COMPLEX MODE SUPERPOSITION ALGORITHM FOR SEISMIC RESPONSES OF NON-CLASSICALLY DAMPED LINEAR MDOF SYSTEM

This paper deals with a complex mode superposition method for the seismic responses of general multiple degrees of freedom (MDOF) discrete system with complex eigenvectors and eigenvalues. A delicate general solution, completely in real value form, for calculat-ing seismic time history response of the MDOF system which cannot be uncoupled by normal modes, is deduced based on the algorithms of the complex superposition method. This solution comprises of two parts which are in relation to the Duhamel integration to sine and cosine function respectively. The related term of the Duhamel integration to sine function is actually the displacement response of the oscillator with corresponding modal frequency and the damping ratio. The other can be transferred into a combina-tion of the displacement and velocity responses of the same oscillator. In order to meet the practical needs of seismic design based on code design spectra for various kinds of structures equipped by viscous dampers, the complex complete quadratic combination (CCQC) method is deduced following similar procedures such as the well-known CQC method, in which a new modal velocity correlation coefficient, together with a new modal displacement-velocity correlation coefficient are involved besides the modal displacement correlation coefficient in normal CQC formula. The new algorithm of CCQC is not only as concise as that of the normal CQC but also has explicit physical meaning. The results obtained from complex mode superposition approaches are discussed and verified in some examples through step by step integration computation under a prescribed earth-quake motion input. From these examplary analyses, it may be pointed that the CCQC algorithm normally yields conservative outcome and that the forced mode uncoupling approach has good approximation even the discussed examplary structures are strongly non-proportional.     

Shock Response from Pseudodynamic Test Using Momentum Equations of Motion

The feasibility of using pseudodynamic techniques to yield shock responses from impulses is studied herein. A direct integration method is often used to solve the force equation of motion in performing a conventional pseudodynamic test. However, this technique might not be applicable to obtain an accurate shock response from an impulse as a load discontinuity occurs at the end of the impulse. This is because this discontinuity will lead to an extra amplitude distortion and the amount of this amplitude distortion is increased with the increase of time step. Hence, a small time step is needed to reduce the extra amplitude distortion and thus the displacement increment for each time step might be smaller than or as small as the resolution of the displacement transducer. As a result, the displacement increment cannot be accurately imposed upon the specimen and the responses will be contaminated by the noise. In addition, the test duration is drastically increased. Alternatively, this difficulty might be overcome if the momentum equation of motion instead of the force equation of motion is solved pseudodynamically. Hence, an external momentum is used in the solution of the momentum equation of motion. Since the external momentum is a resultant of the time integration of the external force the discontinuity problem will automatically disappear. Consequently, reliable shock responses can be obtained from pseudodynamic tests.     

SYSTEM IDENTIFICATION OF FEI-TSUI ARCH DAM FROM FORCED VIBRATION AND SEISMIC RESPONSE DATA

This paper presents the experimental investigation of the Fei-Tsui arch dam using the forced vibration test and its seismic response data. A forced vibration test was conducted on Fei-Tsui dam, this study presents the identified dynamic properties of the dam from these test data. For the identification of dam properties from seismic response data, in order to consider the nonuniform excitation of the seismic input and to describe the global behavior of the dam, the multiple input/multiple output discrete-time ARX model with least square estimation is applied to identify the dynamic characteristics of the dam. The system modal frequency, damping ratio and frequency response function are identified from both the forced vibration and seismic response data. To verify the accuracy of the identification result, comparison between discrete-time ARX model and a frequency domain conditioned spectral analysis was made. Finally, the spatial variation of ground motion across the free-field canyon surface is also studied.     

Dynamic Response of Saturated Soil - Foundation System Acted upon by Vibration

In this study, the response and behavior of machine foundations resting on dry and saturated sand was investigated experimentally. In order to investigate the response of soil and footing to steady state dynamic loading, a physical model was manufactured to simulate steady state harmonic load at different operating frequencies. Total of 84 physical models were performed. The footing parameters are related to the size of the rectangular footing and depth of embedment. Two sizes of rectangular steel model footing were tested at the surface and at 50 mm depth below model surface. Meanwhile the investigated parameters of the soil condition include dry and saturated sand for two relative densities 30% and 80%. The response of the footing was elaborated by measuring the amplitude of displacement by the vibration meter. The response of the soil to dynamic loading includes measuring the stresses inside the soil using piezoelectric sensors as well as measuring the excess pore water pressure using pore water pressure transducers. It was concluded that the maximum displacement amplitude response of the foundation resting on dry sand models is more than that on the saturated sand by about 5.0–10%. The maximum displacement amplitude of footing is reduced to half when the size of footing is doubled for dry and saturated sand. The final settlement (St) of the foundation increases with increasing the amplitude of dynamic force, operating frequency and degree of saturation. Meanwhile, it is reduced with increasing the relative density of sand, modulus of elasticity, and embedding inside soils. The excess pore water pressure increases with increasing the relative density of the sand, the amplitude of dynamic loading and the operating frequency. In contrast, the rate of dissipation of the excess pore water pressure during dynamic loading is more in the case of loose sand.     

