Mode-superposition methods in dynamic analysis of classically and non-classically damped linear systems

Mode-superposition analysis is an efficient tool for the evaluation of the response of linear systems subjected to dynamic agencies. Two well-known mode-superposition methods are available in the literature, the mode-displacement method and the mode-acceleration method. Within this frame a method is proposed called a dynamic correction method which evaluates the structural response as the sum of a pseudostatic response, which is the particular solution of the differential equations, and a dynamic correction evaluated using a reduced number of natural modes. The greater accuracy of the proposed method with respect to the other methods is evidenced through extensive numerical tests, for classically and non-classically damped systems.     

New approaches for non‐classically damped system eigenanalysis

This paper presents three new approaches for solving eigenvalue problems of non‐classically damped linear dynamics systems with fewer calculations than the conventional state vector approach. In the latter, the second‐order differential equation of motion is converted into a first‐order system by doubling the size of the matrices. The new approaches simplify the approach and reduce the number of calculations. The mathematical formulations for the proposed approaches are presented and the numerical results compared with the existing method by solving a sample problem with different damping properties. Of the three proposed approaches, the expansion approach was found to be the simplest and fastest to compute. Copyright © 2005 John Wiley & Sons, Ltd.     

Identification of modal parameters of non‐classically damped linear structures under multi‐component earthquake loading

A method for parametric system identification of classically damped linear system in frequency domain is adopted and extended for non‐classically damped linear systems subjected up to six components of earthquake ground motions. This method is able to work in multi‐input/multi‐output (MIMO) case. The response of a two‐degree‐of‐freedom model with non‐classical damping, excited by one‐component earthquake ground motion, is simulated and used to verify the proposed system identification method in the single‐input/multi‐output case. Also, the records of a 10 storey real building during the Northridge earthquake is used to verify the proposed system identification method in the MIMO case. In this case, at first, a single‐input/multi‐output assumption is considered for the system and modal parameters are identified, then other components of earthquake ground motions are added, respectively, and the modal parameters are identified again. This procedure is repeated until all four components of earthquake ground motions which are measured at the base level of the building are included in the identification process. The results of identification of real building show that consideration of non‐classical damping and inclusion of the multi‐components effect of earthquake ground motions can improve the least‐squares match between the finite Fourier transforms of recorded and calculated acceleration responses. Copyright © 2006 John Wiley & Sons, Ltd.     

New real‐value modal combination rules for non‐classically damped structures

A procedure is presented to determine new modal combination rules (both CQC and SRSS) for non‐classically damped structures. The procedure presented in this paper does not need the solution of any complex eigenvalue problem, in contrast to other methods found in the literature. Thus, the modal combination rules presented here are easily applicable, even by those engineers who are unaccustomed to using complex algebra. Moreover, these formulations show the further advantage of requiring the response spectra only for the target damping ratio value. So the use of approximated formulae, necessary for passing from the response spectrum with the target damping ratio value to other ones, is avoided. Copyright © 2004 John Wiley & Sons, Ltd.     

Modal time history analysis of non-classically damped structures for seismic motions

The step-by-step modal time history integration methods are developed for dynamic analysis of non-classically damped linear structures subjected to earthquake-induced ground motions. Both the mode displacement and mode acceleration-based algorithms are presented for the calculation of member and acceleration responses. The complex-valued eigenvectors are used to effect the modal decoupling of the equations of motion. However, the recursive step-by-step algorithms are still in terms of real quantities. The numerical results for the acceleration response and floor response spectra, obtained with these approaches, are presented. The mode acceleration approach is observed to be decidedly better than the mode displacement approach in as much as it alleviates the so-called missing mass effect, caused by the truncation of modes, very effectively. The utilization of the mode acceleration-based algorithms is, thus, recommended in all dynamic analyses for earthquake-induced ground motions.     

