A modified static procedure for the design of torsionally unbalanced multistorey frame buildings

Seismic building codes include design provisions to account for the torsional effects arising in torsionally unbalanced (asymmetric) buildings. These provisions are based on two alternative analytical procedures for determining the design load for the individual resisting structural elements. A previous study has shown that the linear elastic modal analysis procedure may not lead to conservative designs, even for multistorey buildings with regular asymmetry, when such structures are excited well into the inelastic range of response. The equivalent static force procedure as recommended by codes may also be deficient in accounting for additional ductility demand in the critical stiff-edge elements. This paper addresses the non-conservatism of existing static torsional provisions and examines aspects of element strength distribution and its influence on inelastic torsional effects. A recommendation is made for improving the effectiveness of the code-type static force procedure for torsionally unbalanced multistorey frame buildings with regular asymmetry, leading to a design approach which estimates conservatively the peak ductility demand of edge elements on both sides of the building. The modified approach also retains the simplicity of existing code provisions and results in acceptable levels of additional lateral design strength. It has recently been adopted by the new Australian earthquake code, which is due to be implemented early in 1993.     

Dynamic response of bilinear asymmetric structures

The inelastic behaviour of eccentric single-storey building structures subjected to sinusoidal ground excitation is examined. The Kryloff-Bogoliuboff method is employed to provide approximate solutions in the amplitude-frequency domain. Structural resisting elements are assumed to exhibit bilinear hysteretic behaviour and coupled response is investigated in terms of both system response as well as individual element ductility requirements. In addition to demonstrating the well-known softening property inherent in yielding systems, the importance of the principal parameters governing coupled response is evaluated in a consistent parametric fashion. Within the context of earthquake resistant building design, the results indicate the absence of amplified response when torsional and translational frequencies are close, in contrast to the much emphasized observation of internal resonance for linear elastic structures. Equally important, structural elements located on the stiff edge of eccentric buildings are found to be only marginally affected by the magnitude of the eccentricity, thus indicating that seismic building codes which reduce design requirements for these elements underestimate actual behaviour substantially.     

Additional seismic inelastic deformation caused by structural asymmetry

Using a single mass monosymmetric model, this paper examines the additional seismic inelastic deformations and displacement caused by structural asymmetry of the model. Stiffness eccentricity and resistance eccentricity are used as measures of asymmetry in the elastic and inelastic range respectively. Seven ways of specifying strength distribution among resisting elements are considered, including code provisions from Canada, Mexico, New Zealand and the United States. These specifications are related t o the model resistance eccentricity. It is shown that when torsional shears are included in the strength design of the elements, the structure in general will have small resistance eccentricity, even if it has large stiffness eccentricity in the elastic range. For structures which are designed with allowance for torsional shears, the ductility demands on the elements are similar to those when the structure is symmetrical. However, the edge displacements can be up to three times that if the system is symmetrical. This finding has significant implications in evaluating adequate separation between buildings to avoid the pounding problem during earthquakes.     

Inelastic torsional response of buildings to the 1985 Mexican earthquake: a parametric study

This study aims to determine the influence of torsional coupling on the inelastic response of a series of models representing typical structural configurations in real buildings. The lake bed (SCT) east-west component of the 1985 Mexico City earthquake was employed in the analysis, and is representative of a severe ground motion known to have induced large inelastic structural deformations in a high proportion of those buildings having asymmetrical distributions of stiffness and/or strength. Material non-linearity in lateral load-resisting elements has been defined using a hysteretic Ramberg-Osgood model. Structural eccentricities have been introduced into the building models by (i) asymmetrical distributions of stiffness and/or strength, (ii) asymmetrical configuration of lateral load-resisting elements, or (iii) varying post-elastic material behaviour in the resisting elements. The dynamic inelastic response of these models has been obtained by a numerical integration of the relevant equations of motion, expressed in a non-dimensional incremental form.

In the elastic range, the results correlate well with those of previous studies. In the inelastic range, it is concluded that the peak ductility demand of the worst-affected element increases with the ground excitation level across the range of building periods considered, and that the influence of torsional coupling on the key response parameters is model dependent. Most significantly, the strength eccentricity relative to the centre of mass has been shown to influence the peak edge displacement response more than conventionally employed stiffness eccentricity.     