Assessment of Effects of Reductions of Lateral Stiffness along Height on Buildings Modeled as Elastic Cantilever Shear Beams

A simplified model useful for assessing economic losses due to moderate seismicity events in urban areas has been developed by studying the behavior of buildings before yielding their structural system, allowing for nonuniform stiffness along their height. In particular, buildings are modeled as cantilever shear beams with uniform mass and parabolic reduction of lateral stiffness. This particular stiffness distribution is relevant, as it could be expected to occur in buildings where earthquake action is a critical structural design criterion. The equation of motion governing the dynamic behavior of the proposed model is solved analytically, finding mode shapes in terms of first and second zero-order Legendre functions. The solution is verified by comparing it with results obtained from fine mesh finite element models. The effect of reducing the lateral stiffness is then studied in the first five modes of vibration. Results include modal periods, mode shapes, modal participation factors, and derivatives of mode shapes. In general, it is found that effects of reduction of lateral stiffness in mode shapes are moderate when the lateral stiffness in the free end is smaller than about seventy percent of the lateral stiffness at the fixed end, but become significant for larger reductions. Effects are particularly important for the derivative of the mode shapes, which could play a significant role in estimating interstory drift demands in buildings. Model usefulness is showcased by analyzing a test case where both acceleration and drift demands are assessed by considering uniform beams and beams with parabolic stiffness variation, finding notable improvements by considering the latter.     

Seismic Evaluation of Tall Buildings Using a Simplified but Accurate Analysis Procedure

A simplified analysis procedure for evaluating the nonlinear seismic responses of tall reinforced concrete (RC) buildings is examined in this study. It is called the Uncoupled Modal Response History Analysis (UMRHA) procedure. It can be viewed as an extended version of the classical modal analysis procedure, where the nonlinear response of each vibration mode is first computed, and they are later on combined into the total response of the structure. The procedure requires the knowledge of the modal hysteretic behavior, which can be obtained from a cyclic modal pushover analysis. The responses of four tall buildings in Bangkok to distant large earthquakes are computed by this procedure and compared with those obtained from the Nonlinear Response History Analysis (NLRHA) procedure. These four buildings have different heights—varying from 20 to 44 stories, different configurations of floor plan, and different arrangement of RC walls. The comparison shows that the UMRHA procedure is able to accurately compute the story shears and story overturning moments, floor accelerations, and inter-story drifts of all these tall buildings. The required computational effort is also extremely low compared to that of the NLRHA procedure. Moreover, since the UMRHA procedure computes the response of each individual vibration mode, it provides more understanding and insight into the complex nonlinear seismic responses of these tall buildings.     

宁波保国寺大殿风振性能研究

为了更科学地保护我国传统木构建筑,需要研究我国江南地区传统木构建筑的风振性能,为此,以江南地区最古老、保存最完整的佛教木构建筑——保国寺大殿的主殿为例进行风振性能研究。首先基于三维激光扫描获得的真实可靠的保国寺大殿的几何信息,采用SAP2000有限元软件建立主殿的有限元模型,并进行动力特性分析。再利用MATLAB软件编写脉动风速时程,并将风压作用于有限元模型上分析,得到主殿的风振响应。得到结构模态、位移时程曲线、位移谱曲线、加速度响应,并对时程位移风振系数和规范风振系数进行对比。结构的振动响应主要由前三阶振型控制,在强风荷载作用下,最容易出现的变形就是南北向振动、东西向振动和扭转振动;各个典型节点的最大位移均在侧向位移容许值内;迎风面阑额中心位移最大、屋面上半部次之、屋面下半部及檐柱柱头最小;主殿相连的两个部分的加速度相差较多,屋面和屋身的连接节点应成为日后保国寺大殿的保护重点;时程风振系数变化与规范风振系数的变化规律不同是因为结构横向刚度分布不均匀;时程风振系数是规范风振系数的1.2~1.4倍,这是由于按照规范计算风振系数的过程中只考虑了第一阶振型的影响而忽略其他所有振型的结果;规范风振系数的计算只适用于形状、质量沿高度不变的高耸结构,不适用于类似本研究中的低矮木构传统房屋结构。     

