This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In the last few years, the need to evaluate the seismic performances of buildings on sustaining strong motion has encouraged the development of simplified non-linear static analyses. Several procedures are available today to assess the behavior of plane-frame systems or plan-regular framed buildings suitable for engineering purposes. Less accurate procedures are instead available for irregular structures. This study introduces new tools to assess the seismic performance of irregular structures by using capacity domains and polar spectra. In particular, the capacity domains, plotted in terms of base shear and node control displacements and obtained by means of static non-linear analyses, lead to the evaluation of the direction of least seismic capacity of the investigated structure. The polar spectrum, instead, leads to taking into account the directivity and site effects of seismic events. In particular, the polar spectrum represents the spectral seismic response evaluated for different in-plan directions.

As known, the standard analysis for traditional seismic design is Linear Dynamic Procedure (LDP) with assigned response spectrum (Response Spectrum Analysis). More complex procedures, such as non-linear time history analysis (NDP), are seldom used because of the difficulties in the definition of an accurate hysteretic model to describe the behaviour of the materials under cyclic actions and the choice of a set of accelerograms that describes the real site conditions. Moreover, as it is well-known, NDP analyses present convergence difficulties and require major computational effort.

Non-linear static analysis is instead becoming the key method to evaluate the seismic response of existing structures. This method, which is able to evaluate the collapse mechanism of structures, is suitable for the analysis of existing structures, including the ones not explicitly designed for seismic actions.

However, non-linear static procedures can lead to unsuitable results when applied to irregular structures because of the difficulties in taking into account dynamic lateral-torsional effects and higher mode effects. As it is known, an asymmetric distribution of mass and stiffness, or of strength in plan, leads to high ductility demand for the elements near the soft or weak edge [

This report presents and discusses the main results of the new methodology to assess the seismic response of irregular structures by means of capacity domains which allow us to evaluate the direction of least seismic capacity of the investigated buildings and the polar spectrum which represents the seismic demand in the plan. In particular, the capacity domains, represented in terms of strength and displacement capacities, are evaluated using traditional pushover analyses by applying load distributions, triangular and uniform, at different angles in the plan to search for the least seismic-resistant direction. The polar spectrum, instead, is a new spectral representation which is able to investigate the seismic demand of each in-plane direction. In particular, this tool is based on the spatial spectral surface obtained by the spectral seismic response evaluated for different in-plan directions and the in-plan projection is defined “Polar Spectrum”.

Both polar spectrum and capacity domains allow researchers to investigate the non-linear seismic capacities of irregular structures. As is well-known, indeed, the collapse conditions for a three-dimensional system are also governed by the spatial features of the seismic event. Therefore, the goal of the presented results is to investigate the seismic behavior of irregular structures by considering the least seismic-resistant directions of the investigated structure and the spatial features of the seismic event.

For the scope of this report the benchmark structure proposed in the ReLUIS project (Earthquake Engineering Test Labs Network) is considered [

The structure, designed without seismic actions, is a five-storey L-shaped building with RC moment frames in two orthogonal directions, as represented in

Benchmark structure plan.

A finite-element model has been implemented in Opensees software [

The collapse condition for the generic section is defined in terms of section curvature, evaluated by imposing limit strain values for the concrete fibers (ε_{c} = 0.006 in compression) and for the reinforced steel fibers (ε_{s} = 0.03 in tension).

The main dynamical properties of the model are reported in

Modal properties for the Benchmark Structure.

Mode | T(s) | M%x | M%y | SumX | SumY |
---|---|---|---|---|---|

1 | 1.35 | 10.9% | 52.1% | 10.9% | 52.1% |

2 | 1.22 | 56.9% | 19.6% | 67.8% | 71.8% |

3 | 0.99 | 10.8% | 9.2% | 78.6% | 80.9% |

4 | 0.42 | 0.9% | 8.5% | 79.4% | 89.4% |

5 | 0.38 | 9.4% | 1.9% | 88.8% | 91.2% |

6 | 0.31 | 1.5% | 1.1% | 90.3% | 92.4% |

7 | 0.23 | 0.1% | 4.1% | 90.4% | 96.5% |

8 | 0.20 | 4.2% | 0.5% | 94.6% | 97.0% |

9 | 0.17 | 0.9% | 0.5% | 95.5% | 97.5% |

The modal analysis shows a flexible behavior and, moreover, the plan irregularity leads to latero-torsional behaviour for modes 1 and 2.

In order to assess the non-linear static response of irregular structures it is possible to obtain capacity domains, in terms of control node displacement or base shear. In particular, such domains are obtained from the results of non-linear static procedures (pushover) carried out by varying the in-plan direction of the applied load distribution. The domains, represented in the following, depict the displacement of the control node or the base shear up to collapse.

