Torsional Coupling Effects Research Articles

Damage identification of isolators in base-isolated torsionally coupled buildings

This paper deals with the damage assessment for isolators of base-isolated building systems considering the torsion-coupling (TC) effect by establishing damage indices. The damage indices can indicate the reduction in lateral stiffness of the isolator story as explicit formulas in terms of modal parameters. In addition, the damage location, expressed in terms of the estimated damage index and eccentricities before and after damage, is also presented. Numerical analysis shows that the proposed algorithms are applicable for general base-isolated multi-story TC buildings. A procedure from the analysis of seismic response to the implementation of damage indices is demonstrated by using a numerical case. A system identification technique is employed to extract modal parameters from seismic responses of a building. Results show that the proposed indices are capable of detecting the occurrence of damage and preliminarily estimating the location of damaged isolator.

The effects of lateral–torsional coupling on the nonlinear dynamic behavior of a rotating continuous flexible shaft–disk system with rub–impact

This study investigates the lateral–torsional coupling effects on the nonlinear dynamic behavior of a rotating flexible shaft–disk system. The system is modeled as a continuous shaft with a rigid disk in its mid span. Coriolis and centrifugal effects due to shaft flexibility are also included. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed mode method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work include time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The main objective of the present study is to investigate the torsional coupling effects on the chaotic vibration behavior of a system. Periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for cases with and without torsional effects. As demonstrated, inclusion of the torsional–lateral coupling effects can primarily change the speed ratios at which rub–impact occurs. Also, substantial differences are shown to exist in the nonlinear dynamic behavior of the system in the two cases.

Seismic Response of Torsionally Coupled System with Semi-Active Variable Dampers

The seismic response of single-story, one-way asymmetric building with semi-active variable dampers with two-step viscous damping algorithm is investigated. The response is obtained by numerically solving the governing equations of motion. The effects of superstructure eccentricity ratio, uncoupled time period, and uncoupled torsional to lateral frequency ratio are investigated on peak responses of lateral, torsional, and edge displacements and accelerations, as well as damper forces. It is found that the semi-active dampers reduce the lateral-torsional deformations significantly and effects of torsional coupling on effectiveness of control system are more sensitive to variation of eccentricity and torsional to lateral frequency ratio.

Seismic response of asymmetric systems with linear and non-linear viscous dampers

The seismic response of linearly elastic, single-storey, one-way asymmetric building with linear and non-linear viscous dampers is investigated. The response is obtained by numerically solving the governing equations of motion. The effects of eccentricity ratio, uncoupled lateral time period, ratio of uncoupled torsional to lateral frequency and supplemental damping eccentricity ratio are investigated on peak responses which include lateral, torsional and edge displacements and their acceleration counter parts as well as control forces. To study the effectiveness of dampers, the controlled response of asymmetric system is compared with the corresponding uncontrolled response. Further, to study the effects of torsional coupling, the controlled response of asymmetric system is compared with the corresponding symmetric system. It is shown that the non-linear viscous dampers are quite effective in reducing the responses and the damper force depends on system asymmetry and supplemental damping. Also, the effectiveness of dampers significantly depends on structural and damping eccentricity ratio and torsional to lateral frequency ratio and the effects of torsional coupling are found to be more significant for torsionally flexible and strongly coupled systems. Further, effects of torsional coupling are less for asymmetric systems with non-linear dampers as compared to linear dampers.

Seismic Response of Torsionally Coupled System with Magnetorheological Dampers

The seismic response of linearly elastic, idealized single-storey, one-way asymmetric building with semiactive magnetorheological (MR) dampers with clipped-optimal algorithm is investigated. The response is obtained by numerically solving the governing equations of motion. The effects of eccentricity ratio, uncoupled time period, and ratio of uncoupled torsional to lateral frequency are investigated on peak responses which include lateral, torsional and edge displacements and their acceleration counter parts, base shear, and control forces. To study the effectiveness of control system, the controlled response of asymmetric system is compared with the corresponding uncontrolled response. Further, controlled response of asymmetric system is compared with corresponding symmetric system to study the effects of torsional coupling. It is shown that the implementation of semiactive dampers reduces the deformations significantly. Also, the effects of torsional coupling on effectiveness of semiactive system are found to be more sensitive to the variation of eccentricity and torsional to lateral frequency ratio.

