Contribution Of Higher Modes Research Articles

Improved adaptive spectra-based pushover analysis for bridges

Traditional pushover analysis procedures are based on invariant force distributions, none of which account for the contributions of higher modes to the response. Although adaptive spectra-based pushover analysis describes the effect of higher modes and considers the dynamic properties of structures, it is conceptually complicated and computationally demanding for routine structural engineering applications. This paper proposes an improved adaptive spectra-based pushover analysis procedure that can be employed to analyse bridge structures. This paper compares seismic performance determined by the improved method with pushover analysis using two force distributions and non-linear response history analysis procedures. The results indicate that the improved method provides superior accuracy in estimating the seismic performance of bridge structures.

Additional viscous dampers for civil structures: Analysis of design methods based on effective evaluation of modal damping ratios

Design methods for supplemental dampers aimed at reducing vibrations of civil structures often leave unanswered the following question: starting from the structural inherent damping, the adoption of supplemental dampers which level of damping can achieve? From this perspective, the paper shows a methodology based on the well known state-space representation of dynamical systems to calculate the modal damping ratios of non classical/non proportional damping schemes. The methodology is then adopted to analyse a number of design methods for supplemental damping systems found in the scientific literature. The analysis shows that many design methods yield a similar result of implicitly suggesting a 1st modal damping ratio of about 20%. The same methodology is also adopted to show that the contribution of higher modes is generally negligible in designing a supplemental damping system and that the required damping level can be typically obtained with a limited number of supplemental devices.

A consecutive modal pushover procedure for estimating the seismic demands of tall buildings

The nonlinear static procedure (NSP), based on pushover analysis, has become a favourite tool for use in practical applications for building evaluation and design verification. The NSP is, however, restricted to single-mode response. It is therefore valid for low-rise buildings where the behaviour is dominated by the fundamental vibration mode. It is well recognized that the seismic demands derived from the conventional NSP are greatly underestimated in the upper storeys of tall buildings, in which higher-mode contributions to the response are important. This paper presents a new pushover procedure which can take into account higher-mode effects. The procedure, which has been named the consecutive modal pushover (CMP) procedure, utilizes multi-stage and single-stage pushover analyses. The final structural responses are determined by enveloping the results of multi-stage and single-stage pushover analyses. The procedure is applied to four special steel moment-resisting frames with different heights. A comparison between estimates from the CMP procedure and the exact values obtained by nonlinear response history analysis (NL-RHA), as well as predictions from modal pushover analysis (MPA), has been carried out. It is demonstrated that the CMP procedure is able to effectively overcome the limitations of traditional pushover analysis, and to accurately predict the seismic demands of tall buildings.

A new lateral force distribution formula for base-isolated structures

It has been suggested that the static lateral response procedure for a base-isolated structure presented in the International Building Code (IBC) somewhat overestimates the seismic storey force. In this paper an IBC-equivalent static method for base-shear distribution of seismic isolated structures is evaluated. The results of the equivalent static procedure in the vertical distribution of base shear are compared with results obtained from dynamic linear and nonlinear analysis and the efficiency of its application is studied. In general, the results of the equivalent lateral response procedure in the proposed triangular distribution of base shear are somewhat conservative. Based on linear and nonlinear analysis results of isolated structures, therefore, two new formulae are proposed for the vertical distribution of the base shear. The proposed linear formula is based on the first mode shape of the isolated structure and the nonlinear formula follows three first mode shapes, which imply contribution of higher modes in a more realistic nonlinear analysis. The accuracy of the new formulae is examined by comparing the results obtained from this study with those calculated from linear and nonlinear time history analysis.

Response of suspended beams due to moving loads and vertical seismic ground excitations

In this paper, the vibration of a suspension bridge due to moving loads of equidistant, identical forces and shaken by vertical support motions caused by earthquake is studied. The suspension bridge is modelled as a single-span suspended beam. To conduct the beam vibration with time-dependent boundary conditions, the total response of the suspended beam is decomposed into two parts: the quasistatic component and the dynamic part based on the decomposition method. Since the quasi-static component of the suspended beam under the static action of multiple support motions has been obtained analytically, the remaining dynamic part can be solved using Galerkin’s method. The numerical results indicate that the contribution of higher modes on the maximum acceleration of the suspended beams to moving loads will become significant as the propagation effect and multiple support motions of seismic waves in the subsoil of bridge supports has been taken into account.

