Torsional response of symmetric buildings to incoherent and phase‐delayed earthquake ground motion

This paper studies the effect of coherency loss and wave passage on the seismic torsional response of three‐dimensional, multi‐storey, multi‐span, symmetric, linear elastic buildings. A model calibrated against statistical analyses of ground motion records in Mexico City is used for the coherency function. The structural response is assessed in terms of shear forces in structural elements. Incoherence and wave passage effects are found to be significant only for columns in the ground level of stiff systems. The increase of column shears in the ground level is much higher for soft than for firm soil conditions. For the torsionally stiff systems considered, it is found that incoherent and phase‐delayed ground motions do not induce a significant rotational response of the structure. The use of a code eccentricity to account for torsion due to ground motion spatial variation is assessed. On firm soil, the use of a base shear along with an accidental eccentricity results in highly overestimated shear forces; however, for soft soil conditions, code formulations may result in underestimated shear forces. Copyright © 2003 John Wiley & Sons, Ltd.     

Evaluation of combination rules for maximum response calculation in multicomponent seismic analysis

The existing rules for combining peak response to individual components of ground motion are evaluated. The response values r_e to two horizontal components of ground motion estimated by four multicomponent combination rules—SRSS‐, 30%‐, 40%‐ and simplified‐SRSS‐rules—are compared with the critical response, r_cr, obtained by the CQC3‐rule, which takes into account the direction of the principal ground components with respect to the structural axes and provides the largest response over all possible seismic incident angles. The following results are obtained in the first part of the paper and are valid for any elastic structure and any earthquake design response spectrum: For realistic values of the ratio γ of the design spectra for the two principal components of ground motion the SRSS‐rule estimate lies between 0.79r_cr and 1.00r_cr, the Simplified‐SRSS‐rule estimate lies between 1.00r_cr and 1.26r_cr, the 40%‐rule estimate lies between 0.99r_cr and 1.25r_cr, and the 30%‐rule estimate lies between 0.92r_cr and 1.16r_cr. None of the multicomponent combination rules account for the increase in response of systems if the vibration periods of the two modes that contribute most to the response to the x‐ and y‐components of ground motion are close to each other. Evaluated in the second part of the paper is the accuracy of the multicomponent combination rules in estimating the response of a range of one‐storey systems with (a) symmetrical plan and (b) unsymmetrical plan, and of two multistorey buildings. The SRSS‐rule underestimates the response by up to 16% and the other three rules overestimate it by up to 18%. Although these errors appear to be smaller than the many approximations inherent in structural design, they can be eliminated with very little additional computation by using an explicit formula for the critical response based on the CQC3 rule. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

A modal pushover analysis procedure to estimate seismic demands for unsymmetric‐plan buildings

An Erratum has been published for this article in Earthquake Engng. Struct. Dyn. 2004; 33:1429. Based on structural dynamics theory, the modal pushover analysis (MPA) procedure retains the conceptual simplicity of current procedures with invariant force distribution, now common in structural engineering practice. The MPA procedure for estimating seismic demands is extended to unsymmetric‐plan buildings. In the MPA procedure, the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by non‐linear static analysis using the inertia force distribution for each mode, which for unsymmetric buildings includes two lateral forces and torque at each floor level. These ‘modal’ demands due to the first few terms of the modal expansion are then combined by the CQC rule to obtain an estimate of the total seismic demand for inelastic systems. When applied to elastic systems, the MPA procedure is equivalent to standard response spectrum analysis (RSA). The MPA estimates of seismic demand for torsionally‐stiff and torsionally‐flexible unsymmetric systems are shown to be similarly accurate as they are for the symmetric building; however, the results deteriorate for a torsionally‐similarly‐stiff unsymmetric‐plan system and the ground motion considered because (a) elastic modes are strongly coupled, and (b) roof displacement is underestimated by the CQC modal combination rule (which would also limit accuracy of RSA for linearly elastic systems). Copyright © 2004 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.     

Seismic code analysis of multi-storey asymmetric buildings

Static torsional provisions in most seismic codes require that the lateral force at each floor level be applied at some distance from the reference centre at that floor. However, codes do not specify how to determine the locations of these centres. As a result, several different definitions of the reference centres are being used to implement the code analysis. This investigation examined how the results using various reference centres differ and which of these centres would lead to results that are in agreement with those of dynamic analysis. For this purpose three different buildings ranging form torsionally stiff to torsionally flexible were analysed. It was shown that for the class of buildings studied in this investigation that although the locations of the reference centres were quite different, the results were very similar and nearly independent of the reference centre. Comparison of results calculated from static code equivalent lateral force procedures and results from dynamic response spectrum analyses showed that the static code procedures led to design forces very close (flexible wall) or slightly conservative (stiff wall) when compared to the dynamic analysis for the torsionally stiff building. However, the static code procedures significantly underestimated the design forces of the stiff walls and significantly overestimated the design forces of the flexible walls for the torsionally flexible buildings. © 1998 John Wiley & Sons, Ltd.     

