steel structure optimisation

DESIGN OPTIMIZATION OF
2D STEEL FRAME STRUCTURES
8.1 Objectives
This chapter presents a genetic algorithm for design optimization of multi–bay multi–
storey steel frameworks according to BS 5950 to achieve four objectives. The first is to
ascertain that the developed GA approach can successfully be incorporated in design
optimization in which framework members are required to be adopted from the
available catalogue of standard steel sections. The design should satisfy a practical
design situation in which the most unfavourable loading cases are considered. The
second is to understand the advantages of applying automated design approaches. The
third is to investigate the effect of the approaches, employed for the determination of the
effective buckling length of a column, on the optimum design. Here, three approaches
are tackled and results are presented. The fourth is to demonstrate the effect of the
complexity of the design problem on the developed algorithm. This involves studying
different examples, each of which have different numbers of design variables
representing the framework members. This chapter starts with describing the design
VIII

Design Optimization of 2D Steel Frame Structures 239
procedure for steel frame structures according to BS 5950, then combines this procedure
with the GA to perform design optimization of the steel frame structures.
8.2 Design procedure to BS 5950
In order to correlate between the notations given by BS 5950 and that employed in this
context, the local and global coordinate systems shown in Figure 8.1 are assumed. This
allows us to use the same indices and notations as utilised in BS 5950. Figure 8.2 shows
the coordinate systems combined with a deformed configuration of a framework
Figure 8.1. Local and global coordinate systems
Y′
Z′
X′
Z
Y
X
snh
L
mem
c,Y n′∆
U
mem
c,Y n′∆
X′
sNh
b
1
x
n,
I
max
mem
b
n
δ
11
x
,
I
1s
x
,n
I
1s
x
,N
I
X
Y
Y
Y
Y
YY
YY
YY 1bs
x
+N,N
I
1bs
x
+N,n
I
1b1
x
+N,
I
X
X X
X
X
X
X
X
X
max
mem
b
N
δ
max
1
δ
Z′
Y′
Figure 8.2. Deformed configuration of a framework combined with coordinate systems
1h
1B bNB

BS 5950 recommends that the designer selects appropriate standard sections for
the members of a steel framework in order to ensure a sufficient factor of safety is
achieved. This is accomplished by considering ultimate and serviceability limit states.
In elastic design of rigid jointed multi–storey frameworks, BS 5950 recommends
that a linear analysis of the whole framework is carried out. This was achieved by
utilising the finite element package ANSYS, followed by a design criteria check. This
can be summarised in the following steps.
Step 1. Preparation of data files and these include framework geometry as well as
loading cases.
Step 2. Classification of the framework into sway or non–sway. This is achieved by
applying the notional horizontal loading case. A framework, analysed without including
the effect of cladding, is classified as non–sway if the difference between the upper
)(U
Y mem
c
xn,′
∆ and lower )(L
Y mem
c
xn,′
∆ horizontal nodal displacements of each column
member mem
cn satisfies the following condition:
1
2000
)()(
mem
c
mem
c
mem
c
L
Y
U
Y
≤
∆−∆ ′′
n
n,n,
L
xx
,
mem
c
mem
c 21 N,,,n Λ= . (8.1)
Step 3. Calculation of the effective buckling lengths eff
mem
X, n
L and eff
mem
Y, n
L of columns
and beams. For columns, eff
mem
c
,X n
L is determined according to one of the following three
approaches:
• using the charts from BS 5950 as described in Section 2.6.2.2;

• a more accurate method (SCI, 1988) based on finite element analysis as applied in
Section 7.3.1;
• selection of the conservative (higher) value out of the two approaches.
The effective buckling length eff
mem
b
X, n
L of a beam equals the unrestrained length of
the compression flange that occurs on the underside of a beam (see MacGinley, 1997).
To evaluate )(eff
mem
Y, j,ixL n
of beams and columns, It is presupposed that the lateral
bracing system restrain members from movements out of plane ( Z-X ′′ plane) at their
mid spans. Thus, )(eff
mem
Y, j,ixL n
equals to the half of the length of the member mem
n
L .
Step 4. Calculation of the slenderness ratios )(mem
X,
xn
λ and )(mem
Y, j,ixn
λ of the
member mem
n using
mem
mem
mem
X,
X,
X,
)(
)(
eff
n
r
L n
n
x
x =λ , (8.2)
mem
mem
mem
Y,
Y,
Y,
)(
)(
eff
n
j,i
j,i
r
xL
x
n
n
=λ (8.3)
where mem
X, n
r and mem
Y, n
r are the radius of gyrations of the section about X and Y axes.
Step 5. Check of the slenderness constraints
Sle
mem
n,s
G for each member using
1)(
Sle
mem ≤xn,s
G , s = 1, 2 (8.4)
where
180
)(
)(
mem
mem
X,
1
Sle
x
x
n
n,
G
λ
= and (8.5)
180
)(
)(
mem
mem
Y,
2
Sle j,i
j,in,
x
xG
n
λ
= . (8.6)

