# Time-history output-acceleration accuracy

Measures should be taken to obtain accurate output acceleration from time-history analysis. Acceleration is the second derivative of displacement, and derivative values are less accurate than integrated quantities during numerical analysis.

Guidelines for time-history output-acceleration accuracy are as follows:

• Time steps. For either modal or direct-integration time-history analysis, sufficient output time steps are necessary for accurate representation of the periodic solution. A time increment of one-tenth of the shortest time period of interest should accurately capture response.
• HHT alpha value. For direct-integration time-history analysis, it is often useful to set a non-zero Hilber-Hughes-Taylor (HHT) alpha value (0 ≤ alpha ≤ -1/3) to damp out response from periods shorter than can be captured by the output time step. As stated in the CSI Analysis Reference Manual (Time Integration Parameters, page 362), using alpha = 0 is most accurate, but may permit excessive vibrations in higher frequency modes. A value of alpha = -1/3 tends to remove noise from periods up to about 10 times the time step. Users are advised to experiment, starting with a slightly negative value of -1/24 or -1/48.

• Moving-load analysis. Bridge analysis and design may benefit from a step-by-step moving-load analysis. Automated step-by-step vehicle loading creates separate load patterns at discrete locations along the bridge, amplifying one load pattern while reducing the next. At any instant in time, either one pattern is fully applied, or else two patterns are interpolated. Otherwise, switching one pattern on, then suddenly off while switching the next pattern on, creates an impulse-type load (similar to that from hammer impact) which will excite higher frequencies and ruin acceleration response.
• Inertia. Finally, ensure that mass and inertia are distributed throughout the structure. As a final measure, rotational inertia may also be applied. Displacement load from ground motion effectively applies force and moment to the other end of an element with a restraint support. Since inertia is integral to mathematical formulation, and couples with acceleration, users may smooth acceleration response by applying mass, inertia, and rotational inertia to these elements. If links are located at support points, inertia may still be added, either in the link property or directly at the joint. Rotational inertia may also be added by assigning a mass value between 1/10 md 2 and 1/100 md 2, where m is a characteristic mass located a characteristic distance d from the support.