Application of Seismic Energy Dissipation Approach to Curved Continuous Girder Bridge

Under earthquake action there are irregular rotating displacements in the curved girder bridge, and the effective seismic mitigation method for curved bridge need to be systematically studied. The spatial finite element model is established for a curved continuous girder bridge with double-column piers. Energy dissipation dampers including E-steel dampers, viscous dampers, friction pendulum bearings and lead rubber bearings are set in radial and tangent directions of the curve at the positions of active bearings. The seismic mitigation effectiveness and seismic response characteristics of the curved girder bridge are calculated and analyzed with the four energy dissipation approaches under three dimensional ground motion action. The results indicate that the four energy dissipation approaches can all effectively reduce the bending moments and torsional moments at the bottom of piers, and the bending torsion coupling effect of the curved girder bridge is decreased a bit. The difference of internal force for inside and outside piers is reduced after the dampers set for the curved girder bridge with double-column piers. Relatively, the viscous damper approach can further reduce the whole seismic responses of curved continuous girder bridge, and is an excellent seismic mitigation method suitable for curved girder bridge.

Pseudodynamic response of torsionally unbalanced two‐storey test structure

AbstractTwo one‐way eccentric, two‐storey, one‐by‐one‐bay reinforced concrete (RC) structures are pseudodynamically tested under unidirectional ground motions. Theoretical considerations about the effect of torsional coupling on modal periods and shapes agree with modal results of the test structure, considering member stiffness is equal to the secant stiffness to yielding in skew‐symmetric bending. Modal periods of such an elastic structure are in fair agreement with effective periods inferred from the measured response at the beginning of a test of a thoroughly cracked structure and at the end of the test. A time‐varying stiffness matrix and a non‐proportional damping matrix fitted to the test results may be used to reproduce the measured response approximately by modal superposition and identify the role of the four time‐varying modes. Flexible side columns sustained very large drift demands simultaneously in the two transverse directions and suffered significant but not heavy, damage at lap‐splices. RC‐jacketing of the flexible side columns practically eliminated the static eccentricity between the floor centres of twist and mass as well as the torsional response. Inelastic time‐history analysis with point‐hinge member models, using as elastic stiffness the secant stiffness to yielding and neglecting post‐ultimate‐strength cyclic degradation of resistance in members with plain bars and poor detailing, predicted fairly well the response until the peak displacements and member deformations occurred. After that, it underestimated displacement peaks and the lengthening of the apparent period and missed the gradual drifting of the response towards a permanent offset. Copyright © 2007 John Wiley & Sons, Ltd.

Dynamic stiffness formulation and free vibration analysis of a spinning composite beam

The dynamic stiffness matrix of a spinning composite beam is developed and then used to investigate its free vibration characteristics. Of particular interest in this study is the inclusion of the bending–torsion coupling effect that arises from the ply orientation and stacking sequence in laminated fibrous composites. The theory is particularly intended for thin-walled composite beams and does not include the effects of shear deformation and rotatory inertia. Hamilton’s principle is used to derive the governing differential equations, which are solved for harmonic oscillation. Exact expressions for the bending displacement, bending rotation, twist, bending moment, shear force and torque at any cross-section of the beam, are also obtained in explicit analytical form. The dynamic stiffness matrix, which relates the amplitudes of loads to those of responses at the end of the spinning beam in free vibration is then derived by imposing the boundary conditions. This enables natural frequency calculation of a spinning composite beam at various spinning speeds to be made by applying the Wittrick–Williams algorithm to the resulting dynamic stiffness matrix. The spinning speed at which the fundamental natural frequency tends to zero is the critical speed, which is established for a composite shaft that has been taken from the literature as an example. The results are discussed and some are compared with published ones. The paper concludes with some remarks.

