The CQC rule for modal combination is extended to include the quasi-static contribution of truncated modes and the effects of input narrow-bandedness and cut-off frequency. A simple measure of the error in approximating a high-frequency modal response by its quasi-static contribution is derived. The extended rule is applicable to structures with high-frequency modes and to seismic inputs which may not be regarded as wide band. Numerical examples demonstrate the significance of input bandwidth and cut-off frequency on modal cross-correlation coefficients, and on the error resulting from truncation of high-freqeuncy modes.
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"It follows that to date it is not possible to predict a priori the number of modes necessary for an accurate evaluation of the stochastic response. Moreover, the most efficient approach in the stochastic seismic analysis of large structural systems via modal analysis is nowadays based on the utilization of modal correction methods     . These methods improve the response taking into account approximately the contribution of the neglected higher modes. "
[Show abstract][Hide abstract] ABSTRACT: Dynamic analysis of large linear systems is usually performed adopting the wellknown
modal analysis along with modal truncation of higher modes. However, in the case in which the contribution of higher modes is not negligible, modal correction methods have to be introduced to improve the accuracy of the dynamic response either in the case of deterministic or stochastic excitation. Aim of this paper is to propose a new computationally competitive method for the stochastic analysis of large linear system vibrating under fully non-stationary Gaussian excitations. The method is based on the extension of the mode-acceleration method and its variant, the stochastic mode-acceleration method, for the evaluation of the nongeometric spectral moments of the non-stationary response. Numerical results from the study of a large MDoF structure show the accuracy and the efficiency of the proposed technique.
IOP Conference Series Materials Science and Engineering 07/2010; 10(1). DOI:10.1088/1757-899X/10/1/012201
"In the framework of stochastic analysis the correction methods have been used to evaluate the stochastic response  or the cross-correlation coecients . These methods, since they evaluate the ®rst and second statistical moments of the corrected response, require that the moments or equivalently the cross-covariance of the input are bounded. "
[Show abstract][Hide abstract] ABSTRACT: The role played by the modal analysis in the framework of structural dynamics is fundamental from both deterministic and stochastic point of view. However the accuracy obtained by means of the classical modal analysis is not always satisfactory. Therefore it is clear the importance of methods able to correct the modal response in such a way to obtain the required accuracy. Many methods have been proposed in the last years but they are meaningful only when the forcing function is expressed by an analytical function. Moreover in stochastic analysis they fail for white noise excitation. In the paper a method able to give a very accurate response for both deterministic and stochastic input is presented. This method is based upon the use of Ritz vectors together with the classical modal analysis. Numerical applications for both deterministic and stochastic inputs show the great accuracy of the proposed method.
"It is possible to provide corrections for these errors by directly using the response spectrum ordinates in case of MDOF systems (e.g., see Amini and Trifunac, 1985; Gupta and Trifunac, 1987c), or by using a fictitious PSDF corresponding to an 'equivalent stationary' ground motion process (e.g., see Singh and Chu, 1976; Der Kiureghian, 1981; Shrikhande and Gupta, 1997a; Gupta and Trifunac, 1998a). In the former, simplifications lead to the development of modal combination rules (e.g., see Rosenblueth and Elorduy, 1969; Wilson et al., 1981; Singh and Mehta, 1983; Der Kiureghian and Nakamura, 1993; Gupta, 1996a), wherein the response spectra are used to predict the largest peak response for each mode of the structure, and then these individual modal maxima are combined to give the expected peak response. These simple methods however do not address the inherent uncertainty in the ground motion due to random phasing. "
[Show abstract][Hide abstract] ABSTRACT: Response spectra are commonly used for the estimation of the largest peak response of a linear structural system in a seismic environment. Traditionally, this has been done through the use of appropriate modal combination rules in case of multi-degree-of-freedom systems. While these methods do not consider uncertainty in response due to phasing in seismic waves, those also do not go beyond estimating the largest peak response, and have natural limitation of being applied accurately to only a few types of structural systems. This paper considers a review of alternative methods which have been developed since mid-1970's to give probabilistic estimates of response peaks, while continuing to use the information available through response spectra. These methods have the convenience of being applied in a variety of situations, do not usually suffer from the inaccuracies associated with the use of modal combination rules, and present state-of-the-art methodology in linear seismic response analysis. The limitations of various formulations proposed under these methods are identified, and future directions of required work are suggested.