Kanai-Tajimi Spectrum Research Articles

Nonstationary random analysis of structures with hybrid mass damper system

The building structure is modeled as a linear single-degree-of-freedom system, and an active mass damper (AMD) is also adopted as the control device in order to reduce the structural response due to the external random load. In addition, suppose that the input acceleration of earthquake can be modeled as the product of nonstationary envelope function and narrow-band stationary random process with Kanai-Tajimi spectrum. In order to obtain the nonstationary stochastic response of control system due to the narrow-band nonstationary random process, in consideration of exchanging the active control mode for the passive one, considerable efforts are generally required in the analytical formulation. This paper, therefore, proposes an analytical technique to acquire the response covariance matrix by utilizing the state transition matrix method in state space. The proposed technique is evaluated by comparing the analytical results with those from numerical simulations.

GROUND MOTION DOMINANT FREQUENCY EFFECT ON THE DESIGN OF MULTIPLE TUNED MASS DAMPERS

Utilising the Kanai-Tajimi and Clough-Penzien spectrums and the pseudo-excitation algorithm in the frequency domain, parametric study is performed to examine the effect of the dominant frequency of ground motion on the optimum parameters and effective-ness of multiple tuned mass dampers (MTMD) with identical stiffness and damping coefficient but with unequal mass. The examination of the optimum parameters is con-ducted through the minimisation of the minimum values of the maximum displacement and acceleration dynamic magnification factors of the structure with the MTMD. The optimum parameters of the MTMD include the optimum frequency spacing reflecting the robustness, the average damping ratio and the tuning frequency ratio. Minimisation of the minimum values of the maximum displacement and acceleration dynamic mag-nification factors, nondimensionalised respectively by the maximum displacement and acceleration dynamic magnification factors of the structure without the MTMD, is used to measure the effectiveness of the MTMD. The results indicate that in the two cases where both the total mass ratio is below 0.02 and the total mass ratio is above 0.02, but the dominant frequency ratio of ground motion is below unity (including unity), the earthquake ground motion can be modelled by a white noise. It is worth noting, however, that for the total mass ratio above 0.02, the Kanai-Tajimi Spectrum or Clough-Penzien spectrum needs to be employed to design the MTMD for seismic structures in situations where the dominant frequency ratio of ground motion is beyond unity.

Sensitivity analysis of site effects on response spectra of pipelines

In this paper, a numerical sensitivity analysis of the site effect on dynamic response of pipelines embedded in some idealised soil deposits resting on a halfspace covering a wide range of soil profiles encountered in practice and subjected to vertically propagating shear waves, is presented. The power spectrum of the lateral differential displacement between two distant points due to the site effect is formulated analytically by using an analytical amplification function of a viscoelastic inhomogenous soil profile overlying either a compliant halfspace or a bedrock, represented by a more realistic continuous model. Also, Kanai-Tajimi spectrum parameters are estimated and expressed analytically from the soil profile model. Finally, results in the form of stochastic response spectrum of pipelines, for different key soil and pipeline parameters, are given and discussed.

Transfer-Function-Based Criteria for Decoupling of Secondary Systems

New criteria for decoupling of primary-secondary systems have been recently proposed by the first writer. They are defined by a limited variation of the maximum transfer function of a secondary system that is connected to a single-degree-of-freedom primary-secondary system. In this paper, their sufficiency for decoupling is investigated when the two-degree-of-freedom primary-secondary system is subjected to a filtered white noise process of the Kanai-Tajimi power spectrum. Verifications for sufficiency of the new criteria are extended to a secondary system mounted to a multi-degree-of-freedom primary system. Effects of site condition on the criteria are also studied, by varying the characteristic frequency of the Kanai-Tajimi spectrum. The upper-bound mass ratios resulting from the criteria, above which a coupled analysis is required, are compared with the current decoupling criteria mainly used in the design of nuclear facilities. Revisions of the current decoupling criteria are suggested as a result of this study. A simple procedure is recommended and illustrated by an example to apply the new criteria into the analysis and design of secondary systems.

