Approximations for Natural Frequencies of ...

NOTES

Approximations for Natural Frequencies of Interconnected Walls and Frames Can. J. Civ. Eng. Downloaded from www.nrcresearchpress.com by Hubei university on 06/04/13 For personal use only.

A. RUTENBERG' A N D 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 a H encountered in practice. It is recommended that for the fundamental frequency the flexural beam approximation be used for the lower end of the range ( a H < 7) and the large a H 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 a H 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 a H ( a H < 7 ) , et d'autre part, que dans tous les autres cas, on recoure a la formule asymptotique relike aux valeurs Blevees de a H . De cette f a ~ o nI'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 intensities of the equivalent static forces are functions 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 Technology, Haifa, Israel. Can. J. Civ. Ens., 2, 116 (1975)

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 governing the free vibration of a shear-flexure vertical cantilever beam is given by

subjected to the boundary conditions

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Can. J. Civ. Eng. Downloaded from www.nrcresearchpress.com by Hubei university on 06/04/13 For personal use only.

*(O) = 0

t

in which or2 = GA/EI (i.e. the ratio of shear stiffness to flexural stiffness), A* = w2m/EI, I# = the amplitude of vibration, H = the height of the structure, w = the natural frequency, and m = the mass per unit height. A detailed discussion of the procedure for determining the parameter (Y2 for practical structures is given in Heidebrecht and Stafford Smith (1973). The general solution of Eq. [I] reads [3]

Kz)

+ C2 sinXlz + C, cosh h2z + C4 sinh h2z

= C1 cos hlz

in which

FIG.1. Dependence of

11on aH.

basic frequency parameter may be written in the form

For the flexural beam approximation it is convenient to rewrite Eq. [7] in terms of the ratio hlH/h,H, yielding

and The substitution of the boundarv conditions expressed in ~ q [21 . into ~ q [31Jleads . to the following characteristic equation [6]

In this approximation hlH = h,H, reducing the above to

Igl

i h ~ j l =(hFH)2

[(2l2+

hlH Similarly, .= hsH, foryielding the shear the expression beam approximation (~)2]~oshlH~~~hh2H

+ [::

:I

- --

sin h l H sinh h2H + 2

=

0

When the above equation was solved using a digital computer, it was found that the parameter hlH was only weakly dependent on (YH for a wide range of the latter. Figure 1 shows the results of these computations for the first four modes. It can be seen from this figure that the values of hlH can be approximated either by the eigenvalues of a flexural cantilever (namely hFH = 1.875, 4.694, 7.855, and 10.966 for the first four modes), or by the eigenvalues of a shear cantilever (namely hsH = 1.571,4.712,7.854, 10.966, .... [2n-l]n/2). By substituting Eq. [5] into Eq. [4] the

Before discussing the errors associated with each of the above approximations, it is usefuI to recall that a lower bound for the first eigenvalue and an approximation for the higher frequencies may be obtained by means of Southwell's approximations (Temple and Bickley 1956), for which the relevant expression is given by u 2 N wF2 + ws2 Clll The above may be expressed in terms of the parameters h, hS, and h ~yielding ,


118

CAN. J . CIV. EKiG. VOL. 2, 1975

It can be seen that the above approximation involves both the flexural and shear cantilever eigenvalues in an expression having a similar form to that of Eqs. [9] and [lo]. In fact, if hF is substituted for hs in Eq. 1121, Eq. [9] is obtained and conversely if hs is substituted for h~.,Eq. [lo] is obtained. The percentage of error associated with the approximations given by Eqs. [9], [lo], and [12] for the first three modes is shown in Figs. 2 and 3. For the fundamental mode hF > As, and since Eq. [12] is a lower bound, it gives a better approximation than Eq. [lo] for all values of (YH.However, for values of (YHless than 10, Fig. 2 shows that the flexural beam approximation of Eq. [9] is the most accurate. For the second and higher modes, since h~ and hs are essentially identical, all three approximations yield results of approximately equal accuracy, as shown for the second and third modes in Fig. 3. It is interesting to compare the three approximations with some asymptotic formulae given by Skattum (1971 ), which are also shown in Figs. 2 and 3. For small values of ffH, this reference gives the following expression

FIG.2. Percentage errors in frequency fundamen.! tal mode: Eq. 191, flexural beam approximation; E [lo], shear beam approximation; Eq. [12], Southwell s approximation; Eq. [13], asymptotic for small a H ; Eq. [14], asymptotic for high frequency; Eq. [151, asymptotic for large aH.

0)

I0l

[13]

(AH)'

+

SECOND MODE

I

/

= (AFH)'

(- 1)" - cos (AFH) -nis: AFH

(hFH) ( 4 ' 2(- 1)" + 2 cos (AFH)

in which n is the mode number. It can be seen from Figs. 2 and 3 that this approximation is -10 accurate for only a very small range of (YH, b) THIRD MODE even though this range is increasing in size for FIG.3. Percentage errors in frequency-higher higher modes. A high frequency approximation (Skattum modes. 197 1) is given in the form

1

gives the following approximation and is by definition more accurate for higher modes. The useful range is already quite wide for the second mode, with errors of less than 5 % for (YH< 6.5, while for the third mode the same accuracy limit is reached when (YH = 9.6. For large values of (YH, Skattum (1971)

Figure 2 indicates that the above is a remarkably good approximation for the fundamental mode. Its accuracy is higher than any other approximation for all values of (YH> 7, yielding errors of less than 4%. It is still useful


NOTES

for values of aH > 11 in the second mode but has little practical value for the third mode. Practicing engineers would prefer to have a single simple formula available for the evaluation of natural frequencies. Figures 2 and 3 and the foregoing discussion demonstrate that none of the approximations gives high accuracy for the entire range of aH encountered in practice. Since the fundamental frequency is normally of most interest, it is recommended that the flexural beam approximation of Eq. [9] be used when (YH < 7. This would include shear wall buildings and a high proportion of shear wall frame structures. For higher values of (YH, it is recommended that Eq. [15] be used. This

119

procedure will ensure that errors never exceed 5 % . For higher modes, it is recommended that Eq. [9] be used for all situations with maximum errors as shown in Fig. 3. HEIDEBRECHT, A. C., and STAFFORDSMITH,B. 1973. Approximate analysis of tall wall-frame structures. J. Struct. Div., A.S.C.E., 99 (ST2), February, pp. 199-22 1. K. S. 1971. Dynamic analysis of coupled shear SKATTUM, walls and sandwich beams. Earthquake Engineering Research Laboratory, California Institute of Technology, Pasadena, California, Rep. No. EERL 71-06, pp. 175-182. W. 0. 1956. Rayleigh's princiTEMPLE,G . , and BICKLEY, ple. Dover Publications Inc., New York, N.Y. pp. 121-122.