Contribution Of Higher Modes Research Articles

Dynamic Load Allowances Corresponding to Different Responses in Various Sections of Highway Bridges to Moving Vehicular Loads

Dynamic load allowance (DLA) is an important index used in design and evaluation of dynamic performance of highway bridges to moving vehicular loads. In many codes, DLAs of different responses in various sections are not distinguished. To facilitate a rational design and objective evaluation, the quantitative relationships of DLAs corresponding to different responses in various sections are systematically studied using both theoretical derivation and numerical simulation. According to results of theoretical derivation, for simply supported girder bridges, DLA of deflection is almost 20% higher than that of bending moment in the mid-span section. According to results of numerical simulation, for continuous girder bridges, DLA of negative (upward) deflection is 20% higher than that of positive deflection, and DLA of negative bending moment is 40% higher than that of positive bending moment in critical cases. These significant differences cannot be ignored. Finally, the contributions of high modes to different dynamic responses in various sections are investigated. Based on the recommended approach to considering these differences, the dynamic performance of bridge can be designed and evaluated in a simple yet rotational way.

Design procedure for dynamic response of concrete rectangular liquid storage tanks using generalized SDOF system

In this paper, a design procedure using the generalized single degree of freedom (SDOF) system is proposed for two-dimensional dynamic response of rectangular liquid containing structures (LCS). The proposed model considers the effect of flexibility of tank wall on hydrodynamic pressures and uses the consistent mass approach in dynamic analysis. The contribution of higher modes to the dynamic response of LCS is included in the proposed model. The square root of sum of squares method is proposed for the combination of the first two modes. One set of calculations for a tall tank is presented and compared with the results obtained using the current practice as well as the finite element method reported by the authors in previous investigations. The proposed method using the generalized SDOF system can be simply used in seismic design of LCS.

A Semi-Analytical Approach for the Geometrically Nonlinear Analysis of Skew Plates

The present work concerns the nonlinear dynamic behaviour of fully clamped skew plates at large vibration amplitudes. A model based on Hamilton’s principle and spectral analysis has been used to study the large amplitude free vibration problem, reducing the non linear problem to solution of a set of non-linear algebraic equations. Two methods of solution have been adopted, the first method uses an improved version of the Newton-Raphson method, and the second leads to explicit analytical expressions for the higher mode contribution coefficients to the first non-linear mode shape of the skew plate examined. The amplitude dependent fundamental mode shape and the associated non-linear frequencies have been obtained by the two methods and a good convergence has been found. It was found that the non-linear frequencies increase with increasing the amplitude of vibration, which corresponds to the hardening type effect, expected in similar cases, due to the membrane forces induced by the large vibration amplitudes. The non-linear mode exhibits a higher bending stress near to the clamps at large deflections, compared with that predicted by linear theory. Numerical details are presented and the comparison made between the results obtained and previous ones available in the literature shows a satisfactory agreement. Tables of numerical results are given, corresponding to the linear and nonlinear cases for various values of the skew angle θ and various values of the vibration amplitude. These results, similar to those previously published for other plates, are expected to be useful to designers in the need of accurate estimates of the non-linear frequencies, the non linear strains and stresses induced by large vibration amplitudes of skew plates.

High-frequency force balance technique for tall buildings: a critical review and some new insights

The high frequency force balance (HFFB) technique provides convenient measurements of integrated forces on rigid building models in terms of base bending moments and torque and/or base shear forces. These base moments or forces are then used to approximately estimate the generalized forces of building fundamental modes with mode shape corrections. This paper presents an analysis framework for coupled dynamic response of tall buildings with HFFB technique. The empirical mode shape corrections for generalized forces with coupled mode shapes are validated using measurements of synchronous pressures on a square building surface from a wind tunnel. An alternative approach for estimating the mean and background response components directly using HFFB measurements without mode shape corrections is introduced with a discussion on higher mode contributions. The uncertainty in the mode shape corrections and its influence on predicted responses of buildings with both uncoupled and coupled modal shapes are examined. Furthermore, this paper presents a comparison of aerodynamic base moment spectra with available data sets for various tall building configurations. Finally, e-technology aspects in conjunction with HFFB technique such as web-based on-line analysis framework for buildings with uncoupled mode shapes used in NALD (NatHaz Aerodynamic Loads Database) is discussed, which facilitates the use of HFFB data for preliminary design stages of tall buildings subject to wind loads.

