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m
M
F
m
M
y
yk
ck
LTK
LTk
(23)
+
+
≤
1
P
M
P
Z
ck
bk
yk
Eq. 20 defines the weight of the dome where the minor weight of connections is
ignored. The weight is function of the unit weight
m
r
of the steel section selected for
group
r
in the dome from the standard steel section table.
t
r
is the total number of
members in group
r
and
ng
is the total of groups in the dome.
Eq. 21 defines the displacement limitations that are required for the serviceability
requirements. BS5950 limits the vertical deflections under the unfactored imposed
load to span/360 and horizontal displacements to the height/300.
nd
is the total num-
ber of restricted nodal displacements in the dome.
i
δ
represents one of such dis-
placement and
is its upper bound.
Eqs. 22 and 23 define the cross-sectional capacity and the member buckling resis-
tance checks for compression members with bi-axial moments. These expressions are
given in clauses 4.8.3. and 4.8.3.3 of BS 5950. They ensure that at the points of the
greatest bending moment and axial load, yielding or local buckling does not take
place. In Eq. 22,
F
ck
is the axial compression,
M
xk
is the maximum major axis moment
in the segment length
A
governing
P
cx
and
M
yk
is the maximum minor axis moment
in segment length
A
governing
P
cy
.
M
cxk
and
M
cyk
are the moment capacity of mem-
ber
k
about its major and minor axis.
A
gk
is the gross cross sectional area and
p
y
is the
design strength of the steel grade used for member
k
respectively. Eq. 23 is the simpli-
fied approach for checking the member buckling resistance.
c
P
is the compression
resistance of the member
k
, which is the smaller of
c
P
and
c
P
. These are the com-
pression resistance of the member considering buckling about the major axis and the
minor axis respectively. The compression resistance of a member is equal to the
gross cross sectional area times the compression strength
c
p
which is obtained from
the solution of the Perry-Robertson quadratic equation given appendix C.1. of BS
5950. It is apparent that computation of the compression strength of a member re-
quires its effective length. For simplicity the effective lengths of members are taken
as their lengths.
L
m
is the equivalent uniform moment factor given in the code.
L
M
is the maximum major axis moment in the segment length
L
governing the
buckling resistance moment
δ
iu
M
as explained in clause 4.3.6 of BS5950. For circular
hollow steel sections
M
is equal to moment capacity
M
of the section.
y
Z
is the
section modulus about the minor axis.
nm
is the total number of members in the
dome.
b
5.2 Design Example
The design algorithm presented is used to determine the optimum height and circular
steel hollow section designations for the geodesic dome shown in Figure 5. The
modulus of elasticity is taken as 205kN/mm
2
. The grade of steel adopted is Grade 43.
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