Application of MPA to estimate probability of collapse of structures

This paper develops a modal pushover analysis‐ (MPA) based approximate procedure to quantify the collapse potential of structural systems. The computationally demanding incremental dynamic analysis (IDA) of the structural system is avoided by MPA of the structure in conjunction with empirical equations for the collapse strength ratio for the first‐mode single‐degree‐of‐freedom (SDF) system; higher modes of vibration play essentially no role in estimating the ground motion intensity required to cause collapse of the structure. Presented are collapse fragility curves for 6‐, 9‐, and 20‐story regular special moment‐resisting teel frames computed by the exact and approximate procedures, demonstrating that the MPA‐based approximate procedure requires only a small fraction (1% in one example) of the computational effort inherent in exact IDA and still achieves highly accurate results. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Envelope‐based pushover analysis procedure for the approximate seismic response analysis of buildings

An envelope‐based pushover analysis procedure is presented that assumes that the seismic demand for each response parameter is controlled by a predominant system failure mode that may vary according to the ground motion. To be able to simulate the most important system failure modes, several pushover analyses need to be performed, as in a modal pushover analysis procedure, whereas the total seismic demand is determined by enveloping the results associated with each pushover analysis. The demand for the most common system failure mode resulting from the ‘first‐mode’ pushover analysis is obtained by response history analysis for the equivalent ‘modal‐based’ SDOF model, whereas demand for other failure modes is based on the ‘failure‐based’ SDOF models. This makes the envelope‐based pushover analysis procedure equivalent to the N2 method provided that it involves only ‘first‐mode’ pushover analysis and response history analysis of the corresponding ‘modal‐based’ SDOF model. It is shown that the accuracy of the approximate 16th, 50th and 84th percentile response expressed in terms of IDA curves does not decrease with the height of the building or with the intensity of ground motion. This is because the estimates of the roof displacement and the maximum storey drift due to individual ground motions were predicted with a sufficient degree of accuracy for almost all the ground motions from the analysed sets. Copyright © 2013 John Wiley & Sons, Ltd.     

Seismic response analysis of highway overcrossings including soil–structure interaction

This paper presents a systematic procedure for the seismic response analysis of highway overcrossings. The study employs an elementary stick model and a more sophisticated finite element formulation to compute response quantities. All dynamic stiffnesses of approach embankments and pile groups are approximated with frequency‐independent springs and dashpots that have been established elsewhere. A real eigenvalue analysis confirms the one‐to‐one correspondence between modal characteristics obtained with the three‐dimensional finite element solutions and the result of the simpler stick‐model idealization. A complex eigenvalue analysis yields modal damping values in the first six modes of interest and shows that modal damping ratios assume values much higher than those used by Caltrans. The validity of the proposed method is examined by comparing the computed time response quantities with records from the Meloland Road and Painter Street overcrossings located in southern and northern California, respectively. The proposed procedure allows for inexpensive parametric analysis that examines the importance of considering soil–structure interaction at the end abutments and centre bent. Results and recommendations presented by past investigations are revisited and integrated in comprehensive tables that improve our understanding of the dynamic characteristics and behaviour of freeway overcrossings. The study concludes with a step‐by‐step methodology that allows for a simple, yet dependable dynamic analysis of freeway overcrossings, that involves a stick model and frequency‐independent springs and dashpots. Copyright © 2002 John Wiley & Sons, Ltd.     

Earthquake response analysis considering non-proportional damping

Non-proportional damping may be defined as a form of linear viscous damping which introduces coupling between the undamped modal co-ordinate equations of motion. The standard mode superposition method of earthquake response analysis therefore cannot be employed with non-proportionally damped structures. In this paper, several methods for analysing the dynamic response of non-proportional damped structures are outlined. It is concluded that the most efficient procedure is to express the response in terms of a truncated set of undamped modal coordinates and to integrate directly the resulting coupled equations. The effectiveness of the method is demonstrated by a numerical example.     

Criteria for assessing dynamic collapse of elastoplastic structural systems

Criteria are presented for assessing the propensity for dynamic collapse of elastoplastic structural systems. To detect the onset of dynamic collapse, three key concepts are introduced. First, the equations of motion are linearized and time shifted relative to a reference time in a dynamic process. Second, we study the free vibration of the system after the reference time. Third, we use a modal decomposition of the response. We suggest that dynamic collapse is likely if (1) there exist negative eigenvalues of the Hessian of the total potential energy and (2) the direction of motion is consistent with the loading direction of the elastoplastic material. The direction of motion is determined by using the gradient of total potential energy and the eigenmodes corresponding to the negative eigenvalues. The fidelity of the present criteria is demonstrated through numerical examples for which the non‐linear equations of motion can be integrated exactly. With the present approach, it becomes apparent that p‐delta effects, the tangent stiffness, the internal resistance, and the direction of unloading all play key roles in the dynamic collapse of structures. Copyright © 2000 John Wiley & Sons, Ltd.     