Inelastic seismic response of code-designed multistorey frame buildings with regular asymmetry

Based on an asymmetric multistorey frame building model, this paper investigates the influence of a building's higher vibration modes on its inelastic torsional response and evaluates the adequacy of the provisions of current seismic building codes and the modal analysis procedure in accounting for increased ductility demand in frames situated at or near the stiff edge of such buildings. It is concluded that the influence of higher vibration modes on the response of the upper-storey columns of stiff-edge frames increases significantly with the building's fundamental uncoupled lateral period and the magnitude of the stiffness eccentricity. The application of the equivalent static torsional provisions of certain building codes may lead to non-conservative estimates of the peak ductility demand, particularly for structures with large stiffness eccentricity. In these cases, the critical elements are vulnerable to excessive additional ductility demand and, hence, may be subject to significantly more severe structural damage than in corresponding symmetric buildings. It is found that regularly asymmetric buildings excited well into the inelastic range may not be conservatively designed using linear elastic modal analysis theory. Particular caution is required when applying this method to the design of stiff-edge frame elements in highly asymmetric structures.     

Inelastic seismic response of torsionally unbalanced systems designed using elastic dynamic analysis

This paper evaluates the inelastic seismic response of torsionally unbalanced structural systems with strength distributed using elastic response spectrum analysis. The structural model is a single mass torsionally unbalanced system with lateral load resisting elements spanning in two principal directions. The element strength is distributed based on elastic response spectrum analysis and three different approaches to incorporate accidental torsion are considered: (a) without incorporating accidental torsion; (b) by applying static floor torques; (c) by shifting the location of the centre of mass. The seismic input is bidirectionally applied at the base of the model. It is shown that the inelastic responses depend strongly on the torsional stiffness of the system. For a torsionally stiff system, the torsional response leads to a decrease in the stiff edge displacement; however, for a torsionally flexible system, it tends to increase the stiff edge displacement. Using response spectrum analysis without including accidental torsion may lead to excessive additional ductility demand on the stiff edge element. With accidental torsion effect incorporated, the response spectrum analysis will give a strength distribution such that there will be no excessive additional ductility demands on the lateral load resisting elements.     

The role of overstrength on the seismic performance of asymmetric-plan structures

Uneven distribution of seismic demand in asymmetric-plan structures is a critical concern in earthquake-resistant design. Contemporary seismic design strategies that are based on linear elastic response, single load reduction factor, and uniform ductility demand throughout an asymmetric system generally lead to unsatisfactory performance in terms of realized ductilities and nonuniform damage distribution due to strong torsional coupling associated with asymmetric-plan systems. In many cases, actual nonlinear behavior of the structure displays significant deviation from what is estimated by a linear elastic, force-based seismic design approach. This study investigates the prediction of seismic demand distribution among structural members of a single-story, torsionally stiff asymmetric-plan system. The focus is on the effect of inherent unbalanced overstrength, resulting from current force-based design practices, on the seismic response of code-designed single-story asymmetric structures. The results obtained are utilized to compile unsymmetrical response spectra and uniform ductility spectra, which are proposed as assessment and preliminary design tools for estimating the seismic performance of multistory asymmetric structures. A simple design strategy is further suggested for improving the inelastic torsional performance of asymmetric systems. Providing additional strength to stiff edge members over their nominal design strength demands leads to a more balanced ductility distribution. Finally, seismic responses of several asymmetric case study structures designed with the aid of the proposed strategy are assessed for validating their improved performance.     

Inelastic seismic response of simple eccentric structures

The maximum ductility demand and the edge displacement of a simple single mass eccentric model is evaluated when the system is subjected to ground motions represented by the El Centro 1940 and Taft 1952 earthquake records. The resisting elements are taken to be bilinear hysteretic. It is found that the ductility demand depends to a great extent on the energy content of the ground motions, particularly in the period range beyond the elastic period of the system. Unlike elastic response, the coincidence of uncoupled torsional and lateral frequencies does not lead to exceptionally high inelastic response. An increase by a factor of two in ductility demand is not uncommon for a system with large eccentricity as compared to a symmetrical system. Therefore, system eccentricity has a larger effect on ductility demand than earlier studies indicated. Using Clough's model to allow for stiffness degradation effect, results are found to be within 20 per cent of those calculated based on the bilinear hysteretic model.     

