Earthquake Engineering and Structural Dynamics

doi: 10.1002/eqe.4290050401pmid: N/A

Emery, A. F.; Cupps, F. J.

doi: 10.1002/eqe.4290050402pmid: N/A

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.

Wolf, J. P.

doi: 10.1002/eqe.4290050403pmid: N/A

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.

Warburton, G. B.; Soni, S. R.

doi: 10.1002/eqe.4290050404pmid: N/A

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.

Ramakrishnan, R.; Kunukkasseril, V. X.

doi: 10.1002/eqe.4290050405pmid: N/A

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 forced 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.

Kan, Christopher L.; Chopra, Anil K.

doi: 10.1002/eqe.4290050406pmid: N/A

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.

Zienkiewicz, O. C.

doi: 10.1002/eqe.4290050407pmid: N/A

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.