Base-Isolated Building with Asymmetries Due to the Isolator Parameters

The effects of torsional coupling, due to isolator parameters, on the seismic response of base-isolated buildings are presented. The isolated building is modelled as a single-storey structure mounted on different isolation devices such as elastomeric and sliding systems involving non-linear restoring forces. The governing equations of motion for the uncoupled and torsionally coupled system are derived and solved in time domain by Newmark's method of integration to obtain lateral-torsional displacement response. The displacement response of the isolated system with different combinations of structural configurations, isolation systems and the ratio of uncoupled torsional to lateral frequency of the system is investigated. A comparison of the response of the torsionally coupled base-isolated building is made with the corresponding response obtained from torsionally uncoupled base-isolated building. In addition, a parametric study is conducted to observe the effect of superstructure flexibility on the displacements in torsionally coupled base-isolated building. The eccentricities arising due to the asymmetries in the isolation stiffness and/or yield strength of the isolators are compared with the eccentricity in the system as specified by the Uniform Building Code (UBC 1997). It is observed that the torsional coupling arising due to the dissimilarity in the isolator properties considerably influences the seismic response of the base-isolated building. Effects of superstructure eccentricity are found diminishing when the isolation eccentricities exist. The design bearing displacement suggested by the UBC static formula incorporating accidental torsion is found conservative for the isolation eccentricities arising due to the dissimilarity amongst the isolators.

Optimum properties of multiple tuned mass dampers for reduction of translational and torsional response of structures subject to ground acceleration

The application of multiple tuned mass dampers (MTMD) with identical stiffness and damping coefficient but different mass for suppressing translational and torsional responses is discussed for a simplified two-degree-of-freedom (2DOF) structure, able to represent the dynamic characteristics of general asymmetric structures subject to ground motions. This 2DOF structure is a generalized 2DOF system of an asymmetric structure with predominant translational and torsional responses under earthquake excitations using the mode reduced-order method. Depending on the ratio of the torsional to the translational eigenfrequency, i.e. the torsional to translational frequency ratio (TTFR), of asymmetric structures, the following cases can be distinguished: (1) torsionally flexible structures (TTFR<1.0), (2) torsionally intermediate stiff structures (TTFR=1.0), and (3) torsionally stiff structures (TTFR>1.0). Taking into account the even placement of the MTMD within the width of the asymmetric structure, a careful examination of the effects of the normalized eccentricity ratio (NER) on the performance of the MTMD are carried out with resort to the provided analytical expressions for the dynamic magnification factors (DMF) of both the translational and torsional responses of the asymmetric structure. Extensive numerical simulations have been performed to accurately estimate the dynamic characteristics of the MTMD for asymmetric structures subject to ground acceleration. In the simulations, the dimensionless DMF parameters, bounded between zero and unity, are used as the formal indexes estimating the effectiveness of the MTMD in reducing both the translational and torsional responses of the asymmetric structure. A new basic result is that the NER affects significantly the performance of the MTMD for both torsionally flexible and torsionally intermediate stiff structures; while the influence of the NER is rather negligible on the performance of the MTMD for torsionally stiff structures, thus implying that in such a case the MTMD may be designed by ignoring the effects of torsional coupling. Likewise, the effectiveness and robustness of the MTMD strategies with different layouts are also investigated and demonstrated for the case of mitigating the torsional response of asymmetric structures, thus providing valuable guidance for the MTMD design. Furthermore, the frequency response curves of asymmetric structures without and with both the optimum MTMD and TMD are plotted for the three cases of TTFR as well, consequently obtaining some very useful results.

Seismic performance of multiple tuned mass dampers for soil–irregular building interaction systems

This paper deals with the applicability of multiple tuned mass dampers (MTMD) on the vibration control of irregular buildings modelled as torsionally coupled structures due to base motions considering the soil–structure interaction (SSI) effect. An efficient modal analysis methodology is used to systematically assess the combined soil–structure interaction and torsional coupling effects on asymmetric buildings. This method is implemented in the frequency domain to accurately incorporate the frequency-dependent foundation impedance functions. The performance index of MTMD is established based on the foundation-induced building floor motions with and without the installation of MTMDs. Unlike the traditional MTMD design criteria, the frequency ratio of each MTMD substructure to the controlled structural frequency is independent, in this paper, so that the MTMD with the optimal parameters can actually flatten the transfer functions of building responses. Numerical verifications show that the increase of height–base ratio of an irregular building and the decrease of relative stiffness of soil to structure generally amplify both SSI and MTMD detuning effect, especially for a building with highly torsionally coupled effect. With appropriately enlarging the frequency spacing of the optimal MTMDs, the detuning effect can be reduced. Moreover, the results of numerical investigations also show that the MTMD is more effective than single TMD as the SSI effect is significant.