Behavior of composite rigid frame bridge under bi-directional seismic excitations

Pushover analysis and time history analysis are conducted to explore the bi-directional seismic behavior of composite steel-concrete rigid frame bridge, which is composed of RC piers and steel-concrete composite girders. Both longitudinal and transverse directions excitations are investigated using OpenSees. Firstly, the applicability of pushover analysis based on the fundamental mode is discussed. Secondly, an improved pushover analysis method considering the contribution of higher modes is proposed, and the applicability on composite rigid frame bridges under bi-directional earthquake is verified. Based on this method, an approach to predict the displacement responses of composite rigid frame bridge under random bi-directional seismic excitations by revising the elasto-plastic demand curve is also proposed. It is observed that the developed method yield a good estimate on the responses of composite rigid frame bridges under bi-directional seismic excitations.

H∞ optimization of fluid viscous dampers for reducing vibrations of high-speed railway bridges

This work deals with the optimization of fluid viscous damper systems (FVDs) to reduce the resonant dynamic structural response of high-speed railway bridges by algebraic and numerical approaches. The presented method chooses the objective function based on the H ∞ norm over the frequency band of interest. This function allows taking into account structural damping properties and minimizing simultaneously the structural response associated with multiple modes. Especially, the proposed objective function may also be extended to nonlinear problems to determine optimal parameters of nonlinear fluid viscous dampers which may be an interesting solution in applications where high force levels are expected in the dampers. Finally, the proposed method is validated through numerical simulations. The simulation results show that the optimal FVD coefficients obtained by using the presented method are more exact than those by the previous method. Besides, the effectiveness of the method for solving the problems with the contribution of high modes and the consideration of nonlinear FVDs is also proved.

Damping Effect of Story Isolation Structure Considering Different Positions of Isolation Layer

The multi-mass analysis models of story isolation and general structure are established, major parameters are optimized on story isolation structure, with time-history analysis for isolation layers in different position, the contributions of modes and damping effects of absolute acceleration and story drift about story-isolation structure are analysed. Result indicates that after the parametric optimization, the natural period of story-isolation structure is longer than that of general structure; no matter where the position of isolation layer is, the damping effects are always obviously; the position of isolation layer has direct influence on the damping effect, damping effect gradually reduce when the isolation layer raised, and the contributions of high modes to dynamic responses increase; it's worth noting that infrastructure absolute acceleration magnify to some extent compared with general structure.

Lateral force distribution on base-isolated structures under near-field earthquakes

In the current paper the accuracy of the International Building Code equivalent static method for base-shear distribution is evaluated to determine the response of seismically isolated structures under near-field earthquakes. For this purpose two-, three-, four- and six-storey building models are designed according to the equivalent lateral-response procedure for different elastomeric isolation systems of 2 s, 2·5 s and 3 s for periods of isolated structures and 6%, 8% and 11% for their damping ratio. Different lateral-force resisting systems such as ordinary steel moment frame, ordinary steel concentrically braced frame and ordinary reinforced concrete shear wall are employed as a superstructure and the effects of 13 near-field ground motions on vertical seismic force distribution, and the linear and non-linear properties of isolation systems are investigated. The results of the equivalent static procedure in vertical distribution of base shear are compared with results obtained from dynamic linear and non-linear analysis and the efficiency of their application is studied. Based on linear and non-linear dynamic analysis results of isolated structures, two new formulae are proposed for vertical distribution of base shear for the isolated structures under near field earthquakes. The proposed linear formula is based on the first mode shape of an isolated structure and the non-linear formula follows three first mode shapes which implies contribution of higher modes in more realistic non-linear analyses.

