An alternative approach for assessing eccentricities in asymmetric multistory buildings. 2. Inelastic systems

The approximate method, presented in the companion paper, for assessing modal eccentricities of elastic multistory buildings with simple eccentricity is extended in systems composed by elastoplastic resisting bents. Following the technique of the aforementioned paper for computing modal properties of such buildings by means of an equivalent single‐story system composed of elastic elements, modal capacity curves of these systems may also be drawn when the resisting elements are defined by a bilinear force–displacement (characteristic) curve. The procedure for constructing element‐characteristic curves is based on the methodology presented by the author in an earlier paper, and modal capacity curves of the equivalent single‐story system may be drawn by performing a non‐linear pushover analysis using the inertia force eccentricity of each mode of this system. Therefore, base shears and their eccentricities for the first two modes of vibration of multistory inelastic buildings can be determined as in real one‐story non‐linear systems. The method is illustrated in a 10‐story partial symmetric building, having along the direction of the ground motion three identical, inelastic, coupled wall bents. The structure is analyzed for a strong ground motion, equal to 1·5 × El Centro earthquake excitation, and the results are compared with those obtained from a step‐by‐step non‐linear time history analysis of the discrete member model. Copyright © 2007 John Wiley & Sons, Ltd.     

Ductility factors of symmetrical structures with restricted inelasticity

A simple elasto‐plastic analysis of symmetrical buildings composed of different types of bents is presented. Plasticity in bents is assumed to be restricted to the beams, where significant curvature ductility factors may be attained without loss of strength, and the behaviour of inelastic structures may be traced from elastic to ultimate deflections on the grounds that the external load is monotonically increasing with an invariant shape throughout the loading history. The outlined procedure is based on the concept of the continuous medium and a comprehensive consideration of the structural post‐elastic behaviour of common mixed‐bent‐type buildings is possible by means of parameter variations. Therefore it is suitable for a simple pushover elasto‐plastic analysis in a parametric form. The proposed method of analysis is based on the technique of decomposing a cantilever structural system into two components: a flexural and a shear–flexure subsystem. With this decomposition, not only the axial deformations in the column members of all bents are taken into account in the calculation of deflections, but also different types of bents may be combined in symmetrical buildings. Design charts relating the overall ductility factor with the maximum rotation ductility factor at the ends of the beams are presented for the case of a triangularly distributed external load, together with the formulation that defines the post‐elastic load–deflection curve. These data may be found useful for practising engineers at the stage of a preliminary structural application. Copyright © 2007 John Wiley & Sons, Ltd.     

A simple model for assessing periods of vibration and modal response quantities in symmetrical buildings

A simple mathematical model is proposed for assessing periods of vibration and mode shapes of common cantilever bents used in concrete structures, such as shear walls, coupled walls, rigid frames and wall‐frame assemblies. The bent is treated as a continuum and the proposed model is based on the technique of decomposing a cantilever bent into two complementary subsystems (a flexural and a wall‐frame bent) and on the finding that the use of Dunkerley's formula for calculating natural frequencies yields reasonable results for the first three modes of vibration. The objective in proposing this model is to consider the effect of column axial shortenings in the analysis of structural bents. With this model any cantilever bent may be approximated by a simple incompressible shear–flexure system of equal flexural rigidity, but of equivalent modal shear rigidity. This approach has the advantage that the response of different structural bents may be combined in buildings composed of these bents in any arrangement. All bents are approximated by equivalent shear–flexure models and therefore the complete structure may be analysed by a simple methodology, which has been extensively used in the past. Particularly in symmetrical buildings, frequencies may be determined by a simple formula and modal response quantities by available design charts. A quick estimate of these quantities is of particular importance at the early stages of structural design, prior to a full dynamic analysis. In order to illustrate the application of the proposed model a symmetrical building of varying height, composed of different structural bents, is analysed and comparisons are made with more accurate results obtained by 3D computer dynamic analyses. Copyright © 2006 John Wiley & Sons, Ltd.     

An alternative approach for assessing eccentricities in asymmetric multistory buildings. 1. Elastic systems

A simple hand method for assessing modal eccentricities of multistory structures with partial symmetry is presented. The concept of the modal center of rigidity is introduced and the determination of eccentricities of the first two modal shears is obtained by means of an equivalent one‐story eccentric system. The method is based on Southwell's formula for calculating a lower bound of the fundamental frequency and is applicable to uniform structures composed of different types of bents, such as walls, rigid frames, coupled walls and wall–frame assemblies. These structures do not belong to the special class of proportionate buildings for which dynamic properties can be accurately obtained by analyzing two simpler systems: a multistory torsionally uncoupled counterpart of the actual structure and an associated one‐story coupled system. The accuracy of the proposed method is illustrated in a parametric study that includes common types of mixed‐bent‐type multistory structures and comparisons are made with the results obtained by three‐dimensional dynamic analyses on discrete member models. Copyright © 2007 John Wiley & Sons, Ltd.     

