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.     

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.     

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.     

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.     

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.     

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.     

Laboratory tests of steel simple torsionally unbalanced models

The objective of this work is to obtain estimations of the amplification factors α and δ used for torsion design of buildings, from experiments. For this study, simple one‐storey torsionally unbalanced (TU) steel models were considered. Models consisted of a deck supported on four columns with a selected arrangement of hinges at column ends. Two theoretical structural eccentricities (e = 0.05 and 0.15) were considered. Models were excited with a simple long‐period pendulum consisting of a hanging platform with a forced‐vibration generator on it. Eight models were tested at several excitation levels (frequencies and force magnitudes) in both ranges of behaviour: elastic and inelastic. Experiments were conducted at three frequency ratios of excitation. Registered accelerations of the pendulum platform indicate that the experimental set‐up leads to excitations that resemble narrow‐band seismic ground motions. Frame shear force estimations, based on accelerations recorded at both deck sides, indicate that torsion design factors (α and δ) depend on eccentricity. Estimations of frame shears based on measurements indicate that for normalized eccentricities e ? 0.025, the amplification α can be between 2 and 3; while δ factor resulted between 0.0 and 1.6. 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.     

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.     

Parameter identification of torsionally coupled shear buildings from earthquake response records

This paper presents an efficient procedure to determine the natural frequencies, modal damping ratios and mode shapes for torsionally coupled shear buildings using earthquake response records. It is shown that the responses recorded at the top and first floor levels are sufficient to identify the dominant modal properties of a multistoried torsionally coupled shear building with uniform mass and constant eccentricity even when the input excitation is not known. The procedure applies eigenrealization algorithm to generate the state‐space model of the structure using the cross‐correlations among the measured responses. The dynamic characteristics of the structure are determined from the state‐space realization matrices. Since the mode shapes are obtained only at the instrumented floor (top and first floors) levels, a new mode shape interpolation technique has been proposed to estimate the mode shape coefficients at the remaining floor levels. The application of the procedure has been demonstrated through a numerical experiment on an eight‐storied torsionally coupled shear building subjected to earthquake base excitation. The results show that the proposed parameter identification technique is capable of identifying dominant modal parameters and responses even with significant noise contamination of the response records. Copyright © 2008 John Wiley & Sons, Ltd.     

Response to orthogonal components of ground motion and assessment of percentage combination rules

The effects of horizontal components of ground motion on the linear response of torsionally stiff and torsionally flexible systems, on soft and firm soil conditions, are examined. A one‐story, two‐way asymmetric structural system is used, subjected to uncorrelated ground motion components along their principal directions. Spectral densities for ground accelerations in firm and soft soils are modeled based on recorded data from large intensity Mexican earthquakes. It is shown that for firm soils, in general, these effects are important in the case of torsionally flexible systems that are stiff under translation, or for torsionally stiff systems that are flexible in translation. The percentage combination rules usually specified in seismic design codes are assessed against the dynamic response. Such combination rules can result in overly conservative design forces or underestimated design forces, particularly for torsionally flexible structures. Given the relative magnitude of the response to each ground motion component, it was found that using different percentage values in the combination rules has no significant effect on improving the estimation of the total response. Copyright © 2003 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.     

Elastic earthquake analysis of torsionally coupled multistorey buildings

With the aid of perturbation analysis of vibration frequencies and mode shapes it is shown that any lower vibration mode of a torsionally coupled building may be approximated as a linear combination of three vibration modes of the corresponding torsionally uncoupled system (a system with coincident centres of mass and resistance but all other properties are identical to the actual system): one translational mode along each of the two principal axes of resistance and one mode in torsional vibration. This result provides the motivation for a simpler—relative to the standard—procedure for analysing the response of torsionally coupled multistorey buildings to earthquake ground motion. To illustrate the application and accuracy of this procedure two numerical examples are presented.     

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.     

A macro‐element model for inelastic building analysis

A three‐dimensional model for approximate inelastic analysis of buildings is presented herein. The model is based on a single macro‐element per building storey. The inelastic properties of the model are characterized by the so‐called ultimate storey shear and torque (USST) surfaces. Different algorithms for the construction of these surfaces, as well as their applications in building modelling, are presented and discussed. Two alternative procedures are developed to integrate the force‐deformation constitutive relationship of the macro‐elements. The first one follows the exact trajectory of the load path of the structure on the USST, and the second uses linear programming without ever forming the USST surface. The accuracy of the model and integration procedure is evaluated by means of the earthquake response of single‐storey systems. The model and integration procedure developed is finally used to compute the inelastic response of a seven‐storey R/C building. The results of this investigation show that the model proposed, although approximate, can be effective in estimating the inelastic deformation demand of a building. It also enables the engineer to capture and interpret important features of the three‐dimensional inelastic response of a structure even before performing any inelastic dynamic analysis. Copyright © 2000 John Wiley & Sons, Ltd.     