Estimation of inelastic seismic deformations in asymmetric multistorey RC buildings

Four real buildings with three to six stories, strong irregularities in plan and little engineered earthquake resistance are subjected to inelastic response‐history analyses under 56 bidirectional EC8‐spectra‐compatible motions. The average chord rotation demand at each member end over the 56 response‐history analyses is compared to the chord rotation from elastic static analysis with inverted triangular lateral forces or modal response spectrum analysis. The storey‐average inelastic‐to‐elastic‐chord‐rotation‐ratio was found fairly constant in all stories, except when static elastic analysis is applied to buildings with large higher mode effects. Except for such buildings, static elastic analysis gives more uniform ratios of inelastic chord rotations to elastic ones within and among stories than modal response spectrum analysis, but generally lower than 1.0. With increasing EPA the building‐average inelastic‐to‐elastic‐chord‐rotation‐ratio decreases but scatter in the results increases. Static elastic analysis tends to overestimate the inelastic torsional effects at the flexible or central part of the torsionally flexible buildings and underestimate them at their stiff side. Modal response spectrum analysis tends to overestimate the inelastic torsional effects at the stiff or central part of the torsionally stiff buildings and underestimate them at the flexible side. Overall, for multistorey RC buildings that typically have fundamental periods in the velocity‐sensitive part of the spectrum, elastic modal response spectrum analysis with 5% damping gives on average unbiased and fairly accurate estimates of member inelastic chord rotations. If higher modes are not significant, elastic static analysis in general overestimates inelastic chord rotations of such buildings, even when torsional effects are present. Copyright © 2007 John Wiley & Sons, Ltd.     

Design of torsionally unbalanced structural systems based on code provisions I: Ductility demand

Using a three element single mass model, this paper presents the ductility demands on the elements of torsionally unbalanced systems when subjected to strong earthquake shaking. Torsionally unbalanced systems based on nine structural configurations are considered, ranging from torsionally stiff systems with the centre of rigidity (CR) centrally located to torsionally flexible systems with CR eccentrically located. The strength of the elements is designed based on the Canadian and New Zealand codes, and the Uniform Building Code (UBC) of the United States. It is shown that all three codes can limit the ductility demands on the elements to that of a similar but torsionally balanced system when the system is torsionally stiff. However, substantial additional ductility demands on the element at the stiff edge of the system exist for torsionally flexible systems when the New Zealand code or UBC is used. The large ductility demand is caused by the low strength of the stiff-edge element permitted by these codes.     

Generalized force vectors for multi‐mode pushover analysis of torsionally coupled systems

A generalized multi‐mode pushover analysis procedure was developed for estimating the maximum inelastic seismic response of symmetrical plan structures under earthquake ground excitations. Pushover analyses are conducted with story‐specific generalized force vectors in this procedure, with contributions from all effective modes. Generalized pushover analysis procedure is extended to three‐dimensional torsionally coupled systems in the presented study. Generalized force distributions are expressed as the combination of modal forces to simulate the instantaneous force distribution acting on the system when the interstory drift at a story reaches its maximum value during seismic response. Modal contributions to the generalized force vectors are calculated by a modal scaling rule, which is based on the complete quadratic combination. Generalized forces are applied to the mass centers of each story incrementally for producing nonlinear static response. Maximum response quantities are obtained when the individual frames attain their own target interstory drift values in each story. The developed procedure is tested on an eight‐story frame under 15 ground motions, and assessed by comparing the results obtained from nonlinear time history analysis. The method is successful in predicting the torsionally coupled inelastic response of frames responding to large interstory drift demands. Copyright © 2014 John Wiley & Sons, Ltd.     

基于中美抗震设计规范的双向水平地震效应组合方法的有效性评估

在地震来临时,一般假设建筑结构同时受到两个正交水平方向分量与一个竖向分量的地震动作用。双向水平地震效应组合方法用于估计两个正交水平分量地震动同时作用时结构的内力效应。本文主要对我国与美国抗震设计规范中规定使用的平方和开平方根（SRSS）方法与百分比组合方法的有效性进行了评估。首先,对比了我国与美国规范在考虑双向水平地震效应时的适用情况及相关规定上的异同。以一4层中心支撑-框架结构为工程案例,考虑两国规范在适用情况上的规定,设置了三个结构布置方案。对三个结构布置方案建立有限元模型,选取22组地震动,开展了动力时程分析。提出了针对SRSS方法与百分比组合方法的评估指标,基于时程分析结果,发展了双向水平地震效应组合的概率性评估方法。评估结果表明:SRSS方法与百分比组合方法用于平面扭转不规则结构的设计较为保守。在简化组合规则的适用条件上,美国规范对平面扭转不规则结构不进行考虑有一定的合理性。建议我国规范对中心支撑-框架结构中含双向受压柱的设计要求考虑双向水平地震效应组合。     

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.     