Investigation of story ductility demands of inelastic concrete frames subjected to repeated earthquakes

The present study focuses on the influence of repeated earthquakes on the maximum story ductility demands of three-dimensional inelastic concrete frames. A comprehensive assessment is conducted using generic frames with 3-, 6-, 12-, and 18-story structures. Each is assumed to have behaviour factors of 1.5, 2, 4, and 6 referring to Eurocode 8. Stiffness and strength degrading hysteresis rule to represent reinforced concrete structure is considered in the plastic hinge of members. Twenty ground motions are selected, and single, double, and triple events of synthetic repeated earthquakes are considered. Some interesting findings are provided showing that repeated earthquakes significantly increase the story ductility demand of inelastic concrete frames. On average, relative increment of maximum story ductility demand is experienced 1.4 and 1.3 times when double and triple events of repeated earthquakes are induced, respectively. Empirical relationships are also provided to predict these increments where their efficiency is presented examining characteristic 3- and 8-story reinforced concrete buildings.     

Effect of transverse load-resisting elements on inelastic earthquake response of eccentric-plan buildings

The accurate evaluation of code torsional provisions for plan-eccentric structures exhibiting inelastic response relies on the adoption of appropriate systems defining both the torsionally balanced (reference) and torsionally unbalanced cases. Whilst a considerable number of analytical studies of this problem have been presented in the literature, inconsistencies have arisen in their conclusions. It is evident from a review of previous studies that one factor contributing significantly to these discrepancies arises in the definition of the structural layout. An issue of particular importance is whether the transverse load-resisting elements oriented perpendicular to the assumed (lateral) direction of earthquake loading should, for purposes of realism, be included in model definitions. Given the diverse approaches in the existing literature, clarification of this issue is required in order to advance the understanding of inelastic torsional response behaviour and to assist the interpretation and comparison of previous studies. This paper aims to provide such clarification, based on analyses of a series of models defined rigorously according to code design provisions. Such models have been subjected to both uni- and bi-directional ground motion input. It is concluded that for the flexible-edge element, accurate estimates of additional ductility demand arising from torsional effects may be obtained from uni-directional models (in which both the transverse elements and the corresponding earthquake component are neglected) only for medium-period to long-period systems. Such estimates may be over-conservative for short-period systems, which constitute a large proportion of systems for which code static torsional provisions are utilized. It is further concluded that models incorporating the transverse elements but analysed under uni-directional lateral loading may underestimate by up to 100% the torsional effects in such systems, but are reasonably accurate for medium- and long-period structures.     

Effective strength eccentricity concept for inelastic analysis of asymmetric structures

A large number of parameters affect the inelastic response of an eccentric system. The centre of resistance no longer remains constant due to continuous loading and unloading of various elements of the system in and out of the inelastic region. The objective of this study is to develop the concept of strength eccentricity for asymmetrical structures excited well into the inelastic region in the event of a severe earthquake. A single mass monosymmetric three-element system is selected. The torsional flexibility parameter is varied so as to obtain nine eccentric configurations covering the entire spectrum of such systems. These models were subjected to S00E component of the El Centro earthquake of 1940 which is considered to be a very severe earthquake. The concept of effective strength eccentricity is introduced which is defined as strength eccentricity at the point of intersection of ductility ratio curves of REE and FEE. It is proposed to provide additional strength in the elements of a torsionally unbalanced system independent of the design eccentricity. Empirical equations are proposed to account for design eccentricity and additional strength as well as distribution of the design strength. The proposed formulation is compared with the torsional provisions of UBC 1991, NBCC 1990 and NZC 1992. It is concluded that additional ductility demand on the REE and FEE designed based on the proposed formulation is almost nil as compared to that given by the three codes. The NZC and UBC require maximum ductility on the REE. The increase in overall strength of the system is least for the proposed formulation as compared to that given by the three codes. NBCC consistently requires the maximum total strength ratio.     

