effectiveness of the n2 method for the seismic analysis of … · 2017-04-13 · effectiveness of...

effecTiveness of The n2 meThod for The seismic AnAlysis of sTrucTures wiTh differenT hysTereTic behAviourG. rinaldin1, c. Amadio2, m. fragiacomo1

1Department of Architecture, Design and Urban Planning, University of Sassari, Alghero, Italy2Department of Engineering and Architecture, University of Trieste, Italy

Introduction. The seismic design is usually carried out via linear or non-linear static analyses using design spectra, which are obtained on the basis of ductility considerations and/or equivalent damping of the structure. The introduction of a behaviour factor or an equivalent damping ratio is necessary to allow the elastic design of structures, taking into account the dissipative capacity of the structure due to the seismic actions. This factor reduces the elastic spectrum to a design spectrum, which can be used to find the design acceleration to be applied to the analysed building. The most common methods adopted in seismic engineering are: (i) the N2 method (Fajfar, 1999, 2000), which reduces the elastic spectrum by a behaviour factor q for the verification of the structure in a linear or non-linear analysis (this method was adopted by Eurocode 8 [CEN 2003]); and (ii) the overdamped spectrum method (Freeman, 1978) [that was adopted by the US code ATC-40 (ATC, 1996)], which is based on the use of an elastic spectrum calculated for an appropriate equivalent damping.

In a pushover analysis, both methods compare the capacity of a structure with the demand of an earthquake ground motion. From this type of non-linear static analysis, the capacity of the structure is represented by a force-displacement curve, calculated on an equivalent SDOF system, while the demand curve is represented by the elastic spectra reduced by q for the N2 method or by the overdamped elastic spectra. While the overdamped spectrum method is directly related to the shape of the hysteretic response, which depends on the hysteretic behaviour of the structure and the material, the N2 method works on a procedure based on ductility. A study conducted on the former method (Krawinkler, 1994) identified a fundamental flow: the relationship between the hysteretic energy dissipation at the maximum displacement and the equivalent viscous damping is not justified. In order to clarify and validate the N2 and overdamped spectrum methods, in this work inelastic spectra are calculated via non-linear time-history analyses and compared with the ones provided by the two methodologies.

N2 method. The behaviour factor q in this method is calculated using the equations (Fajfar, 2000):

(1)

(2)

where µ=δu/δy is the ductility value, T is the natural vibration period of the structure, and TC the period value at the end of the constant acceleration part of the elastic spectrum.

Overdamped spectrum method. For the overdamped spectrum method, the equivalent damping ratio, in the hypothesis of an equivalent viscous elastic behaviour of a SDOF system in a harmonic motion, depends upon the energy dissipated in a cycle ED and the elastic deformation energy in a half cycle ES0 as reported in Eq. (3).

(3)

A previous investigation (Fragiacomo et al., 2006) demonstrated that the use of the N2 method leads to the best accuracy for elasto-plastic SDOF systems, with and without stiffness degradation and with bilinear or Clough hysteretic behaviour. In this work, different hysteretic

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models are employed, namely flag-shape behaviour with different energy dissipation, and slip-type behaviour. The former hysteretic model is typical of masonry slender structures (Amadio et al., 2011), whilst the latter one represents in general all the wooden structures with metal fasteners (Rinaldin et al., 2013). The aim is to evaluate whether the aforementioned methods proposed by current seismic design codes are accurate also for these hysteretic models and different dissipative capacity of the structure.

Inelastic spectra. To calculate the inelastic spectra, 15 Italian recorded earthquake ground motions have been considered. They are consistent with the Eurocode 8 (EC8) spectrum with their average value.

The adopted EC8 elastic spectrum has been calculated for a ground with the characteristics presented in Tab. 1, while the complete list of employed seismic records is reported in Tab. 2.

Tab. 1 – Characteristics of elastic spectrum according to EC8.

Ground type S β0 k1 k2TB[s]

TC[s]

TD[s]

A 1.0 2.5 1.0 2.0 0.15 0.40 2.0

Tab. 2 – Seismic record description.

Code Record Date Component ag Sd,max Sa,max

[g] [cm] [g]

01 0032 Codroipo 06/05/76 N-S 0.066 8.3 0.21702 0038 Tolmezzo 06/05/76 N-S 0.366 7.5 1.060

03 0143 Buia 11/09/76 E-W 0.110 3.5 0.271

04 0143 Buia 11/09/76 N-S 0.234 7.3 0.639

05 0152 Forgaria C. 15/09/76 E-W 0.218 7.0 0.858

06 0153 S. Rocco 15/09/76 E-W 0.135 7.7 0.513

07 0156 Buia 15/09/76 E-W 0.094 5.1 0.287

08 0156 Buia 15/09/76 N-S 0.109 9.4 0.327

09 0168 Forgaria C. 15/09/76 N-S 0.352 7.8 1.058

10 0169 S. Rocco 15/09/76 E-W 0.251 9.9 0.679

11 0169 S. Rocco 15/09/76 N-S 0.131 7.3 0.315

12 0301 Patti 15/04/78 N-S 0.071 2.6 0.271

13 0302 Naso 15/04/78 E-W 0.132 2.1 0.503

14 0350 Cascia 19/09/79 E-W 0.210 7.0 0.59615 0636 Calitri 23/11/80 E-W 0.175 18.6 0.595

Hysteretic models. Three hysteresis models were used: the former one presents a fully dissipative cycle (Fig. 1a), the second one is a flag-shape model (Fig. 1a) and the latter one represents slip-type systems (Fig. 1b).

The flag-shape model was studied with three cycle amplitudes, characterised by three different CF coefficients (Fig. 1a), as reported in Eq. (4).

(4)

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The coefficients CF were chosen with the aim to investigate systems with different energy dissipation, where the case with represents a non-linear elastic system, namely a system that practically does not dissipate energy.

The case with represents a poor dissipative structural behaviour [slender masonry panels subjected to rocking collapse mechanism (Amadio et al., 2011)], while the case with represents a dissipative capacity intermediate from the previous case and the full dissipative one. This last case is typical of timber frames with hybrid beam-column and wall-foundation joints where unbonded prestressed tendons are used to re-centre the structure at the end of an earthquake event in this way minimizing residual damage, and energy dissipaters are placed in parallel to provide dissipation (Priestley et al., 1999). Finally, the slip-type model (Fig. 1b) is typical of timber structures, where significant pinching effect due to the plasticization of metal fasteners and crushing of the timber at the interface with the fastener occurs (Foliente et al., 1995).

Adopted software. Each inelastic spectrum was calculated with a software written in Fortran language and purposely developed for a SDOF system. Three versions of this program have been developed, the first for the fully dissipative behaviour, the second one for the flag-shape hysteretic behaviour, and the third one for the slip-type systems. As input data, the software requires a seismic record and the desired ductility value of the SDOF systems. Acceleration, velocity, displacement and energy spectra are returned as output. The program uses the bisection method to evaluate iteratively the yielding force Fy of the SDOF in order to match the desired ductility value µ provided by the user as an input.

The software carries out a time-history non-linear dynamic analysis with the seismic record given as input. The program uses the well-known Newmark integration method (Chopra, 2011). The considered SDOF system has unitary mass; in this way the stiffness can be easily calculated for each considered natural vibration period [Eq. (5)].

(5)

The assumed damping ratio is 0.05 (5%) for all cases.Results for SDOF systems. Results were mainly obtained in terms of acceleration and

displacement spectra. The ductility levels investigated are the typical values of 2 (low), 4 (medium) and 6 (high). The spectra obtained from the fully dissipative system are denoted with “EP”, the ones calculated with the flag-shape hysteretic models with “FS” and the ones for the slip-type model with “ST”. The EC8 elastic spectra are compared with the inelastic spectra

Fig. 1 – Adopted hysteretic models: “fully-dissipative” and flag-shape (left, a), and slip-type (right, b).