The analyses are carried out by applying two load distributions along the height of the structure. The first one is a triangular distribution, taking into consideration the inertial masses and the height above the ground. The second one considers only the floor masses. The forces are applied at beam-column joints, proportionally to the joints mass.

For each applied load distribution, 24 capacity curves have been obtained by rotating the direction of action in the x-y plane, 15° at each step. The control node is assumed to be the centre of mass on the top floor in all the pushover analyses carried out.

Using the described limit values for section curvature, the global collapse is assumed when the first primary element exceeds its limit. Each pushover curve is bi-linearized according to the FEMA 356 procedure [

The capacity domains are compared to the non-linear dynamic response to recorded earthquakes, selected among those proposed by ReLUIS [

The main results, for the considered accelerogram, are represented in the following. In particular,

In order to evaluate the applicability of the capacity domains to accurately describe the collapse state of a structure, incremental dynamic analyses have been performed. For both the considered in-plan dispositions of the structures, 0° and 90°, the analyses have been carried out considering the Iceland 2000 earthquake (code 06334, maximum PGA equal to 7.07 m/s2). The

(

(

(

As it is apparent, the collapse is attained in the direction of about 255° independently of plan orientation of benchmark structure. Moreover, the collapse occurs in correspondence with the capacity domain for triangular load distribution, and the collapsed element, column b-1 (

As a further comparison between static and dynamic responses,

Results show that the collapse mechanism is a global one, being the displacement path carried out by NDP well described by the one obtained by NSP with triangular load distribution. Moreover, as is apparent, the collapse as well as when read with the NDP procedure always intercepts the capacity domains corresponding to the triangular load distribution (

(

Comparison between Incremental Dynamical Analysis and static non linear analyses-In plan orientation of benchmark structure 0°.

In particular, the figures show the comparison between pushover curves obtained by static non linear analyses in the least resistance direction considering the uniform and triangular load distributions and the pushover curves obtained by incremental dynamic analyses considering several criteria for the graphical representation. Regarding the representation of the time history data, several proposals can be found in the literature [

The results (

Comparison between Incremental Dynamical Analysis and static non linear analyses-In plan orientation of benchmark structure 90°.

As is known, seismic events generally present strong directivity effects due to site geo-morphological conditions, the source mechanism and the path from the source to the site of the seismic waves.

To take into account the directivity and site effects it is possible to consider the polar spectrum, a new tool, proposed by the authors, which is able to investigate each in-plan direction. In particular, this tool is based on the spatial spectral surface obtained by the spectral seismic response evaluated for different in-plan directions. The in-plan projection of this surface is defined “Polar Spectrum”.

Polar Spectrum in terms of pseudo-acceleration for the South Iceland aftershock (code 006334).

The analysis of the polar spectrum shows that the considered seismic event presents a peak demand along the direction (75°–255°) regardless of the vibration period value.

The comparison between the polar spectrum of the considered seismic event and the results of Non-Linear Dynamic Procedure (NDP) is shown clearly, as the collapse of the benchmark structure is controlled in this case by the directivity effects more than by the seismic capacity of the analyzed structure.

The possibility of evaluating the worst structural response moving from observation of the polar spectrum suggests performing dynamic analyses with the structure oriented so that its least capacity direction (assessed by means of the capacity domains) is aligned with the maximum seismic demand (evaluated by means of the polar spectrum).

Dynamic analyses results, in plan orientation of the benchmark structure 105°, NDP collapse amplification factor 0.30.

The obtained results show that when the least seismic resistance direction is aligned with the maximum seismic demand, the minimum collapse amplification factor is reached.

In the field of seismic engineering, an existing open issue is the need to define a feasible procedure to evaluate the seismic performance of irregular structures. In this article, new tools, devised in order to assess the non-linear behaviour of an irregular building and the spectral seismic demand in the plan, have been presented. In particular, capacity domains are evaluated by means of pushover analyses by varying the load distribution in the plan and lead to the search for less seismic resistant directions. The polar spectra lead to full assessment of the spatial features of the seismic demand and represent an opportunity to investigate thoroughly the seismic demand and the event characteristics, being able to clearly evaluate directivity and site effects.

The results, for the investigated structure and the considered earthquake, show that the collapse condition, assessed by means of non linear dynamic procedures, turns out to be coherent with the capacity domains based on the results of traditional non linear procedures carried out by varying the angle of the application of the load distribution.

The non-linear dynamic analyses have shown that the directional components of the seismic event govern the collapse direction. Therefore, the worst condition for an investigated structure is obtained by aligning its least resistance capacity direction, evaluated on the basis of the capacity domains, with the maximum response demand direction identified by the polar spectrum.

Moreover, the study proposes a new criterion to compare incremental dynamic and static non linear analysis which seems to supply the best results for all investigated cases.