Probabilistic Seismic Design of Pile Foundations in Non Liquefiable Soil by Response Spectrum Approach

The behavior of pile foundations in non liquefiable soil under seismic loading is considerably influenced by the variability in the soil and seismic design parameters. Hence, probabilistic models for the assessment of seismic pile design are necessary. Deformation of pile foundation in non liquefiable soil is dominated by inertial force from superstructure. The present study considers a pseudo-static approach based on code specified design response spectra. The response of the pile is determined by equivalent cantilever approach. The soil medium is modeled as a one-dimensional random field along the depth. The variability associated with undrained shear strength, design response spectrum ordinate, and superstructure mass is taken into consideration. Monte Carlo simulation technique is adopted to determine the probability of failure and reliability indices based on pile failure modes, namely exceedance of lateral displacement limit and moment capacity. A reliability-based design approach for the free head pile under seismic force is suggested that enables a rational choice of pile design parameters.     

Considering Soil-Nonlinearity Effect on Seismic Performance Evaluation of Structures based on Pushover Analysis

According to the most of current seismic codes, nonlinear soil behavior is commonly ignored in seismic evaluation procedure of the structures. To contribute on this matter, a pushover analysis method incorporating the probabilistic seismic hazard analysis (PSHA) is proposed to evaluate the effect of nonlinear soil response on seismic performance of a structure. The PSHA outcomes considering soil nonlinearity effect is involved in the analysis procedures by modifying the site-specific response spectrum. Results showed that incorporation of nonlinear soil behavior leads to an increase in displacement demand of structures which should accurately be considered in seismic design/assessment procedure. Results of implemented procedure are confirmed with the estimated displacement demand including soil-structure interaction (SSI).     

Effect of Infill Walls on the Drift Behavior of Reinforced Concrete Frames Subjected to Lateral-Load Reversals

Four-story, single-bay, 1/5 scaled reinforced concrete frames were tested with and without infill walls. Frames were subjected to pseudo-static cyclic loading. In addition, impact hammer measurements were made to obtain the natural frequencies and modal shapes at certain drift levels. It was observed that infill walls cause major changes on both the stiffness and the drift behavior of the frames. Effect of observed changes can be either advantageous or disadvantageous depending on failure mode. Results showed that the distribution of drift that is based on the mode shapes has higher local concentrations than the distribution observed under forced static conditions.     

Transient Response of Concrete Gravity Dam Considering Dam-Reservoir-Foundation Interaction

This article deals with the finite element analysis of dam with and without fluid-structure, soil-structure and soil-structure-fluid interaction. A two-dimensional direct coupling methodology is proposed to obtain the response of dam-reservoir-foundation system considering fluid-structure and soil-structure interaction simultaneously. The displacement based finite element technique is used to formulate the dam and foundation. The reservoir is modeled by pressure based finite element to reduce the degree of freedoms and there by the computational cost. The responses of dam, reservoir, and foundation with and without fluid-structure, soil-structure and soil-structure-fluid interaction are compared to study the influence of reservoir and soil foundation on the behavior of these respective sub systems. The fundamental frequency of individual sub system decreases with the consideration of coupling effect among these sub systems. On the comparison of the responses of dam, it is observed that the displacement and principal stresses are increased if the effect of reservoir and foundation are considered and the worst responses were observed when both the fluid-structure and soil-structure interaction effects are considered simultaneously. The magnitude and distribution of stresses within the foundation change with the consideration of soil-structure-fluid interaction. Similar to wstresses in the foundation, the hydrodynamic pressure within the reservoir also gets magnified due to interaction effects. The velocity distribution within the reservoir becomes distorted when the fluid-structure and soil-structure-fluid interaction are considered.     

