Showing papers in "Earthquake Engineering & Structural Dynamics in 1977"

Improved numerical dissipation for time integration algorithms in structural dynamics

Abstract: A new family of unconditionally stable one-step methods for the direct integration of the equations of structural dynamics is introduced and is shown to possess improved algorithmic damping properties which can be continuously controlled. The new methods are compared with members of the Newmark family, and the Houbolt and Wilson methods.

Soil-pile interaction in horizontal vibration

Abstract: Interaction between soil and an elastic pile vibrating horizontally is theoretically examined. The soil is modelled as a linear, viscoelastic layer overlying rigid bedrock. The pile is assumed to be vertical and point bearing. This study utilizes the definition of soil resistance presented in a preceding paper.1 A direct solution is developed which yields closed form formulas for pile displacement, stiffness and damping. A parametric study clarifies the role of the parameters involved, illustrates the interaction between the soil and the pile and shows the stiffness and damping properties of the soil-pile system for typical values of the governing parameters.

A new look at the newmark, houbolt and other time stepping formulas. A weighted residual approach

Abstract: The Newmark time stepping algorithm which was introduced in 1959, using constants γ and β which average the integration process, can be rederived as the most general finite element-weighted residual algorithm involving three consecutive sets of displacements. This derivation is much simpler than that involved originally in the New-mark presentation, and indicates a very wide range of possibilities of approximation. The application of the process to four point (cubic) algorithms leads to another family of formulas of which the Houbolt algorithm is a particular case. The use of the generalized expressions in the context of first-order equations is indicated and shows how some new, as well as some of the old, formulas can be developed.

Resistance of soil to a horizontally vibrating pile

Abstract: The resistance of a soil layer to steady horizontal vibration of an elastic pile is theoretically investigated. The pile is assumed to be vertical and of circular cross-section. The soil is modelled as a linear viscoelastic layer with hysteretic material damping. A closed form solution is obtained for the resistance of the soil layer to the motion. This resistance depends on shear modulus of soil, frequency, pile slenderness, material damping and Poisson's ratio. A parametric study of the effect of these parameters is included. The soil layer resistance is expressed in a form which can be used directly in the solution of the soil-pile interaction problem which is treated in a subsequent paper. The approach also applies for rigid deeply embedded footings.

Errors in response calculations for non‐classically damped structures

Abstract: The classical normal mode method of determining response is extremely useful for practical calculations, but depends upon the damping matrix being orthogonal with respect to the modal vectors. Approximations that allow the method to be used when this condition is not satisfied have been suggested; the simplest approach is to neglect off-diagonal terms in the triple matrix product formed from the damping and modal matrices. In this paper the errors in response caused by this approximation are determined for several simple structures for a wide range of damping parameters and different types of excitation. Based on these results a criterion, relating modal damping and natural frequencies, is formulated; if this is satisfied, the errors in response caused by this diagonalization procedure are within acceptable limits.

Seismic design spectra and mapping procedures using hazard analysis based directly on oscillator response

Abstract: The calculation of design spectra for building sites threatened by seismic ground motion is approached by considering the maximum responses of linearly elastic oscillators as indicators of ground motion intensity. Attenuation functions describing the distribution of response as a function of earthquake magnitude and distance are derived using 68 components of recorded ground motion as data. With a seismic hazard analysis for several hypothetical building sites, the distributions of maximum oscillator responses to earthquakes of random magnitude and location are calculated, and spectra are drawn to indicate the maximum responses associated with specified probability levels. These spectra are compared to design spectra calculated from published methods of amplifying peak ground motion parameters. The latter spectra are found to be inconsistent in terms of risk for building sites very close and very far from faults. A ground motion parameter defined to be proportional to the maximum response of a 1 Hz, 2 per cent damped linearly elastic oscillator is investigated; this parameter, in conjunction with peak ground acceleration, is found to lead to risk-consistent design spectra. Through these two parameters, a design earthquake magnitude and design hypocentral distance are defined, for a specified building site and risk level. The use of these parameters in the seismic hazard mapping of a region is illustrated.

