Civil Engineering Reference
In-Depth Information
3.2 FREQUENCY ANALYSIS
The dynamic problem of beam-columns of thin-walled cross-section has been
investigated by several scientists. Garland [1940] applied the Rayleigh-Ritz
method to cantilever beams and derived a simple approximate solution but
assumed infinitely great stiffness in one of the principal directions. Gere and Lin
[1958] set up the complete system of governing differential equations but only
produced a solution for beam-columns on pinned supports. Their difficulty in
providing a solution for other support systems might have lain with handling
pure torsional vibration. Gere [1954] had earlier published the differential equation
of pure torsional vibration but only presented a solution for the simply supported
case. Several approximate solutions have been recommended for the pure torsional
vibration of cantilevers [Southwell, 1922; Gere, 1954; Kollár, 1979; Goschy, 1981]
but the exact solution has not been produced and the accuracy of the approximate
procedures has not been investigated either. The exact solution to pure torsional
vibrations is presented in this section while the results of a comprehensive
accuracy analysis regarding the approximate solutions mentioned above are
available elsewhere. For lack of the exact solution to pure torsional vibration,
solutions to coupled vibrations [Kollár, 1979; Rosman, 1980 and 1981; Vértes,
1985; Goschy, 1981] could only be approximate.
The procedure presented in this section can also be used for a simplified
dynamic analysis of buildings in seismic zones, where one of the most important
input data is the fundamental frequency of the building [Eurocode 8, 1996; Zalka,
1988].
Assuming uniformly distributed mass, the vibrations of the building are defined
by the simultaneous partial differential equations [Gere and Lin, 1958]:
