Showing papers in "Earthquake Engineering & Structural Dynamics in 1986"

Random vibration of hysteretic systems under bi-directional ground motions

Abstract: A random vibration method is developed for the response analysis of hysteretic structural systems under stochastic two-dimensional earthquake excitations. The biaxial hysteretic restoring force is modelled by coupled non-linear differential equations; the response statistics are obtained using the equivalent linearization technique. The validity of the proposed model is appraised using available biaxial loading tests of reinforced concrete columns. A parametric study was performed to examine the significance of the effects of biaxial interaction under earthquake excitations. A practical method to evaluate the extreme response statistics is also presented.

Response of a rigid foundation to a spatially random ground motion

Abstract: Methode d'obtention de la reponse dynamique d'une fondation rigide etendue sur un demi-espace elastique lorsqu'elle est soumise a un mouvement du sol variant spatialement incluant a la fois des effets deterministes et aleatoires. Description des resultats numeriques pour une fondation carree rigide et pour un mouvement du sol caracterise par une fonction de coherence spatiale particuliere. Les resultats obtenus indiquent que le mouvement du sol spatialement aleatoire produit des effets similaires aux effets deterministes du passage d'une onde incluant la reduction des composantes de translation de la reponse aux hautes frequences et la creation d'un balancement

Modal analysis of non-classically damped linear systems

Abstract: SUMMARY A critical, textbook-like review of the generalized modal superposition method of evaluating the dynamic response of nonclassically damped linear systems is presented, which it is hoped will increase the attractiveness of the method to structural engineers and its application in structural engineering practice and research. Special attention is given to identifying the physical significance of the various elements of the solution and to simplifying its implementation. It is shown that the displacements of a nonclassically damped n-degree-of-freedom system may be expressed as a linear combination of the displacements and velocities of n similarly excited single-degree-of-freedom systems, and that once the natural frequencies of vibration of the system have been determined, its response to an arbitrary excitation may be computed with only minimal computational effort beyond that required for the analysis of a classically damped system of the same size. The concepts involved are illustrated by a series of exqmples, and comprehensive numerical data for a three-degree-offreedom system are presented which elucidate the effects of several important parameters. The exact solutions for the system are also compared over a wide range of conditions with those computed approximately considering the system to be classically damped, and the interrelationship of two sets of solutions is discussed.

Damping in tall buildings—a mechanism and a predictor

Abstract: Only when the data base is of sufficient quality can a theoretical model of any process be justified. Such a data base has recently been assembled in the field of damping in tall buildings. It is now possible to specify a damping value appropriate to a particular building vibrating at a specified amplitude which is well correlated with values which are measured in practice. The mechanism on which this predictor is based suggests that the predictor will be applicable to other types of structure as well. As yet the data base for these other structures does not exist, but some circumstantial evidence is encouraging.

Approximate dynamic model of embedded foundation in time domain

Abstract: A discrete model to represent the unbounded soil (halfspace) in a soil–structure interaction analysis in the time domain is developed. For each dynamic degree of freedom of the foundation node, the discrete model consists of a mass M0 which is attached to a rigid support with a spring K and with a damper C0. In addition, a free node with the mass M1 is introduced, which is connected to the foundation node with a damper C1. All coefficients are frequency-independent. The discrete model is semi-empirical. It is based on a semi-infinite truncated cone, whereby, after enforcing the static stiffness, the remaining parameters are modified to achieve an optimal fit of the dynamic-stiffness coefficient in the frequency domain. The spring K is equal to the static stiffness. The coefficients appearing in the equations for the dampers C0, C1 and the masses M0, M1 are specified (assuming a homogeneous halfspace) for the disc, the embedded cylinder, the rectangle (also embedded) and the strip. A square on a layer whose stiffness increases with depth resting on a homogeneous halfspace is also treated. For an embedded foundation, eccentricities arise. Material damping increases the damper C0 and the mass M0.

Steady‐state harmonic response of a rigid plate bearing on a liquid‐saturated poroelastic halfspace

Abstract: Soil–structure interaction problems are typically modelled by assuming subgrade behaviour to be either elastic or viscoelastic. Herein, compliance functions that may be used to solve soil–structure interaction problems are evaluated by treating the subgrade as a liquid-saturated poroelastic material whose behaviour is governed by Biot's theory. The compliances are evaluated for the harmonic rocking and vertical motions of rigid permeable and impermeable plates bearing on a poroelastic halfspace. Comparisons are made with elastic solutions which assume the subgrade to be either completely drained or undrained. Also, solid and fluid contact stresses are reported for the poroelastic case and compared to the solid contact stresses for the elastic cases.

