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(
)(
)
2
γ
γ
π
/
k
−
36
π
/
k
1
2
=
(11)
(
)
(
)
6
γ
+
γ
tan
π
/
k
1
2
where
k
is the effective length factor and
γ
and
γ
are the relative stiffness ratio
1
2
for the compression member which are given as:
∑
∑
I
/
A
I
/
A
c
2
c
2
c
1
c
1
γ
=
,
γ
=
(12)
1
∑
2
∑
A
I
/
I
/
A
b
1
b
1
b
2
b
2
The subscripts
c
and
b
refer to the compressed and restraining members respec-
tively and the subscripts 1 and 2 refer to two ends of the compression member under
investigation. The solution of the nonlinear equation (Eq. 11) for
k
results in the ef-
fective length factor for the member under consideration. Eq. 11 has the following
form for non-swaying frames.
2
γ
γ
π
γ
+
γ
⎛
π
/
k
⎞
2
tan(
π
/
2
k
)
⎛
⎞
⎛
⎞
1
2
1
2
⎜
⎜
⎝
⎟
⎟
⎠
⎜
⎝
⎟
⎠
+
⎜
⎝
⎟
⎠
1
−
+
=
1
(13)
4
k
2
tan(
π
/
k
)
π
/
k
Eq. 10 defines the moment capacity check for beams. The details of the computa-
tion of these are given in [24]. Further to the above restrictions the geometric con-
straints that are necessary for the constructability of a steel frame are also considered.
These geometric constraints are such that the beam section at each beam-column con-
nection at each storey should be less than or equal to the flange width of column sec-
tion and the depth and the mass per meter of column section at each storey and at each
beam-column connection should be less than or equal to width and mass of the
column section at the lower storey.
3.2 Design Example
Three-bay, fifteen-storey frame shown in Figure 1 is designed by optimum design al-
gorithm described in section 2. The dimensions of the frame and the loading are
shown in the figure. The frame is subjected to gravity loading of 12.4kN/m on the
beams of roof level and 27kN/m on the beams of each floor. The lateral loading is the
wind loading that is computed according to British Code considering 45m/s wind
speed and 6m frame spacing. The modulus of elasticity is 200kN/mm
2
. Frame con-
sists of 105 members that are collected in 12 groups. Columns in every story made
out of the same section. The beams of roof and intermediate floors are considered to
be two different groups as shown in the figure. The allowable inter-storey drift is
11.7mm while the lateral displacement of the top storey is limited to 176.7mm. The
strength capacities of steel members are computed according to BS5950.
Two design pools, one for the beams of the frame and the other for the columns of
the frame are established before initiating the search for the optimum design. Among
the steel sections list of BS 5950 [25], 64 Universal Beam (UB) sections starting from
914×419×388 to 254×102×28 and 32 Universal Column (UC) sections starting from
356×406×634 to 152×152×23 are selected to constitute two discrete sets which are
used as design pools.
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