Exact dynamic stiffness matrix of non-symmetric thin-walled beams on elastic foundation using power series method

Exact dynamic element stiffness matrix for the flexural–torsional free vibration analysis of the shear deformable thin-walled beam with non-symmetric cross-section on two-types of elastic foundation is newly presented using power series method based on the technical computing program Mathematica. For this, the shear deformable beam on elastic foundation theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This beam includes the shear deformation effects due to the shear forces and the restrained warping torsion and due to the coupled effects between them, and rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. And then equations of motion and force–deformation relations are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components and the exact dynamic element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element solutions using the isoparametric beam elements. Particularly the influences of the coupled shear deformation on the vibrational behavior of non-symmetric beam on elastic foundation are investigated.

Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects

A general theory is proposed for the shear deformable thin-walled beam with non-symmetric open/closed cross-sections and its exact dynamic and static element stiffness matrices are evaluated. For this purpose, an improved shear deformable beam theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This includes the shear deformations due to the shear forces and the restrained warping torsion and due to the coupled effects between them, rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. Governing equations and force–deformation relations are derived from the energy principle and a system of linear eigenproblem with non-symmetric matrices is constructed based on 14 displacement parameters. And then explicit expressions for displacement parameters are derived and the exact dynamic and the static stiffness matrices are determined using force–deformation relationships. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with other numerical solutions available in the literature and results using the thin-walled beam element and the shell element. Particularly the influences of the coupled shear deformation on the vibrational and the elastic behavior of non-symmetric beams with various boundary conditions are investigated.

Random response of a rotating composite blade with flexure–torsion coupling effect by the finite element method

A finite element model is employed to investigate the mean square response of a damped rotating composite blade with flexure–torsion interaction under stationary or non-stationary random excitation. The effects of transverse shear deformation and rotary inertia are considered. The finite element model can satisfy all the geometric and natural boundary conditions of a thick blade. The blade is considered to be subjected to white noise, band-limited white noise or filtered white noise excitation. The numerical results indicate that the increment of rotational speed will reduce the mean square response. It is also found that the mean square response decreases when the low natural frequency of base decreases. Inversely, the mean square response increases when the high natural frequency of base decreases. It is also shown that the fiber orientations have a significant effect on the mean square response of an orthotropic blade under random excitations. Moreover, the flexure–torsion coupling effect on the mean square response is changed by different fiber orientations.

Seismic response of torsionally coupled structures with active control device

AbstractA study is carried out for investigating the effectiveness of active control systems for one‐storey torsionally coupled structures. The governing equations of motion of the system are derived. The control forces are obtained using the classical linear optimal control algorithm. The stochastic seismic response of a torsionally coupled controlled system is investigated under important parametric variations such as: uncoupled lateral time period, system eccentricity and ratio of uncoupled torsional to lateral frequency of the system. In order to investigate the effectiveness of the control device, the response of the system with control device is compared with the corresponding system without control device. Further, the controlled response of the torsionally coupled system is compared with the corresponding response of the uncoupled system. It is shown that the control mechanism is also equally effective in reducing the seismic response of a torsionally coupled system. In addition, it is to be noted that the effectiveness of the active control device is overestimated when it is designed without considering the effects of torsional coupling.

