A yield displacement distribution‐based approach for strength assignment to lateral force‐resisting elements having strength dependent stiffness

Recent studies have shown that for many lateral force‐resisting elements (LFRE) stiffness is dependent on strength, and as a result strength assignment to these elements would affect both the strength and stiffness distributions in a structure. Consequently, stiffness distribution cannot be considered known prior to strength assignment. This paper presents a yield displacement distribution‐based strength assignment strategy that does not require the knowledge of stiffness distribution prior to strength assignment. It is shown that structural systems with their center of rigidity (CR) and center of strength (CV) located on the opposite sides of the center of mass (CM) will have small torsional responses under seismic excitation. Copyright © 2003 John Wiley Sons, Ltd.     

Design of torsionally unbalanced structural systems based on code provisions II: Strength distribution

This paper presents the results of an analytical study of the strength distribution of lateral load resisting elements in torsionally unbalanced systems designed based on codified torsional provisions. It is shown that the element strength can be expressed conveniently as the element strength of a similar but torsionally balanced system multiplied by a strength factor. This strength factor depends on three system parameters, namely, the location of the element relative to the centre of rigidity, and the torsional stiffness and eccentricity of the structure. In addition, it depends on the design coefficients of the code specified design eccentricity expressions. The influence of each of these factors on the element strength distribution is discussed. A new set of values for the design coefficients is proposed. By means of examples, it is shown that the proposed torsional provision is an improvement over those suggested in the National Building Code of Canada and the New Zealand code.     

Re-evaluation of seismic torsional provisions

Traditionally, seismic torsional provisions have been evaluated based on the assumption that the strength of the lateral load resisting elements can be adjusted without changing their stiffness. There is an important class of elements that a change of their lateral strength implies a corresponding change of stiffness, as exemplified by reinforced concrete flexural walls. This would imply that when torsional provisions are applied to adjust the strengths of these elements, the stiffness distribution, and also the eccentricity of the system, will change. This paper re-evaluates the consequences of applying the torsional provisions of the Uniform Building Code (UBC, 1997) and also the Eurocode (Eurocode 8, 1994) to single mass eccentric systems supported by elements having such characteristics. In conjunction with the results based on the traditional assumption, the effectiveness of the two provisions to mitigate torsional effects is discussed from a broader perspective. Copyright © 1999 John Wiley & Sons, Ltd.     

偏心结构扭转振动研究中几个基本参量的讨论

本文根据工程振动实测观察及结构地震模拟实验中的模态测量结果指出，通常的确定性弹性分析方法中，直接由构件的几何尺寸确定其刚度并进而确定结构的自振特性，进行结构的弹性反应分析和设计，其结果只能代表结构处于完全理想弹性状态时对外界激励的一种理论估计。由此提出了比较切合实际的基于强度及屈服变形的构件刚度的确定方法，进而指出在偏心结构的扭转振动研究中，结构的刚度偏心实际上依赖于强度偏心，并不能随研究者的意愿随意调整。     

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.     

Torsionally coupled dynamic performance analysis of asymmetric offshore platforms subjected to wave and earthquake loadings

The dynamic equations of motion of asymmetric offshore platforms under three different environmental conditions:seismic action,wave action and their combination are established in this paper. In establishing these motion equations,three typical eccentricity types including mass eccentricity,rigidity eccentricity and their combination were considered,as are eccentricities that occur un-idirectionally and bi-directionally. The effects of the eccentricity type,the dynamic characteristics and the environmental conditions on the torsional coupling response of platforms are investigated and compared. An effort has also been made to analyze the inffluence of accidental eccentricity on asymmetric platforms with different eccentricity in two horizontally orthogonal directions. The results are given in terms of non-dimensional parameters,accounting for the uncoupled torsional to lateral frequency ratio. Numerical results reveal that the eccentricity type has a great inffluence on the torsionally coupled response under different environmental conditions. Therefore,it is necessary to consider the combination of earthquake and wave action in the seismic response analysis of some offshore platforms.     