Mode-superposition methods in dynamic analysis of classically and non-classically damped linear systems

Abstract: Mode-superposition analysis is an efficient tool for the evaluation of the response of linear systems subjected to dynamic agencies. Two well-known mode-superposition methods are available in the literature, the mode-displacement method and the mode-acceleration method. Within this frame a method is proposed called a dynamic correction method which evaluates the structural response as the sum of a pseudostatic response, which is the particular solution of the differential equations, and a dynamic correction evaluated using a reduced number of natural modes. The greater accuracy of the proposed method with respect to the other methods is evidenced through extensive numerical tests, for classically and non-classically damped systems.

Transient elastodynamic analysis of three‐dimensional problems by boundary element method

Abstract: An advanced implementation of the direct boundary element method applicable to transient problems involving three-dimensional solids of arbitrary shape and connectivity is presented. The work first focuses on the formulation of the method, followed by a discussion of the fundamental singular solutions. Subsequently, a family of isoparametric boundary elements is introduced, along with the necessary numerical integration techniques as well as the solution algorithm. Numerical examples are presented, which demonstrate the unconditional stability and high accuracy of this dynamic analysis technique.

Rotational components of the surface ground motion during an earthquake

Abstract: Under the assumption of an immovable point source for the earthquake energy release, a mathematical model is developed to compute the components of the free field surface rotation consistent with assigned translational components. Experimental evidence on these quantities is offered by the strong-motion instrument arrays. However, lack of adequate identification of the different wave contributions prevents in general a direct measurement of rotational and torsional components, allowing only a qualitative assessment of just how large these components might be. The only exceptions are the records of SMART-1, which offer the entire information to derive rotations. The available data of strong-motion instrument arrays for this problem are summarized in Table I. They are in qualitative agreement with the suggestions of the mathematical model. Earthquake response spectra for the rotational components are developed for a few selected cases, in particular a rotational spectrum consistent with the prescription of USNRC Regulatory Guide 1.60. The correlations between the translational and rotational components of the ground motion are also shown.

Inferred displacements, velocities and rotations of a long rigid foundation located at el centro differential array site during the 1979 imperial valley, california, earthquake

Abstract: Digitally recorded acceleration time histories of the 1979 Imperial Valley earthquake by the five elements of the El Centro Differential Array are used to estimate the induced torsional and rocking vibrations of long structures in the near field of strong earthquakes. The underlying assumptions in these estimates are high rigidity of the foundation relative to the surrounding soil and its negligible mass. Based on these analyses, the peak rotation amplitude of an 18 m long foundation within 5 km of the fault is approximately 1 × 10−3 rad, or nearly 0·06 degrees, and decreases rapidly for larger foundation dimensions for either the rocking or torsional mode. The strongest impulses of rotations coincide with arrival of the S phase. This paper examines the sensitivity of the resulting peak displacement and particle velocity amplitudes to the foundation dimension. It is inferred that the base averaging effects of displacement and velocity fields at frequencies contributing to peak values (f < 1 Hz) are negligibly small. Rocking and torsional amplitudes for the specific site and geometry considered here show a tendency to reduce with the increase of foundation dimensions.

Floor response spectrum method for seismic analysis of multiply supported secondary systems

Abstract: Fundamental principles from structural dynamics, theory of random processes and perturbation techniques are used to develop a new method for seismic analysis of multiply supported secondary subsystems, such as piping attached to primary structures. The method provides a decoupled analysis of the secondary subsystem wherein the response is given in terms of response spectra associated with the attachment points (‘floors’). In order to account for correlations between modal responses and between support motions, an extension of the conventional floor response spectrum, denoted crossoscillator, cross-floor response spectrum, is introduced. Important effects of tuning, interaction, non-classical damping and spatial coupling, which are inherent characteristics of combined primary–secondary systems, are included through the extended spectra. An efficient method for generation of the extended spectra directly in terms of a ground response spectrum is developed. Numerical comparisons with exact results are used to examine the accuracy of the proposed method and to demonstrate the importance of the characteristics mentioned above. In all cases examined, the proposed method shows excellent agreement with exact results. By accounting for the effect of interaction, the proposed method leads to more realistic and economical design criteria for secondary subsystems.