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obtained by applying the reduction factor q proposed by Fajfar in the N2 approach, and with the overdamped spectra, computed with a proper damping factor.

The equivalent damping ratio to be used in the overdamped spectrum method is reported in Eq. (6).

(6)

According to Eq. 6, the damping ratios related to the plastic energy dissipated, are given by Eqs. 7a,b,c for the different cases investigated:

(7a)

(7b)

(7c)

Eqs. 7a, 7b and 7c are referred to fully dissipative, flag-shape and slip-type behaviour, respectively.

Some alternative formulations to Eqs. 7b and 7c were developed, because they were found to be too conservative. Priestley et al. (2007) proposed the use of the equations provided in the following for the equivalent damping ratio of EP [Eq. (8)] and FS systems with [Eq. (9)]:

(8)

(9)

The following spectra are compared:• The average elastic spectrum of the 15 recorded earthquakes, and the EC8 elastic

spectrum;• The EC8 design spectrum obtained from the elastic EC8 spectrum using the behaviour

factor q, in accordance with the N2 approach [Eqs. (3) and (4)];• The EC8 overdamped spectra, obtained using the reduction factor ,

with in %, evaluated using both Eqs. 7a and 8 for the EP cases, Eqs. 7b or 9 for the FS case;

• The average numerical spectrum obtained by integrating in time the 15 selected earthquake ground motions for a SDOF system characterized by a given ductility value µ.

The results show a good agreement between the N2 design spectrum and the computed inelastic one, for every type of hysteretic cycle and ductility value. On the contrary, the over-damped spectrum method overestimates the inelastic spectrum in all cases. In general, better accuracy is obtained only for low ductility values and great amount of dissipated energy. More-over, this difference is greater for low periods.

For the sake of completeness, also the total input energies have been calculated in the same analyses, including the separate contributions (elastic, hysteretic, damping, and kinetic).

Reported results are referred to the case with µ = 2: Fig. 2a shows the spectra in ADRS format obtained with two hysteretic laws, EP (fully dissipative) and FS with CF = 0.01, which represent the limit conditions for analysed SDOF systems.

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N2 method gives only one inelastic spectrum, and this does not depend on the hysteretic law adopted; in Fig. 2a the calculated inelastic spectrum is close to that curve, while the overdamped spectrum method gives too conservatives acceleration curves. These last are, on the contrary, dependent on the dissipated energy amount. For the ST case, the numerical results have shown an intermediate response between EP and FS systems. In general, the differences of the studied systems can be further investigated with their energy spectra. Figs. 2b and 3a,b depict the energy spectra.

It can be observed that, while for the EP behaviour there is a great percentage of hysteretic energy (Fig. 3a), for the FS behaviour the SDOF moves with higher velocity and the dissipation is mainly due to the 5% damping. The hysteretic energy percentage is quite different: the FS SDOF system dissipates far less hysteretic energy than the EP (Fig. 3a) and ST (Fig. 3b) ones. Conversely, the FS SDOF system shows a great percentage of kinetic and viscous energy to balance the poor dissipative capacity with higher velocity of motion. Obviously, in this case the viscous energy cannot be correctly evaluated considering an equivalent damping ratio determined on the basis of the dissipated energy. In fact, for SDOF systems with the same ductility level, the hysteretic dissipation was found not to markedly affect the structural response, as can be seen on the inelastic spectra reported in Fig. 2a. Instead, the N2 method, which considers the effects of the ductility in all cases, provides a sufficiently accurate evaluation of the structural response.

Conclusions. In the paper the accuracy of the N2 method and of the overdamped spectrum method is investigated through a series of non-linear analyses conducted on SDOF systems. Three different hysteretic laws were considered, and different levels of dissipative capacity were analysed.

It was found that the overdamped spectrum method overestimates in general the acceleration, particularly when the dissipation is limited. Conversely, the design spectrum obtained in accordance with the N2 method, reducing the elastic spectrum by the behaviour factor q on

Fig. 2 – Average ADRS spectra for EP and FS SDOF systems with µ = 2 (a); average energy spectra for FS behaviour with CF = 0.01 (b).

