17.10. POST26 - Response Spectrum Generator (RESP)

Given a motion as output from a transient dynamic analysis, POST26 generates a response spectrum in terms of displacement, velocity, or acceleration.

A response spectrum is generated by imposing the motion of the point of interest on a series of single-mass oscillators over a period of time and calculating the maximum displacement, velocity, or acceleration. This is illustrated in Figure 17.14: Single Mass Oscillators.

In Figure 17.14: Single Mass Oscillators, the following definitions are used:

M_i = mass of oscillator i

C_i = damping of oscillator i

K_i = stiffness of oscillator i

u_i = motion of oscillator i

u_b = motion of point of interest

Figure 17.14: Single Mass Oscillators

In the absence of damping, the natural frequency of an oscillator i is:

(17–137)

The basic equation of motion of the oscillator can be given as a one degree of freedom (DOF) version of Equation 15–5:

(17–138)

where:

a dot (

) over a variable = derivative with respect to time

, the relative motion of oscillator i, is defined by:

(17–139)

The damping is given by:

(17–140)

where:

C_cr,i =

= critical damping coefficient

Equation 17–137 through Equation 17–140 are combined to give:

(17–141)

Equation 17–141 is solved when the input time-history is the acceleration (inputType = 1).

For a displacement input u_b (inputType = 0), the equation is rewritten as:

(17–142)

where:

is the velocity of the point of interest. It is the derivative of the displacement input.

Both equations are solved using Newmark integration scheme. See Description of Structural and Other Second Order Systems for more details.

Depending on the spectrum type (specType), the output spectrum values are the following maximum quantities:

the relative displacement:
the relative velocity:
the absolute acceleration:
the pseudo-velocity:
the pseudo-acceleration:

17.10.1. Time Step Size

The time step size (Δt) is selected in the following way. If data is from a full transient analysis (ANTYPE,TRANS with TRNOPT,FULL):

Δt = input time step size (input as DTIME on RESP command)

or if no input is provided:

(17–143)

where:

f_max = highest value of frequency table (table input using LFTAB on the RESP command

The transient data from full transient analysis (ANTYPE,TRANS with TRNOPT, FULL analysis) is taken from the next available time step used in the analysis. This can cause a decrease in accuracy at higher frequencies if Δt is less than the time step size of the input transient.