Environmental Engineering Reference
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where all stresses are in MPa. For axial compression plus bending:
r
a
F
a
þ
r
b
F
b
1
ð
10
:
28
Þ
Equations
10.26
and
10.27
apply to ''manufactured'' sections rather than those
''fabricated from plates…'' .
To estimate the buckling stress for the back leg of the tower, assume the leg is
vertical and supports one-third of the turbine and tower mass. Thus f
a
is given by:
h
i
Þ
2
Þ
9
:
81
=
3p 0
:
0605
2
0
:
0605
0
:
00365
r
a
¼
4
170
þ
640
ð
ð
¼
7
:
87 MPa
ð
10
:
29
Þ
Again
the
axial
load
due
to
turbine
and
tower
weight
is
small.
Using
f
b
= 156 MPa from the FEA analysis, and Eq.
10.28
gives
r
a
F
a
þ
r
b
F
b
¼
7
:
87
255
þ
156
255
¼
0
:
643
ð
10
:
30
Þ
so the member is safe from buckling. The Eurocode 3 [
5
] equations in Annex D of
EN 1993-1-6:2007 are more complex. The critical linear meridional (axial in this
case) buckling stress is given by
r
Rcr
¼
0
:
605EC
x
t
=
r
ð
10
:
31
Þ
where r is the mid-radius of the member. The use of (
10.31
) for studying the
buckling of large towers is described in Sect. 7.9.3 of Burton et al. [
15
]. To fix the
value of C
x
, it is first necessary to calculate the dimensionless length parameter x,
defined as
x
¼
l
.
p
ð
D
t
Þ
t
=
2
ð
10
:
32
Þ
For the bottom back leg in Fig.
10.7
, l = 1487.5 mm, r = 28.425 mm, and
t = 3.65 mm, so x = 146. Since x [ 0.5r/t, the leg is a ''long cylinder'' and the
remaining unknown in (
10.31
)is
h
i
0
:
6
;
1
þ
0
:
2
C
x
;
b
1
2x
t
r
C
x
¼
max
ð
10
:
33
Þ
where C
xb
depends on the boundary conditions as shown in Table D.1 of Euro-
code 3. Taking the ends to be ''clamped'', that is radially, meridionally, and
rotationally restrained, the leg is subject to BC 1 at both ends, and C
xb
= 6. Thus
C
x
= 0.6 from (
10.31
).
For elastic buckling, the maximum stress from (
10.31
), must be multiplied by
the ''imperfection factor'' a, which supposedly allows for manufacturing errors
and makes these correlations so valuable. It is given by
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