Code-oriented floor acceleration spectra for building structures

Abstract

In the seismic design and assessment of acceleration-sensitive equipment installed in buildings, floor acceleration spectra, which are based on an uncoupled analysis of the structure and the equipment, are usually used. However, in order to obtain an “accurate” determination of floor spectra, a complex and quite demanding dynamic response-history analysis is needed. Recently a method for the direct generation of floor acceleration spectra from ground motion spectra, taking into account the dynamic properties of the structure, has been developed and validated. It is based on the theory of structural dynamics, in combination with empirically determined values for the amplification factors in the resonance region. The method can be used for both elastic and inelastic multi-degree-of-freedom structures and equipment modelled as an elastic or inelastic single-degree-of-freedom oscillator. In the case of inelastic primary structures, the method is coupled with the pushover-based N2 method. The variant of the method which is presented in this note is intended for practical applications, e.g. for implementation in guidelines and codes, and it represents a simplified version of the original method. In addition to some simplifications, the option of taking into account the inelastic response of the equipment was added. In the note, the method is summarized, and all the formulae needed for the calculation of floor acceleration spectra are provided. A description of all steps of the analysis, together with all the relevant numerical data, is presented in a test example.

Appendices

Appendix 1

In this appendix selected results of a very recent unpublished study by the authors are presented, which justify the preliminary proposal for taking into account the inelastic behaviour of the equipment (Sect. 2.2.3). The study was performed in a similar way as the study reported by Vukobratović and Fajfar (2015). The floor acceleration spectra in Figs. 6 and 7 represent mean values of spectra obtained by response-history analysis of an elastic and inelastic SDOF primary structure with the period T _p = 0.3 s and with 5% damping. In the case of the inelastic structure, stiffness degrading hysteretic behaviour was used. The ductility demand μ amounted to 2.0. Equipment was modelled as a SDOF system with ideal elasto-plastic behaviour. Thirty ground motions with the mean acceleration spectrum close to the Eurocode 8 (2004) Type 1 spectrum for soil type B normalized to peak ground acceleration PGA = 0.35 g were used for response-history analysis (see Vukobratović and Fajfar 2015). The results in Fig. 6 demonstrate a substantial decrease of floor acceleration spectra (with an exception of rigid equipment) for both elastic and inelastic primary structures. The floor spectra in Fig. 7 show (1) that the beneficial effect of the equipment damping is, in the case of inelastic equipment, relatively small, and (2) that the floor acceleration spectrum proposed in this note (simulating inelastic equipment by elastic equipment with 10% damping) corresponds very well to the more accurate spectrum for equipment ductility μ _s = 1.5 and 1% equipment damping, whereas it is conservative for larger damping and/or larger ductility.

Fig. 6

Floor acceleration spectra for the elastic and inelastic (elasto-plastic) SDOF equipment (ξ _s = 1%) mounted on the elastic and inelastic (stiffness degrading, μ = 2.0) SDOF primary structure (T _p = 0.3 s, ξ _p = 5%)

Fig. 7

Comparison of floor acceleration spectra for the inelastic (elasto-plastic) SDOF equipment (μ _s = 1.5, different damping ratios) with the proposed direct spectra for inelastic equipment. The SDOF primary structure (T _p = 0.3 s, ξ _p = 5%) is elastic and inelastic (stiffness degrading, μ = 2.0)

Appendix 2

In this appendix, the floor acceleration spectra obtained by the proposed direct method are compared with the spectra determined by the formula provided in Eurocode 8 (2004) (denoted as EC8) which, in the notation used in this note, reads

$$A_{\rm s} = \frac{PGA}{{q_{\rm s} }}\left[ {\frac{{3\left( {1 + z/H} \right)}}{{1 + \left( {1 - T_{\rm s} /T_{{\rm p},1} } \right)^{2} }} - 0.5} \right]$$

(6)

where z is the height of the non-structural element (equipment) above the level of application of the seismic action (foundation or top of a rigid basement), H is the building height measured from the foundation or from the top of a rigid basement, and q _s is the behaviour factor of the element (equipment) which amounts to 1.0 or 2.0. Other parameters are defined in the main part of this note. The value of the A _s must not be taken less than PGA/q _s.

The EC8 formula applies to an average case. It does not take into account the differences in the ductility of the primary structure and in the damping of equipment. The higher mode effects and the frequency content of ground motion are also not considered. For these reasons, a comparison with the direct spectra is difficult since it strongly depends on the input data. Nevertheless, in order to illustrate the order of magnitude of differences, two cases are presented.

The comparison for the case of the SDOF primary structure discussed in Appendix 1 is shown in Fig. 8. The direct spectra are determined for ductilities of the primary structure μ = 2.0 and μ = 4.0. Equipment is elastic with 3% damping, and inelastic. The EC8 spectra apply to q _s = 1.0 and q _s = 2.0. The comparison for the SPEAR building is presented in Fig. 9. The direct spectra are shown for μ = 2.4. For equipment, the same parameters as in the case of the SDOF structure are used. The results demonstrate that the differences between the EC8 and proposed direct spectra depend on the period of the equipment. Note that the differences may grossly increase for more extreme cases.

Fig. 8

A comparison of the proposed spectra and EC8 spectra for the SDOF structure from Appendix 1

Fig. 9

A comparison of the proposed spectra and EC8 spectra for the SPEAR building