A modal correction method for non-stationary random vibrations of linear systems

The computational effort in determining the dynamic response of linear systems is usually reduced by adopting the well-known 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 been introduced to improve the accuracy of the dynamic response, for both deterministic and stochastic input. In the latter case the random response is usually corrected via various methods determined as rough extensions of methods originally proposed for deterministic input. Consequently the efficiency of the correction methods is not suitable, from both theoretical and computational points of view. In this paper, a new approach to cope with the non-stationary response of linear systems is presented. The proposed modal correction method provides a correction term determined as a pseudo-stationary contribution of the equation governing either first-order or second-order statistics. Owing to the fact that no truncation criteria are well established for random vibration study, the proposed modal correction method offers a suitable vehicle for determining very accurately the stochastic response of MDOF linear systems under Gaussian stationary and non stationary excitation as evidenced in the numerical applications.

Determining the contribution of higher modes to the impedance of magnetic microwires measured using the waveguide method

A method of experimental determination of the contribution of higher modes to the impedance of microwires is described, which provides an increase in the accuracy of impedance measurements by the waveguide technique developed previously. The proposed method is based on measurements of the reflection coefficient of a wire with the known impedance.

Vertical distributions of lateral forces on base isolated structures considering higher mode effects

Base isolation technology has been accepted as a feasible and attractive way in improving seismic resistance of structures. The seismic design of new seismically isolated structures is mainly governed by the Uniform Building Code (UBC-97) published by the International Conference of Building Officials. In the UBC code, the distribution formula of the inertial (or lateral) forces leads to an inverted triangular shape in the vertical direction. It has been found to be too conservative for most isolated structures through experimental, computational and real earthquake examinations. In this paper, four simple and reasonable design formulae, based on the first mode of the base-isolated structures, for the lateral force distribution on isolated structures have been validated by a multiple-bay three-story base-isolated steel structure tested on the shaking table. Moreover, to obtain more accurate results for base-isolated structures in which higher mode contributions are more likely expected during earthquakes, another four inertial force distribution formulae are also proposed to include higher mode effects. Besides the experimental verification through shaking table tests, the vertical distributions of peak accelerations computed by the proposed design formulae are in good agreement with the recorded floor accelerations of the USC University Hospital during the Northridge earthquake.

A correction method for dynamic analysis of linear continuous systems

A method to improve the dynamic response analysis of continuous classically damped linear system is proposed. As in fact usually, following a classical approach, a reduced number of eigenfunctions are accounted for and the response is evaluated by integrating the uncoupled differential equations of motion in modal space, neglecting the contribution of high frequency modes (truncation procedure). Here, starting from the given system, it is proposed to set up two differential equations governing the motion of two new continuous systems: the first one contains only the first m non-zero eigenvalues of the given system and the second one contains the remainder non-zero ~-m eigenvalues. By summing up the response of the two aforementioned systems it is proven that the global response is recovered. Then, the response of the first system is evaluated by classical dynamic methods (Duhamell integral or step-by-step integration procedures), while the particular solution of the second one is obtained in series form that is shown to converge quickly.

EARTHQUAKE FLOOR SPECTRA FOR UNRESTRAINED BUILDING COMPONENTS

This paper introduces the use of elastic displacement floor spectra to model the seismic performance behavior of unrestrained components. The displacement spectrum has the attribute of tracking period-shifts associated with "rocking" behavior. Analyses of the displacement floor spectra in high-rise buildings demonstrate the significance of contributions by the higher modes of vibration. Significantly, such higher mode contributions could not always be captured by established code procedures wherein the seismic demand on a building floor is obtained by linear interpolation between the seismic demand at the roof and ground level. The significant influence of the higher modes in high-rise buildings also precludes the use of simple static models to predict seismic demand. Results are also presented herein to demonstrate the sensitivity of the displacement floor spectrum to a multitude of modeling uncertainties.

Quantum Vacuum Contribution to the Momentum of Dielectric Media

Momentum transfer between matter and electromagnetic field is analyzed. The related equations of motion and conservation laws are derived using relativistic formalism. Their correspondence to various, at first sight self-contradicting, experimental data (the so-called Abraham-Minkowski controversy) is demonstrated. A new, Casimir-like, quantum phenomenon is predicted: contribution of vacuum fluctuations to the motion of dielectric liquids in crossed electric and magnetic fields. Velocities of about 50 nm/s can be expected due to the contribution of high frequency vacuum modes. The proposed phenomenon could be used in the future as an investigating tool for zero fluctuations. Other possible applications lie in fields of microfluidics or precise positioning of micro-objects, e.g., cold atoms or molecules.

