and Table 2.9, respectively. The beams in these frames are assumed to be uniform and sufficiently slender so that shear, axial, and torsional deformations can be neglected. The method of analysis of these frames is given in Yang and Sun (1973). The vibration is assumed to be in the plane of the frame, and the results are presented for portal frames with pinned and fixed bases.

If the beam is rigid and the columns are slender and uniform, but not necessarily identical, the natural fundamental frequency of the frame can be approximated using the following formula of Robert (1979):

where M is the mass of the beam, M\ is the mass of the ┬┐th column, and EI is the flexural rigidity of the ith column. The summation refers to the sum of all columns, and i must be greater than or equal to 2. Additional results for frames with inclined members are discussed in Chang (1978).

TABLE 2.9 Fundamental Frequencies of Portal Frames in Symmetrical Mode of Vibration

First symmetric in-plane mode

E2I2, m2 i

First symmetric in-plane mode

0 0

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