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The Gmth Windows Application

Analyzing Ground Motion Characteristics In Accelerograms

Ground Motion Time Histories Of Earthquakes

Program Functions

Response spectrum

Power Spectral Density

Time History Properties

Input File

Output Files

Properties

Using The Application

The Applications Opening Window

The Initialize Menu

Ground Motion Time Histories Of Earthquakes

            The time history of the ground motion of an earthquake is a description of the earthquake vibration as a function of time. Acceleration, velocity and displacement of the ground motion are related to each other through the frequency of the ground motion. Given the time history of one of these components the other two   of ground motion the time histories of the other two components can be derived through the mathematical processes of integration or differentiation. However, because of the difficulty in determining the slopes in the high frequency waveforms differentiation of the displacement and velocity time histories does not provide accurate results. Integration of accelerograms is, therefore, preferred for deriving velocity and displacement time histories. A digitized accelerogram can be easily integrated numerically to get the other two time histories of ground motion. The computation procedure, which is used to derive velocity and displacement from a given accelerogram is describe below:

            Let a₀, v₀ and x₀ be the values of the ground acceleration, velocity and displacement at time t. Let a₁ be the ground acceleration at time (t + h), where h is a small time interval during which the ground motion may be considered linear.

 Thus a (t + h) = a (t) + a (t + h –h) = a₀ + a h

At time t the acceleration can be written as

dv/dt =a(t)  = a₀ +a (t-t);   t < t < t +h

Here v is the velocity. Integrating with the initial condition v(t) =v₀ one gets:

                   t

v(t) = v₀ +  ò  [a₀ + a (t  -t)]dt

                    t             

                                   t 

v(t) = v₀  +a₀(t -t) +a ò  (t  -t)]dt

                                    t  

 v(1) =v₀ +a0h +(1/2) ah²

         = v₀+(h/2)(a₀+a₀+ah)

           =v₀+(h/2)(a₀+a₁)

To obtain the displacement one writes

dx/dt = v₀ +a₀(t-t) +(1/2) a(t-t)²

Integrating with the initial condition, namely x(t)=x₀ one gets

x (t) = x₀ + v(t -t) + (1/2)a₀(t-t)² + (1/2).(1/3) a(t-t)³

= x₀ + v₀h + (1/2)a₀h² + (1/6) ah ³

        = x₀ + v₀h + (h²/6)(3a₀ + ah)

x (1) = x₀ + v₀h + (h²/6)( 2a₀ + a₁)

The figures below show a synthetic accelerogram and the ground velocity  and displacement time histories derived from the acceleration time history using the equations given above.

Program Functions

(1) Deriving velocity and displacement time histories of ground motion from the accelerogram.
(2) Determining peak values of acceleration, velocity and displacement, number of zero crossings per second in the accelerogram, value of ad/v2 and duration of strong motion. These parameters have referred to as properties in the application.
(3) Computing response spectrum for a specified value of damping.
(4) Computing power spectral density of the ground motion.

(5) Displaying the results on the monitor screen, and storing these in ASCII files (output files) for further use

Response spectrum

Damping is Specified by the user as percent of critical.

Response spectrum is computed at the following frequencies (Hrtz)

Power Spectral Density

Power spectral density (PSD) is used to quantify the frequency content of the ground motion, and computing  is an important step setp in determining, and adjusting, the spectral compatibility of  ground motion. It is briefly described here.

            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.

Time History Properties  

            The time history of the ground motion of an earthquake is a description of the earthquake vibration as a function of time. Acceleration, velocity and displacement of the ground motion are related to each other through the frequency of the ground motion. Given the time history of one of these components the other two   of ground motion the time histories of the other two components can be derived through the mathematical processes of integration or differentiation. However, because of the difficulty in determining the slopes in the high frequency waveforms differentiation of the displacement and velocity time histories does not provide accurate results. Integration of accelerograms is, therefore, preferred for deriving velocity and displacement time histories. A digitized accelerogram can be easily integrated numerically to get the other two time histories of ground motion. The computation procedure, which is used to derive velocity and displacement from a given accelerogram is describe below:

            Let a₀, v₀ and x₀ be the values of the ground acceleration, velocity and displacement at time t. Let a₁ be the ground acceleration at time (t + h), where h is a small time interval during which the ground motion may be considered linear.

 Thus a (t + h) = a (t) + a (t + h –h) = a₀ + a h

At time t the acceleration can be written as

dv/dt =a(t)  = a₀ +a (t-t);   t < t < t +h

Here v is the velocity. Integrating with the initial condition v(t) =v₀ one gets:

                   t

v(t) = v₀ +  ò  [a₀ + a (t  -t)]dt

                    t             

                                   t 

v(t) = v₀  +a₀(t -t) +a ò  (t  -t)]dt

                                    t  

 v(1) =v₀ +a0h +(1/2) ah²

         = v₀+(h/2)(a₀+a₀+ah)

           =v₀+(h/2)(a₀+a₁)

To obtain the displacement one writes

dx/dt = v₀ +a₀(t-t) +(1/2) a(t-t)²

Integrating with the initial condition, namely x(t)=x₀ one gets

x (t) = x₀ + v(t -t) + (1/2)a₀(t-t)² + (1/2).(1/3) a(t-t)³

= x₀ + v₀h + (1/2)a₀h² + (1/6) ah ³

        = x₀ + v₀h + (h²/6)(3a₀ + ah)

x (1) = x₀ + v₀h + (h²/6)( 2a₀ + a₁)

The figures below show a synthetic accelerogram and the ground velocity  and displacement time histories derived from the acceleration time history using the equations given above.  

Input File

The accelerpgram to be analyzed is store in an ASCII file resoding in the directory of the application. The file has the extension .ths (or .THS).

The first line in the file is the number of points (digitized values of groun acceleration) in ground motion time history. In the file shown below this number is 1000. The number on the second line (0.02) is the sampling interval of dozitization. This is followed by the acceleration values (these values are in terms og 'g' -the acceleration due to gravity. 

Output files

Using The Application

The application is initiated by either double clicking the Gmth icon, or by typing Gmth on the command line. The application opens with the window shown below:

The Applications Opening Window

The Initialize Menu

View Menu

Acceleration Time History

Velocity Time History

Displacement Time History

Response Spectrum

Power Spectral Density

Properties