Elastic earthquake analysis of torsionally coupled multistorey buildings

Abstract: With the aid of perturbation analysis of vibration frequencies and mode shapes it is shown that any lower vibration mode of a torsionally coupled building may be approximated as a linear combination of three vibration modes of the corresponding torsionally uncoupled system (a system with coincident centres of mass and resistance but all other properties are identical to the actual system): one translational mode along each of the two principal axes of resistance and one mode in torsional vibration. This result provides the motivation for a simpler—relative to the standard—procedure for analysing the response of torsionally coupled multistorey buildings to earthquake ground motion. To illustrate the application and accuracy of this procedure two numerical examples are presented.

Earthquake response of steel‐framed multistorey buildings with set‐backs

Abstract: A study is made of the dynamic behaviour of multistorey steel rigid-frame buildings with set-back towers. The effects of set-backs upon the building frequencies and mode shapes are examined. Then the effects of set-backs on seismic response are investigated by analysing the response of a series of set-back building frame models to the El Centro ground motion. Finally, the computed responses to the El Centro earthquake are compared with some code provisions dealing with the seismic design of set-back buildings. The conclusions derived from the study include the following: 1.The higher modes of vibration of a set-back building can make a very substantial contribution to its total seismic response; this contribution increases with the slenderness of the tower. 2.Some of the important response parameters for the tower portion of a set-back building are substantially larger than for a related uniform building. 3.For very slender towers, the transition region between the tower and the base may be subjected to very large storey shears.

Contact stresses and ground motion generated by soil‐structure interaction

Abstract: A study has been made of the dynamic contact stresses that the foundation of a nine-storey reinforced concrete building exerts on the soil during forced vibration tests. The effects of the flexibility of the foundation on the contact stress distribution and on the force-displacement relationship for the foundation have been examined in an attempt at testing several simplifying assumptions commonly used in soil-structure interaction studies. Comparisons of calculated and observed ground displacements induced by soil-structure interaction in the immediate neighbourhood of the building have also been presented.

Seismic response due to travelling shear wave including soil‐structure interaction with base‐mat uplift

Abstract: The seismic response due to a travelling shear wave is investigated. The resulting input consists of a translational-and a torsional-acceleration time history, which depend on the ratio of the wavelength to the dimension of the footing. A nuclear reactor building is used for illustration. The combined result of the translational and torsional elastic response (the latter arises even in an axisymmetric structure) will not, in general, be larger than that encountered in the case of a spatially uniform earthquake. If the footing slips or becomes partially separated from the soil, a non-linear dynamic analysis has to be performed to determine the response. Substantial motions in all three directions will take place. The peak structural responses and the floor-response spectra are found to be highly non-linear for high acceleration input values.

Dynamic properties of asymmetric wall‐frame structures

Abstract: An approximate method is proposed for evaluation of the natural frequencies and mode shapes of uniform asymmetric wall-frame structures. An exact solution is first given for the case in which the coefficient matrix of the dynamic equilibrium equations satisfies certain conditions. Using perturbation analysis, the method is then applied to the more general case in which these conditions are only approximately satisfied. A numerical example is presented to illustrate the technique.

Frequency analysis of coupled shear wall assemblies

Abstract: The finite strip procedure is used to predict the free vibration response of both planar and non-planar coupled shear wall assemblies. The solid walls are considered as vertical cantilever strips and a comparison is made between modelling the spandrel beams as discrete beams and as an equivalent continuum with orthotropic plate properties. It is shown that both approaches lead to essentially the same frequencies. The effects of vertical inertial forces and shear deflection are included, and structures considered may have properties that vary with height. The method presented appears to be more versatile than previously published techniques and numerical comparisons with existing methods indicate the predicted results to be accurate.