Approximate analysis of symmetrical structures consisting of different types of bents

An approximate analysis is presented for calculating the deflections of individual cantilever bents, under lateral loading, as a sum of deflections of two complementary subsystems: a flexural and a shear–flexure subsystem. The analysis accounts for axial deformations in the vertical members of bents, since it is based on the continuum approach as is used in the coupled wall theory, and is also applicable to other types of cantilever bents used in concrete structures, such as rigid frames and wall frame assemblies. The fact that the deflection equation of a cantilever bent may be decomposed into two components makes possible the development of an approximate method for estimating the deflections of uniform plan‐symmetrical buildings composed of different structural bents. The method provides a rapid estimate of deflections and load distribution in such multi‐bent structures and therefore it is appropriate for preliminary structural design. At this stage, it is desirable for the practising engineer to have a quick estimate of the maximum response, even if the actual sizes of the structural elements are not yet known and only assumptions can be made about the relative stiffnesses of the major structural elements. The proposed analysis, as based on distributed parameter formulations, has the advantage of providing a deep insight into the structural behaviour of high‐rise structures. Its accuracy is evaluated by comparing the approximate results with those obtained by stiffness matrix analyses. Copyright © 2007 John Wiley & Sons, Ltd.     

Simplified dynamic analysis of eccentric buildings with a setback. 2: the effect of stiffness irregularity

The effect of stiffness irregularity is investigated in eccentric buildings with a setback. Stiffness irregularity is assumed to be created when some of the lateral load resisting bents are curtailed at the top of the base structure. The methodology suggested in the companion paper for assessing vibration frequencies, base shears and torques is extended to include the effect of curtailed bents. Their contribution is evaluated by an indirect method after the modal stiffness of the symmetrical counterpart structure is determined and the modal contribution of the full height bents is assessed by Southwell's approach. A number of numerical examples, representing common types of building structures, are presented to illustrate the procedure and provide an insight into the response of such setback buildings. The results are compared with more accurate results obtained by three‐dimensional dynamic analyses. Copyright © 2009 John Wiley & Sons, Ltd.     

Modal eccentricities of asymmetric structures

Modal eccentricities of monosymmetric structures are given by a simple hand method, which is based on the concept of the continuum medium. The method is applicable to structures consisting of any combination of walls, rigid frames and coupled wall bents that are uniform over the height of the structure. These structures do not belong to the special class of proportionate buildings for which the dynamic properties can be obtained approximately from the properties of an equivalent single‐storey torsionally coupled building, in combination with those of a torsionally uncoupled multistorey structure. In the proposed method the equation of motion of each bent of the structure is approximated by that of an incompressible wall‐frame system (model) of equal flexural rigidity, but of an equivalent modal shear rigidity, which accounts for axial deformations in the vertical members of the bent. With this model any uniform structure composed of different types of bents may be analysed by standard methods and dynamic properties may be determined by simple and rapid means. Modal eccentricities are given in respect to the centre of flexural rigidities, which is easily determined in uniform structures, and as the method is based on the continuous approach, a deep insight into the structural behaviour is possible by consideration of parameter variations. Approximate expressions are also given for frequencies and modal base shears, as these properties play a key role in the response spectrum analysis. In order to illustrate the accuracy of the proposed method comparisons are made with accurate results obtained by 3D computer dynamic analyses on discrete member models. Copyright © 2006 John Wiley & Sons, Ltd.     

