NOTES Approximations for Natural Frequencies of Interconnected Walls and Frames A. RUTENBERG' AND A. C. HEIDEBRECHT Departmetzt of Civil Engirzeeritlg and Engineering Mechanics, McMaster University, Hamilton, Ontario L8S 4L7 Received July 3 1, 1974 Accepted'October 7, 1974 Several approximate formulae for the determination of natural frequencies of interconnected walls and frames with uniform mass and stiffness throughout their height are compared and the range of their applicability discussed. It is found that none of the approximations give high accuracy for the entire range of the stiffness parameter aH encountered in practice. It is recommended that for the fundamental frequency the flexural beam approximation be used for the lower end of the range (aH < 7) and the large aH asymptotic formula be used elsewhere. This ensures that the error never exceeds 5%. For the higher modes there is a choice between three approximations, the maximum error associated with each being practically identical: 10% in the second mode and 5% in the third. L'article compare plusieurs formules approximatives servant B la determination des frequences propres d'un systeme murs-portiques relies entre eux et de masse et de rigidit6 uniformes sur toute la hauteur; on discute aussi leur domaine d'application. I1 ressort qu'aucune des formules approchtes ne fournit une precision &levee pour toute la plage des valeurs du paramktre de rigiditi aH rencontrees en pratique. Pour I'btude de la frequence fondamentale, les auteurs recommandent, d'une part, que I'approximation "poutre flechie" soit exploitee pour la zone infkrieure de la gamme des aH (aH < 7), et d'autre part, que dans tous les autres cas, on recoure a la formule asymptotique relike aux valeurs Blevees de aH. De cette f a ~ o n I'erreur ne depasse jamais 5%. Pour les modes superieurs, on dispose de trois approximations donnant lieu B une erreur maximale a toutes fins pratiques identique: 10% pour le deuxikme mode, 5% pour le troisieme. [Traduit par la Revue] When considering the effect of earthquake loading on tall building structures, the inten- sities of the equivalent static forces are func- tions of the natural frequencies of horizontal vibration. The evaluation of these dynamic properties for structures consisting of flexural walls and frames may be based on a single mathematical model consisting of uniform flexural and shear vertical cantilever beams which are constrained to have identical horizontal deflections throughout their height. The applicability of this model requires the satisfaction of the following assumptions: (i) the structure is symmetrical about the line of loading; (ii) axial strains in frame columns have negligible effect on stiffness; (iii) mass and stiffness are uniformly distributed along 'On leave from Technion, Israel Institute of Tech- nology, Haifa, Israel. building height; and (iv) the foundations are rigid. The determination of "exact" frequencies for such structures (Heidebrecht and Stafford Smith 1973; Skattum 1971) requires the use of a digital computer, which may render the method impracticable for preliminary analysis. The aim of the present note is to discuss the accuracy of several alternative approximate formulae for the determination of the natural frequencies of interconnected walls and frames so that engineers may have more confidence in the application of these approximate formulae. It has been shown (Heidebrecht and Stafford Smith 1973) that the differential equation gov- erning the free vibration of a shear-flexure vertical cantilever beam is given by subjected to the boundary conditions Can. J. Civ. Ens., 2, 116 (1975) Can. J. Civ. Eng. Downloaded from www.nrcresearchpress.com by Hubei university on 06/04/13 For personal use only.