## Question

**In dynamic analysis, what mass should be used to represent the self-weight of a structure?**

## Answer

**While the lumped mass model only considers the mass related to displacement, the consistent mass model also takes into account the rotational mass along with the mass related to displacement.**

The mass related to displacement is a physical quantity of mass that we commonly know. Rotational mass is a concept of additional mass calculated based on the distance from the rotation center (shear center). The mass participation factor is calculated as the ratio of the total mass to the cumulative mass of each mode, where the total mass is the physical quantity of mass that we commonly know.

When calculating the sum of mass for each mode, the lumped mass model only considers the mass related to displacement, while the consistent mass model considers the mass related to both displacement and rotational components. Therefore, when calculating the mass participation factor, both the lumped mass model and the consistent mass model apply the same value, which can result in a mass participation factor exceeding 100% in the consistent mass model.

To verify if the dynamic behavior of a structure has been sufficiently considered in eigenvalue analysis, a widely used criterion is to check if the mass participation factor is at least 90%. Therefore, if the mass participation factor is checked using this criterion, the mass participation factor of the consistent mass model should not be used. It is recommended to check the mass participation factor using the lumped mass model and then analyze the eigenvalue results using the consistent mass model.

The consistent mass model has the advantage of providing more accurate eigenvalue results for structures that are sensitive to rotation. If two bridges have cross-sections as shown in (A) and (B) in the figure below, the mass related to displacement is the same for both cases, but the rotational mass is greater for the B. If this bridge is a curved bridge, it can be expected that the effect of torsion will be even greater for the bridge for the B. Therefore, for a curved bridge with the B cross-section, using the consistent mass model rather than the lumped mass model can provide more accurate eigenvalue results.