Soil‐structure interaction at higher frequency factors

Abstract: Approximate solutions to the forced vibrations of a rigid circular plate attached to the surface of an elastic halfspace are presented for large values of the frequency factor. These results are important when solving soil-structure interaction problems when such problems involve high-frequency factors. This situation arises when high-frequency components of earthquakes are associated with a relatively rigid foundation of a large base and located on a soft terrain. Similar situations occur in cases of blast loadings and impact and in the foundations of large high-speed machinery. These solutions are used to solve the problem of the motion of a rigid mass on an elastic halfspace subjected to steady state and transient horizontal accelerations. From these results, it is deduced that a large rigid mat foundation located on soft terrain significantly attenuates input accelerations and consequently may be useful as the foundation of large massive rigid structures such as nuclear power station. /Author/

Antiplane dynamic soil-bridge interaction for incident plane SH-waves

Abstract: The analysis of dynamic soil-bridge interaction has been performed in three steps. These are: the analysis of input motions, the force-displacment relationships for the foundations, the dynamic analysis of the structure itself, i.e. the bridge. Based on the exact solution of the first two steps, the dynamic interaction of a simple two-dimensional bridge model erected on an elastic half-space has been investigated for a single span case. The two-dimensional model under study consists of an elastic shear girder bridge supported by two rigid abutments and rigid foundations which have a circular cross-section and are welded to the half-space. It has been shown that the dynamic interaction depends on: the incidence angle of plane SH-waves, the ratio of the rigidity of the girder and the soil, the ratio of the girder mass to the mass of the rigid abutment-foundation system, the span of the bridge. The dynamic response of the girder and the effect of the radiative damping in the half-space on the interaction of the girder have been studied. /Author/

Free vibrations of open-section shear walls

Abstract: The coupled torsional-flexural vibration of open-section shear walls, braced by connecting beams at each floor level, is analysed on the basis of Vlasov's theory of thin-walled beams. The basic dynamic equations and boundary conditions are derived from Hamilton's principle, and a numerical solution obtained by the Ritz-Galerkin method. In addition to the primary torsional and flexural inertias, secondary effects due to rotatory and warping inertia forces have also been taken into account. The method is suitable for both rigid and flexible base conditions. A series of numerical examples is presented in which analytical results are compared with available experimental data, and the effects of secondary inertia forces, base flexibility and connecting beams upon the vibration characteristics of such shear walls are examined for two different structural forms.

Comparative analysis of the seismic risk in sites of different seismicity

Abstract: Four sites (a, b, c, d) are considered of differing seismicity. The calculations, for two different buildings on each site, consisted of: 1.The total monetary cost as a function of the design seismic coefficient c. 2.The marginal cost of a saved life, ΔD/ΔL, also depending on c. 3.The ratios between the design seismic coefficients in the various sites, cd/ca, cd/cb, cd/cc, that lead to the same marginal cost ΔD/ΔL in all sites. The marginal cost ΔD/ΔL turns out to be highly sensitive to uncertainties, but the other ratios are scarcely influenced by the uncertainties met in analysing the problem. This would suggest the adoption of a rationalized principle for comparing the levels of severity in the Codes for sites with differing degrees of seismicity. If a criterion for evaluating indirect cost due to damage or collapse is fixed for each building and each site the values of c corresponding to minimum monetary cost are not strongly influenced by the said uncertainties. This would make it possible to look into the problem of the minimum levels of severity in the Codes. Of course, this assumes that the levels of severity should always be higher than those corresponding to the minimum monetary cost.