Midbroken reinforced concrete shear frames due to earthquakes. A hysteretic model to quantify damage at the storey level

A nonlinear hysteretic model for the response and local damage analyses of reinforced concrete shear frames subject to earthquake excitation is proposed, and, the model is applied to analyse midbroken reinforced concrete (RC) structures due to earthquake loads. Each storey of the shear frame is represented by a Clough and Johnston hysteretic oscillator with degrading elastic fraction of the restoring force. The local damage is numerically quantified in the domain [0,1] using the maximum softening damage indicators which are defined in closed form based on the variation of the eigenfrequency of the local oscillators due to the local stiffness and strength deterioration. The proposed method of response and damage analyses is illustrated using a sample 5 storey shear frame with a weak third storey in stiffness and/or strength subject to sinusoidal and simulated earthquake excitations for which the horizontal component of the ground motion is modeled as a stationary Gaussian stochastic process with Kanai-Tajimi spectrum, multiplied by an envelope function.

Seismic response variability of hybrid-controlled bridges

The present author, collaborating with others, developed earlier a prototype hybrid base isolation system using friction-controllable sliding bearings and demonstrated its effectiveness in circumventing the shortcomings of passive counterparts by means of shaking table tests and numerical analysis under the 1940 El Centro earthquake record linearly scaled to various levels of intensity. The purpose of the present paper is to show that the same hybrid system exhibits a similar degree of effectiveness over a practical range of parameters that define the ground acceleration characteristics. These parameters include spectral shape, dominant frequency, and duration of the ground motion. For this purpose, artificial earthquake ground acceleration histories representing different sets of these parameter values are generated using a modified Kanai-Tajimi spectrum. The bridge responses under these earthquakes are evaluated numerically when the bridge is isolated by passive and hybrid sliding systems. The results demonstrate that the hybrid system indeed more effectively isolates the bridge under a variety of ground motion characteristics.

Response of MDOF systems to nonstationary random excitation

A procedure for calculating the dynamic response of a multi-degree-of-freedom (MDOF) system to nonstationary non-white vector-valued random excitation is developed utilizing a modal time-domain approach. The nonstationary excitation is represented by an envelope-modulated stationary random process that may be expressed as a product of a stationary random process with a deterministic envelope modulating function. The response of the coupled system is obtained at any time interval using the modal impulse-response function. The envelope intensity function is approximated by a staircase unit step function. This feature facilitates the use of envelope functions of any arbitrary shape. This formulation permits computations through a reduction in the multiplicity of the integrals involved in the evaluation of the response covariance matrices. The modal response covariance matrices are transformed to the associated response matrices in physical coordinates utilizing a mode displacement method. The peak structural response values at any level on the structure are obtained by following the evolutionary distribution of the extreme values. The coupled lateral-torsional response of a ten-storey unsymmetrical building subjected to a ground motion in any arbitrary direction is considered for illustrating the procedure developed here. The stationary component of the ground motion is represented by the Kanai-Tajimi spectrum, which is modulated by a typical trapezoidal envelope intensity function to include the nonstationary feature. The response estimates are in good agreement with a modal-state space Markov vector approach. The computationally efficient procedure presented here holds the promise of becoming a convenient design office tool.

Design charts for random vibrations of elasto-plastic oscillators subjected to Kanai-Tajima spectra

Abstract The yielding frame of Fig. 1 is approximated by a single degree of freedom elasto-plastic oscillator in a plastic hinge model. A probabilistic theory with effective random loading of the time-invariant linear structure is used to calculate the stochastic response. After a short review of this theory of effective earthquake loading, nondimensional results of the quasi-stationary elasto-plastic response are presented in the graphical form of design charts, where the statistical response measures are given as functions of similarity numbers, and the influence of the characteristic ground parameters of a Kanai-Tajimi power spectrum representation is studied.

Stochastic estimation of the nonlinear response of earth dams to strong earthquakes

The paper develops a simple and rational piece-wise linear random vibration procedure to estimate statistics of the nonlinear hysteretic response of earth dams to strong nonstationary stochastic excitation characterized by a Kanai-Tajimi spectrum. The method utilizes published data relating soil modulus and damping to the level of induced shear strain and incorporates a realistic inhomogeneous shear beam model of the dam. It is shown that the method is technically superior and computationally simpler than currently used iterative linear deterministic methods of analysis. Results are presented in the form of variation with time and distribution with depth from the crest of statistical measures of displacements, accelerations, shear strains and seismic coefficients on potentially sliding masses. The influence of the intensity and frequency characteristics of the excitation on the nonlinear response of a broad range of earth dams is graphically illustrated and a set of normalized graphs are presented whereby one can readily obtain preliminary estimates of key response quantities to be experienced by an earth dam during a potential earthquake motion of known rms acceleration and predominant frequency.