Online Manuals RFEM 6 / RSTAB 9 | Dynamic Analysis

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2023-11-14

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Overview

Add-ons for Dynamic Analysis

Harmonic Response Analysis

Pushover Analysis

Documentation

Spectral Analysis Settings

A spectral analysis setting (SLS) specifies the rules according to which the resulting forces and deformations are calculated. A standard analysis type is preset. You can adjust this type or create further spectral analysis settings at any time.

Dialog Box "New Spectrum Analysis Settings"

The "Spectral Analysis Settings" dialog box controls the settings of the analysis type selected in the "List" on the left.

Modal Combination Method

In this section, you can specify the method by which the results of the modal analysis are superimposed, as well as which results are prepared for output.

Combination Rule for Periodic Responses

The list provides three options for combining the results from various mode shapes of the model. The modal results of a response spectrum analysis are usually superimposed quadratically. This superposition is the first step of the dynamic combination.

Selecting Combination Rule for Periodic Responses

SRSS – Square Root of Sum of Squares

The standard form of the SRSS rule combines the maximum results quadratically. The algebraic signs are lost in the process. The combination rule is as follows:

E_{SRSS} = \sqrt{E_{1}^{2} + E_{2}^{2} + . . . + E_{p}^{2}}

Standard SRSS Rule

The combined results E_SRSS result from the modal responses E_p from p mode shapes of the model.

The SRSS rule is only allowed for models where the neighboring natural frequencies T_i < T_j deviate by more than 10%. Therefore, the following condition must be met:

\frac{T_{i}}{T_{j}} \leq 0.90

T_i, T_j

Adjacent mode shapes

Condition for Application of SRSS Rule

If this is not the case, the CQC rule must be applied.

CQC – Complete Quadratic Combination

The CQC rule is defined as follows:

E_{CQC} = \sqrt{\sum_{i = 1}^{p} \sum_{j = 1}^{p} E_{i} \cdot ε_{ij} \cdot E_{j}}

ε_ij	Correlation coefficient

Standard CQC Rule

Using the correlation coefficient ε_ij, the damping and the angular frequencies are determined:

ε_{ij} = \frac{8 \cdot \sqrt{D_{i} \cdot D_{j}} \cdot (D_{i} + r \cdot D_{j}) \cdot r^{\frac{3}{2}}}{{(1 - r^{2})}^{2} + 4 \cdot D_{i} \cdot D_{j} \cdot r \cdot (1 + r^{2}) + 4 \cdot ({D_{i}}^{2} + {D_{j}}^{2}) \cdot r^{2}}

D_i, D_j	Damping values
r	Quotient of the natural angular frequencies (ω_j/ ω_i)

Correlation Coefficient (CQC Rule)

You can enter the damping values D_i in the Selection of Modes tab of the "Load Cases and Combinations" dialog box in the Damping column (see also the image Specifying Damping for CQC Rule).

If the viscous damping value D is the same for all mode shapes, the correlation coefficient ε is simplified as follows:

ε_{ij} = \frac{8 \cdot D^{2} \cdot (1 + r) \cdot r^{\frac{3}{2}}}{{(1 - r^{2})}^{2} + 4 \cdot D^{2} \cdot r \cdot {(1 + r)}^{2}}

D	Damping value
r	Quotient of the natural angular frequencies (ω_j/ ω_i)

Correlation Coefficient (CQC Rule, Simplified: D_i = D_j)

In this case, enter the damping value D directly in the Damping for CQC Rule section of the dialog box.

Absolute Sum

It is also possible to superimpose modal combinations conservatively by using the absolute sum. The absolute sum is obtained as follows:

E_{AbsSum} = \sum_{i = 1}^{p} |E_{i}|

Absolute Sum (Modal Superposition)

Tip

For more information and mathematical derivations of the combination rules, see Gupta [1], Wilson [2], and Der Kiureghian [3].

Using Equivalent Linear Combination

In the case of the quadratic combination according to the SRSS or CQC rule, the direction of the excitation and thus the sign of the result are lost. Therefore, the results are always given as maximum values in the positive as well as in the negative direction. Corresponding internal forces, such as a corresponding moment at the maximum axial force, are lost. This can be avoided by modifying the SRSS and CQC rule: The formulas are not written as a root, but as a linear combination. This rule was introduced by Katz [4].