An incremental mode-superposition for non-linear dynamic analysis

This paper uses an incremental mode-superposition procedure to compute the inelastic dynamic response of multi-degree-of-freedom systems. A damping matrix proportional to the instantaneous properties is used throughout the analysis. The non-linear response of several shear type plane and space frames with elastic-plastic and bilinear column properties subjected to ground excitation was computed by both the incremental mode-superposition and the direct integration of the coupled equations of motion. When all modes are considered, the responses computed by the incremental mode-superposition are identical to those from the direct integration. Fewer modes can also be used to compute the response with reasonable accuracy by performing the modal truncation for each time increment. The study shows that incorporating instantaneous damping in non-linear dynamic analysis is relatively simple and requires less computational time than the direct integration.     

Closed‐form solution for seismic response of adjacent buildings with linear quadratic Gaussian controllers

Closed‐form solution for seismic response of adjacent buildings connected by hydraulic actuators with linear quadratic Gaussian (LQG) controllers is presented in this paper. The equations of motion of actively controlled adjacent buildings against earthquake are first established. The complex modal superposition method is then used to determine dynamic characteristics, including modal damping ratio, of actively controlled adjacent buildings. The closed‐form solution for seismic response of the system is finally derived in terms of the complex dynamic characteristics, the pseudo‐excitation method and the residue theorem. By using the closed‐form solution, extensive parametric studies can be carried out for the system of many degrees of freedom. The beneficial parameters of LQG controllers for achieving the maximum response reduction of both buildings using reasonable control forces can be identified. The effectiveness of LQG controllers for this particular application is evaluated in this study. The results show that for the adjacent buildings of different dynamic properties, if the parameters of LQG controllers are selected appropriately, the modal damping ratios of the system can be significantly increased and the seismic responses of both buildings can be considerably reduced. Copyright © 2001 John Wiley & Sons, Ltd.     

Dynamic response of an elastic-viscoplastic system in modal co-ordinates

A dynamic analysis of elastic–viscoplastic systems, incorporating the modal co-ordinate transformation technique, is presented. The formulation results in uncoupled incremental equations of motion with respect to the modal co-ordinates. The elastic–viscoplastic model adopted allows the analysis not to involve yielding regions and loading/unloading processes. An implicit Runge–Kutta scheme together with the Newton–Raphson method are used to solve the non-linear constitutive equations. Stability and accuracy of the numerical solution are improved by utilizing a local time step sub-incrementing procedure. Applications of the analyses to multi-storey shear buildings show that good results can be obtained for the maximum displacement response by including only a few lower modes in the computation, but the prediction of the ductility factor response tends to underestimate the peak values when too few modes are used. In addition, stable and valid results can be obtained even with a sizable time step increment.     

Seismic behaviour of asymmetric buildings with supplemental damping

This paper investigates the response of asymmetric‐plan buildings with supplemental viscous damping to harmonic ground motion using modal analysis techniques. It is shown that most modal parameters, except dynamic amplification factors (DAFs), are affected very little by the plan‐wise distribution of supplemental damping in the practical range of system parameters. Plan‐wise distribution of supplemental damping significantly influences the DAFs, which, in turn, influence the modal deformations. These trends are directly related to the apparent modal damping ratios; the first modal damping ratio increases while the second decreases as CSD moves from right to left of the system plan, and their values increase with larger plan‐wise spread of the supplemental damping. The largest reduction in the flexible edge deformation occurs when damping in the first mode is maximized by distributing the supplemental damping such that the damping eccentricity takes on the largest value with algebraic sign opposite to the structural eccentricity. Copyright © 2000 John Wiley & Sons, Ltd.     