Earthquake analysis of arch dams including dam–water interaction, reservoir boundary absorption and foundation flexibility

Abstract: The available substructure method and computer program for the steady-state, harmonic response analysis of arch dams, including the hydrodynamic effects, are extended to consider flexibility of the foundation rock and to include Fourier synthesis of harmonic responses to obtain the earthquake response of arch dams. By efficient evaluation of hydrodynamic terms, interpolation of frequency response functions and more efficient computer programming, the computational costs for analysing arch dams have been reduced by an order of magnitude relative to the available procedure.

On the dynamic response of rigid body assemblies

Abstract: Examen de la reponse dynamique de structures ayant un degre significatif de mouvement de corps rigide durant un seisme. On choisit un assemblage de prismes rigides bidimensionnels soumis a des contraintes cinematiques auto-imposees comme modele representatif de ce type de structure. Le phenomene de separation ou de soulevement a ete souvent observe entre une structure et sa base dans de nombreux seismes

A comparison of time‐domain transmitting boundaries

Abstract: A non-linear interaction analysis with a (generalized) non-linear structure and a linear unbounded soil is analysed in the time domain, based either on the sub-structure method, which involves global convolution integrals, or on the direct method with local boundary conditions. Alternatively, the hybrid frequency–time-domain method of analysis, which is an iterative scheme, could be used. Approximate local boundary conditions to model the wave propagation towards infinity on the artificial boundary used in the direct method of non-linear soil–structure-interaction analysis to be performed in the time domain are examined. A semi-infinite rod supported elastically, which exhibits the same properties as certain unbounded soils such as dispersion and a cut-off frequency, is used for the investigation. For a transient excitation, the superposition boundary with frequent averaging, the well-known viscous damper and the extrapolation algorithm lead to good accuracy. Moving the artificial boundary further away from the structure (or more precisely, increasing the ratio of the distance of the artificial boundary to the wave length) improves the accuracy.

Optimum design and earthquake-response constrained design of elastic shear buildings

Abstract: The necessary and sufficient conditions for global optimality and the closed form solution are derived for a problem of minimizing a weighted sum of storey stiffnesses of a shear building subject to a fundamental frequency constraint. A set of higher frequency formulae is derived for optimally designed shear buildings with equal masses. It is shown that, if the weight coefficients are regarded as parameters for adjustment, the distribution of the SRSS estimates of the maximum interstorey drifts can be adjusted so as to coincide with a specified one. The base shear–fundamental period formulae and the base shear–design drift formulae are derived semi-analytically for shear buildings with equal masses on the basis of the optimum design formulae and the SRSS estimates of the maximum interstorey drifts. Some numerical examples are presented in order to illustrate the validity of the proposed method of earthquake-response constrained design.

A model for system identification of degrading structures

Abstract: This paper presents a physically motivated model for the gross dynamic behaviour of degrading hysteretic structures subjected to damaging earthquake motions. The model is intended to provide a vehicle for gaining insight into the nature of the non-linear response of such structures and to be used in system identification studies. A concept is presented for identifying the model parameters based on examination of the gross restoring force characteristic of a real structure. The model is applied to data from two damaged structures: the Bank of California building in the 1971 San Fernando earthquake and the Imperial County Services building in the 1979 Imperial County earthquake.

Frequency response functions for arch dams: Hydrodynamic and foundation flexibility effects

Abstract: La reponse dynamique des barrages-voutes en beton a un mouvement harmonique du sol est influencee par l'interaction entre le barrage et l'eau retenue et par l'absorption des ondes hydrodynamiques dans les alluvions et sediments a la limite de la retenue. Etude de ces effets et de ceux dus a la flexibilite de la roche de fondation. L'etude est faite sur le barrage de Morrow Point

Structural response for six correlated earthquake components

Abstract: The paper examines the effect on the structural response of the inevitable correlation which exists between the six earthquake components acting along a set of structural axes. The rotational components are expressed in terms of the spatial derivatives of the translational components. For the calculation of response, modal analysis is employed so that ground response spectra can also be used as seismic input. A methodology is developed to obtain the maximum mean square response which can occur in a structure, irrespective of its orientation with respect to the impinging seismic waves. The application of this methodology for the calculation of design response is advocated, especially for asymmetric structures. For the assumed model of seismic wave motion, the numerical results show a significant contribution to the response from the rotational components. This contribution is, however, expected to be reduced by structural foundation averaging and interaction effects. Further studies with more complete models of seismic wave motions, and their interaction with structural foundations, are thus warranted for a realistic evaluation and characterization of the rotational inputs for design purposes.

