G.17.3.5 Response Time History
STAAD.Pro is equipped with a facility to perform a response history analysis on a structure subjected to time varying forcing function loads at the joints and/or a ground motion at its base. This analysis is performed using the modal superposition method. Hence, all the active masses should be modeled as loads in order to facilitate determination of the mode shapes and frequencies. Refer to G.17.3.2 Mass Modeling for additional information on this topic. In the mode superposition analysis, it is assumed that the structural response can be obtained from the "p" lowest modes. The equilibrium equations are written as
Using the transformation
The equation for {P(t)} reduces to p
separate uncoupled equations of the form
These are solved by the Wilson- θ method which is an unconditionally stable step by step scheme. The time step for the response is entered by you or set to a default value, if not entered. The qis are substituted in equation 2 to obtain the displacements {x} at each time step.
Time History Analysis for a Structure Subjected to a Harmonic Loading
The results are the maximums over the entire time period, including start-up transients. So, they do not match steady-state response.
Definition of Input in STAAD.Pro for the above Forcing Function
As can be seen from its definition, a forcing function is a continuous function.
However, in STAAD.Pro, a set of discrete time-force pairs
is generated from the forcing function and an analysis is performed using these
discrete time-forcing pairs. What that means is that based on the number of cycles
that you specify for the loading, STAAD.Pro will generate
a table consisting of the magnitude of the force at various points of time. The time
values are chosen from this time '0' to n×tc in steps of STEP
where n is the number of cycles and tc is the duration of
one cycle. STEP
is a value that you may provide or may choose the
default value that is built into the program. STAAD.Pro
will adjust STEP
so that a 1/4 cycle will be evenly divided into
one or more steps. See TR.31.4 Definition of Time History Load for a list of input
parameters that need to be specified for a Time History Analysis on a structure
subjected to a Harmonic loading.
The relationship between variables that appear in the STAAD.Pro input and the corresponding terms in the equation shown above is explained below.
- F0 = Amplitude
- ω = Frequency
- ϕ = Phase
For control/dependent specifications, the forces applied at dependent dof will be ignored. Apply them at the control instead.