Strength reduction factor for MDOF soil–structure systems

In most of the seismic design provision, the concept of strength reduction factor has been developed to account for inelastic behavior of structures under seismic excitations. Most recent studies considered soil–structure interaction (SSI) in inelastic response analysis are mainly based on idealized structural models of single degree‐of‐freedom (SDOF) systems. However, an SDOF system might not be able to well capture the SSI and structural response characteristics of real multiple degrees‐of‐freedom (MDOF) systems. In this paper, through a comprehensive parametric study of 21600 MDOF and its equivalent SDOF (E‐SDOF) systems subjected to an ensemble of 30 earthquake ground motions recorded on alluvium and soft soils, effects of SSI on strength reduction factor of MDOF systems have been intensively investigated. It is concluded that generally, SSI reduces the strength reduction factor of both MDOF and more intensively SDOF systems. However, depending on the number of stories, soil flexibility, aspect ratio and inelastic range of vibration, the strength reduction factor of MDOF systems could be significantly different from that of E‐SDOF systems. A new simplified equation, which is a function of fixed‐base fundamental period, ductility ratio, the number of stories, structure slenderness ratio and dimensionless frequency, is proposed to estimate strength reduction factors for MDOF soil–structure systems. Copyright © 2012 John Wiley & Sons, Ltd.     

Equivalent lateral force procedure for the seismic story drift design of tall frames using a hand‐calculated approach

The basic theory of the cantilever moment distribution method and the application of this method to conduct the equivalent lateral force procedure for the seismic design of tall frames are discussed in detail. Deflection–rotation formulas are introduced in this paper to determine the relative lateral story deflections and the joint rotations of laterally loaded rigid frames. An example is presented to conduct the seismic story drift design of a multistory, multi‐bay, moment‐resisting reinforced concrete frame using the cantilever moment distribution method and the deflection–rotation formulas developed in this paper. The hand‐calculated approach presented in the example can be used as a rapid and accurate method to determine the story drift for any laterally loaded multistory, multi‐bay rigid frames that are composed of identical single‐bay symmetrical bents. The following are the advantages of using this rapid approach: (1) this approach can be carried out by using hand calculations only (without the use of computer computations); (2) the results obtained from this approach are as accurate as those derived from the traditional moment distribution and the slope deflection methods; and (3) the results obtained from this approach can be used to verify the accuracy of the results obtained from computer computations. Copyright © 2005 John Wiley & Sons, Ltd.     

钢筋混凝土梁式构件抗爆分析的改进等效单自由度方法

爆炸荷载作用下钢筋混凝土梁式构件可能会出现多种破坏模式，主要有弯曲破坏、直剪破坏和弯剪联合破坏等。现有结构构件抗爆分析方法主要考虑爆炸荷载下结构构件的弯曲破坏，对直剪破坏和弯剪联合破坏研究较少，尤其缺少快速准确的抗爆分析方法。基于等效单自由度模型理论，改进了钢筋混凝土梁抗爆分析的直剪单自由度方法，提出了直剪、弯剪联合破坏的判定准则，给出了两种破坏模式下钢筋混凝土梁动态响应的计算方法。进一步考虑爆炸荷载作用下钢筋混凝土梁的弯曲破坏、直剪破坏和弯剪联合破坏，提出钢筋混凝土梁式构件抗爆分析的改进等效单自由度方法，并给出了其分析步骤。该方法可以直接评估爆炸荷载下钢筋混凝土梁式构件的破坏模式，并计算其动态响应。通过有限元软件LS-DYNA数值模拟了不同爆炸工况下钢筋混凝土梁式构件的动态响应和破坏模式，与改进的等效单自由度方法的计算结果进行了对比，验证了提出的改进等效单自由度分析方法的正确性。     

弹塑性MDOF系统地震输入能量研究

基于地震作用下弹塑性SDOF(Single Degree of Freedom)和弹性MDOF(Multi-degree of Freedom)系统能量输入研究,本文分析了弹塑性MDOF系统能量输入规律。以实测地震记录为输入,采用弹塑性时程分析方法,计算了一阶初始周期≤3s的5、10、20、30层双线性滞回剪切层模型,不同阻尼比、承载力降低系数情况下4800个结构和一阶初始周期>3s的40层结构能量输入,与弹塑性SDOF系统的输入能量谱进行对比。研究表明:一阶周期在3s以内的弹塑性MDOF系统输入能量EI可用相同阻尼比、初始周期、承载力降低系数的弹塑性SDOF系统计算结果近似确定;一阶周期大于3s的弹塑性MDOF系统,可采用MPA方法等效为多个弹塑性SDOF系统,按照振型叠加法计算总输入能。     

Equivalent linearization parameters of soil-MDOF structure systems subjected to pulse-like earthquakes

