In a multi-modal response spectrum analysis, it is important to determine a sufficient number of eigenvalues of the structure and to consider their dynamic responses. Regulations such as EN 1998‑1 [1] and other international standards require activation of 90% of the structural mass. This means: to determine so many eigenvalues that the sum of the effective modal mass factor is greater 0.9.
As gravity loads act on a structure, lateral displacement occurs. In turn, a secondary overturning moment is generated as the gravity load continues to act on the elements in the laterally displaced position. This effect is also known as "P-Delta (Δ)". Sec. 12.9.1.6 of the ASCE 7-16 Standard and the NBC 2015 Commentary specify when P-Delta effects should be considered during a modal response spectrum analysis.
The modal results of a response spectrum analysis are combined with quadratic combination rules, and in RF‑/DYNAM Pro, the SRSS and the CQC rules are available. The default setting modifies the quadratic expressions into equivalent linear combinations. The advantage of this option is that the corresponding internal forces keep their signs and are often much smaller, compared to the standard SRSS or CQC rules. The standard SRSS and CQC rules are on the conservative side and the "equivalent linear combinations" are recommended.
The response spectrum analysis is one of the most frequently used design methods in the case of earthquakes. This method has many advantages. The most important is the simplification: It simplifies the complexity of an earthquake to such an extent that an analysis can be carried out with reasonable effort. The disadvantage of this method is that a lot of information is lost due to this simplification. One way to mitigate this disadvantage is to use the equivalent linear combination when combining the modal responses. This article explains this option by describing an example.
In order to consider inaccuracies regarding the position of masses in a response spectrum analysis, standards for seismic design specify rules that have to be applied in both the simplified and multi-modal response spectrum analyses. These rules describe the following general procedure: The story mass must be shifted by a certain eccentricity, which results in a torsional moment.
The National Building Code of Canada (NBC) 2015 Article 4.1.8.7 provides a clear procedure for earthquake methods of analysis. The more advanced method, the Dynamic Analysis Procedure in Article 4.1.8.12, should be used for all structure types except those that meet the criteria set forth in 4.1.8.7. The more simplistic method, the Equivalent Static Force Procedure (ESFP) in Article 4.1.8.11, can be used for all other structures.
Modal analysis is the starting point for the dynamic analysis of structural systems. You can use it to determine natural vibration values such as natural frequencies, mode shapes, modal masses, and effective modal mass factors. This outcome can be used for vibration design, and it can be used for further dynamic analyses (for example, loading by a response spectrum).
Seismic Analysis in RFEM 6 is possible using the modal analysis and the response spectrum analysis add-ons. As a matter of fact, the general concept of the earthquake analysis in RFEM 6 is based on the creation of a load case for the modal analysis and the response spectrum analysis, respectively. The standard groups for these analyses are set in the Standards II tab of the model’s Base Data.
The steady state for periodically excited structures can be determined by means of the modal analysis in the DYNAM Pro - Forced Vibrations add-on module. This is an advantage if only the structure's steady state is of interest. Instead of a complete solution of the equation of motion, only a special solution is displayed.
RF-/DYNAM Pro - Equivalent Loads allows you to determine the loads due to equivalent seismic loads according to the multi‑modal response spectrum method. In the example shown here, this was done for a multi‑mass oscillator.