Torsional response of nonductile structures with soft‐first‐storey

In this paper, torsional response of nonductile structures with soft‐first‐storey subjected to bidirectional ground motions is studied using a simplified two‐storey model with two‐way eccentricities. The stiffness ratio of second storey to first storey is varied to create different levels of soft‐first‐storey effect, while the stiffness eccentricity is varied to create torsional effects. Different overstrength ratios are used in the simplified models to study the response of structure with different structural capacity. Hysteretic model with strength deterioration and stiffness degradation properties is used to capture the deterioration of element stiffness and strength. Ductility capacity of 2.0 is used as the models are for nonductile structures. In general, displacement amplification of irregular model with respect to regular model increases as stiffness ratio increases, while no consistent trend of changes in displacement amplification is found with increase in stiffness eccentricity. It is found that the displacement amplification due to only soft‐first‐storey effect can be conservatively taken as 1.5. Coupling of torsional and soft‐first‐storey effects is more significant in affecting the displacement amplification of elements at flexible side. The trend of changes in displacement amplification of elastic system is similar to that of inelastic system. The displacement amplification of elements at the flexible side is larger than that at the stiff side. The elements at the flexible side in the direction of shorter uncoupled lateral period have larger displacement response than those in the orthogonal direction. Ductility demand–capacity curves subsequently constructed can be used to approximately assess the seismic performance of existing structures and as guidelines for designing structures in Singapore to withstand the maximum credible earthquake considering the coupling of torsional and soft‐first‐storey effects. Copyright © 2014 John Wiley & Sons, Ltd.     

Predicting torsion‐induced lateral displacements for pushover analysis: Influence of torsional system characteristics

The increasing popularity of simplified nonlinear methods in seismic design has recently led to many proposals for procedures aimed at extending pushover analysis to plan asymmetric structures. In terms of practical applications, one particularly promising approach is based on combining pushover analysis of a 3D structural model with the results of linear (modal) dynamic analysis. The effectiveness of such procedure, however, is contingent on one fundamental requirement: the elastic prediction of the envelope of lateral displacements must be conservative with respect to the actual inelastic one. This paper aims at verifying the above assumption through an extensive parametric analysis conducted with simplified single‐storey models. The main structural parameters influencing torsional response in the elastic and inelastic range of behaviour are varied, while devoting special attention to the system stiffness eccentricity and radius. The analysis clarifies the main features of inelastic torsional response of different types of building structures; in this manner, it is found that the above‐mentioned method is generally suitable for structures characterized by moderate to large torsional stiffness, whereas it cannot be recommended for extremely torsionally stiff structures, as their inelastic torsional response almost always exceeds the elastic one. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimation of seismic demands on isolators in asymmetric buildings using non‐linear analysis

A procedure based on rigorous non‐linear analysis is presented that estimates the peak deformation among all isolators in an asymmetric building due to strong ground motion. The governing equations are reduced to a form such that the median normalized deformation due to an ensemble of ground motions with given corner period T_d depends primarily on four global parameters of the isolation system: the isolation period T_b, the normalized strength η, the torsional‐to‐lateral frequency ratio Ω_θ, and the normalized stiffness eccentricity e_b/r. The median ratio of the deformations of the asymmetric and corresponding symmetric systems is shown to depend only weakly on T_b, η, and Ω_θ, but increases with e_b/r. The equation developed to estimate the largest ratio among all isolators depends only on the stiffness eccentricity and the distance from the center of mass to the outlying isolator. This equation, multiplied by an earlier equation for the deformation of the corresponding symmetric system, provides a design equation to estimate the deformations of asymmetric systems. This design equation conservatively estimates the peak deformation among all isolators, but is generally within 10% of the ‘exact’ value. Relative to the non‐linear procedure presented, the peak isolator deformation is shown to be significantly underestimated by the U.S. building code procedures. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Seismic behavior of asymmetric RC wall buildings: principles and new deformation‐based design method

