Seismic loading: Mass in the seismic analysis

The mass can be taken in three ways in the seismic analysis. The setting Mass in the analysis offers 3 options:

• Participation mass only
• Missing mass in modes
• Residual mode

In case Participating mass only is selected, then only the participating mass from the computed modes is taken into account in the seismic analysis.

In case Missing mass in modes is selected, then the missing mass is calculated and assigned to the known modes (i.e. modes selected by the user in the analysis).

In case Residual mode is selected, then the missing mass is taken in the seismic analysis as an extra pseudo-mode which represents the weight of the missing mass. The modal result of this mode is computed by a static equivalent load case.

Remark: The used method is displayed in the linear calculation protocol.

Missing mass in modes: Redistribute missing masses to the known modes

The aim of the method is to smooth the missing mass to the known modes and then compute modal deformations and corresponding modal forces. Modal superposition is then carried out normally, using the rule SRSS, CQC or MAX.

Missing masses are assigned to known eigenmodes. Suppose we have determined nFreq eigenmodes, where

(5.18)

k - is direction

M_ef,k - is effective mass

The missing mass can now be written as

(5.19)

The ratio between the effective mass and the missing mass is

(5.20)

Now we can write these formulas

	(5.21)

Then

(5.22)

Residual mode: Add rigid behaviour as an additional pseudo-mode

The aim of the method is to evaluate the missing mass as an extra mode, which is computed as an equivalent static load case. The static load represents the weight of the missing mass under the zero period acceleration. Modal superposition is then carried out normally according to the selected rule SRSS, CQC or MAX.

The missing mass is computed in each node as difference between total mass and effective mass.

(5.23)

k - is direction

i - is node

j - is mode

M_ef,k,i - is effective mass, direction k, node i

Missing mass can now be written as

(5.24)

A static load case of weight is computed, which is handled in the same way as a “real” mode.

For each direction k, selected in the seismic load case settings, the amplitude of static load is computed as:

(5.25)

Sa_k,ZPA is the value of acceleration for the rigid mode, a.k.a. Zero Period Acceleration

^MissM_k,i is the missing mass in direction k, node i

Afterwards, the extra mode is combined according to the selected rule SRSS, CQC or MAX.

Remarks

• In case of CQC superposition, it is assumed that there is no correlation between the residual mode and the other modes (i.e. SRSS is applied for the residual mode).
• The direction of the static equivalent loads is the same as the direction selected by user in the spectrum direction.
• If the seismic load is defined in 2 or 3 global directions, then the static equivalent mass in the residual mode is computed in the global 2 or 3 defined directions with respect to the directions defined in the input.
• The value of ZPA is taken as the last value of the selected response spectrum (i.e. highest frequency defined in the spectrum). It is of the responsibility of the user to define the spectrum accordingly.