In seismic design codes and documents, the period and damping ratio of a linear fixed–base single degree-of-freedom (SDOF) system are tuned to return the dynamic response of a given nonlinear fixed–base SDOF structure through a so-called equivalent linearization method. The period and damping ratio of the equivalent linear system are referred to as equivalent or effective linear parameters (ELPs). These ELPs may vary due to higher modes of the superstructure, the soil-structure interaction (SSI), or the pulse effects of the earthquake ground motions. Hence, this paper focuses on an extension of the equivalent linearization concept to nonlinear soil-multi-story structure systems under pulse-like earthquakes where the whole system is substituted with an equivalent fixed-base SDOF oscillator. To achieve this goal, the underlying soil is assumed as a homogeneous half-space medium and the superstructure is modeled as a nonlinear multiple degrees-of-freedom (MDOF) shear building assuming bilinear behavior for each story. Non-dimensional frequency and aspect ratio parameters are used to enforce the SSI effects in the system. The seismic responses are determined under an extensive ensemble of 91 pulse-like earthquakes for two different ranges of fixed-base structure-to-pulse period ratios, and the ELPs are determined for a range of soil-structure systems throughout a statistical framework. It is found that higher modes have decreasing effects on the ELPs of soil-MDOF structure systems, while the effects of the strain-hardening ratio of each story are insignificant. In addition, the impulsive nature of the ground motions strongly affects the ELPs, and this influence is far more pronounced for structure-to-pulse period ratios between 0.8 and 1.2, i.e., very close to the resonance of the first mode of the superstructure.     

An adaptive numerical scheme based on the Craig–Bampton method for the dynamic analysis of tall buildings

A new numerical scheme is proposed to perform a nonlinear dynamic analysis for tall buildings. The structural components (beams and columns) of tall buildings gradually enter the inelastic phase under strong seismic excitation. Because the distribution of nonlinear components is initially unknown due to the randomness of earthquake inputs, a group of linear and nonlinear substructures are automatically figured out during the time‐history analysis of a structure. Then a modified Craig–Bampton method is proposed to condense the DOFs of the linear substructures in modal coordinates at each time step while keeping the governing equation of the nonlinear substructure in physical coordinates. The dominant modes of the linear substructures are selected to capture the main dynamic characteristics of the structure. The time step integration analysis is used to solve the governing equation of the structures in hybrid coordinates. A 20‐story building is employed as the numerical simulation test to validate the feasibility and effectiveness of the proposed numerical scheme. This scheme provides a new method for the nonlinear dynamic analysis of tall buildings with acceptable simulation accuracy and high computational efficiency.     

Prediction of nonlinear seismic demands of high‐rise rocking wall structures using a simplified modal pushover analysis procedure

This study presents a simplified analysis procedure for the convenient estimation of nonlinear seismic demands of high‐rise rocking wall structures. For this purpose, the displacement modification approach used in the nonlinear static procedure of ASCE/SEI 41‐13 is adopted. However, in the current study, this approach is extended to every significant vibration mode of the structure whereas the displacement modifying coefficients for different modes are calculated using the typical flag‐shaped hysteresis behavior of rocking walls. The parameters of this hysteresis behavior are selected to represent rocking walls with a practical range of energy dissipation capacity and postgap‐opening stiffness. The computed peak inelastic‐to‐elastic displacement ratios are presented as mean spectra, which can be used for the convenient estimation of pushover target displacement for every significant vibration mode. The accuracy of proposed procedure is examined using the seismic demands obtained from the nonlinear response history analysis of a 20‐story case study rocking wall structure. Furthermore, a modal decomposition technique is used to determine the individual modal seismic demands. The proposed procedure is found to predict both the combined and the individual modal demands with a reasonable accuracy and can serve as a convenient analysis option for the design and performance evaluation of high‐rise rocking wall systems.     

Modelling shear–flexure interaction in equivalent frame models of slender reinforced concrete walls

Quasi‐static cyclic tests on reinforced concrete (RC) walls have shown that shear deformations can constitute a significant ratio of the total deformations when the wall is loaded beyond the elastic regime. For slender RC walls that form a stable flexural mechanism, the ratio of shear to flexural deformations remains approximately constant over the entire range of imposed displacement ductilities. This paper proposes a method for incorporating shear‐flexure interaction effects in equivalent frame models of slender RC walls by coupling the shear force‐shear strain relationship to the curvature and axial strain in the member. The suggested methodology is incorporated in a finite element consisting of two interacting spread inelasticity sub‐elements representing flexural and shear response, respectively. The element is implemented in the general finite element code IDARC and validated against experimental results of RC cantilever walls. In a second step, it is applied in inelastic static and dynamic analyses of tall wall and wall‐frame systems. It is shown that ignoring shear‐flexure interaction may lead to erroneous predictions in particular of local ductility and storey drift demands. Copyright © 2013 John Wiley & Sons, Ltd.     

