8.0       LOAD-TIME FUNCTIONS

 

            There must be at least one load-time function.  If no load-time function is specified, the default is a constant = 1.0.  The load-time function may be generated by using various procedures as defined hereafter, or may be specified by defining at each time instant its function value.  In that latter case, the load-time function is defined by NLS pairs of time instants and function values, where NLS is the number of load steps defined on the control command.  A schematic of a typical load-time function is shown in Figure 8.0.1.  The time instants must be in ascending order (i.e. t(j+1)  t(j), 1  j  NLS). Load step intervals need not be equal and need not be the same from one load-time function to another.  As shown in Figure 8.0.1, the load-time function is assumed to behave in a piecewise linear fashion between data points.  For values of t outside the interval [t(1), t(NLS)] the G's are defined by constant extrapolation (i.e. G[t] = G[t(1)] for all t  t(1); and G[t] = G[t(NLS)] for all t  t(NLS)).  As an example of the use of this feature, we may take the case in which the load-time function is constant throughout the duration of the analysis.  In this case, we may set NLS = 1 and simply read in one data point to define G(t).  For instance, the nodal load at time t is defined to be:

 

F(X, t) =  F(X, i) * G[j, t]           with   j = load_time_function (i) ,  i = Load_case

 

Element consistent loads (e.g. pressure, gravity, etc.) are also multiplied by load-time functions.  The load case number is defined in the element group data (see Chapter 9).  Input for a load-time function is described below.

 

 

Figure 8.0.1 Schematic Representation of a Load-Time Function

 

 

8.1       Load Time Functions

 

LOAD_TIME_FUNCTION

 

 

            LOAD_TIME_FUNCTION          function_type = type ,      etc...

                 < etc..., terminate with a blank record >

 

 

Create a load-time function.  Two main options are available.  The load-time function may be read in directly as a list (optionally from another file) or may be generated.

 

Note    Variable Name                       Type     Default         Description

 

            Load_time_function_number  integer      [1]        Load time function number

 

            Function_type                             list         [*]        Function type

               piecewise_linear

               piecewise_constant

               function_formula

               acceleration_spectrum

               design_spectrum

 

• Piecewise Linear and Piecewise Constant Cases

            Time_offset                                real       [0.0]       Time offset

(1)        File_name                                 string    [none]     File name (optional).  Name must be
                                                                                             enclosed in quotation marks.

            Load time function data must then follow in the form:

               < Time_instant (i), Load_time_function_value (i) >  ( i = 1, number_of_load_steps )

               < etc..., terminate with a blank record >

 

• Function Formula Case

(2)        A_1 , A_2 , ... , A_7                  real       [0.0]       Formula coefficients

 

• Acceleration Spectrum Case

(3)        Duration                                     real      [10.0]      Duration (in time units: T)

            Cut_off_freq                              real      [10.0]      Cut-off frequency (in Hertz: 1/T)

            Max_acceleration                       real      [0.20]      Maximum acceleration (in units: L/T2)

            Damping_ratio                           real      [0.05]      Damping ratio; > 0.0 and < 1.0

            Seed                                           real  [7654321]  Seed for random number generation;

                                                                                             > 100,000 and < 100,000,000

            Spectral_ordinate_1                   real       [0.5]       Spectral ordinate S1 at period T1

            Spectral_abscissa_1                    real   [0.15625]   Spectral abscissa T1

            Spectral_ordinate_2                   real       [0.5]       Spectral ordinate S2 at period T2

            Spectral_abscissa_2                    real      [0.40]      Spectral abscissa T2

            Spectral_ordinate_3                   real       [0.2]       Spectral ordinate S3 at period T3

            Spectral_abscissa_3                    real       [1.0]       Spectral abscissa T3

            Power_exponent                        real       [1.0]       Power exponent m

(4)        Rise_time_t1                               real       [2.0]       Rise time t1

            Decay_time_t2                            real       [5.0]       Decay time t2

            Decay_rate_c                             real       [0.4]       Decay rate c

 

(cont'd)

(cont'd)

 

Note    Variable Name                       Type     Default         Description

 

• Design Spectrum Case

(5)        Damping_ratio                           real      [0.05]      Damping ratio; > 0.0 and < 1.0

            Max_acceleration                       real      [0.20]      Maximum acceleration (in units: L/T2)

            Spectral_ordinate_1                   real       [0.5]       Spectral ordinate S1 at period T1

            Spectral_abscissa_1                    real   [0.15625]   Spectral abscissa T1

            Spectral_ordinate_2                   real       [0.5]       Spectral ordinate S2 at period T2

            Spectral_abscissa_2                    real      [0.40]      Spectral abscissa T2

            Spectral_ordinate_3                   real       [0.2]       Spectral ordinate S3 at period T3

            Spectral_abscissa_3                    real       [1.0]       Spectral abscissa T3 

            Power_exponent                        real       [1.0]       Power exponent m

 

 

EXAMPLE

         Load_Time_Function   /

            load_time_function_number = 1 /

            function_type = piecewise_linear

                 0.0 , 1.0                                                # at time = 0.0, function = 1.0

                 1.0 , 5.0                                                # at time = 1.0, function = 5.0

 

Notes/

(1)        This option allows the load-time function to be read in from a data file separate from the main input file.  The default (i.e., filename is left empty) assumes that the load-time function is contained in the main input file.

 

(2)        The load time-function is generated by using the following formula:

 

G(t) = A1 + A2t

If (A3A4  0.0) then:

 

 

and when t  A7 then:

 

 

where the parameters A1,..., A7 are input.

 

(3)               The load time-function is assumed to be an acceleration time history.  The acceleration time history is generated by using the acceleration response spectrum shown
in Figure 8.1.1, following the procedure proposed by Vanmarcke et al. (1976) and reported in [1].  The implementation of the procedure was performed in cooperation with Dr. G. Deodatis at Princeton University in the Fall of 1992.  Note that for the UBC (1994) spectrum, S1 = 2.5 Amax, S2 = S1 and S3T3 = S2T2 (i.e., m=1).

 

 

Figure 8.1.1 Acceleration Spectrum

 

 

(4)               The acceleration time history envelope is given by the following formula (Jennings, 1968):

                      

                                        

      

 

as illustrated in Figure 8.1.2

Figure 8.1.2

 

(5)               The load time-function is an acceleration spectrum as shown in Figure 8.1.1.  Used for spectral analysis (see Section 12.2).

 

 

Reference / Bibliography

 

1. - Gasparini, D.A.  and E.H. Vanmarcke,"Simulated Earthquake Motions Compatible with Prescribed Response Spectra", MIT Report No. R76-4, (Jan. 1976).

 

 

 

8.2       Filtering Request

 

FILTER

 

 

            FILTER       load_time_function_number = ltime,  etc...

 

 

This allows input load-time functions to be filtered.

 

Note    Variable Name                             Type     Default       Description

 

            Load_time_function_number     integer       [0]           Load-time function number

 

            Low_cut_off_frequency               real       [0.05]         Low cut-off frequency

                                                                                                       (in Hertz) (unit: 1/T)

 

            High_cut_off_frequency              real    [1/(2*Dt)]     High cut-off frequency

                                                                                                      (in Hertz) (unit: 1/T)

 

            Transition_band_width                 real     [low/2]      Transition band width

                                                                                                      (in Hertz) (unit: 1/T)

 

 

 

 

 

 

Notes . .