The seismic design of multi‐story buildings asymmetric in plan yet regular in elevation and stiffened with ductile RC structural walls is addressed. A realistic modeling of the non‐linear ductile behavior of the RC walls is considered in combination with the characteristics of the dynamic torsional response of asymmetric buildings. Design criteria such as the determination of the system ductility, taking into account the location and ductility demand of the RC walls, the story‐drift demand at the softer (most displaced) edge of the building under the design earthquake, the allowable ductility (ultimate limit state) and the allowable story‐drift (performance goals) are discussed. The definition of an eccentricity of the earthquake‐equivalent lateral force is proposed and used to determine the effective displacement profile of the building yet not the strength distribution under the design earthquake. Furthermore, an appropriate procedure is proposed to calculate the fundamental frequency and the earthquake‐equivalent lateral force. A new deformation‐based seismic design method taking into account the characteristics of the dynamic torsional response, the ductility of the RC walls, the system ductility and the story‐drift at the softer (most displaced) edge of the building is presented and illustrated with an example of seismic design of a multi‐story asymmetric RC wall building. Copyright © 2005 John Wiley & Sons, Ltd.     

Earthquake response of asymmetric building with MR damper

Plan asymmetric buildings are very susceptible to earthquake induced damage due to lateral torsional coupling, and the corners of these systems suffer heavy damage during earthquakes. Therefore, it is important to investigate the seismic behavior of an asymmetric plan building with MR dampers. In this study, the effectiveness of MR damper-based control systems has been investigated for seismic hazard mitigation of a plan asymmetric building. Furthermore, the infl uence of the building parameters and damper command voltage on the control performance is examined through parametric study. The building parameters chosen are eccentricity ratio and frequency ratio. The results show that the MR damper-based control systems are effective for plan asymmetric systems.     

摩擦摆基础隔震上部偏心结构地震反应影响因素分析

对上部结构存在偏心的摩擦摆基础隔震结构进行了水平双向地震作用下的地震反应分析,研究了上部结构偏心距和抗扭刚度对结构地震反应的影响。分析表明:上部结构偏心距对上部结构和隔震层的位移反应和加速度反应均有较大影响,即使在上部结构偏心距较小时,其对结构地震反应仍有一定程度的影响;上部结构的抗扭刚度对上部结构的加速度反应影响较小,而对上部结构的位移反应影响较大;上部结构的抗扭刚度对隔震层的加速度反应和位移反应影响较小。因而,对于上部结构存在偏心的摩擦摆基础隔震结构,应减小上部结构偏心距并增大其抗扭刚度以减小摩擦摆基础隔震结构的扭转反应。     

Seismic reliability analysis of buildings with torsional eccentricities

A study is presented of the influence of stiffness and strength eccentricities on the inelastic torsional response of buildings under the action of two simultaneous orthogonal horizontal ground motion components. Asymmetric buildings were obtained from their respective symmetric systems and were characterized by their stiffness and strength torsional eccentricities in both orthogonal directions. Based on the results of inelastic response of both building types (symmetric and asymmetric), the seismic reliability functions are determined for each system, and their forms of variation with different global system parameters are evaluated. Illustrative examples are presented about the use of this information for the formulation of seismic design criteria for in‐plan asymmetric multistory systems, in order to attain the same reliability levels implicit for symmetric systems designed in accordance with current seismic design codes. Copyright © 2014 John Wiley & Sons, Ltd.     