Systematic assessment of irregular building–soil interaction using efficient modal analysis

AbstractIn this paper an efficient methodology applying modal analysis is developed to assess systematically the combined soil–structure interaction and torsional coupling effects on asymmetric buildings. This method is implemented in the frequency domain to accurately incorporate the frequency‐dependent foundation impedance functions. For extensively extracting the soil–structure interaction effects, a diagonal transfer matrix in the modal space is derived. A comprehensive investigation of asymmetric building–soil interaction can then be conveniently conducted by examining various types of response quantities. Results of parametric study show that the increasing height‐to‐base ratio of a structure generally amplifies its translational and torsional responses. Moreover, both the translational and torsional responses are reduced for the case where the two resonant frequencies are well separated and this reduction is enhanced with the decreasing values of the relative soil stiffness and the height‐to‐base ratio. The most noteworthy phenomenon may be the fact that the SSI effects can enlarge the translational response if the structure is slender and the two resonant frequencies are very close. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Torsional Displacements in Base-Isolated Buildings

An analytical study of the effects of torsional coupling on the seismic response of a base-isolated building is presented. The isolated structure is modeled as a rigid deck supported on axially inextensible bearings. The governing equations of motion for the coupled lateral-torsional response of the system are derived. The eccentricity in the system is that specified by the Uniform Building Code (UBC). The displacement response of the isolated system with different combinations of building configuration, isolation damping, and the ratio of uncoupled torsional to lateral frequency of the system is investigated. The response of the isolated structure under a variety of near-fault and other earthquake ground motions is compared to that obtained by use of response spectrum analysis. In the response spectrum analysis the accuracy of several modal combination rules is evaluated. It is shown that torsional coupling can influence the response of the isolated structure, but if the layout of the isolation bearings is such that the torsional frequency is larger than the lateral frequency, the effect is reduced and the usual modal combination rules work well. It is also shown that in this case, the UBC static formula for the additional isolator displacements due to torsion is conservative.

Improved formulation for spatial stability and free vibration of thin-walled tapered beams and space frames

A consistent finite element formulation is presented for the free vibration and spatial stability analysis of thin-walled tapered beams and space frames. The kinetic and potential energies are derived by applying the extended virtual work principle, introducing displacement parameters defined at the arbitrarily chosen axis and including second order terms of finite semitangential rotations. As a result, the energy functional corresponding to the semitangential rotation and moment is obtained, in which the elastic strain energy terms are coupled due to axial–flexural–torsional coupling effects but the potential energy due to initial stress resultants has a relatively simple expression. For finite element analysis, cubic polynomials are utilized as the shape functions of the two-noded Hermitian space frame element. Mass, elastic stiffness, and geometric stiffness matrices for the unsymmetric thin-walled cross-section are precisely evaluated, and load-correction stiffness matrices for off-axis concentrated and distributed loadings are considered. In order to illustrate the accuracy and practical usefulness of this formulation, finite element solutions for the free vibration and lateral–torsional buckling problems of thin-walled tapered beam-columns and space frames are presented and compared with available solutions.

単層一軸偏心線形系の地震応答における動的捩れ効果の指標

The indices of the effects of the dynamic torsional coupling are investigated, using the simplest analytical model: a single story linear system with an uniaxial eccentricity. The main results are as follows: The formulae of the peak response under simplified input ground motions were derived in an explicit form. Based on them, the fundamental properties of the dynamic torsional coupling were investigated and two simple indices were proposed. These indices can indicate rationally the effects of the torsional coupling whether the eccentricity is large or small. For example, they cover the torsional coupling induced by the small eccentricity that is inherent in the dynamic one.

Seismic Response of Asymmetric Structures to Multiple Ground Motions

If ground motion is not uniform over a building's base, or if the building is asymmetric, torsional response occurs. This paper investigates the effect of torsional coupling induced by both multiple ground excitations and building asymmetry on earthquake forces. The system studied is an idealized single-story two-degrees-of-freedom building model with four supports. Spatially varying ground excitations at the supports are used. Thirty sets of spatially correlated ground-motion time histories are simulated for the analysis. Each set of the simulated time histories are compatible with an empirical coherency loss function of recorded motions, and they are also individually compatible with the Newmark-Hall response spectrum with 5% damping and normalized to 0.5 g . The mean and standard deviations of base shear and torque corresponding to the 30 sets of simulated time histories are calculated. Results are presented in terms of nondimensional parameters, accounting for both the eccentricity and multiple-ground...