The ASCE 7-22 Standard [1], Sect. 12.9.1.6 specifies when P-delta effects should be considered when running a modal response spectrum analysis for seismic design. In the NBC 2020 [2], Sent. 4.1.8.3.8.c gives only a short requirement that sway effects due to the interaction of gravity loads with the deformed structure should be considered. Therefore, there may be situations where second-order effects, also known as P-delta, must be considered when carrying out a seismic analysis.
This article presents the basic concepts in structural dynamics and their role in the seismic design of structures. Great emphasis is given to explaining the technical aspects in an understandable way, so that readers without deep technical knowledge can gain an insight into the subject.
The National Building Code of Canada (NBC) 2020 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.
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 earthquakes so far that the design can be performed with reasonable effort. The disadvantage of this method is that a lot of information is lost due to this simplification. One way to moderate this disadvantage is to use the equivalent linear combination when combining the modal responses. This article explains this option by describing an example.
The modal relevance factor is a result of the linear stability analysis and qualitatively describes the degree of participation of individual members in a specific mode shape.
The “Modal Analysis” add-on in RFEM 6 allows you to perform modal analysis of structural systems, thus determining natural vibration values such as natural frequencies, mode shapes, modal masses, and effective modal mass factors. These results can be used for vibration design, as well as for further dynamic analyses (for example, loading by a response spectrum).
The dynamic analysis in RFEM 6 and RSTAB 9 is divided into several add-ons. The Modal Analysis add-on is a prerequisite for all other dynamic add-ons, since it performs the natural vibration analysis for member, surface, and solid models.
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).
In RFEM 6, seismic analysis can be done by using the Modal Analysis and the Response Spectrum Analysis add-ons. Once the spectral analysis has been performed, it is possible to use the Building Model add-on to display story actions, interstory drifts, and forces in shear walls.
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 dialog box for editing load or result combinations is a non-modal dialog box. This means that after you open this dialog box, you can edit the combinations outside the dialog box as well. For manually defining or editing a combination, a separate dialog box can be opened parallel to the "Edit load cases and combinations" dialog box.
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.
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.
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.
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.
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.
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 the activation of 90% of the structural mass. This means: to determine so many eigenvalues that the sum of the effective modal mass factors is greater than 0.9.
In RF-/DYNAM Pro – Equivalent Loads, a signed result option in accordance with the dominant eigenmode has been available since version X.06.3039. For the modal combination of results corresponding to the single eigenvalues, a quadratic combination rule has to be used. In RFEM and RSTAB, the SRSS and the CQC rule are available. Only results, not loads, are allowed to be combined directly. The reason is the mode shapes, which are arbitrarily scaled and signed.
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.
Result combinations exported from RF‑/DYNAM Pro – Equivalent Loads are generated by superimposing the results from the individual modal responses. For this, the SRSS rule can be used as "equivalent linear combination". When result combinations are used in RF‑/STEEL, two options are available for calculating stresses. In the first option, the results from the result combinations are used directly. This is done line by line, for each maximum and minimum controlling internal force. In the second option, stresses are determined from the individual load cases. The quadratic superposition rule is then performed again in RF-/STEEL.
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.
With the latest version of RF‑/DYNAM Pro, you can exclude the mode shapes from the seismic design. In most standards, there are provisions to exclude the effective modal mass factors that are too small. The "Mode Shapes" tab offers the option to specify the factor and to disregard the mode shapes automatically.