平面不规则结构非弹性扭转地震反应研究进展

不规则建筑结构在侧向地震荷载作用下由于质量中心和刚度中心的不重合导致平扭耦联反应的发生,使得结构构件的变形需求分布在结构平面内并不一致,从而产生附加的强度和变形需求。尽管不规则建筑结构在地震作用下的扭转问题一直受到研究学者的关注和研究,并取得了很多显著的成果,但仍然存在着一些争议,有待于继续深入研究。本文从结构的分析模型、影响参数及地震动输入等方面回顾总结了平面不规则建筑结构在地震作用下非弹性扭转的研究进展,结合当前的研究工作指出今后研究的发展方向。     

Effect of orthogonal inplane structural elements on inelastic torsional response

The elastic torsional stiffness of a structure has important influence on the seismic response of an asymmetric structure, both in the elastic and inelastic range. For elastic structures it is immaterial whether the stiffness is provided solely by structural elements in planes parallel to the direction of earthquake or by a combination of such elements in parallel and orthogonal planes. The issue of how the relative contribution of structural elements in orthogonal planes affects the torsional response of inelastic structures has been the subject of continuing study. Several researchers have noted that structural elements in orthogonal planes reduce the ductility demands in both the flexible and stiff edge elements parallel to the earthquake. Some have noted that the beneficial effect of structural elements in orthogonal planes is more pronounced when such elements remain elastic. These issues are further examined in this paper through analytical studies on the torsional response of single-storey building models. It is shown that, contrary to the findings of some previous studies, the torsional response of inelastic structures is affected primarily by the total torsional stiffness in the elastic range, and not so much by whether such stiffness is contributed solely by structural elements in parallel planes or by such elements in both parallel and orthogonal planes. Copyright © 1999 John Wiley & Sons, Ltd.     

Effects of stiffness degradation on ductility requirements for multistorey buildings

The effect of stiffness degradation in reinforced concrete structural members on the inelastic response of multistorey buildings to earthquakes is investigated. In particular, the following question is examined. How do the ductility requirements for multistorey systems with degrading stiffness behaviour compare with those for structures with ordinary bilinear hysteretic property? Inelastic dynamic responses of two idealized multistorey buildings, one having a long and the other a relatively short fundamental period, to an ensemble of twenty simulated earthquakes representative of moderately intense ground motions in California at moderate epicentral distances on firm ground, are analysed for ordinary bilinear hysteretic behaviour and for bilinear hysteretic behaviour with stiffness degradation property. The conclusions deduced from the results of this investigation include the following (1) It is, in general, not possible to predict the maximum response of a degrading stiffness system from results for the corresponding ordinary bilinear system (2) The differences in ductility requirements due to stiffness degradation are generally smaller than those associated with probabilistic variability from one ground motion to another (3) Stiffness degradation has little influence on the ductility requirements for flexible buildings, but it leads to increased ductility requirements for stiff buildings.     

Torsional balance of plan‐asymmetric structures with frictional dampers: analytical results

This investigation deals with the torsional balance of the earthquake response and design of elastic asymmetric structures with frictional dampers. Plan asymmetry leads to an uneven lateral deformation demand among structural members and to unbalanced designs with larger capacities in some resisting planes. Frictional dampers are capable of controlling lateral‐torsional coupling by placing the so‐called empirical center of balance (ECB) of the structure at equal distance from all edges of the building. This rule is developed for single‐story systems with linear and inelastic behavior. However, recently obtained theoretical and experimental results demonstrate that this rule carries over to multistory structures. Results show that the peak displacement demand at the building edges and that of resisting planes equidistant from the geometric center may be similar if the damper is optimally placed. It is also shown that torsional amplification of the edge displacements of arbitrary asymmetric structures relative to the displacement of the symmetric counterparts are approximately bound by a factor of 2. Furthermore, frictional dampers are equally effective in controlling lateral‐torsional coupling of torsionally flexible as well as stiff structures. Copyright © 2005 John Wiley & Sons, Ltd.     

Equivalent ductility factors,taking into account low-cycle fatigue

During strong earthquakes, the deformation capacity of structures is reduced due to the cumulative damage caused by cyclic load reversals. In the paper, equivalent (reduced) ductility factors have been proposed, which take into account this effect. They are based on different failure hypotheses. Ductility reduction due to low-cycle fatigue is controlled by a dimensionless parameter γ, which is a function of dissipated hysteretic energy, maximum displacement and the natural frequency of the structural system, and which has been proved to be a relatively stable quantity in the whole period range. If approximate values for γ are used, the determination of equivalent ductility is very simple, and thus appropriate for design purposes. The formulae for equivalent ductility factors include damage indices, and permit the designer to choose acceptable level of structural damage explicitly. As an example, equivalent ductility factors have been used to construct inelastic acceleration spectra, which are proportional to strength demand, for the El Centro 1940 SOOE record. The results have been compared with the ‘exact’ spectra obtained by non-linear dynamic analysis.     

