# 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

[m]{x''} + [c]{x'} + [k]{x} = {P(t)}

Using the transformation

The equation for {P(t)} reduces to "p" separate uncoupled equations of the form

q''_{i} + 2 ξ_{i}ω_{i}q'_{i} + ω_{i}^{2}q_{i} = R_{i}(t) |

= | ||

= | ^{th} mode. |

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 q_{i}s 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

F(t) = F_{0}sin(ωt + ϕ) |

= | ||

_{0} | = | |

= | ||

= |

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.

- F
_{0}= Amplitude - ω = Frequency
- ϕ = Phase

For control/dependent specifications, the forces applied at dependent dof will be ignored. Apply them at the control instead.