Supplementary remarks

The procedures presented in the previous sections for the frequency analysis concentrate on the dominant characteristics of the structure while items whose influence on the dynamic response is normally small are neglected. Under certain circumstances, however, some phenomena and characteristics of usually secondary importance may increase their effect and therefore they may be no longer negligible. Such cases are discussed briefly in this section.

(a) Compressive forces

In analysing the lateral vibrations of simply supported beams subjected to concentrated end-forces and cantilevers under their uniformly distributed weight, Timoshenko [1928] took into account the effect of axial forces. His results showed that the axial compressive forces reduce the value of the frequencies of natural vibration by the factor of

where F is the magnitude of the axial load and F_cr is the corresponding critical load.

The reduction factor is considered exact for simply supported beams subjected to concentrated end-forces and can be used as reasonable approximation for cantilevers under uniformly distributed axial load.

Timoshenko’s results can be generalized and the same approach can be applied to the coupled vibrations of thin walled cantilevers of open cross-section. Taking into account the effects of the uniformly distributed axial forces, the relevant formulae can be modified. This leads to the following formulae for the first basic frequencies

which now take into consideration the effect of the axial forces. Critical loads N_cr,X, N_cr,Y and N_cr,ϕ are the basic critical loads and are obtained from the stability analysis as described in section 3.1.1.

(3.79)

(3.80)

(3.81)

(b) Shear deformation

When relatively short bracing elements (compared to the cross-sectional dimensions) are analysed and higher frequencies are also needed, the effect of the shear deformations can be of considerable magnitude. The deformations increase, resulting in a reduction in the frequencies. Various sources offer excellent treatment of this problem. Detailed mathematical background is given by Bishop and Price [1977] and Capuani, Savoiva and Laudiero [1992]. Closed form solutions are presented by Huang [1969], Timoshenko, Young and Weaver [1974] and Capuani, Savoiva and Laudiero [1992]. According to the investigations, the shear deformation may have considerable effect on the higher frequencies but only slightly modifies the fundamental frequency. As the effect of shear deformation is considered of secondary importance, it can be neglected when the fundamental frequency is calculated.

(c) Damping

Most frequency analyses including the one presented here ignore the fact that due to movements of a structure, energy absorption occurs through friction, air resistance and viscous behaviour resulting in damping. The omission of these effects leads to overestimated frequencies. The inclusion of damping characteristics in the governing equations would make the analysis much more complex. The effect of damping in multistorey buildings is not significant, as a rule, as the natural frequencies are not highly affected by the degree of damping.

The problem is therefore often sidestepped in structural engineering design by estimating the damping forces separately. This can be done either theoretically or using test data [Littler, 1993]. Different forecast models are available for damping and vibration periods of buildings [Lagomarsino, 1993]. Damping coefficients are also provided in papers and monographs [Fintel, 1974; Hart and Vasudevan, 1975; Irwin, 1984; MacLeod, 1990; Goschy, 1990], which then may be used for estimating the damped frequency. A simple estimation [Fintel, 1974] is obtained using the formula

which is derived for single-degree-of-freedom systems. In formula (3.82), f is the frequency when damping is neglected and f_d represents the frequency when damping is taken into account. Coefficient ß is the damping ratio relative to the critical value of the damping coefficient, i.e. to the value of damping which would just cause an initial displacement to decay to zero without any oscillation. The value of coefficient ß ranges from about 0.02 to 0.20 for most civil engineering structures; the effect of damping is usually well under 10%.

(3.82)

(d) Approximate methods

The accuracy of some approximate formulae for pure torsional vibrations is investigated in detail in [Zalka, 1994b]. Many approximate methods have been published for predicting the fundamental natural frequency of buildings [Ellis, 1980; Goldberg, 1973; Goschy, 1990]. Detailed evaluation of their accuracy is available [Ellis, 1980; Jeary and Ellis, 1981] and it is not the intention of this section to do further research in this area.

However, having evaluated a number of numerical examples, it is worth emphasizing one important point. Most approximate methods are one-parameter (height of building) or two-parameter (height and width of building) methods offering a simple formula for the fundamental frequency. They do not take into consideration the nature of the mode of vibration and ignore the possibility of mode coupling. The well-known formula

is perhaps the most characteristic example. It produces surprisingly good approximations for lateral frequencies. However, when torsion (and/or mode coupling!) plays an important role, the formula may produce totally incorrect results.

The relative success of the one-parameter formulae is due to the fact that they are based on the height, one of the three most important dynamic characteristics.

The mass and the bending stiffness of the building do not vary much in relative terms and can be represented fairly well by a single constant, e.g. 46, in formula (3.83).

The situation is different with torsion. The mass of the building is still easily predictable and the Saint-Venant torsional stiffness plays a very little role.

However, the effect of the warping stiffness can be significant. Its value depends on the arrangement of the bracing elements to a great extent. Using the same size of bracing elements producing the same bending stiffness and the same mass, the warping stiffness can be increased, or decreased, by orders of magnitude. (Figure 4.13 in Chapter 4 shows an example for maximizing the warping stiffness of a bracing system with given bracing elements.)

(3.83)

4