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Spectrum Compatible Accelerograms

It is possible to synthesize an accelerogram, which has certain known characteristics of ground motion, e.g. peak ground acceleration, power spectral density, duration of strong motion, shape of the strong motion window function and, above every thing else, a specified response spectrum. Design response spectra (DRS) represent a hypothetical earthquake. No real accelerogram will, therefore, posses the characteristics of a specified design response spectrum (SDRS). Also, the accumulated data set of recorded accelerograms from strong earthquakes is not large enough to allow the use of any real accelerogram in design. An accelerogram to meet the requirements of design has to be compatible with the design response spectrum. This means that the time history response spectrum (THRS) should be sufficiently close to the SDRS at all frequencies. In fact, for the sake of conservatism, the THRS ordinates are required to remain above those of the SDRS at all frequency points, except for a few. At frequencies, where the THRS go below the SDRS, the deviation should remain small (5-10%).    In earlier days of aseismic design artificial accelerograms were synthesized by modifying an existing record by changing the amplitudes and rates of zero crossings in selected ranges to bring these closer to the required levels. Contracting and / or expanding the record along the two axes were used to obtain the desired compatibility. With improved computing facilities, it became easier and faster to generate artificial accelerograms on digital computers. Synthesizing the ground motion as a stationary random process, and modifying it by adding to it a window process to bring it close to representing a ground motion time history achieve this. (The window process ensures that the synthesized time history has a beginning and an interval of strong motion, which decays gradually. )The synthesized accelerogram is written as:  

a(t) = I(t). SA_n Sin (nw_nt + j_n)

Here I(t) describes the windowing function. I(t) is equal to 0 before t=0 and after t=t_d  where t_d  is the duration of the accelerogram. The amplitudes A_n and phases j_n are then adjusted, in an iterative process, to match the THRS with the SDRS.

1. Gasparini,A.D. and Vanmarcke, E.H.(1976). Simulated Earthquake Ground Motion Compatible with Prescribed Response Spectra. Report No. 2 (Publication No. R76-4), Deptt. Civil Engineering. MIT, Massachusettes, USA.

2. Levy, S. and Wilkinson,J.P.D.(1976). Generation of Artificial Time Histories Rich in all Frequencies from given Response Spectra. Nuclear Engineering. and design. Vol. 38, pp 241-251.

Synthesizing Spectrum Compatible Accelerograms

     Spectrum compatible accelerograms  are required for dynamic analysis of the structures and for generating response spectra for different floor levels. In the earlier days of aseismic design such accelerograms were obtained by modifying a recorded accelerogram (by suitably adjusting its time and amplitude scales). However, digital computers allow easy generation of  synthetic accelerograms  , and achieving a higher level of compatibility. A brief outline of the principle of generating spectrum compatible accelerograms is given below.

            Any periodic function can be represented as a sum of sinusoids in the form:

X (t) = S A_n Sin (w_nt+j_n)  ......   (1)

                 Here A_n is the amplitude and j_n is the phase of the n^th sinusoid. For fixed amplitudes A_n different ground motion time histories with similar appearances, but different details, can be generated. For X(t) the amplitudes An are related to the power spectral density (see Frequency content of accelerograms 5.4 ) of the time series at angular frequency wn  by the following relationship:

G(w_n).Dw = A_n²/2  ...  (2)

As n becomes large the total power in the series is given by:

¥

òG (w). dw = SG(w_n).Dw = S A_n²/2   ... (3)

0                   n                     n

 Equation (1) represents a stationary time series, in which the power does not vary with time. Earthquake signals are of transient nature. They have an onset, a rise time during which the signal builds up, duration of strong motion and, finally, a signal decay time in which the signal reaches the average ground level. The transient character of the strong motion signal can be simulated by multiplying X(t) with a window function I(t) describing the character of the transient. Thus the artificial ground motion time history can be written as:

a (t) = I (t). S  An Sin (w_nt + j_n) .... (4)

                  n

Matching the response spectrum of the accelerogram with a target spectrum is done in several cycles by comparing the calculated response spectrum with the target response spectrum at selected frequencies[1] ^,[2]. In the computation procedure the power spectral density (PSD) at the i^th angular frequency is modified as follows:

G(w)_i+1 = G(w)_i . S_v(w)/S_v⁽ⁱ⁾(w)  ... (5)

where S_v(w) is the spectral velocity in the target response spectrum, S_vⁱ(w)  is the spectral velocity and G(w)_i the PSD at frequency  w in the i^th iteration. G(w)_{i+1 is the modified PSD,} for (i+1)^th iteration, if another iteration is required.  The following table lists out the frequency intervals in different frequency ranges, which have been recommended for comparing spectra for matching[3].