Alternative method to locate centre of rigidity in asymmetric buildings

Response of asymmetric buildup under earthquake excitation often involves lateral vibration coupled with torsional vibration. Floor slab is, in general, assumed as rigid along the in‐plane direction. Building code provisions to account for the torsional effect in static force procedure are based on centre of rigidity or shear centre of the building. A convenient procedure is developed here to locate the centre of rigidity or shear centre, which can be implemented, using any standard building analysis software. The procedure is applicable for orthogonal as well as non‐orthogonal building systems and accounts for all possible definitions of static eccentricity to compute the design response. An irregular building is analysed to illustrate the proposed methodology. Significant variation in member force resultants is observed due to different definitions of static eccentricity. Finally, a mathematical proof is presented to substantiate the applicability of the proposed procedure to a non‐orthogonal building. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

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

This investigation deals with the measured seismic response of a six‐storey asymmetric structural model with frictional dampers. Its main objective is to experimentally prove the concept of weak torsional balance for mass‐ and stiffness‐eccentric model configurations. The goal is to control the torsional response of these asymmetric structures and to achieve, if possible, a weak form of torsional balance by placing the so‐called empirical centre of balance (ECB) of the structure at equal distance from the edges of the building plan. The control of the dynamic response of asymmetric structures is investigated herein by using steel–teflon frictional dampers. As expected from theory, experimental results show that the mean‐square and peak displacement demand at the flexible and stiff edges of the plan may be similar in magnitude if the dampers are optimally placed. Frictional dampers have proven equally effective in controlling lateral‐torsional coupling of torsionally flexible as well as stiff structures. On the other hand, it is shown that impulsive ground motions require larger frictional capacities to achieve weak torsional balance. Copyright © 2006 John Wiley & Sons, Ltd.     

Seismic behavior of single‐story asymmetric‐plan buildings under uniaxial excitation

The critical parameters that influence the nonlinear seismic response of asymmetric‐plan buildings are identified by evaluating the effects of different asymmetries that may characterize the structure of a building as well as exploring the influence of the ground motion features. First, the main findings reported in the literature on both the linear and nonlinear dynamic response of asymmetric‐plan buildings are presented. The common findings and the conflicting conclusions reached in different investigations are pointed out. Then, the results of comprehensive nonlinear dynamic analyses performed for evaluating the seismic response of systems characterized by different strength and stiffness configurations, representative of a large class of asymmetric‐plan buildings, are reported. Findings from the study indicate that the building response changes when moving from the linear to the nonlinear range, so that the seismic behavior of asymmetric‐plan buildings, apart from the source of asymmetry, can be always classified as irregular. Additionally, it was observed that as the seismic demands cause amplification of system nonlinearity with increasing earthquake intensity, the maximum displacement demand in the different resisting elements tends to be reached with the same deformed configuration of the system. The resultant of the seismic forces producing such a maximum demand is located at the center of resistance and corresponds to the collapse mechanism of the system that provides the maximum lateral strength in the exciting direction of the seismic action. Copyright © 2008 John Wiley & Sons, Ltd.     

On the inelastic torsional response of single‐storey structures under bi‐axial excitation

An attempt has been made to explore the general trends in the seismic response of plan‐asymmetric structures without any restrictions imposed by a particular code. Systems with structural elements in both orthogonal directions under bi‐directional excitation were studied. Idealized single‐storey models with bi‐axial eccentricity were employed. The systems were torsionally stiff and, in the majority of cases, mass‐eccentric. The main findings are: in general, inelastic torsional response is qualitatively similar to elastic torsional response. Quantitatively, the torsional effect on the flexible side, expressed as an increase of displacements due to torsion, decreases slightly with increasing plastic deformation, unless the plastic deformations are small. The response on the stiff side generally strongly depends on the effect of several modes of vibration and on the influence of the ground motion in the transverse direction. These influences depend on the structural and ground motion characteristics in both directions. Reduction of displacements due to torsion, typical for elastic torsionally stiff structures, usually decreases with increasing plastic deformations. As an additional effect of large plastic deformations, a flattening of the displacement envelopes in the horizontal plane usually occurs, indicating that torsional effects in the inelastic range are generally smaller than in the elastic range. The dispersion of the results of inelastic torsional response analysis is generally larger than that of elastic analysis. Copyright © 2005 John Wiley & Sons, Ltd.