On the inelastic seismic response of asymmetric buildings under bi‐axial excitation

The elastic and inelastic seismic response of plan‐asymmetric regular multi‐storey steel‐frame buildings has been investigated under bi‐directional horizontal ground motions. Symmetric variants of these buildings were designed according to Eurocodes 3 and 8. Asymmetric buildings were created by assuming a mass eccentricity in each of the two principal directions. The torsional response in the elastic and inelastic range is qualitatively similar with the exception of the stiff edge in the strong direction of torsionally stiff buildings and the stiff edge in the weak direction of torsionally flexible buildings. The response is influenced by the intensity of ground motion, i.e. by the magnitude of plastic deformation. In the limiting case of very strong ground motion, the behaviour of initially torsionally stiff and initially torsionally flexible buildings may become qualitatively similar. A decrease in stiffness due to plastic deformations in one direction may substantially influence the behaviour in the orthogonal direction. The response strongly depends on the detailed characteristics of the ground motion. On average, torsional effects are reduced with increasing plastic deformations, unless the plastic deformations are small. Taking into account also the dispersion of results which is generally larger in the inelastic range than in the elastic one, it can be concluded that (a) the amplification of displacements determined by the elastic analysis can be used as a rough estimate also in the inelastic range and (b) any favourable torsional effect on the stiff side of torsionally stiff buildings, which may arise from elastic analysis, may disappear in the inelastic range. The conclusions are limited to fairly regular buildings and subject to further investigations. Copyright © 2005 John Wiley & Sons, Ltd.     

Damage‐dependent vulnerability curves for existing buildings

Seismic behavior of damaged buildings may be expressed as a function of their REsidual Capacity (REC), which is a measure of seismic capacity, reduced by damage. REC can be interpreted as the median value of collapse vulnerability curves. Its variation owing to damage is a useful indication of increased building vulnerability. REC reduction, indicating the lowering of seismic safety after an earthquake (performance loss, PL), represents an effective index for assessing the need of seismic repair/strengthening after earthquakes. The study investigates the applicability of a pushover‐based method in the analysis of damaged structures for the case of existing under‐designed RC buildings. The paper presents a systematization of the procedure in an assessment framework that applies the capacity spectrum method based on inelastic demand spectra; furthermore, the vulnerability variation of a real building is investigated with a detailed case study. The behavior of damaged buildings is simulated with pushover analysis through suitable modification of plastic hinges (in terms of stiffness, strength and residual drift) for damaged elements. The modification of plastic hinges has been calibrated in tests on nonconforming columns. The case study analysis evidenced that, for minor or moderate damages, the original structural displacement capacity was only slightly influenced, but the ductility capacity was significantly reduced (up to 40%) because of the increased structure deformability. This implied performance loss in the range 10%–20%. For severe damages the PL ranged between 41% and 56%. Local mechanism types exhibit PL nearly double with respect to global mechanism types. Copyright © 2012 John Wiley & Sons, Ltd.     

结构抗震设计中的强度折减系数研究

借助于单自由度弹塑性动力时程分析程序，对延性结构的强度折减系数进行了研究，在统计平均和回归分析的基础上，建立了平均强度折系数的函数形式，本文所建立的平强度折减系数函数，从理论上明确了结构具有延性对弹性地震力的折减关系，研究成果可供结构抗震设计规范采纳应用。     

剪切型结构的抗震强度折减系数研究

为了研究剪切型结构抗震强度需求的变化规律,本文基于单自由度体系的非线性时程分析,研究了不同场地条件下延性折减系数与位移延性系数和结构自振周期的关系;采用修正等效单自由度体系位移延性折减系数的方法,研究了剪切型多自由度体系的延性折减系数;以基于中国建筑抗震规范设计的代表不同抗震能力要求的RC框架结构为分析对象,通过静力弹塑性分析,研究了RC框架结构的体系超强能力。分析结果表明场地类别、位移延性水准和结构振动周期对单自由度体系的延性折减系数有显著的影响;多自由度体系的抗震延性折减系数明显比其相应的等效单自由度体系的抗震延性折减系数小;RC框架结构的超强系数一般随结构楼层数的增加而减小,随抗震设防烈度的增大而减小,内框架的超强系数比边框架的超强系数大。