Effectiveness of modified pushover analysis procedure for the estimation of seismic demands of buildings subjected to near-fault ground motions having fling step

Abstract. Near-fault ground motions with long-period pulses have been identified as being critical in the design of structures. These motions, which have caused severe damage in recent disastrous earthquakes, are characterized by a short-duration impulsive motion that transmits large amounts of energy into the structures at the beginning of the earthquake. In nearly all of the past near-fault earthquakes, significant higher mode contributions have been evident in building structures near the fault rupture, resulting in the migration of dynamic demands (i.e. drifts) from the lower to the upper stories. Due to this, the static nonlinear pushover analysis (which utilizes a load pattern proportional to the shape of the fundamental mode of vibration) may not produce accurate results when used in the analysis of structures subjected to near-fault ground motions. The objective of this paper is to improve the accuracy of the pushover method in these situations by introducing a new load pattern into the common pushover procedure. Several pushover analyses are performed for six existing reinforced concrete buildings that possess a variety of natural periods. Then, a comparison is made between the pushover analyses' results (with four new load patterns) and those of FEMA (Federal Emergency Management Agency)-356 with reference to nonlinear dynamic time-history analyses. The comparison shows that, generally, the proposed pushover method yields better results than all FEMA-356 pushover analysis procedures for all investigated response quantities and is a closer match to the nonlinear time-history responses. In general, the method is able to reproduce the essential response features providing a reasonable measure of the likely contribution of higher modes in all phases of the response.

Effect of higher modes on the seismic response and design of moment-resisting RC frame structures

Recently, extensive research has been conducted regarding higher-mode effects on the response of multi degree-of-freedom (MDOF) systems. The research has been focused mainly on structures with a lateral force resisting system consisting of slender walls, since these types of buildings are expected to be mostly affected by higher-mode phenomena according to structural dynamics, and simplified expressions have been proposed for slender-wall structures to account for higher-mode response in estimating shear forces. Current seismic design practice assumes the same reduction factor for all modes, even though there is strong evidence that inelasticity affects higher modes of vibration unequally. Additionally, simplified design methods are based only on the fundamental mode of vibration neglecting the effect of higher modes or considering them as elastic. In this paper, higher-mode contributions on the overall response of a nine-storey moment resisting frame (MRF), for which a domination of the first mode is expected, are investigated. The accuracy of a modified Modal Response Spectrum Analysis (mMRSA) method and other available methods is evaluated by comparing the results with the ones of the nonlinear response history analysis. Modal behaviour (reduction) factors are directly calculated for the first three modes and the validity of common assumptions is examined. The assessment of the methods is not restricted to deformations, but is extended to storey inertial forces and shears as well, which have attracted less interest from structural engineers, even though they are considered responsible for numerous structural and non-structural failures during major recent earthquakes and are critical for the design of several structures, such as precast buildings. The results suggest that the storey inertial forces and accelerations at all storeys and shear forces at higher storeys are significantly underestimated by methods neglecting or non-properly accounting for higher modes, even for first-mode dominated structures. The contribution of higher modes depends on the ground motion characteristics, the overstrength associated with each mode and the response quantity examined.

Dynamic behavior of a building structure tested with base and mid-story isolation systems

The discrepancies of dynamic behavior in base-isolated and mid-story isolated buildings are investigated experimentally in this study. In a mid-story isolated building, the isolation system is incorporated into the mid-story rather than the base of the building. The shaking table test results disclose that with the same design of the isolated superstructure and isolation system, a mid-story isolated building has smaller fundamental modal quantities than a base-isolated building. Besides, the seismic response at the superstructure is mainly attributed to the fundamental mode response, while the higher mode responses contribute significantly to the seismic responses at the substructure below the isolation system. Therefore, the modal response spectrum analysis appropriately considering the contribution of higher modes, rather than the equivalent lateral force procedure only considering the fundamental mode, is suggested for the preliminary design of mid-story isolated buildings prior to performing nonlinear responses history analyses.

