A note on rotational components of earthquake motions on ground surface for incident body waves M. D. TRIFUNAC Department of Civil Engineering, University of Southern California, Los Angeles, California 90007 This paper shows that the Fourier amplitude spectra of rocking and torsional components of strong shaking on ground surface can be derived exactly in terms of (1) wavelength of incident waves, (2) Fourier amplitude spectra of vertical (for incident P and SVwaves) or horizontal (for incident SH waves) ground motion, and (3) the angle of incidence of plane body waves, 00. Application of these results in earthquake engineering is discussed. INTRODUCTION Rocking and torsional components of strong earthquake ground motion are beginning to attract the attention of engineering and research communities as it is becoming evident that those motions may contribute significantly to the overall response of structures to strong earthquake ground shaking 1 4. With further development of strong motion instruments to record rotational components of strong motion 5, it will become possible to examine these experimentally. However, it may take a long time before sufficient data are gathered to facilitate reliable and detailed empirical studies. Therefore, in the interim, it is useful to explore the possibility of estimating them in terms of the corresponding translational components of strong motion, which have been recorded and studied more extensively. The purpose of this paper is to show how the rocking and torsion of strong motion at ground surface associated with incident plane P, SV and SH waves can be determined exactly from the known translational components of ground motion there. Comparison of the torsional results presented here with those presented and reviewed by Nathan and MacKenzie 6 will show the nature of the previous approximate analyses. Rotations of surface ground motion are derived by applying the curl operator on the vector components of displacements associated with incident waves. For incident body P, SV and SH waves, the corresponding surface displacements have been studied extensively and are available in classical literature on wave propagation TM. It is noted, however, that many papers on this subject are either in error or incomplete. Typically, reflection of S V waves past the critical angle and the associated phase delays are rarely presented in detail. Such results are essential for interpretation of incident motion and serve as a basis for computing rotational components of strong shaking. Therefore, for completeness of this presentation, these classical results 7 are briefly summarized. The aim of this note is to provide an analytical basis for estimation of rotational components of stron~ ground motion. To see how these can be used in engineering applications, one can study papers by Trifunac s and by Wong and Trifunac 9. The first paper shows how different wave types can be associated with a train of waves corresponding to strong ground shaking, and how through dispersion analysis for the site geology, one can determine the phase and group velocities for surface waves, the arrival times of body waves, and their relative contributions to the complete motion. The second paper ,;hows how to construct artificial accelerograms empirically but using the physical nature of wave propagation as in Trifunac 8. Finally, by using the results presented here in the superposition model employed by Wong and Trifunac 9, one can compute artificial accelerograms of ground rotations and their spectra. INCIDENT P-WAVES Figure 1 shows the coordinate system (xl, x2) and the incident and reflected rays associated with plane P wave reflecting offthe free boundary of the elastic homogeneous and isotropic half-space (x2~<0). Without loss of generality, it is assumed here that the incident and reflected rays are in the plane x3=O. Particle motion amplitudes and their assumed positive directions in the planes perpendicular to the respective rays are then given by A0, A1 and A1 where Ao and A~ are associated with incident and reflected P waves, while A2 corresponds to the reflected SVwave. For this excitation and coordinate system, the only non-zero components of motion (at 02 A REFLCT P-WAVE INCIDENT P-WAVE REFLECTED SV- WAVE ~xt Figure 1. Coordinate system)or incident p-wave 0261-7277/82/010011~09$2.00 © 1982 CML Publications Soil Dynamics and Earthquake Engineering, 1982, Vol. 1, No. 1 11