Rocking damage‐free steel column base with friction devices: design procedure and numerical evaluation

Earthquake‐resilient steel frames, such as self‐centering frames or frames with passive energy dissipation devices, have been extensively studied during the past decade, but little attention has been paid to their column bases. The paper presents a rocking damage‐free steel column base, which uses post‐tensioned high‐strength steel bars to control rocking behavior and friction devices to dissipate seismic energy. Contrary to conventional steel column bases, the rocking column base exhibits monotonic and cyclic moment–rotation behaviors that are easily described using simple analytical equations. Analytical equations are provided for different cases including structural limit states that involve yielding or loss of post‐tensioning in the post‐tensioned bars. A step‐by‐step design procedure is presented, which ensures damage‐free behavior, self‐centering capability, and adequate energy dissipation capacity for a predefined target rotation. A 3D nonlinear finite element (FE) model of the column base is developed in abaqus . The results of the FE simulations validate the accuracy of the moment–rotation analytical equations and demonstrate the efficiency of the design procedure. Moreover, a simplified model for the column base is developed in OpenSees . Comparisons among the OpenSees and abaqus models demonstrate the efficiency of the former and its adequacy to be used in nonlinear dynamic analysis. A prototype steel building is designed as a self‐centering moment‐resisting frame with conventional or rocking column bases. Nonlinear dynamic analyses show that the rocking column base fully protects the first story columns from yielding and eliminates the first story residual drift without any detrimental effect on peak interstory drifts. The study focuses on the 2D rocking motion and, thus, ignores 3D rocking effects such as biaxial bending deformations in the friction devices. The FE models, the analytical equations, and the design procedure will be updated and validated to cover 3D rocking motion effects after forthcoming experimental tests on the column base. Copyright © 2017 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.     

Modified predictor–corrector numerical scheme for real‐time pseudo dynamic tests using state‐space formulation

This paper deals with an explicit numerical integration method for real‐time pseudo dynamic tests. The proposed method, termed the MPC‐SSP method, is suited to use in real‐time pseudo dynamic tests as no iteration steps are involved in each step of computation. A procedure for implementing the proposed method in real‐time pseudo dynamic tests is described in the paper. A state‐space approach is employed in this study to formulate the equations of motion of the system, which is advantageous in real‐time pseudo dynamic testing of structures with active control devices since most structural control problems are formulated in state space. A stability and accuracy analysis of the proposed method was performed based on linear elastic systems. Owing to an extrapolation scheme employed to predict the system's future response, the MPC‐SSP method is conditionally stable. To demonstrate the effectiveness of the MPC‐SSP method, a series of numerical simulations were performed and the performance of the MPC‐SSP method was compared with other pseudo dynamic testing methods including Explicit Newmark, Central Difference, Operator Splitting, and OS‐SSP methods based on both linear and non‐linear single‐degree‐of‐freedom systems. Copyright © 2004 John Wiley & Sons, Ltd.     

Conditional spectrum‐based ground motion record selection using average spectral acceleration

The use of a seismic intensity measure (IM) is paramount in decoupling seismic hazard and structural response estimation when assessing the performance of structures. For this to be valid, the IM needs to be sufficient;that is, the engineering demand parameter (EDP) response should be independent of other ground motion characteristics when conditioned on the IM. Whenever non‐trivial dependence is found, such as in the case of the IM being the first‐mode spectral acceleration, ground motion selection must be employed to generate sets of ground motion records that are consistent vis‐à‐vis the hazard conditioned on the IM. Conditional spectrum record selection is such a method for choosing records that are consistent with the site‐dependent spectral shape conditioned on the first‐mode spectral acceleration. Based on a single structural period, however the result may be suboptimal, or insufficient, for EDPs influenced by different period values, for example, peak interstory drifts or peak floor accelerations at different floors, potentially requiring different record suites for each. Recently, the log‐average spectral acceleration over a period range, AvgSA, has emerged as an improved scalar IM for building response estimation whose hazard can be evaluated using existing ground motion prediction equations. Herein, we present a recasting of conditional spectrum record selection that is based on AvgSA over a period range as the conditioning IM. This procedure ensures increased efficiency and sufficiency in simultaneously estimating multiple EDPs by means of a single IM. Copyright © 2017 John Wiley & Sons, Ltd.