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Missing any Modal Mass?

Example Demonstrating Missing Mass Method | FEA Consultant
July 12, 2016 By: Peter Barrett

Unlike tall buildings and other relatively flexible systems, nuclear power plants and other stiff structures may have important natural vibration modes at high frequencies. In most cases, it is not practical to calculate these high-frequency modes, which are typically not excited by the seismic or instructure motion, with sufficient accuracy to warrant the effort. If only the lower modes are included in the dynamic analysis, the mass associated with the high frequency modes are excluded from (i.e., is “missing” from) the dynamic analysis. In this case it is still important to account for the residual rigid response that has significant natural vibration modes at higher frequencies. Ignoring the residual rigid response in these cases may result in underestimation of element forces, moments and associated stresses. It is also important to account for the contribution to support reactions of mass that is apportioned to system support points. The residual rigid response of the missing mass modes can be calculated using the Missing Mass method as documented by Kennedy (Ref. 1) and summarized in the ANSYS theory manual shown in Figure 1. Use of the Missing Mass method for calculating the contribution of high frequency modes and support reactions is acceptable for both response spectrum analysis and modal superposition time history analysis.

Figure 1: Excerpt From ANSYS Theory Manual on Missing Mass Calculation

An example seismic response spectrum analysis of a piping system (Figure 2) is used to demonstrate the efficiency of utilizing the missing mass method to account for the high frequency response and support mass. 

Figure 2: Example Piping System to Demonstrate Missing Mass Method (Total Mass = 8.25)

Analyses are performed with increasing number of modes included in the analysis, with and without the missing mass method active. Figure 3 illustrates the X-direction spectrum analyzed. 

Figure 3: X direction Response Spectrum Applied to Base Supports

Comparisons of the mass excited, peak stress, and reaction forces (locations shown in Figure 4) are illustrated in Table 1.

Figure 4: Nodal Support Locations

A review of Table 1 shows:
•    The missing mass method produces reasonable results for peak stress with as few as 5 modes included in the modal analysis. 20 modes were required if the missing mass method was not used.
•    In the example, the reaction forces are computed correctly only with the missing mass method active, even when 200 modes are used. (Attachment forces can also be correctly computed without using the missing mass method if a support block is modeled, instead of connecting the structure directly to ground)

I hope you found this blog interesting. I would be interested in how others have incorporated the missing mass method into their dynamic analyses.

(1) R.P. Kennedy, “Position Paper on Response Combinations,” Report No. SMA 12211.02-R2-0, March 1984. Published in “Report of the U.S. Regulatory Commission Piping Review Committee: Evaluation of Other Dynamic Loads and Load Combinations,” NUREG-1061, Vol. 4, December 1984, Washington, DC, available through ADAMS under Accession No ML11343A0343.