EVALUATION OF THE INFLUENCE OF VERTICAL STIFFNESS AND STRENGTH IRREGULARITIES ON THE SEISMIC RESPONSE OF A 9-STOREY STEEL FRAME

A methodology based on Incremental Dynamic Analysis (IDA) is proposed for evaluating the response of structures with single-storey vertical irregularities. Using the well-known 9-storey LA9 steel frame, the objective is to study the effect of the irregular distribution of stiffness and strength along its height. This is achieved by means of IDA, i.e., by performing nonlinear time history analysis for a suite of twenty ground motion records scaled to several intensity levels. The reference and each modified structure are hence forced to show the complete spectrum of behavior from elasticity to final global instability, allowing the estimation of capacities for the full range of limit-states and enabling a straightforward comparison without needing to “tune” the structures to the same fundamental period and/or the yield base shear. Using the bootstrap method confidence intervals are calculated and hypothesis testing is performed for changes in the median and the dispersion of capacity for each limit-state. Thus, it becomes possible to isolate the effect of irregularities from any possible record-to-record variability. In conclusion, the proposed methodology enables a full-range performance evaluation using a highly accurate analysis method that pinpoints the effect of any source of irregularity for every limit-state. INTRODUCTION A large portion of building structures are in some sense vertically irregular. Some buildings have been initially designed so, e.g., in the case of a soft first-storey, or a large entrance lobby. Others have become so by accident, for example due to inconsistencies or even errors during the construction process, while the rest have been rendered irregular during their lifetime because of rehabilitation or change of use. Therefore, it is essential for structural engineers to obtain a better understanding of the seismic response of structures with vertical irregularities. This need has been recognized by current seismic guidelines (e.g. FEMA-356 [1], Eurocode standards [2]). Such guidelines contain a number of criteria in order to identify vertically irregular buildings and determine whether their effect is “significant”. Naturally, the question that arises is when should this effect be considered “significant”? Several researchers have attempted to provide an answer, the most recent and comprehensive efforts including the work of Valmundsson and Nau [3], Al-Ali and Krawinkler [4] and Chintanpakdee and Chopra [5]. Valmundsson and Nau [3] evaluated the applicability of simplified elastic analysis methods suggested by seismic design code procedures on irregular structures. The definitions of seismic design codes for regular and irregular structures for mass, stiffness and strength were examined with respect to the response obtained from inelastic time-history analysis. Al-Ali and Krawinkler [4] followed by Chintanpakdee and Chopra [5] performed systematic investigations on the effect of vertical irregularities. Both studies were based on simple single-bay frames of 10 and 12 stories respectively. Al-Ali and Krawinkler [4] used a strong-beam-weak-column philosophy as opposed to Chintanpakdee and Chopra [5] who adopted a more realistic strong-column-weak-beam philosophy. In both studies time history analyses were performed with bins of 15 and 20 records, respectively. Despite the different approaches used, all efforts reached relatively compatible conclusions. However, several issues were left open. The above mentioned studies focused mostly on the influence of irregularities to the seismic demands rather than capacities. This in fact constitutes a broadband comparison that encompasses several limit-states of the structure, rather than considering each limit-state, or level of structural response separately. Furthermore, in order to enable the comparison of different buildings, both Al-Ali and Krawinkler [4] and Chintanpakdee and Chopra [5] proposed to modify, or to “tune” the alternative designs, when necessary, in order to match the fundamental period and/or the yield base shear of the reference regular frame. This approach is clearly limited to relatively simple structural systems and cannot be extended to the comparison of realistic design alternatives. Finally, the influence of the record-to-record variability was not taken into account. Thus we have no evidence to determine how much of the observed changes can be attributed to the irregularities or to the natural randomness in the seismic loading. To answer these questions we examine the LA9 9-storey steel frame (Figure 1) designed for a Los Angeles site (Foutch and Yun [6]) using Incremental Dynamic Analysis (IDA) (Vamvatsikos and Cornell [7]). The effects of single-storey stiffness and strength vertical irregularities are compared and quantified against the response of the reference regular structure by means of IDA using a suite of twenty ground motion records. This enables us to perform a comparison across the full spectrum of structural performance without needing to “tune” our structures. In order to isolate the effect of irregularities from any possible recordto-record variability and to obtain evidence that show the influence of the modifications to the original structure the bootstrap method [8] is adopted. For each limit-state, confidence intervals are calculated and hypothesis testing is performed for the median and the dispersion of capacity, providing evidence of whether the differences in capacities are due to the vertical irregularities or the randomness in the seismic loading. STRUCTURAL MODELS The structure considered is a 9-storey steel moment resisting frame with a single-storey basement and a fundamental period of sec (Figure 1). It has been designed for a Los 25 . 2 * 1 = T Angeles site, following the 1997 NEHRP (National Earthquake Hazard Reduction Program) provisions (Foutch and Yun [6]). A centerline model with fracturing connections was formed using the OpenSEES [9] platform. It allows for plastic hinge formation at the beam ends while the columns are assumed to remain elastic. The fracturing connections are modelled as rotational springs with 1% strain hardening and a strength drop to 60% of the plastic moment capacity at ten times the yield rotation. Similar behaviour was assumed for both positive and negative moments. Geometric nonlinearities in the form of P-∆ effects were considered while the effect of internal gravity frames has also been incorporated (Figure 1). Figure 1: LA9 frame model. Theoretically, an infinite number of vertically irregular designs can be obtained by selecting different properties and varying their distribution along the height. The irregularities considered are limited to the stiffness and the strength of the 9-storey frame which are modified separately for single stories, along the height of the building. Typical stiffness irregularity cases are the cases of soft stories or when elements of the lateral-force-resisting system such as braces are present on one storey but they are not present on adjacent stories. In many practical cases, strength changes occur together with stiffness, e.g., when the crosssection of a member is changed where both the moment of inertia and the plastic moment capacity are modified, however there are cases where strength only modifications may be encountered for example when changing the steel yield strength without modifying the crosssections. The storey properties are modified by upgrading or degrading the properties of all of the storey’s members, i.e., the beams and the supporting columns, by a single modification factor. In our case one modification factor, equal to 2, was considered. Therefore, for the upgraded stiffness cases, the stiffness of all members of that storey is multiplied by 2, while for the degraded cases it is divided by 2. Stiffness irregularity cases are denoted as “KI”, while strength irregularity cases as “SI”. Each of the two types of irregularities was applied to each of the nine stories of the frame separately for a total of 18 cases: case “(n) KI” refers to the modification of the stiffness of the n-th storey and case “(n) SI” denotes a strength modification of the n-th storey. 0 0.04 0.08 0.12 0.16 0.2 0 0.5 1 1.5 2 2.5 3 Maximum interstorey drift ratio, θ max S pe ct ra l a cc el er at io n, S a( T 1 , 5% )