Discrete crack modelling for dynamically loaded, unreinforced concrete structures

Abstract: The dynamic response of unreinforced concrete structures is studied taking account of initiation, extension, closing and reopening of so-called discrete cracks. The computational procedure is based on the finite-element method and is at present restricted to two-dimensional situations. The discrete cracks are simulated by separation of originally adjacent finite elements. An equivalent tensile-strength criterion is used for the initiation and extension of the cracks which are assumed to propagate perpendicularly to the principal tensile stress. If this direction does not coincide with the interelement boundaries of the finite-element mesh, the latter is automatically altered. Between elements being separated by a crack special ‘crack elements’ are introduced, which take account of the stress transfer by aggregate interlock. The equations of motion are integrated numerically using an explicit formulation. The procedures outlined are demonstrated on a simplified cross-section of a concrete gravity dam subjected to horizontal earthquake excitation.

Modal time history analysis of non-classically damped structures for seismic motions

Abstract: The step-by-step modal time history integration methods are developed for dynamic analysis of non-classically damped linear structures subjected to earthquake-induced ground motions. Both the mode displacement and mode acceleration-based algorithms are presented for the calculation of member and acceleration responses. The complex-valued eigenvectors are used to effect the modal decoupling of the equations of motion. However, the recursive step-by-step algorithms are still in terms of real quantities. The numerical results for the acceleration response and floor response spectra, obtained with these approaches, are presented. The mode acceleration approach is observed to be decidedly better than the mode displacement approach in as much as it alleviates the so-called missing mass effect, caused by the truncation of modes, very effectively. The utilization of the mode acceleration-based algorithms is, thus, recommended in all dynamic analyses for earthquake-induced ground motions.

Dynamic stiffness of rigid rectangular foundations on the half-space

Abstract: The dynamic soil–structure interaction of a rigid rectangular foundation with the subsoil represents a mixed-boundary value problem. This problem is formulated in terms of a system of coupled Fredholm integral equations of the first kind. The subsoil is modelled by a homogeneous, linear-elastic and isotropic half-space which is perfectly bonded to the rigid, rectangular foundation. An approximate solution for the resultant loads between the foundation and the half-space due to a unit forced displacement or rotation is obtained using the Bubnov–Galerkin method. Using this method the displacement boundary value conditions are exactly satisfied and the contact stress distributions between the foundation and the half-space are approximated by series expansions of Chebyshev polynomials. This method provides a simple means of studying the soil-structure interaction of rectangular foundations with different inertia properties.

Study of the earthquake response of pine flat dam

Abstract: The earthquake response of Pine Flat Dam is examined by a study of time history responses computed for a large set of earthquake ground acceleration records whose time axes have been systematically varied. Linear elastic behaviour is assumed. Topics considered include an investigation of the importance of the presence of water, water compressibility and the vertical component of ground motion; an evaluation of the accuracy of the lumped, added mass representation of the water; and a determination of the intensity of earthquake required to initiate non-linear behaviour in both the dam and water.

Attenuation of peak ground acceleration, velocity and displacement based on multiple regression analysis of japanese strong motion records

Abstract: This paper presents the result of multiple regression analysis of 197 sets of two orthogonal horizontal strong motion acceleration records. They were obtained at 67 free field sites in Japan from 90 earthquakes with focal depth less than 60 km. Because sensitivity of the Japanese SMAC accelerograph is appreciably low at the high frequency range, instrument correction was performed on the original data. Each pair of two orthogonal horizontal components was combined in the time domain to get the maximum peak ground motions in the horizontal plane. The records were classified into three groups due to subsoil condition. With the use of multiple regression analysis, empirical formulae of attenuation of the maximum peak ground acceleration, velocity and displacement were proposed for three subsoil conditions.

