Determination of Sensitivity Coefficient to Investigate the Need for Second-Order Analysis in Dynamic Analyses

Technical Article on the Topic Structural Analysis Using Dlubal Software

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Technical Article

To evaluate whether it is also necessary to consider the second-order analysis in a dynamic calculation, the sensitivity coefficient of interstory drift θ is provided in EN 1998‑1, Sections 2.2.2 and 4.4.2.2. It can be calculated and analyzed using RFEM 6 and RSTAB 9.
The coefficient θ is calculated as follows:$$\mathrm\theta\;=\;\frac{\displaystyle{\mathrm P}_\mathrm{tot}\;\cdot\;{\mathrm d}_\mathrm r}{{\mathrm V}_\mathrm{tot}\;\cdot\;\mathrm h}\;$$

Sensitivity Coefficient θ

The sensitivity coefficient θ is defined as follows [1]: $$\mathrm\theta\;=\;\frac{\displaystyle{\mathrm P}_\mathrm{tot}\;\cdot\;{\mathrm d}_\mathrm r}{{\mathrm V}_\mathrm{tot}\;\cdot\;\mathrm h}\;$$ where

  • θ = interstory drift sensitivity coefficient
  • Ptot = the total gravity load at and above the story considered in the seismic design situation
  • dr = design interstory drift, evaluated as the difference of the average lateral displacements dS at the top and bottom of the story under consideration; for this, the displacement is determined using the linear design response spectrum with q = 1.0
  • Vtot = total seismic story shear determined using the linear design response spectrum
  • h = interstory height

In the following, the procedure for calculating the sensitivity coefficient is shown using the example of a reinforced concrete building with a main floor and six upper floors.

The Modal Analysis, the Response Spectrum Analysis, and the Building Model add-ons are used for calculating the sensitivity coefficient. After modeling the building, the stories must be defined by means of the Building Model add-on.

Then, a calculation according to the response spectrum analysis is required.

If the input is complete, you can analyze the results after the calculation. You can find important properties for the dynamic analysis in the results table of the spectral analysis. In the "Results by Stories" tab, you can also check the sensitivity coefficient of the individual stories.

After calculating the sensitivity coefficient of the interstory drift, a case distinction regarding the relevance of the second-order analysis is made according to EN 1998‑1, Sections 2.2.2 and 4.4.2.2 [1]:

Case Distinction for Second-Order Analysis

1. Sensitivity coefficient θ ≤ 0.1

The second-order analysis does not have to be considered.

2. Sensitivity coefficient 0.1 < θ ≤ 0.2

The second-order analysis may be taken into account approximately by a factor equal to 1 / (1 − θ).

3. Sensitivity coefficient 0.2 < θ ≤ 0.3

The second-order analysis must be considered directly. You can do this by adjusting the geometric stiffness matrix in RFEM 6 and RSTAB 9. Find detailed information in the following technical article: KB 001867, "Considering Second-Order Analysis in Dynamic Analyses in RFEM 6 and RSTAB 9".

4. Sensitivity coefficient 0.3 < θ

The design of the structure is invalid and requires revision.

Author

Thomas Eichner, M.Sc.

Thomas Eichner, M.Sc.

Product Engineering & Customer Support

Mr. Eichner is responsible for the development of products for dynamic analysis and provides technical support for our customers.

Keywords

Second-Order Analysis Dynamics Dynamic Analysis Dynamic Calculations Response Spectrum Analysis Sensitivity Coefficient P-Δ Effect

Reference

[1]   Eurocode 8: Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings; EN 1998‑1:2004/A1:2013
[2]   Eurocode 0: Basis of structural design: EN 1990:2002
[3]   Werkle, H. (2008). Finite Elemente in der Baustatik (3rd ed.). Wiesbaden: Vieweg & Sohn.

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  • Updated 01/11/2024

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