mu] to [E subscript I] ratio, computed on SDOF systems having elasto-perfectly-plastic behavior and a [xi] of 0.05, and subjected to 20 synthetic ground motions with a frequency content and strong motion duration similar to El Centro N-S. As shown, this ratio is practically independent of the period (T) of the SDOF systems and fairly independent of the displacement ductility ratio (mu) for [mu is greater than or equal to 3]. If the [E subscript H mu] to [E subscript I] ratio can be estimated through simple analytical expressions (which according to the experience of the author is possible), it is conceivable to estimate, during the Numerical Phase, the [E subscript H mu] demands in the structure using a design [E subscript I] spectra and the above mentioned ratio. Similar conclusions can be derived, from Figure 3b, about the stability and possible use of the EH, to E, ratio during the Numerical Phase. Figure 3b was obtained from SDOF systems having a [xi] of 0.20 and subjected to the same 20 ground motions.
Although not illustrated in this paper, there also exist stable and relatively simple relationships between the above energy demands and the maximum demands of displacement, relative velocity and absolute acceleration. As stated above, it is conceivable to use a design E, spectra to estimate or account for these demands during the Numerical Phase. Thus, it is worthwhile to study, in more detail, the above relationships and their possible use in formulating a flexible, transparent, simple and concise numerical methodology that helps the designer establish the design parameters of the earthquake-resistant structure at hand.
Once the design parameters have been established, the designer can proceed to size the structural members and carry out an elastic analysis to estimate their strength demands. The elastic analysis should be carried out using strength design spectra, obtained according to the design parameters, representative of all the limit states under consideration. The designer can then proceed to the preliminary detailing of the structure which consists in the determination of the longitudinal and transverse reinforcement, and its detailing. Finally, it is necessary for the designer to establish if the design is satisfactory or not. One of the most critical aspects of the revision of the design is to determine if the actual maximum and cumulative deformation capacities of the structure are consistent with those established as design parameters. For this purpose, the nonlinear pushover and time-history analyses of the structure are useful. It is important to note that currently, not enough information and analytical tools are available to estimate, in a reliable manner, the real deformation capacity of RC members or to carry out nonlinear analysis.
The exact behavior of a structure to a particular ground motion is not as important, for design purposes, as the estimation of its average response to a set of ground motions that are representative of the design ground motion. Thus, predicting the exact response of a structure is not essential to its performance-based design. Actually, it is not possible to implement in practice performance-based design procedures. It is still necessary to undertake ambitious experimental, analytical and field research programs.
Finally, the reader is encouraged to seek out other publications where the problems of current earthquake-resistant design procedures and the implementation of a performance-based numerical methodology are discussed in detail (Bertero y Bertero 1992, Teran-Gilmore 1996). The use of a performance-based numerical methodology is illustrated, in these publications, for the design of RC buildings with 2, 10 and 30 stories.
Bertero, R.D. and Bertero, V.V. 1992; "Tall reinforced concrete buildings: conceptual earthquake--resistant design methodology;" Report No. UCB/EERC-92/16; University of California at Berkeley.
Park, Y.J., Ang, A.H. and Wen Y.K. 1987; "Damage-limiting aseismic design of buildings;" Earthquake Spectra, Vol. 3, Number 1, pp. 1-26; California.
Teran-Gilmore, A. 1996; "Performance-based earthquake-resistant design of framed buildings using energy concepts;" Ph.D. Dissertation, Department of Civil Engineering; University of California at Berkeley.
Vision 2000 Committee 1995; "Performance-based seismic engineering of buildings;" Structural Engineers Association of California.
