Estructura y propuesta de plan de implementación tecnológica

In Chapter 1, it is shown that the load and the response can have a simple proportional relationship by using the concept of dynamic stiffness. However, in that case, the excitation is sinusoidal. In the above section, the case in which the excitation or forcing function is deterministic was discussed. In this case, the convolution integral can be used to determine the responses. For the topic of random excitation mentioned in this chapter, mathematical tools, such as correlation functions and power

spectral density function, were used to analyze the force and the response. Obtaining the structural responses has not yet been discussed.

Under earthquake loading, there cannot be an analytical relationship between the loading and the response, as in the case of harmonic excitation. Although the convolution integral can be used to obtain the response, the forcing function of the earthquake ground motion is unknown.

Furthermore, even under a given amplitude of earthquake loading, the amplitude of the response, in most cases, cannot be completely determined. Yet, to seek the deterministic relation-ship between the input level and the structural response is a necessity in earthquake engineering.

Consequently, the concept of the response spectrum has been developed and is widely used (e.g., see Chopra 2006).

In earthquake engineering, for design convenience, the input levels are often treated as determin-istic, although the seismic ground motion is random. This is because although earthquake excitation is random, it is expected that the peak amplitude will not exceed a certain level under some statisti-cal rules. Therefore, a table that lists the deterministic level is possible by having prior knowledge of the regional history of earthquakes and the site conditions. A more sophisticated table will also consider the importance of the structure to be designed. The specifics of how to obtain such a table is beyond the scope of this book. However, with this prior knowledge, it is possible to conclude that the earthquake considered during an aseismic design is bounded. The upper bound is the design earthquake level, e.g., 0.4 (g). By using the idea of a bounded forcing function, it can be stated that one of the major characteristics of aseismic design, especially for an earthquake protective system, is that under bounded input excitation, the output bound for the response of the system is very dif-ficult to determine.

In the case of earthquake response of an SDOF system,

mx cx kx f+ + = = −mx_g (2.301)

Note that x, x, and x are relative displacement, velocity, and acceleration with units of (m), (m/s), and (m/s²), respectively, and xg is the ground acceleration with units of meters per square second, as mentioned previously.

Mathematically speaking, the amplitude, or the bound of the forcing function f, is taken as |mxg| (N or kN) in Equation 2.301, where f is the earthquake loading. The bound of the response x, or the amplitude of x₀, cannot be specified if f is random. Under any given earthquake record, it is not dif-ficult to solve Equation 2.301 for the peak response. However, for any given earthquake record, the forcing function is no longer random. Therefore, even if all the available records are used as input, Equation 2.301 will not give the peak response because the “next” earthquake ground motion is unknown.

However, for a practical design of the structural capacity of earthquake resistance, engineers need to have deterministic numbers, not only in terms of the input level, but more importantly, in terms of the response amplitude. To meet this requirement, a methodology for using the design response spectrum was introduced by using existing records of various earthquakes to establish the upper bound of an SDOF system. Therefore, it can provide the needed force for design, if the structure to be built can be approximated by an SDOF system.

Specifically, the design response spectrum is generated in several steps. First, a proper group of earthquake records is selected. The selection is based on certain criteria, e.g., high peak value of ground acceleration (PGA) and/or velocity (PGV) and/or displacement (PGD). Other criteria include the duration of an earthquake, the distance of signal pickup to earthquake epicenter, specific site conditions, etc. For convenience, let the number of these records be N.

Second, these records are normalized or scaled to a standard level. For example, all of the records will have the same level of PGA. Since each record has different peak amplitude, unscaled records will not have uniform input levels. This is the main reason for record scaling. In earlier days, PGA

was often used for the scaling standard. However, many papers reported that PGA might not be a good factor, because peak acceleration might not directly relate to structural damage. In damping design, this statement must be carefully evaluated. In many cases, the use of additional dampers is intended to avoid serious damage to the structure. Note that structural damage may not only occur when large floor-drift happens, but may also occur in several other combinations of different responses. In this sense, using damage to a structure (which may occur during an earthquake with/

without supplemental damping) may not be a proper criterion.

Third, the scaled records are used as forcing functions to excite a series of SDOF systems. Each of the systems will have the governing equation described in Equation 2.301 with a fixed damping ratio and the natural frequency or period will be varied. Therefore, with any period T_i, the convolu-tion integral can be used to compute the corresponding responses. In so doing, all the numbers of peak values of the responses, say, the displacements denoted by x_ij, can be obtained. Here, the first subscript i stands for the period T_i, while the second subscript j means that the response is calculated by using the j^th record. By collecting all the data and plotting the results in Cartesian coordinates with the X-axis as the period and the Y-axis as the peak amplitudes, a group of response spectra can be obtained. Figure 2.17 shows such spectra under 11 records, whose amplitudes are all scaled to be 0.4 (g) (Naeim and Kelly 1999, for these 11 records).

Fourth, at each period T_i, the mean value xi and the standard deviation σi of these peak responses are computed. Here, subscript i stands for the statistics taken in accordance with the i^th period T_i. The sum of xi+ σi is taken as the raw data for the statistical response spectral value x_i (m). That is,

In Equation 2.302, N stands for the number of records used. In this way, all x_i are positive, which are the functions of T_i, with reasonable resolution of T_i, and the response spectrum is obtained. Here, T_i is the i^th natural period. Note that for convenience, subscript n is omitted from T_ni here and in the following text.

Fifth, since all of the quantities x_i form a nonsmooth curve, this result is not convenient to use.

Therefore, further measurements are taken to smooth the curve, which may be the envelope of all x_i or other measures, which are referred to as the design spectra.

Response spectra, input 0.4 (g) 2.5

FIGURE 2.17  Earthquake response spectra, ξ = 5%.

Recall the concept of ensemble average introduced in Section 2.3, which is an average of the random process at a specific time. The reason for using the ensemble average is that the random process is not ergodic. Now, it is seen that the earthquake responses are nonergodic so that the ensemble average is used instead of the temporal average. From Equation 2.302, index i implies that the average is taken when there is a fixed period T_i. That is, Equation 2.302 is actually an ensemble average of period, instead of time.