Improved earthquake resistant design of eccentric steel buildings

Recently, a design modification has been proposed for eccentric, torsionally stiff, braced steel buildings, designed according to the current Eurocodes 3 and 8, that improves noticeably their inelastic response under the action of design level earthquakes. The improvement consists in a more uniform distribution of ductility demands throughout the building. In the present paper, a similar, though differently derived, modification is applied to torsionally flexible eccentric buildings and their response is again evaluated under pairs of design earthquake motions. A substantial improvement of their inelastic response is also observed, similar to what had been obtained for torsionally stiff buildings. The new approach is also tested with torsionally stiff buildings and leads to similarly satisfactory results. Thus it may be recommended for general application.     

Importance of seismic design accidental torsion requirements for building collapse capacity

Seismic ground motions induce torsional responses in buildings that can be difficult to predict. To compensate for this, most modern building codes require the consideration of accidental torsion when computing design earthquake forces. This study evaluates the influence of ASCE/SEI 7 accidental torsion seismic design requirements on the performance of 230 archetypical buildings that are designed with and without accidental torsion design provisions, taking building collapse capacity as the performance metric. The test case archetypes include a broad range of heights, gravity load levels, and plan configurations. Results show that the ASCE/SEI 7 accidental torsion provisions lead to significant changes in collapse capacity for buildings that are very torsionally flexible or asymmetric. However, only inconsequential changes in collapse capacity are observed in the buildings that are both torsionally stiff and regular in plan. Therefore, the study concludes that accidental torsion provisions are not necessary for seismic design of buildings without excessive torsional flexibility or asymmetry. Copyright © 2013 John Wiley & Sons, Ltd.     

The extended consecutive modal pushover procedure for estimating the seismic demands of two-way unsymmetric-plan tall buildings under influence of two horizontal components of ground motions

This paper aims to extend the consecutive modal pushover (CMP) procedure for estimating the seismic demands of two-way unsymmetric-plan tall buildings subjected to bi-directional seismic ground motions taking the effects of higher modes and torsion into account. Multi-stage and single-stage pushover analyses are carried out in both X and Y directions. Inelastic seismic responses obtained by multi-stage and single-stage pushover analyses for X and Y directions are combined using the SRSS combination scheme. The final seismic responses are determined by enveloping the combined results of multi-stage and single-stage pushover analyses. To evaluate the accuracy of the proposed procedure, it is applied to two-way unsymmetric-plan tall buildings which include torsionally stiff and torsionally flexible systems. The results derived from the CMP procedure are compared with those from nonlinear response history analysis (NL-RHA), as a benchmark solution. Moreover, the advantages of the proposed procedure are demonstrated by comparing the results derived from the CMP to those from pushover analysis with uniform and fundamental effective mode distributions. The proposed procedure is able to accurately predict amplification or de-amplification of the seismic displacements at the flexible and stiff edges of the two-way unsymmetric-plan tall buildings by considering the effects of higher modes and torsion. The extended CMP procedure can accurately estimate the peak inelastic responses, such as displacements and storey drifts. The CMP procedure features a higher potential in estimating plastic hinge rotations at both flexible and stiff sides of unsymmetric-plan tall buildings under bi-directional seismic excitation when compared to the uniform and fundamental effective mode force distributions.     

Late resonant response at Texcoco, Valley of Mexico, during distant earthquakes

Recordings of the ground motion induced by two shallow (15–25 km deep), distant (300 and 605 km) earthquakes made on deep, soft lacustrine sediments at Texcoco, Valley of Mexico, show a late monochromatic response at 0.48 Hz. Data from a strong-motion recorder array show that this late response is consistent with slow (60 m/s group velocity) Rayleigh waves generated near the 6 km distant soft/stiff soil interface of the ex-lake surface margin. It is concluded that the excitation of local Rayleigh waves in soft soil deposits by arriving earthquake ground motion provides one mechanism to explain the prolonged duration of resonant motion on soft soils, and hence the extreme damage often associated with soft soils responding to distant 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.     