Non‐linear analysis of a collapsed reinforced concrete chimney

AbstractDuring the Ismit (Kocaeli) earthquake of 17 08 1999, a 115 high reinforced concrete chimney or heater stack, located at the Tüpras Refinery, collapsed. The falling debris cut 63 pipes, which contributed to interrupted production for more than 14 months. This stack was designed and constructed according to international standards and is representative of similar structures at refineries throughout the world, including those in earthquake‐prone regions. It was distinguished from similar stacks at the site by a much larger rectangular opening for the flue duct, circumscribing a horizontal arc of about 50°. The opening was located about 1/3 of the height above the base and appeared to be the region of initiation of the collapse.The investigation is focused on the dynamic response of the stack due to an earthquake motion recorded at a nearby site. In this paper the results of a response spectrum analysis of the Tüpras stack and a generic U.S. stack are summarized. Then, an on‐linear static analysis of the collapsed stack is presented using a demand–collapse comparison. The demand is represented by an acceleration–displacement response spectrum based on the recorded motion as well as some smoothed adaptations typical of design spectra, while the capacities are calculated from pushover curves using a non‐linear reinforced concrete finite element analysis.Results are presented that show the effects of the hole and the orientation of the motion with respect to the hole. Also higher‐mode contributions to the pushover pattern are considered. The results confirm that the stack could readily fail under the considered earthquake and are also consistent with the debris pattern. Copyright © 2003 JohnWiley & Sons, Ltd.

A Comparison of ESAC and FK Methods of Estimating Phase Velocity Using Arbitrarily Shaped Microtremor Arrays

Focusing on a comparison of the accuracy of the extended spatial autocorrelation (ESAC) method and the frequency–wavenumber spectrum (FK) method, we carried out short-period microtremor measurements of arbitrarily shaped array configurations at a site with a well-known velocity structure at shallow depths. Using both techniques, we measured phase velocities for frequencies between 2.5 Hz and 13.5 Hz and compared the results with the theoretical Rayleigh-wave-dispersion characteristics of both the fundamental and the first higher modes calculated from the PS logging data. Next, we tried to estimate the S -wave velocity structure to a depth of 43 m by fitting the theoretically calculated phase velocities to the experimental data, taking into consideration both the higher mode contributions as well as the fundamental mode. The theoretical dispersion characteristics were successfully fitted to the results analyzed by the ESAC method but not by the FK method. The estimated S -wave velocity structure from the ESAC method results was in good agreement with the PS logging profile. We also confirmed, using the model, that the S -wave transfer function due to vertical incidence coincided with that from the PS logging data. As a result of the study, we conclude that the ESAC method gives more accurate results than the FK method in determining the Rayleigh-wave phase velocity from records of short-period microtremors using arbitrarily shaped array configurations. The ESAC method can also provide a better estimate of the S -wave velocity structure and site amplification than the FK method.

Higher-mode effect on the seismic responses of buildings with viscoelastic dampers

In conventional modal analysis procedures, usually only a few dominant modes are required to describe the dynamic behavior of multi-degrees-of-freedom buildings. The number of modes needed in the dynamic analysis depends on the higher-mode contribution to the structural response, which is called the higher-mode effect. The modal analysis approach, however, may not be directly applied to the dynamic analysis of viscoelastically damped buildings. This is because the dynamic properties of the viscoelastic dampers depend on their vibration frequency. Therefore, the structural stiffness and damping contributed from those dampers would be different for each mode. In this study, the higher-mode effect is referred to as the response difference induced by the frequency-dependent property of viscoelastic dampers at higher modes. Modal analysis procedures for buildings with viscoelastic dampers distributed proportionally and non-proportionally to the stiffness of the buildings are developed to consider the higher-mode effect. Numerical studies on shear-type viscoelastically damped building models are conducted to examine the accuracy of the proposed procedures and to investigate the significance of the higher-mode effect on their seismic response. Two damper models are used to estimate the peak damper forces in the proposed procedures. Study results reveal that the higher-mode effect is significant for long-period viscoelastically damped buildings. The higher-mode effect on base shear is less significant than on story acceleration response. Maximum difference of the seismic response usually occurs at the top story. Also, the higher-mode effect may not be reduced by decreasing the damping ratio provided by the viscoelastic dampers. For practical application, it is realized that the linear viscous damping model without considering the higher-mode effect may predict larger damper forces and hence, is on the conservative side.

Generalized receptance-based method for accurate and efficient modal synthesis

Modal synthesis plays an important role in efficient dynamic analyses of large structural assemblies. However, the numerical accuracy and the computational efficiency which existing modal synthesis methods have achieved to date are not quite satisfactory. This paper presents a new generalized receptance-based modal synthesis method which is numerically very accurate and computationally very efficient. The method employs receptance data computed at just few frequency values to formulate a derived equivalent eigensys tem from which required eigenvalues and eigenvectors of interest can be solved. It provides mathematical generalization for existing mode-based modal synthesis methods since modal data are essentially those receptance data at resonant frequencies. A normalization procedure is also presented which can be used to mass normalize computed mode shapes which are often required in practice. Compensation for the contribution of higher uncomputed modes to receptance data at lower frequency of interest is made to improve analysis accuracy by using lower calculated modes and known system matrices. Numerical results are presented to demonstrate the effectiveness of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.