Simple Models for Inelastic Seismic Analysis of Asymmetric Multistory RC Buildings

A simple stick model is presented for the inelastic seismic analysis in 3D of two-way eccentric multistory RC buildings. It has 3 DoFs per floor, point hinges at the ends of the vertical elements connecting floors, elastic story stiffness derived from the corresponding story force-interstory deformation relations of the elastic 3D structure under inverted-triangular floor loading (by torques for torsional stiffness, by horizontal forces for the lateral ones), story yield forces derived from the total resistant shear of the story vertical elements, but no coupling between lateral and torsional inelasticity. It is evaluated on the basis of comparisons of response histories of floor displacements to those from full nonlinear models in 3D of four actual buildings. Alternative locations of the story vertical element with respect to the floor mass center are examined: (a) the floor “center of twist” of the elastic 3D building under inverted-triangular floor torques; (b) the story “effective center of rigidity,” through which application of inverted triangular lateral forces does not induce twisting of floors; (c) the centroid of the secant stiffness of the story vertical members at yielding and (d) the centroid of the lateral force resistance of story vertical elements. Among alternatives (a)–(d), the floor “center of twist” provides the best agreement with floor displacement response-histories from full 3D nonlinear models. This means that the static eccentricity that matters for torsional response may be taken as that of the floor “center of twist.” The center of resistance comes up as the second-best choice.     

Seismic Response of Base-Isolated Benchmark Building with Variable Sliding Isolators

The seismic response of base-isolated benchmark building with variable sliding isolators like variable friction pendulum system (VFPS), variable frequency pendulum isolator (VFPI), and variable curvature friction pendulum system (VCFPS), along with conventional friction pendulum system (FPS), was studied under the seven earthquakes. The earthquakes are applied bi-directionally in the horizontal plane ignoring vertical ground motion component. The shear type base-isolated benchmark building is modeled as three-dimensional linear elastic structure having three degrees of freedom at each floor level. Time domain dynamic analysis of the benchmark building was carried out with the help of constant average acceleration Newmark-Beta method and nonlinear isolation forces was taken care by fourth-order Runge-Kutta method. The base-isolated benchmark building is investigated for uniform isolation and hybrid isolation in combination with laminated rubber bearings through the performance criteria and time history response of important structural response parameters like floor accelerations, base displacement, etc. It is observed that variable sliding isolators performed better than conventional FPS due to their varying characteristic properties which enable them to alter the isolator forces depending upon their isolator displacements thus improves the performance of the structure. The VFPS efficiently controls large isolator displacements and VFPI and VCFPS improve super structural response on the cost of isolator displacement. It is also observed that the hybrid isolation is relatively better in comparison to the uniform isolation for the benchmark building.     

Experimental Identification of the Dynamic Characteristics of a Flexible Rocking Structure

This article presents the results of free vibration and earthquake excitation tests to investigate the dynamic behavior of freely rocking flexible structures with different geometric and vibration characteristics. The primary objective of these tests was to identify the complex interaction of elasticity and rocking and discuss its salient effects on the rocking and vibration mode frequencies, shapes and excitation mechanisms. The variability of response is discussed, including critical investigation of the repeatability of the tests. It was found that the variability in energy dissipation and energy transfer to vibrations at impact may lead to significantly different responses to almost identical excitations.     

THE γ-FUNCTION PSEUDODYNAMIC ALGORITHM

Practical applications of the γ-function dissipative explicit method to pseudodynamic tests are thoroughly investigated herein. Detailed implementation of this pseudodynamic algorithm is schematically sketched. Numerical experiments and verification tests strongly indicate that the γ-function dissipative explicit method can effectively filter out the spurious participation of high frequency responses while the lower mode responses can be obtained very accurately. In addition, error propagation analysis also shows that the γ-function dissipative explicit method possesses much better error propagation characteristics when compared to the Newmark explicit method. Thus, this pseudodynamic algorithm is very suitable for the test system where the responses are dominated by the low frequency modes and the high frequency responses are of no interest.     

Multiple Degrees of Freedom Positioning Correction for Hybrid Simulation

This study proposes a high-precision positioning correction method for multiple degree-of-freedom loading units in hybrid simulation. These loading units can impose inaccurate displacements to the specimens due to the elastic deformation at the reaction wall or connections. To compensate for these displacement errors, an online correction method adjusts the displacement command by the difference between the target and achieved displacement. This correction method also accompanies an accurate 6DOF monitoring system to detect the displacement errors. Two examples of hybrid simulation tests are provided to demonstrate the precise displacements attained on the specimens through this control method.     

