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190 200 210 220 230 240 250 260 (cm2/column) FIGURE 27.35 Reinforcement in shell and supports for prototype tower.

190 200 210 220 230 240 250 260 (cm2/column) FIGURE 27.35 Reinforcement in shell and supports for prototype tower.

is often investigated, at least to determine the positions of the nodal lines and areas of particularly intensive vibrations. In any case the first step is to carry out a free vibration analysis. This analysis represents the modes of free vibration associated with each natural frequency, f or its inverse the natural period T, as the product of a circumferential mode proportional to sin n6 or cos n6 and a longitudinal mode along the z-axis [9,10]. Some results are shown in Figure 27.41, which demonstrate the lowest vibration modes of a tower without and then with holes for flue gas inlets. Such inlets require great strengthening of the injection area. Further representative results are shown in Figure 27.42 and Figure 27.43, as discussed below.

As an illustration, the cooling tower from Figure 27.4 is again considered. Some key circumferential and longitudinal modes for a fixed-base boundary condition are shown in Figure 27.42. Also, the effects of different cornice stiffnesses are demonstrated. This model may be regarded as preliminary in that the relatively soft column supports are not properly represented, but it illustrates the salient characteristics of the modes of vibration. Most interesting are the frequency curves in Figure 27.43 for the first ten harmonics. Note that the natural frequencies decrease with increasing n until a minimum is reached whereupon they increase, a characteristic behavior for cylindrical-type shells. Also, the stiffening of the cornice tends to raise the minimum frequency, which is desirable for resistance to dynamic wind. Longitudinally, the cornice stiffness effect is significant for odd modes only, as shown in Figure 27.42.

Deformation, m

FIGURE 27.36 Load-displacement diagrams for load combination D + lW.

Deformation, m

FIGURE 27.36 Load-displacement diagrams for load combination D + lW.

Deformation

FIGURE 27.37 Displacement plot for load combination D + 1W (exaggeration: 40).

Deformation

FIGURE 27.37 Displacement plot for load combination D + 1W (exaggeration: 40).

Specifically for earthquake effects and other coherent excitations, only the first mode participates in a linear analysis for uniform horizontal base motion and the respective values for n — 1 should be entered into the design response spectrum.

Results from a seismic analysis of a cooling tower are presented in Figures 27.44 to 27.47. The cooling tower of Figure 27.4 is subjected to a horizontal base excitation based on Figure 27.44, leading to

Inner face

Outer face

Wind

Inner face

Outer face

Wind

Wind

Inner face

Outer face

Inner face

Outer face

Wind

Crack pattern at D + AT45 + 2.3 W, wc> 0.05 mm FIGURE 27.38 Different crack patterns for load combination D + lW.

a first circumferential mode (n — 1) response. A response spectrum analysis provides the lateral displacements w of the tower axis, the meridional forces n22, and the shear forces n12, as shown on the indicated figures. In general, cooling tower shells have proven to be reasonably resistant against seismic excitations, but obviously the most critical region is the connection between the columns and the lintel as portrayed in Figure 27.48.

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