Peter Fajfar

Capacity spectrum method based on inelastic demand spectra

By means of a graphical procedure, the capacity spectrum method compares the capacity of a structure with the demands of earthquake ground motion on it. In the present version of the method, highly damped elastic spectra have been used to determine seismic demand. A more straightforward approach for the determination of seismic demand is based on the use of the inelastic strength and displacement spectra which can be obtained directly by time-history analyses of inelastic SDOF systems, or indirectly from elastic spectra. The advantages of the two approaches (i.e. the visual representation of the capacity spectrum method and the superior physical basis of inelastic demand spectra) can be combined. In this paper, the idea of using inelastic demand spectra within the capacity spectrum method has been elaborated and is presented in an easy to use format. The approach represents the so-called N2 method formulated in the format of the capacity spectrum method. By reversing the procedure, a direct displacement-based design can be performed. The application of the modified capacity spectrum method is illustrated by means of two examples. Copyright

The N2 method for the seismic damage analysis of RC buildings

A comprehensive, though relatively simple, non-linear method for the seismic damage analysis of reinforced concrete buildings (the N2 method) has been elaborated. The basic features of the method are: the use of two separate mathematical models, application of the response spectrum approach and of the non-linear static analysis, and the choice of a damage model which includes cumulative damage. The method yields results of reasonable accuracy provided that the structure oscillates predominantly in the first mode. Three variants of a seven-storey building have been used as illustrative examples for the application of the method. Four different types of the analysis, with different degrees of sophistication, have been performed in order to estimate the influence of several assumptions and approximations used in the N2 method.

TORSIONAL EFFECTS IN THE PUSHOVER-BASED SEISMIC ANALYSIS OF BUILDINGS

The general trends of the inelastic behaviour of plan-asymmetric structures have been studied. Systems with structural elements in both orthogonal directions and bi-axial eccentricity were subjected to bi-directional excitation. Test examples include idealised single-storey and multi-storey models, and a three-storey building, for which test results are available. The response in terms of displacements was determined by nonlinear dynamic analyses. The main findings, limited to fairly regular and simple investigated buildings, are: (a) The amplification of displacements determined by elastic dynamic analysis can be used as a rough, and in the majority of cases conservative estimate in the inelastic range, (b) Any favourable torsional effect on the stiff side, which may arise from elastic analysis, may disappear in the inelastic range. These findings can be utilised in the approximate pushover-based seismic analysis of asymmetric buildings, e.g. in the N2 method. It is proposed that the results obtained by pushover analysis of a 3D structural model be combined with the results of a linear dynamic (spectral) analysis. The former results control the target displacements and the distribution of deformations along the height of the building, whereas the latter results define the torsional amplifications. The proposed approach is partly illustrated and evaluated by test examples.

SOFT STOREY EFFECTS IN UNIFORMLY INFILLED REINFORCED CONCRETE FRAMES

A large number of multi-storey reinforced concrete frame buildings with masonry infill walls, which were uniformly distributed over the height of the building, collapsed in the 1999 Kocaeli (Turkey) earthquake, due to complete failure of the first storey or the bottom two stories. In the paper it is demonstrated that a soft storey mechanism is formed in such structural systems if the intensity of ground motion is above a certain level. It is likely that collapse will occur if the global ductilities of the bare frames, as well as the ductilities of the structural elements, are low, and if the infill walls are relatively weak and brittle.

The extended N2 method considering higher mode effects in both plan and elevation

The extended N2 method has been developed, which takes into account higher mode effects both in plan and in elevation. The extension is based on the assumption that the structure remains in the elastic range when vibrating in higher modes. The seismic demand in terms of displacements and storey drift can be obtained by combining the results of basic pushover analysis and those of standard elastic modal analysis. In the paper, the proposed procedure was summarized and applied to a test example which represents an actual 8-storey reinforced concrete building. The results obtained by the extended N2 method were compared with the results of nonlinear response history analysis and basic N2 analysis without the consideration of higher modes. The extended N2 method was able to provide fair, conservative estimates of response in the case of the test example. In comparison to the basic N2 method, the prediction of seismic demand was greatly improved in the upper part of the building and at the flexible edges.

A measure of earthquake motion capacity to damage medium-period structures*

The expression I=v g t D 0.25 is proposed as an instrumental measure of earthquake ground motion capacity to damage structures with fundamental periods in the medium-period (velocity-controlled) region. Only two of the basic ground motion parameters which can be routinely predicted in the design procedure (peak ground velocity and the duration of strong shaking) are included in the formula. Expressions for determining the bounds of the medium-period region are also proposed as a function of the basic ground motion parameters. A total of 40 records having very different characteristics, including extremely short and long duration, were used in the statistical study. The new intensity parameter has been evaluated by using inelastic and elastic relative displacement and input energy spectra.

