Tower Design and Manufacture - Small Wind Turbines: Analysis, Design, and Application - page 210

Fig. 10.3 Maximum stress
along tower for the IEC SLM
load cases calculated by
tapered_oct_tower.m and
finite element analysis (FEA)
results for load case H

250

FEA for Load Case H
Load Case H
Load Case A
Load Case B

200

150

100

50

0

0 5 10 15 20

Height from tower base (m)

the natural frequency. Methods A and B give natural frequencies that lie either side
of the FEA value. This value is matched by Method B when the tower contribution
is 0.77m t , but there is no deeper justification for this value.
Using the ASCE buckling conditions gives a maximum CF of 0.623 whereas
the linear buckling factor from FEA implies a much lower CF. This difference is to
be expected since the former is based on experiments on actual octagonal towers.
If the SLM partial safety factors are applied to T max , the maximum CFs
increase. From Table 9.7 , the load safety factor is 3.0, and from Table 9.8 , the
''fully characterised'' material safety factor is 1.1. Their product is 3.3 which is
close to twice the inverse of CF = 0.6. Therefore the SLM factors can be
incorporated into tapered_oct_tower.m by doubling T max to 4400 N. With this
thrust, the maximum CF for buckling increases to 0.832 and for stress to 0.768 and
the tower would not be safe.
Holmes [ 3 ] gives an approximate equation, his (E4), for the lowest natural
frequency of a circular tapered tower of varying thickness is

1 = 2

k
2ph 2

EI h
q t A h

n 1 ¼ ð 10 : 18 Þ

where

h i 1

Þ 2 = 3 k ¼ 1 : 9 exp 4d 0 = d h ð Þ þ 6 : 65 0 : 9 þ t 0 = t h ð ð 10 : 19 Þ

which reduces to the exact formula for untapered towers of constant thickness

1 = 2

n 1 ¼ 0 : 56
h 3 = 2

EI
m t

ð 10 : 20 Þ

when t 0 = t h and d 0 = d h . The comparison with the FE result shows that the
approximation is reasonably accurate for this octagonal tower. It is emphasized
again, that FEA should be undertaken to accurately determine the natural fre-
quencies and that the approximate methods used here are just that: approximate.
One important use of these calculation is to see whether there is likely to be any
resonance

caused by blade passing frequencies matching the lowest natural