Response of circular bridge decks to moving vehicles

Abstract: This paper discusses the dynamic response of a curved bridge deck to a moving vehicle. The bridge deck is idealized as a set of annular sector plates and circular rings rigidly jointed together. On the basis of classical plate and ring theories a method has been developed to obtain the response to a moving vehicle idealized as a spring mass system. After obtaining the normal modes and frequencies and establishing the orthogonality conditions, the problem of the forces motion of the deck is solved by the method of spectral representation. Numerical results have been presented to illustrate the effect of several vehicle and bridge parameters on the response. /Author/

Ground vibrations due to seismic detonation of a buried source

Abstract: The problem of vibrations at the free surface of an elastic half-space due to detonation of a buried source is studied here from the viewpoint of the geophysical seismic technique employed in oil exploration. The fundamental assumption of the theory, therefore, is that the pressure at the source is impulsive and a Dirac delta function of time. The depth of the source below the surface of the medium is considered in the present theory as an additional parameter which has hitherto made the buried source problem formidable and, therefore, has limited almost all previous works to the relatively easy problem of surface blasting. An exact formulation is presented by dividing the half-space into a stratum above the level of the buried source and a half-space below it. For boundary conditions at the interface, it is assumed that the vertical displacement and shear stress at the level of the source are continuous while the direct stress is discontinuous. A numerical evaluation of the contour integration in the analysis is presented for the components of vibrations at any point of the free surface for a Poissonian medium. Finally, the analysis provides a theoretical justification for the interesting results obtained from field experiments recorded in an earlier work.

The essex earthquake of 1884

Abstract: The largest British earthquake to occur in several hundred years took place in Essex, in the south east of England, in 1884. From contemporary accounts and a recent visit to the area affected by the earthquake, the intensity of the earthquake was assessed. It is estimated that the peak accelerations generated may have been as much as 0.1 g. This would be of consequence to some modern structures, designed without earthquake provisions.

Stresses in column-supported hyperboloidal shells subject to seismic loading

Abstract: The computation of stresses within a finite element displacement method analysis of a shell of revolution is considered. The common procedure of applying the kinematic and constitutive laws to the displacement functions is examined and justified for models where the displacements are represented by high-order polynomial expansions. Also, two alternative computational formats within this technique are explored. The influence of the column-supported base condition on a hyperboloidal shell of revolution is studied with respect to the stresses calculated from a response spectrum analysis. These studies emphasize the importance of accurately modelling the base region of a column-supported shell such as a hyperbolic cooling tower.

Seismic risk analysis of discrete system of sites

Abstract: Two efficient schemes have been developed for the analysis of discrete systems of sites. Both schemes have the same objective of finding the probability of simultaneous failure of any number of sites belonging to a given system of sites subject to threats from a given set of earthquake sources with known seismic history. In the first scheme, systems with deterministic site resistances can effectively be analysed using a non-linear transformation of variables. In the second scheme, systems with random site resistances can be analysed. To overcome the computational difficulties involved in the analysis, a new set of simple recursive formulas has been developed and used effectively. Based on these two schemes, two efficient computer programs were prepared and used to perform a parametric study on a system of nine actual or contemplated nuclear power plants in New England. The results have shown that the problem is very sensitive to the coefficient of variation of the resistances and not so sensitive to the mean resistances.

Safety of structural systems with reserve elements for earthquake protection

Abstract: Some results of approximate analyses of the safety of earthquake protection systems with reserve elements are presented. Systems with one and several reserve elements are considered. The overshoot random vibration approximation is used for analysis. A numerical example is given. It is shown that the failure probability of structures with reserve elements is considerably lower and the safety is much higher when compared with similar characteristics of structures without reserve elements.

Finite difference computation of the dynamic motion of cylindrical shells including the effect of rotatory inertia and transverse shear

Abstract: The accuracy of finite difference computations for the dynamic motion of cylindrical shells, including transverse shear and rotatory inertia, has been assessed by comparison with Fourier series solutions. The finite difference models were based upon either the differential equations or upon control volume concepts, with the results of the latter being less sensitive to mesh size and in better agreement with the Fourier series results. A frequency analysis of all of the finite difference algorithms and the Fourier series demonstrated that for shells with clamped ends the finite difference spectra had a very considerable gap which closed slowly as the mesh size decreased, with the spectra being most complete for the best algorithm. A new algorithm was based upon these spectra and is shown to yield good results while permitting increases in the computational speed by the factor of mesh size/shell thickness.