Simple push-over analysis of asymmetric buildings

A simple method for the non-linear static analysis of complex building structures subjected to monotonically increasing horizontal loading (push-over analysis) is presented. The method is designed to be a part of new methodologies for the seismic design and evaluation of structures. It is based on the extension of a pseudo-three-dimensional mathematical model of a building structure into the non-linear range. The structure consists of planar macroelements. For each planar macroelement, a simple bilinear or multilinear base shear-top displacement relationship is assumed. By a step-by-step analysis an approximate relationship between the global base shear and top displacement is computed. During the analysis the development of plastic hinges throughout the building can be monitored. The method has been implemented into a prototype computer program. In the paper the mathematicalmodel, the base shear-top displacement relationships for different types of macroelements, and the step-by-step computational procedure are described. The method has been applied for the analysis of a symmetric and an asymmetric variant of a seven-storey reinforced concrete frame-wall building, as well as for the analysis of a complex asymmetric 21-storey reinforced concrete wall building. The influence of torsion on structural behaviour is discussed.

A PROCEDURE FOR ESTIMATING INPUT ENERGY SPECTRA FOR SEISMIC DESIGN

In this paper, the damage potential of an earthquake ground motion is evaluated in terms of the total power of the acceleration of the ground motion. By assuming an appropriate spectral shape for the input energy spectrum, and using the well-known Parseval theorem for evaluating the total power of a random signal, the peak amplification factor for the equivalent input energy velocity spectrum can be determined. It is shown that the peak amplification factor for the input energy spectrum depends on the peak-ground-acceleration to peak-ground-velocity ratio and duration of the strong motion phase of the ground motion. Values for the equivalent input energy velocity amplification factor vary from about 2 to 10 for most of the recorded ground motions used in this study. Although a considerable scatter of data is observed in this study, the peak amplification factor predicted by the Fourier amplitude spectrum of the ground acceleration provides a fairly good estimate of the mean value of the peak input energy compared to that determined from inelastic dynamic time history analyses, particularly for systems with high damping and low lateral strength. The peak amplification factor derived in this paper provides a more consistent approach for estimation of seismic demand when compared to an earlier empirical expression used for the formulation of duration-dependent inelastic seismic design spectra, even though only a slight difference in the required lateral strength results from the use of the new formula.

A method for the direct determination of approximate floor response spectra for SDOF inelastic structures

Floor response spectra, which are used for the seismic design of equipment, are often based on the assumption that the behaviour of a structure and its equipment is linearly elastic. Significant reductions in the peak values of floor acceleration spectra can be achieved if inelastic behaviour of the structure is taken into account. This paper presents the most important results of an extensive parametric study of floor acceleration spectra, taking into account inelastic behaviour of the structure, and linear elastic behaviour of the equipment. The structure and the equipment were modelled as single-degree-of-freedom systems. The influences of the input ground motion, ductility, hysteretic behaviour and the natural period of the structure, as well as that of damping of the equipment, have been studied. A simple practice-oriented method for direct determination of floor acceleration spectra from an inelastic spectrum for the structure and an elastic spectrum for the equipment is proposed and validated. In this method, the floor response spectra in the resonance region, where the natural period of the equipment is close to the natural period of the structure, are based on the empirical values obtained in the parametric study, whereas the spectra in the pre- and post-resonance regions are based on the principles of dynamics of structures. The method is intended for a quick estimation of approximate floor acceleration spectra.

A NON-PARAMETRIC APPROACH TO ATTENUATION RELATIONS

Abstract A non-parametric multidimensional regression method is proposed for the prediction of seismic ground motion parameters. The main features which distinguish the method from standard regression procedures are: (1) The relationship between the input and output variables is not selected a priori by a prediction law, (2) an arbitrary number of input variables Can be taken into account, provided that an appropriate data base exists, and (3) the computational procedure is very simple. The results can be easily updated when new information becomes available. The method has been applied for the derivation of attenuation relations by using a combination of databases compiled by other researchers. In the majority of the cases discussed in this paper, the method was used for the prediction of horizontal peak ground acceleration as a function of magnitude and distance. In some cases, ground conditions were also taken into account. Some results on the attenuation relations of peak ground velocity and displacem...