The equivalent linear combination for the superposition according to the SRSS rule is as follows:

E_{SRSS} = \sum_{i = 1}^{p} f_{i} \cdot E_{i} where f_{i} = \frac{E_{i}}{\sqrt{\sum_{j = 1}^{p} E_{j}^{2}}}

Modified SRSS Rule – Equivalent Linear Combination

The equivalent linear combination for the superposition according to the CQC rule is as follows:

E_{CQC} = \sum_{i = 1}^{p} f_{i} \cdot E_{i} where f_{i} = \frac{\sum_{j = 1}^{p} ε_{ij} \cdot E_{j}}{\sqrt{\sum_{i = 1}^{p} \sum_{j = 1}^{p} E_{i} \cdot ε_{ij} \cdot E_{j}}}

Modified CQC Rule – Equivalent Linear Combination

Detailed information about the superposition of modal responses in the response spectrum analysis using the equivalent linear combination can be found in Knowledge Base 001823.

Signed Results Using Dominant Mode

The function is still under development and therefore not accessible.

Save Results of All Selected Modes

By default, the program only shows the results of the envelope of the directional components or the results of the individual directions X, Y, and Z that have a Response Spectrum assigned. If you activate the "Save results of all selected modes" check box, the intermediate results of all selected mode shapes (see Selection of Modes) will also be output for the individual directions X, Y, and Z, to which you have assigned a response spectrum.

Combination of Directional Components

In this section, you can specify how to combine the results from different excitation directions. The combination of directional components is the second step of dynamic combinations.

Selecting Combination Rule for Directional Components

SRSS (Square)

The internal forces from different excitation directions can be combined quadratically according to the SRSS rule. It is applied with i = 1 ... p for the excitation directions X, Y, and Z. The SRSS rule for the combination of directions can also be performed as an equivalent linear combination.

Info

The SRSS combination option is only available if deactivating the Signed results using dominant mode option.

Scaled Sum (100%/30% or 100%/40%)

The combination of the directional components can also be performed as a user-defined scaled sum, as specified in EN 1998‑1 [5] 4.3.3.5.1(3), for example. Define the percentage value in the text box. With the 100%/30% rule, the combination is as follows:

E_{Ed} = 1, 0 \cdot E_{EdX} \oplus 0, 3 \cdot E_{EdY} \oplus 0, 3 \cdot E_{EdZ}

E_{Ed} = 0, 3 \cdot E_{EdX} \oplus 1, 0 \cdot E_{EdY} \oplus 0, 3 \cdot E_{EdZ}

E_{Ed} = 0, 3 \cdot E_{EdX} \oplus 0, 3 \cdot E_{EdY} \oplus 1, 0 \cdot E_{EdZ}

100/30 Rule

Absolute Sum

You can also combine the directional components conservatively with the absolute values. The absolute sum is obtained as follows:

E_{Ed} = 1, 0 \cdot E_{EdX} \oplus 1, 0 \cdot E_{EdY} \oplus 1, 0 \cdot E_{EdZ}

Absolute Sum (Combination of Directions)

Considering Independent Directions in Envelope Results

The function is still under development and therefore not accessible.

Damping for CQC Rule

This dialog section is available if you have specified a CQC rule for the superposition of the modal analysis results.

Defining Damping for CQC Rule

The Lehr's damping values D_i are required for the CQC rule. They can be specified as equal (constant) or defined differently for each mode shape of the model.

Constant for Each Mode

If the damping is the same for all mode shapes, you can specify the damping value D here.

Different for Each Mode

You can enter the damping values D_i required for the calculation of the correlation coefficient ε_ij in the Selection of Modes tab of the "Load Cases & Combinations" dialog box in the Damping column.

Setting Different Damping for Each Mode

References

Gupta, A.K. (1992). Response Spectrum Method in Seismic Analysis and Design of Structures. CRC Press.
Wilson, E. L., Der Kiureghian, A., & Bayo, E. P. (1981). A replacement for the SRSS method in seismic analysis. Earthquake Engineering & Structural Dynamics, 9(2), 187–192.
Kiureghian, A. D. (1981). A response spectrum method for random vibration analysis of MDF Systems. Earthquake Engineering & Structural Dynamics, 9(5), 419–435.
Katz, C. (2009). Anmerkung zur Überlagerung von Antwortspektren. D-A-CH Mitteilungsblatt.
EN 1998‑1. (2013). Eurocode 8: Design of Structures for Earthquake Resistance – Part 1: General Rules, Seismic Actions and Rules for Buildings; EN 1998‑1:2004/A1:2013.

Parent Chapter

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