Effective eccentricity for inelastic seismic response of buildings

Abstract: Generalisation du concept d'excentricite effective pour les systemes non elastiques en vue d'obtenir des directives pour l'estimation du comportement dynamique d'un systeme monosymetrique soumis a des excitations unidirectionnelles du sol. Comparaison des reponses non elastiques de 3 modeles de structure ayant les memes reponses elastiques globales. Choix d'un de ces modele pour evaluer l'influence de l'asymetrie

Effect of vertical motion on friction‐driven isolation systems

Abstract: The effect of the vertical component of ground motion on the horizontal response of a sliding system is studied. Soil–foundation interaction is included. The results indicate that the effect of vertical motion is significant in the cascs of harmonically excited foundations.

Analysis of the observed seismic response of a highway bridge

Abstract: Strong-motion accelerograms obtained on the San Juan Bautista 156/101 Separation Bridge during the 6 August 1979 Coyote Lake, California, earthquake are used to examine the response of this multiple-span bridge to moderate levels of earthquake loading. Although the bridge was not damaged, the records are of significant engineering interest as they are the first to be recorded on a highway bridge structure in North America. A technique of system identification is used to determine optimal modal parameters for linear models which can closely replicate the observed time-domain seismic response of the bridge. Time variations in frequency and damping in the horizontal response are identified using a moving-window analysis. A three-dimensional finite element model is developed to study the bridge response in detail. The first two horizontal modal frequencies computed from this model are in excellent agreement with information obtained during the system identification analysis provided the finite element model's expansion joints are locked, preventing relative translational motions from occurring across the joints. Locking is confirmed by the observed seismic deformations of the structure in the fundamental mode. Fundamental vertical frequencies of the individual spans, predicted by the finite element model, are in very good agreement with ambient vibration test data.

Seismic shear vibration of embankment dams in semi‐cylindrical valleys

Abstract: Solution analytique rigoureuse de la reponse en cisaillement lineaire lateral de ces barrages. Expressions algebriques sous forme fermee appartenant a la fois aux oscillations induites a la base et aux oscillations libres et etudes comparatives et parametriques intensives pour elucider les effets importants de la geometrie du canyon (forme et elancement) sur la reponse dynamique. Etude de l'etat stationnaire harmonique, des accelerations par cisaillement du barrage et comparaison avec les valeurs obtenues a l'aide d'une analyse tridimensionnelle d'autres geometries de canyons

Dynamic modelling of axisymmetric foundations

Abstract: A boundary impedance matrix to characterize a homogeneous half-space with a cylindrical cavity is formulated by the global Ritz method. This method, which is basically the same as the indirect boundary element method, uses a very limited number of distributed sources to generate Ritz vector sets representing the far-field impedances. The resulting impedance matrix is fully-coupled, complex-valued, frequency-dependent and can be combined with any compatible finite element near field for carrying out soil–structure interaction analyses. The numerical results obtained by this method for surface supported and embedded foundations are very close to corresponding published results obtained by different procedures. Combining the boundary impedance matrix with a finite element near field provides an hybrid model which is effective and efficient in carrying out dynamic soil–structure interaction analyses.

A hybrid method for three‐dimensional problems of dynamics of foundations

Abstract: A method is developed applicable to problems of dynamics of arbitrary-geometry foundations. Finite elements are employed in the near field in order to obtain a discrete solution. In the far field, a semidiscrete solution is synthesized from modes also calculated by the finite element method. The solutions are matched by applying the stationarity condition of a functional. Examples of application are presented in order to verify the validity and illustrate the use of the method.

Non-linear soil–structure interaction analysis based on the boundary-element method in time domain with application to embedded foundation

Abstract: The various boundary-element methods, well established in the frequency domain, are developed in the time domain for a foundation embedded in a layered halfspace. They are the weighted-residual technique and the indirect boundary-element method, based on a weighted-residual equation, and the direct boundary-element method based on a reciprocity equation, both equations involving time and space. In the indirect approach, formulating the weighted-residual equation over the last time step only results in the truncated indirect boundary-element formulation which requires a reduced computational effort. In all cases, convolution integrals occur. The truncated indirect boundary-element method leads to a highly reliable algorithm, as is verified when a linear analysis in the time domain is compared to the corresponding one in the frequency domain. This boundary-element formulation, which is non-local in space and time, represents a rigorous generally applicable method taking into account a layered halfspace in a non-linear soil-structure interaction analysis. As an example, the non-linear soil-structure interaction analysis of a structure embedded in a halfspace with partial uplift of the basemat and separation of the side wall is investigated.