The Influence of Higher Modes and Support Gaps on the Seismic Response of Piping Systems Containing Snubbers

In this study of flexible piping systems (0.6 to 6.6 Hz), it was shown that the fundamental mode was not always the most significant mode of vibration. This study also concluded that variations in support condition do not significantly influence response in terms of nodal stress, fundamental frequency, and contribution of higher modes. In terms of support stiffness, the ratio of the magnitude of the gap (the total of all the mechanical tolerances and deadbands in the snubber support system) to the magnitude of the support nodal displacement was found to be more important than the actual magnitude of the gap. Additionally, when the magnitude of the nodal displacement exceeded the gap magnitude by a factor of about five or more, support stiffness was found to remain relatively constant.

The contribution of high order Zernike modes to wavefront tilt

The contribution of high order Zernike modes on average phase gradient is investigated. The resulting temporal power spectra are compared to data collected using the tip/tilt servo at the Sydney University Stellar Interferometer (SUSI). The low frequency part of the power spectrum follows a − 2 3 power law which is in agreement with other similar work. Due to the effect of higher order terms, the high frequency power law for average phase gradient becomes − 11 3 rather than the − 17 3 predicted for the ‘tilt’ component of the Zernike representation of the wavefront.

Rayleigh wave scattering at a basin type heterogeneity

Rayleigh waves are investigated numerically on a subsurface model composed of a layer over a substratum, and a basin embedded in the layer. The basin elastic parameters are varied to obtain three laterally heterogeneous models of different velocity contrast between the basin and the layer. The wave incident upon the basin is defined to be a pure fundamental mode having wavelengths comparable to the basin dimensions. Analysis of the wave transmitted reveals mode conversion, i.e., higher mode excitation which is increasingly important with increased velocity contrast. In terms of (rate of transport of) energy, the contribution of the first higher mode (being the only higher mode in the analysis interval) is relatively small, which differs from earlier results for the case of Love waves. To enable direct comparison, a fundamental Love mode having spectral properties analogous to the Rayleigh wave, is propagated across the subsurface models of the present paper. Hence, the discrepancy between the two waves basically arises from the structure of the subsurface transfer functions. The fraction of modal energy transmitted across the basin proves a sensitive indicator of the velocity contrast, showing similarity in amount and frequency dependence for Rayleigh and Love waves. In terms of modal amplitudes, conventionally considered in practice, the first higher mode contribution (due to mode conversion) is roughly comparable to the fundamental mode contribution, for both transmitted waves. It implies that the amplitude spectrum of the transmitted surface wave is modulated in the horizontal direction. The effect adds to the spectral distortion due to reflection at the basin. Hence, standard interpretation based upon a laterally homogeneous model, of measurements in corresponding regions, may be significantly biased.

Shear and Overturning Moment for Earthquake-resistant Building Design

The provisions of the present Canadian and U.S. seismic codes are examined with respect to their requirements on the distribution of seismic forces and overturning moments throughout the heights of building structures. Assuming the code design base shears to be correct, it is shown that the prescribed distribution of lateral forces may underestimate the shear forces in the upper stories of the building. It is further shown that the contribution of the higher modes is not directly related to the height-to-width ratio of the building, but rather to its fundamental period of vibration. By idealizing the building as an average uniform cantilever deflecting partly in a shear mode and partly in a bending mode and by taking a root-sum-square combination of the modal responses to a standard seismic acceleration spectrum, the envelope of the resultant maximum shears throughout the building height is obtained; the maximum overturning moments throughout the building height are obtained in a similar manner. By comparing these with the cantilever overturning moments computed from the shear envelope referred to above, the moment reduction factors applicable to the various levels of a building are calculated. New empirical relationships are then developed to approximate the shape of the maximum shear and overturning moment envelopes.

A Study on the Higher Decay Modes of Resonance Radiation Transfar

An eigenvalue problem for decay modes of resonance radiation proposed by Holstein is solved by a variational method and some higher decay modes are obtained by rigorous calculations of matrix elements when enclosure geometry is an infinite slab and the line-shape is due to Doppler broadening. Holstein's intergo-differential equation is extended to include the effect of external radiation and the contributions of higher modes to output radiation are calculated after stationary external radiation irradiated perpendicularly to one of the surfaces of the slab is abruptly cut off. This result is not consistent with the strange experimental results for mercury 2537 A radiation obtained by Mochizuki and Fowler–advancement of the relative phase of the output radiation when the intensity of external radiation is modulated sinusoidally–and it is concluded that the reason for their results should be found in some unknown mechanisms.