Plasmonic band structures and optical properties of subwavelength metal/dielectric/metal Bragg waveguides

In this paper, we applied the band structure theory to investigate the plasmonic band (PB) structures and optical properties of subwavelength metal/dielectric/metal Bragg waveguides in the near infrared range with either dielectric or geometric modulation. The Bloch wave vector, density of states, slowdown factor, propagation length and transmittance are calculated and analyzed. Both the modulations are in favor of manipulating surface-plasmon-polariton (SPP) waves. For the dielectric modulation, the PB structure is mainly formed by SPP modes and possesses a "regular pattern" in which the bands and gaps have a relatively even distribution. For the geometric modulation, due to the strong transverse scattering, the contributions of higher modes have to be considered and the gap widths have a significant increase compared to the dielectric modulation. A larger slowdown factor may emerge at the band edge; especially for the geometric modulation, the group velocity can be reduced to 1/100 of light, and negative group velocity is observed as well. While inside the bands, the slowdown factor is smaller and the bands are flat. The contribution of each eigenmode to the PB structure is analyzed.

MOVING LOAD ANALYSIS OF SUBMERGED FLOATING TUNNELS

The concept of floating submerged tunnels becomes increasingly attractive idea to cross the straits. The structural solution in these bridges includes buoyancy force on tunnel body plus tension in mooring tethers. This paper investigates the effect of submergence on the dynamic response of submerged floating tunnels due to moving load. The inertial effect of the fluid on the submerged structure is accounted for by evaluating the added mass in deep and shallow waters. Then the effect of moving load velocity on the dynamic amplification factor for mid span displacement is evaluated. The results show that while the inertial effect of fluid reduces the critical velocity, closely spaced tethers provides a means to increase this velocity and could be used to control the moving load effect. Although increasing the tether stiffness increases the critical velocity, at the same time it results in the escalation of the impact factor. It is also found that for tunnels of long in length the contribution of higher modes in the response becomes more and more important and closeness to the seabed in shallow strait crossings can result in the amplification of the moving load effect.

Response Measurement Accuracy for Off-Resonance Excitation in Atomic Force Microscopy

Displacement measurement in atomic force microscopy (AFM) is most commonly obtained indirectly by measuring the slope of the AFM probe and applying a calibration factor. Static calibration techniques operate on the assumption that the probe response approximates single mode behavior. For off-resonance excitation or different operating conditions the contribution of higher modes may become significant. In this paper, changes to the calibrated slope-displacement relationship and the corresponding implications on measurement accuracy are investigated. A model is developed and numerical simulations are performed to examine the effect of laser spot position, tip mass, quality factor and excitation frequency on measurement accuracy. Free response conditions and operation under nonlinear tip-sample forces are considered. Results are verified experimentally using a representative macroscale system. A laser spot positioned at a nominal position between x = 0.5 and 0.6 is determined to minimize optical lever measurement error under conditions where the response is dominated by contributions from the first two modes, due to excitation as well as other factors.

ANALYTIC SOLUTIONS TO THE OSCILLATORY BEHAVIOR AND PRIMARY RESONANCE OF ELECTROSTATICALLY ACTUATED MICROBRIDGES

In this paper, the vibration and primary resonance of electrostatically actuated microbridges are investigated, with the effects of electrostatic actuation, axial stress, and mid-plane stretching considered. Galerkin's decomposition method is adopted to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. The homotopy perturbation method (a special case of homotopy analysis method) is then employed to find the analytic expressions for the natural frequencies of predeformed microbridges, by which the effects of the voltage, mid-plane stretching, axial force, and higher mode contribution on the natural frequencies are studied. The primary resonance of the microbridges is also investigated, where the microbridges are predeformed by a DC voltage and driven to vibrate by an AC harmonic voltage. The methods of homotopy perturbation and multiple scales are combined to find the analytic solution for the steady-state motion of the microbeam. In addition, the effects of the design parameters and damping on the dynamic responses are discussed. The results are shown to be in good agreement with the existing ones.

