Simplified approaches for Arias Intensity correction of synthetic accelerograms

Abstract

For the generation of synthetic accelerograms, spectrum compatibility is usually emphasized. However, it is well known that the correct evaluation of the seismic response depends on suitable seismic inputs. For example, Arias Intensity, that measures the energy of an earthquake and which attracts more and more attention in probabilistic seismic analysis, cannot be ignored. Thus, simplified methods which could generate both spectrum-compatible and energy-compatible accelerograms are required. This study focuses on the correction of Arias Intensity when generating artificial synthetic accelerograms for given specific earthquake records. Two simple and efficient approaches are proposed. The first approach introduces an energy-compatible algorithm to the spectrum-compatible model, which enables the generated accelerograms to match both the target response spectrum in the frequency domain and Arias Intensity in the time domain. The second approach refers to an empirical way in which empirical envelope shape functions are directly defined based on the energy distribution profile of given earthquake records. The two approaches are validated using various earthquake records, their performance is proven satisfactory and their application is straightforward in the relative fields of earthquake engineering.

Appendices

Appendix 1: Commonly used envelope functions

1.1 Exponential envelope shape, Liu (1969)

$$\begin{aligned} q(t)=a_{0}(\mathrm {e}^{-\alpha t}-\mathrm {e}^{-\beta t}) \end{aligned}$$

(10)

The envelope shape proposed by Liu (1969) (with selected parameters $\alpha = 0.2$ and $\beta = 0.6$) is shown in Fig. 14.

Fig. 14

Envelope shape proposed by Liu (1969). $\alpha = 0.2$ and $\beta = 0.6$

1.2 Piece-wise envelope shape Jennings et al. (1968)

In 1986, Jennings et al. (1968) propose a piece-wise envelope shape as, Eq. (11).

$$\begin{aligned} q(t) = {\left\{ \begin{array}{ll} \left( \frac{t}{T_{1}}\right) ^2 &{} 0 \le t \le T_{1} \\ 1.0 &{} T_{1}< t \le T_{2} \\ \mathrm {e}^{-\alpha (t-T_{2})} &{} T_{2}< t \le t_{d}\\ \end{array}\right. } \end{aligned}$$

(11)

where, $T_{1}$ and $T_{2}$ denote the start and end times of an earthquake motion; $\alpha$ is the model parameter and $t_{d}$ is the total duration of an earthquake. The envelope shape with selected parameters ($T_{1}=3.0$, $T_{2}=8.0$ and $\alpha =0.2$) is shown in Fig. 15.

Fig. 15

Envelope shape proposed by Jennings et al. (1968). $T_{1}=3.0$, $T_{2}=8.0$ and $\alpha =0.2$

1.3 Gamma envelope shape proposed by Saragoni and Hart (1974)

Saragoni and Hart (1974) proposed a ‘gamma’ function to simulate the temporal nonstationary. The ‘gamma’ function is proportional to the gamma probability density which is the reason for the name. The ‘gamma’ function reads:

$$\begin{aligned} q(t)=\alpha _{1}t^{\alpha _{2}-1}\mathrm {e}^{-\alpha _{3}t}, ~~~~t\ge 0 \end{aligned}$$

(12)

The ‘gamma’ has its maximum equals to $\alpha _{1}\mathrm {e}^{1-\alpha _{2}}[(\alpha _{2}-1)/\alpha _{3}]^{\alpha _{2}-1}$, when t is $(\alpha _{2}-1)/\alpha _{3}$. The ‘gamma’ envelope shape with selected parameters ($\alpha _{1}=0.4618$, $\alpha _{2}=3.0$ and $\alpha _{3}=0.5$) is shown in Fig. 16.

Fig. 16

Gamma envelope shape proposed by Saragoni and Hart (1974). $\alpha _{1}=0.4618$, $\alpha _{2}=3.0$ and $\alpha _{3}=0.5$

Appendix 2: Matlab subroutines

1.1 Matlab subroutine for computing energy distribution shapes of earthquake records

1.2 Matlab subroutine for finding the envelope shapes of earthquake records