Accidental torsion in buildings due to base rotational excitation

This investigation is concerned with accidental torsion in buildings resulting from rotational excitation (about a vertical axis) of the building foundations as a result of spatially non-uniform ground motions. Because of this accidental torsion, the displacements and deformations in the structural elements of the building are likely to increase. This increase in response is evaluated using actual base rotational excitations derived from ground motions recorded at the base of 30 buildings during recent California earthquakes. Accidental torsion has the effect of increasing the building displacements, in the mean, by less than 5 per cent for systems that are torsionally stiff or have lateral vibration periods longer than half a second. On the other hand, short period (less than half a second) and torsionally flexible systems may experience significant increases in response due to accidental torsion. Since the dependence between this increase in response and the system parameters is complex, two simplified methods are developed for conveniently estimating this effect of accidental torsion. They are the ‘accidental eccentricity’ and the ‘response spectrum’ method. The computed accidental eccentricities are much smaller than the typical code values, 0.05bb or 0.1b, except for buildings with very long plan dimensions (b ≥ 50 m). Alternatively, by using the response spectrum method the increase in response can be estimated by computing the peak response to each base motion independently and combining the peak values using the SRSS rule.     

Seismic performance of three-dimensional frame structures with underground stories

This paper investigates the seismic performance of moment-resisting frame steel buildings with multiple underground stories resting on shallow foundations. A parametric study that involved evaluating the nonlinear seismic response of five, ten and fifteen story moment-resisting frame steel buildings resting on flexible ground surface, and buildings having one, three and five underground stories was performed. The buildings were assumed to be founded on shallow foundations. Two site conditions were considered: soil class C and soil class E, corresponding to firm and soft soil deposits, respectively. Vancouver seismic hazard has been considered for this study. Synthetic earthquake records compatible with Vancouver uniform hazard spectrum (UHS), as specified by the National Building Code of Canada (NBCC) 2005, have been used as input motion. It was found that soil–structure interaction (SSI) can greatly affect the seismic performance of buildings in terms of the seismic storey shear and moment demand, and the deformations of their structural components. Although most building codes postulate that SSI effects generally decrease the force demand on buildings, but increase the deformation demand, it was found that, for some of the cases considered, SSI effects increased both the force and deformation demand on the buildings. The SSI effects generally depend on the stiffness of the foundation and the number of underground stories. SSI effects are significant for soft soil conditions and negligible for stiff soil conditions. It was also found that SSI effects are significant for buildings resting on flexible ground surface with no underground stories, and gradually decrease with the increase of the number of underground stories.     

Static versus modal analysis: influence on inelastic response of multi-storey asymmetric buildings

The paper investigates the influence of design procedures on the seismic response of multi-storey asymmetric buildings. To this end, some structures are designed according to methods based on either static or modal analysis, with or without design eccentricities. The seismic response of these systems is determined by means of inelastic dynamic analyses and the design is thoroughly examined in order to explain the results of the dynamic analyses. Attention is basically focused on the ability of design methods to prevent asymmetric buildings from experiencing ductility demands much larger than those of the corresponding torsionally balanced systems. Numerical analyses underline that while design procedures based on either static or modal analysis are suitable for the design of torsionally rigid structures only those based on modal analysis lead to the satisfactory performance of torsionally flexible buildings. Furthermore, the study highlights the qualities of a design method proposed by the Authors. Its application does not require any explicit calculation of design eccentricities and leads to proper seismic response of both torsionally rigid and flexible asymmetric buildings.     

Effect of irregular structural configuration on floor acceleration demand in hill‐side buildings

A suite of reinforced‐concrete frame buildings located on hill sides, with 2 different structural configurations, viz step‐back and split‐foundation, are analyzed to study their floor response. Both step‐back and split‐foundation structural configurations lead to torsional effects in the direction across the slope due to the presence of shorter columns on the uphill side. Peak floor acceleration and floor response spectra are obtained at each storey's center of rigidity and at both its stiff and flexible edges. As reported in previous studies as well, it is observed that the floor response spectra are better correlated with the ground response spectrum. Therefore, the floor spectral amplification functions are obtained as the ratio of spectral ordinates at different floor levels to the one at the ground level. Peaks are observed in the spectral amplification functions corresponding to the first 2 modes in the upper portion of the hill‐side buildings, whereas a single peak corresponding to a specific kth mode of vibration is observed on the floors below the uppermost foundation level. Based on the numerical study for the step‐back and split‐foundation hill‐side buildings, simple floor spectral amplification functions are proposed and validated. The proposed spectral amplification functions take into account both the buildings' plan and elevation irregularities and can be used for seismic design of acceleration‐sensitive nonstructural components, given that the supporting structure's dynamic characteristics, torsional rotation, ground‐motion response spectrum, and location of the nonstructural components within the supporting structure are known, because current code models are actually not applicable to hill‐side buildings.