Vibration of vertical rectangular plate in contact with water on one side

In this paper, the vibratory characteristics of a rectangular plate in contact with water on one side are studied. The elastic plate is considered to be a part of a vertical rectangular rigid wall in contact with water, the edges of which are elastically restrained and parallel to those of the rigid wall. The location and size of the plate on the rigid wall may vary arbitrarily. The water with a free surface is in a rectangular domain infinite in the length direction. The effects of free surface waves, compressibility of the water and the hydrostatic water pressure are neglected in the analysis. An analytical‐Ritz method is developed to analyse the interaction of the plate–water system. First of all, by using the method of separation of variables and the method of Fourier series expansion, the exact expression of the motion of water is derived in the form of integral equations including the dynamic deformation of the plate. Then the Rayleigh–Ritz approach is used to derive the eigenfrequency equation of the system via the variational principle of energy. By selecting beam vibrating functions as the admissible functions of the plate, the added virtual mass incremental (AVMI) matrices for plate vibration are obtained. The convergency studies are carried out. The effects of some parameters such as the depth and width of water, the support stiffnesses, location and aspect ratio of the plate and the plate–water size and density ratios on the eigenfrequencies of the plate–water system are investigated. Several numerical examples are given. The validity of AVMI factor approach is also confirmed by comparing the AVMI factor solutions with the analytical‐Ritz solutions. The results show that the approach presented here can also be used as excellent approximate solutions for rectangular plates in contact with water of infinite width and/or infinite depth. Copyright © 2000 John Wiley & Sons, Ltd.     

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 response of torsionally coupled base isolated structures

An analytical study of the seismic response of typical base isolated structures mounted on rubber bearings is presented. Isolated buildings are liable to have closely spaced lower modes of vibration with small eccentricity between centres of mass and rigidity. The isolated structure is modelled as a rigid deck with lumped masses supported on axially inextensible elastomeric rubber bearings. This simplified system has three degrees of freedom (dof), two translations and one rotation in the horizontal plane. The Green's functions for the displacement response of the 3 dof system are derived for both undamped and damped cases with small and large eccentricities. The small eccentricity case is taken from a specific isolated building, while the large eccentricity case arises from the 5 per cent accidental eccentricity which is required by various seismic codes. An interaction equation for normalized displacements is established for an idealized flat velocity spectrum or hyperbolic acceleration spectrum. An isolated building on rubber bearings would have its fundamental period fall into this range of a design spectrum. Numerical results for the specific building subjected to the El Centro earthquake of 1940 are presented. Both the time history and the response spectrum modal superposition analysis were performed. In the response spectrum analysis, the Complete Quadratic Combination (CQC) showed superiority over the Square Root of the Sum of Squares (SRSS) in estimating maximum responses. It is concluded that the effect of torsional coupling on the transient response of base isolated structures is insignificant, due to the combined effect of the time lag between the maximum translational and torsional responses and the influence of damping in the isolation system which for elastomeric bearings can be as high as 8 to 10 per cent.     

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.     

An alternative approach for computing seismic response with accidental eccentricity

Accidental eccentricity is a non-standard assumption for seismic design of tall buildings. Taking it into consideration requires reanalysis of seismic resistance, which requires either time consuming computation of natural vibration of eccentric structures or finding a static displacement solution by applying an approximated equivalent torsional moment for each eccentric case. This study proposes an alternative modal response spectrum analysis(MRSA) approach to calculate seismic responses with accidental eccentricity. The proposed approach, called the Rayleigh Ritz Projection-MRSA(RRP-MRSA), is developed based on MRSA and two strategies:(a) a RRP method to obtain a fast calculation of approximate modes of eccentric structures; and(b) an approach to assemble mass matrices of eccentric structures. The efficiency of RRP-MRSA is tested via engineering examples and compared with the standard MRSA(ST-MRSA) and one approximate method, i.e., the equivalent torsional moment hybrid MRSA(ETM-MRSA). Numerical results show that RRP-MRSA not only achieves almost the same precision as ST-MRSA, and is much better than ETM-MRSA, but is also more economical. Thus, RRP-MRSA can be in place of current accidental eccentricity computations in seismic design.