Table- : Suggested Frequencies for calculation of Response Spectra.

Frequency Content Of An Accelerogram

            For a given accelerogram a (t) its Finite Fourier Transform (FFT) at angular frequency w is written as: FS (w)  = [U²+V²] ^½     …..  (1) 

           T

U  = ò a(t) Sin (wt).dt  ……. (1a)

        0

       T

U = òa(t) Cos (wt).dt  ……. (1b)

       0

Here T is the duration of strong motion. The Fourier amplitude spectrum FS (w) is written as the square root of the sum of squares of the real and imaginary parts, U and V, of F (w). FS (w) has the units of velocity. For an undamped SDOF system subjected to the acceleration a(t) at the base the response of the SDOF system and the Fourier amplitude are closely related. The equation of motion of the undamped SDOF system is written as:

d²/dt²  +  w_n x =-a(t)  ……. (2)  

Here x (t) is the relative displacement of is mass and w_n is its natural frequency. The steady state response of the system is given by

                      t

x(t) = (1/w_n) ò -a(t)Sin wn (t-t)dt  ….. (3)

                     0

The right hand side is known as Duhamel’s integral.[1]

Its relative velocity is given by

             t

dx/dt = ò a(t)Cos(wt)dt ……  (4)

            0

The total energy per unit mass of the system, E(w_n), at the end of time T can be written as the sum of the kinetic and the strain energies:  

E (w_n) = (1/2)[(dx/dt)²] +(1/2) w_n ²x²(T) …… (5)  

Substituting the values of x(T) and  (dx/dt) in Equation (6) gives the relationship between E (w_n) and FS (w_n).  

E (w_n)  =  (1/2)[FS (w_n)]²  ……    (6)

Thus at the end of the strong motion the total energy per unit mass is one half of the square of the Fourier amplitude at that frequency. Expanding Equation (5), denoting the maximum relative velocity (spectral velocity) of the system SV (w_n) and assuming that the maximum velocity occurs at time t_v gives

                      t_v                             t_v

SV (w_n) = [  ò  a(t) Sin w_ntdt}²+{ò  a(t) Cos w_n tdt}2]^1/2 ……. (7)

                    0                                0

A comparison between equations (1) and (7) shows that, for zero damping the maximum relative velocity is equal to the Fourier amplitude when t_v = T.  If the maximum relative displacement SD (w_n) occurs at time t = t_d, and a pseudo velocity PSV (w_n) is defined as the product of the natural frequency and the maximum relative displacement, SD (w_n),   

PSV (w_n) = w_n. SD (w_{n)  …….} (8)  

It can, then, be seen that the pseudo velocity is equal to the Fourier amplitude if td = T. While SV (w_n) and FS (w_n) are closely related, in general, the Fourier amplitude is less than SV (w_n) and PSV (w_n). If w₀ is the  highest frequency in a(t) then the inverse Fourier transform relationship is written as:

                   w₀  

a (t) = (1/T)ò F(w)dw   ……    (9)

                  0

The intensity or total energy of a(t) is given by

      T

I = ò  a² (t) dt    …….  (10)

      0

Using  the Perseval’s  theorem on Fourier transforms the intensity can also be written as  

I =(1/p)ò[F (w)]² dw ……. (11)  

The intensity per unit time is, then, written as

                T

G = (1/T) ò  a² (t) dt   =  =(1/pT) ò[F (w)]².dw  …. (12)

                0

The right hand side represents the mean square vale of the acceleration, a(t).

In the frequency domain the power spectral density (PSD) is defined as  

G (w) = (1/pT)[F(w)]²  ……. (13)  

This gives the relationship between the PSD and the mean square acceleration r

       w₀

r = ò G (w) dw   …… (14)

       0

PSD is, normally, represented as the product of a normalized PSD function G^<n>(w), which has a unit area under the w, G(w) curve, and a mean square acceleration. Thus

G (w) = r. G^<n>(w)…. (15)

PSD is an important characteristic determining the frequency content of the accelerogram.