Fig. 3 - Average energy spectra for EP behaviour (a) and for ST behaviour (b) with µ = 2.

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the base of the ductility, gives satisfactory results for inelastic response of a SDOF system, independently of its energy dissipation capacity.

Analysing the energy spectra for each type of hysteretic behaviour, the results indicate that a SDOF system with low dissipation capacity moves faster than a SDOF system with higher dissipation capacity. Furthermore, it was observed that the acceleration of the structure reduces as the ductility increases. The phase where the system moves with reduced stiffness (plastic phase) is therefore of primary importance for the overall dynamic behaviour of the system. In the case of SDOF systems with the same ductility level, the type of hysteretic cycle was found not to markedly affect the structural response, as expected by the N2 method.

As future development, MDOF structures must be investigated to confirm the results obtained from SDOF systems.referencesAmadio C, Rinaldin G, Macorini L.: 2011: An equivalent frame model for nonlinear analysis of unreinforced masonry

buildings under in-plane cyclic loading, Proceedings of ANIDIS conference, Bari, Italy.ATC: 1996: Seismic Evaluation and Retrofit of Concrete Buildings, Products 1.2 and 1.3 of the Proposition 122 Seismic

Retrofit Practices Improvement Program, California Seismic Safety Commission. In: Report No. SSC 96-01, ATC-40, Applied Technology Council, Redwood City, California.

CEN, European Committee for Standardisation TC250/SC8; 2003: Eurocode 8: Design Provisions for Earthquake Resistance of Structures, Part 1.1: General rules, seismic actions and rules for buildings, PrEN1998-1.

Chopra A.K.: 2011: Dynamics of Structures, Theory and Applications to Earthquake Engineering, Fourth Edition, Prentice Hall, 2011, ISBN-10: 0132858037.

Fajfar P.: 1999: Capacity spectrum method based on inelastic spectra. In: Earthquake Engineering and Structural Dynamics, 28, pp. 979-993.

Fajfar, P.: 2000: A Nonlinear Analysis Method for Performance Based Seismic Design, In: Earthquake Spectra, Vol. 16, No. 3, pp.573-592.

Foliente G.C.: 1995: Hysteresis Modeling of Wood Joints and Structural Systems, Journal of Structural Engineering, Vol. 121, No. 6, June 1995, pp. 1013-1022, ASCE. DOI: 10.1061/(ASCE)0733-9445, 121:6(1013).

Fragiacomo M., Amadio C., and Rajgelj S.: 2006: “Evaluation of the structural response under seismic actions using non-linear static methods.” In: Earthquake Engineering & Structural Dynamics, Vol. 35 No. 12, pp. 1511-1531.

Freeman S.A.: 1978: Prediction of Response of Concrete Buildings to Severe Earthquake Motion, In: Douglas McHenry International Symposium on Concrete and Concrete Structures, SP-55, pp. 589-605, American Concrete Institute, Detroit.

Krawinkler H.: 1994: “New trends in seismic design methodology”, Proc. 10th Eur. Conf. Earthquake Engng., Vol. 2, Vienna, Balkema, Rotterdam, Vol. 2, 1995, pp. 821, 830.

Priestley M.J.N., Calvi, M.C., and Kowalsky, M.J.: 2007: Displacement-Based Seismic Design of Structures IUSS Press, Pavia, 670 pp.

Priestley, M.J.N., Sritharan, S., Conley, J. R., Pampanin, S.: 1999: Preliminary results and conclusions from the PRESSS five-story precast concrete test-building. PCI Journal, Vol 44(6), pp. 42-67.

Rinaldin G., Amadio C., Fragiacomo M.: 2013: A component approach for the hysteretic behaviour of connections in cross-laminated wooden structures. In: Earthquake Engineering and Structural Dynamics, Wiley Online Library, DOI: 10.1002/eqe.2310.

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