Effectiveness of modified pushover analysis procedure for the estimation of seismic demands of buildings subjected to near‐fault earthquakes having forward directivity

AbstractNear‐fault ground motions with long‐period pulses have been identified as being critical in the design of structures. These motions, which have caused severe damage in recent disastrous earthquakes, are characterized by a short‐duration impulsive motion that transmits large amounts of energy into the structures at the beginning of the earthquake. In nearly all of the past near‐fault earthquakes, significant higher mode contributions have been evident, resulting in the migration of dynamic demands (i.e., drifts) from the lower to the upper stories. Due to this, the static nonlinear pushover analysis (PA) (which utilizes a load pattern proportional to the shape of the fundamental mode of vibration) may not produce accurate results when used in the analysis of structures subjected to near‐fault ground motions. The objective of this paper was to improve the accuracy of the pushover method in these situations by introducing a new load pattern into the common pushover procedure. Several PAs are performed for six existing reinforced concrete buildings that possess a variety of natural periods. Then, a comparison is made between the PA results (with four new load patterns) and those of FEMA‐356 with reference to nonlinear dynamic time‐history analyses. The comparison shows that, generally, the proposed pushover method yields better results than all FEMA‐356 PA procedures for all investigated response quantities, and is a closer match to the nonlinear time‐history responses. In general, the method is able to reproduce the essential response features providing a reasonable measure of the likely contribution of higher modes in all phases of the response. Copyright © 2009 John Wiley & Sons, Ltd.

Direct Displacement Based Seismic Design of High Rise Structures with Strengthened Stories (I) - Theory

High rise structures with strengthened stories are widely used nowadays. A rational seismic design procedure for this kind of structure is thus necessary. Based on mode theory and direct displacement based seismic design of multi-story buildings, this paper proposed a new direct displacement based seismic design procedure. In the proposed method, each mode of the high rise structure is equivalent to a single degree of freedom (SDOF) system. Seismic response of each mode is calculated and adds up to consider the contributions of higher modes. Considering the characteristic of strengthened stories, two stories above the strengthened stories are taken as the “key stories”, whose performance indicates the performance of the building. The proposed procedure is logical, simple and can serve for reference of actual design.

Studies and Improvements on Modal Pushover Analysis and Application on Bridge

Because the traditional pushover analysis can not take the contributions of higher modes into account, To overcome this limitation, a modal pushover analysis procedure (MPA) is proposed by some researchers, which can involve the combination of multi-mode contributions to response. In this paper, much improvement on MPA procedure is made with consideration of the changes of seismic response after structural yielding and anew distribution of inertia forces. The method is verified by one example of bridge structure. It is concluded that the improvement part-sectionalized MPA presented in this paper has high accuracy.

A new misfit function for multimodal inversion of surface waves

Higher-mode contribution is important in surface-wave inversion because it allows more information to be exploited, increases investigation depth, and improves model resolution. A new misfit function for multimodal inversion of surface waves, based on the Haskell-Thomson matrix method, allows higher modes to be taken into account without the need to associate experimental data points to a specific mode, thus avoiding mode-misidentification errors in the retrieved velocity profiles. Computing cost is reduced by avoiding the need for calculating synthetic apparent or modal dispersion curves. Based on several synthetic and real examples with inversion results from the classical and the proposed methods, we find that correct velocity models can be retrieved through the multimodal inversion when higher modes are superimposed in the apparent dispersion-curve or when it is not trivial to determine a priori to which mode each data point of the experimental dispersion curve belongs. The main drawback of the method is related to the presence of several local minima in the misfit function. This feature makes the choice of a consistent initial model very important.

Stochastic analysis of large structural systems under fully non-stationary input

Dynamic analysis of large linear systems is usually performed 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 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 non-geometric 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.