Optimal Control of Structures under Earthquakes Including Soil-Structure Interaction

It is well known that the soil-structure interaction (SSI) changes the dynamic response of a structure supported on flexible soil. The analysis of optimally controlled SSI systems has certain difficulties due to the nature of the SSI and the optimal control problem. In this paper, a two-step iteration-based numerical algorithm is proposed to handle optimally controlled SSI systems under earthquakes. First, the optimal control forces are obtained by using a fixed-base system. Then, the optimal control forces are converted to the frequency domain by the Fourier transform technique to be used in the equations of the SSI system. The lateral displacement and the rocking of the foundation are obtained from the equations of the SSI system containing the optimal control forces in the frequency domain. The lateral displacement and rocking of the foundation are then converted to the time domain by the inverse Fourier transform technique, and the lateral accelerations and the rocking accelerations of the foundation are obtained by the forward finite difference method. During the second step, the optimal control forces are calculated again by using the lateral acceleration and the rocking acceleration of the foundation along with the earthquake ground motion. Using the method explained above, the optimal control forces obtained in the time domain are used in the equations of the soil-structure system from which the behavior of foundation and structure is obtained. In the final section of the paper, a numerical study is conducted for a controlled structure supported on flexible soil.     

Dynamic Analysis and Seismic Response of Planar Circular Arches with Variable Cross-Section

The aim of this paper is to investigate the dynamic response of planar circular arches with variable cross-section subjected to seismic ground motions. Arches have a wide range of application (e.g. bridges, roofs) thanks to their capacity to span large areas by resolving vertical actions into compressive stresses and confining tensile stresses. The full understanding of their dynamic response is a challenging technical and computational problem, especially when seismic loading is considered. For example, the assumption of axial inextensibility simplifies the differential equations but overestimates the vibration frequencies, especially those of shallow arches since axial forces are of paramount importance (as opposed to beams). In lieu of the above, our formulation incorporates the effect of axial extension, and the arches are modeled using a new generic curved beam model that includes both axial (tangential) and transverse (normal) to the arch centerline deformations, and is able to account for variable mass and stiffness properties, as well as elastic support or restraint. The resulting dynamic governing equations of the circular arch are formulated in terms of the displacements, and solved using an efficient integral equation method. Three circular arches with variable rectangular cross-section are analyzed in order to investigate their dynamic properties and seismic performance. Using both time history and modal analysis useful conclusions are drawn with regard to the contribution of each mode on the calculation of different response quantities.     

Completeness Verification of Complex Response Spectrum Method for Underdamped and Overdamped Multiple-Support Systems Regarding the Decoupled Damping as Mathematical Parameter without Physical Meaning

The completeness of the complex response spectrum method for both formally underdamped and overdamped modes is theoretically proved, and the physical meanings of the decoupled modes as well as involved parameters are recognized and clarified in this paper. For the system with relatively large non-classical damping, the eigenvalue pairs generated by the complex mode decomposition method are real and the so-called modal damping ratios are larger than unity. In this paper, we firstly clarified that the decoupled modes are virtual and the so-called modal frequency and damping ratio are mathematical parameters that have no physical meaning. Then, the completeness of the complex response spectrum method for both formally underdamped and overdamped modes is rigorously proved by allowing the “damping frequency” to be an imaginary number. For the virtually overdamped modes, Duhamel integral involved in the calculation for formally underdamped modes automatically convert to hyperbolic Duhamel integral. A numerical example taken from the published literature is given to verify this method. Structural responses for the system with coupled damping under multi-support seismic excitations are further analyzed and numerical results indicate the accuracy of complex response spectrum method.     

Seismic Displacement Demands on Skewed Bridge Decks Supported on Elastomeric Bearings

The unseating of decks is one of the most prevalent failure modes of bridges after earthquake events, as observed in the 2010 Chile Earthquake. Damaged bridges in Chile often had skew angles and were supported on elastomeric bearings. Similar bridge construction practices with decks supported on elastomeric bearings are also common in the central and eastern U.S. (CEUS). The seismic displacement demands on skewed bridges are more complicated than those on bridges without skew angles due to the coupling of translational modes with the rotational mode of vibration. The study presented in this article seeks to understand the seismic response of skewed bridge decks supported on elastomeric bearings. The scope of the study is limited to one- and two-span bridges, which constitute a large portion of bridge inventory in the CEUS. The vibration modes of skewed bridge decks are derived in closed form and the modes are compared when the gaps between the bridge deck and the abutment are open and when one of the gaps is closed due to seismic excitation. Nonlinear response history analyses are carried out to understand the effects of vertical ground motion, skew angles, aspect ratios, and different ground motion types on the seismic displacement demand in these cases. Amplification factors that approximate the increase in the displacement demand due to the skew angle are proposed.