线杆系多层多间框架非线性地震反应的简化计算

本文将多层多间线杆系框架遵循一定规则简化为多层单柱线杆系型框架,从而简化了多层乡间线杆系框架的非线性地震反应分析,所获得的计算结果,无论是层间相对位移或楼层绝对位移,都比较令人满意。     

设置BRB桥梁排架墩基于位移抗震设计方法

基于“保险丝”和“损伤控制”的抗震设计理念,提出在桥梁双柱式排架墩中通过设置屈曲约束支撑(BRB)以提高其横桥向抗震性能的构想。首先从获得“抗震能力”的角度对设置BRB桥梁排架墩的抗震设计参数进行系统性分析,推导出与剪跨比和墩柱间距与直径(边长)比相关的BRB核心段最大和最小长度取值范围|再从求解“地震需求”角度建立了设置保险丝的(SDOF)主结构体系弹塑性反应谱基本方程,分析该体系的非线性地震反应一般规律|然后,基于体系的“抗震能力”和“地震需求”,发展了设置BRB的桥梁排架墩基于位移的抗震设计方法,并结合一个具体桥梁排架墩实例说明建议设计方法的可行性。     

A modified response spectrum analysis procedure to determine nonlinear seismic demands of high‐rise buildings with shear walls

The standard response spectrum analysis (RSA) procedure prescribed in various design codes is commonly used by practicing engineers to determine the seismic demands for structural design purpose. In this procedure, the elastic force demands of all significant vibration modes are first combined and then reduced by a response modification factor (R) to get the inelastic design demands. Recent studies, however, have shown that the response of higher vibration modes may experience much lower level of nonlinearity, and therefore, it may not be appropriate to reduce their demand contributions by the same factor. In this study, a modified RSA procedure based on equivalent linearization concept is presented. The underlying assumptions are that the nonlinear seismic demands can be approximately obtained by summing up the individual modal responses and that the responses of each vibration mode can be approximately represented by those of an equivalent linear SDF system. Using 3 high‐rise buildings with reinforced concrete shear walls (20‐, 33‐, and 44‐story high), the accuracy of this procedure is examined. The inelastic demands computed by the nonlinear response history analysis procedure are used as benchmark. The modified RSA procedure is found to provide reasonably accurate demand estimations for all case study buildings.     

Estimation of inelastic displacement factor of soil‐shallow‐foundation MDOF systems incorporating higher modes effect

This study attempts to investigate the higher‐mode effects on the constant‐ductility inelastic displacement factors of multi‐degree‐of‐freedom (MDOF) systems considering soil‐structure interaction. These factors were computed for 12,600 two‐dimensional superstructure models of shear buildings and their corresponding equivalent single‐degree‐of‐freedom (ESDOF) systems under 26 ground motions recorded on alluvium and soft soil. An intensive parametric study was carried out for a wide range of non‐dimensional parameters, which completely define the problem. The underlying soil is considered as a homogeneous half‐space based on the concept of cone model. The higher‐mode effects were then investigated through defining the ratio of inelastic displacement factor of MDOF system to that of the corresponding ESDOF one. The influence of soil‐structure interaction key parameters, fundamental period, ductility ratio, the number of stories, and dispersion of the results are evaluated and discussed. Results indicate that as the base becomes very flexible, unlike to the fixed‐base systems, in which the defined ratios are greater than unity, using the inelastic displacement factors of ESDOF models for MDOF ones would result in a remarkable overestimation of maximum inter‐story displacement demand. A new expression is proposed to estimate the ratio of inelastic displacement factor of MDOF soil‐structure systems to that of SDOF counterpart.     

Assessment of inelastic response of buildings using force‐ and displacement‐based approaches

In recognition of the increasing importance of accurate seismic vulnerability assessment, this paper deals with procedures and the application of inelastic acceleration and displacement spectra in the seismic assessment of buildings. An identification procedure is outlined, whereby an equivalent single degree of freedom (SDOF) system is devised to represent the building. The SDOF system characteristics (stiffness, strength, post‐peak force response and ductility) are readily evaluated from observation of the seismic response of buildings and simple mechanics. The characteristics are then tuned using measurements from instrumented buildings. Based on the earthquake scenario and structural response characteristics, appropriate inelastic acceleration and displacement spectra are selected and used to ‘predict’ the response. Comparison between the measured and predicted responses for the five buildings studied in the paper confirm the feasibility of the procedure and the realism of the results. Copyright © 2000 John Wiley & Sons, Ltd.