Step 6. Analysis of the framework under each loading case q to obtain the normal force,
shearing forces and bending moments for each member.
Step 7. Check of the strength requirements for each member mem
n under the loading
case q as follows:
a) Determination of the type of the section of the member (e.g. slender, semi–compact,
compact or plastic).
b) Evaluation of the design strength mem
y n,
p of the member.
c) Check of the strength constraints )(
Str
mem x
q,
n,r
G depending on whether the member is
in tension or compression. This stage contains four checks (r = 4) for each member
under each loading case q. The strength constraints, which are local capacity, overall
capacity, shear capacity and the shear buckling capacity, should satisfy
1)(
Str
mem ≤x
q,
n,r
G , r = 1, 2, 3, 4, and q = 1,2, Q,Λ (8.7)
where the local capacity
+
+
=
members
comprissonfor
)(
)(
)()(
)(
(8.8)
members
tensionfor
)(
)(
)()(
)(
)(
mem
mem
memmem
mem
mem
mem
memmem
mem
mem
CX
X,
yg,
CX
X,
ye,
1
Str
j,ij,ij,i
j,ij,ij,i
q,
xM
M
xpxA
F
xM
M
xpxA
F
G
n,
q
n
n,n
q
n
n,
q
n
n,n
q
n
n,
xx
xx
x
where )(mem xq
n
F is the axial force, )(mem
X,
xq
n
M is the moment about the major local
axis (x) at the critical region of the member under consideration, )(mem
y, j,ixp n
is the
design strength of the member and )(mem
CX j,in,
xM is the moment capacity of the

member section about its major local axis (X). The effective area and gross area of the
section of the member under consideration )(and)( memmem
g,e, j,ij,i xAxA nn
are equal.
For each member, the overall capacity )(
Str
mem
2
x
q,
n,
G is determined by
+
=
members
comprissonfor
)(
)()(
)()(
)(
(8.9)
memberstensionfor
)(
)()(
)(
mem
memmem
memmem
mem
mem
memmem
mem
b
X,
Cg,
b
X,
2
Str
x
xxx
x
xx
x
n,
q
n
q
n
n,j,in
q
n
n,
q
n
q
n
n,
M
Mm
xpxA
F
M
Mm
G
j,i
q,
where )(mem xq
n
m is the equivalent uniform factor and is calculated as discussed in
Chapter 2 for each loading case (q). )(mem
b
xn,
M is the buckling resistance moment.
The shear capacity )(
Str
mem
3
x
q,
n,
G is computed by
)(
)(
)(
mem
mem
mem
Y,
Y,
3
Str
j,in
q
n
n,
xP
F
G
q, x
x = (8.10)
where )(mem
Y, j,in
xP is the shear capacity of the member, and )(mem
Y,
xq
n
F is the critical
shear force under the specified loading case (q).
Each member should also satisfy the shear buckling constraint )(
Str,
mem
4
x
q
n,
G if
)(63
)(
)(
,
,
,
ji
ji
ji
x
xt
xd
ε≥ . (8.11)
Hence, )(
Str
mem
4
x
q,
n,
G is computed by
)(
)(
)(
mem
mem
mem
cr,
Y,
4
Str
j,in
q
n
n,
xV
F
G
q, x
x = (8.12)

where )(mem
cr, j,in
xV is the shear resistance of the member section.
d) For a sway structure, the notional horizontal loading case is considered, this is
termed sway stability criterion.
Step 8. Checks of the horizontal and vertical nodal displacements. These are known as
serviceability criteria
1)(
Ser
mem ≤xn,t
G , t = 1, 2 and 3. (8.13)
This is performed by:
a) Computing the horizontal nodal displacements due to the unfactored imposed loads
and wind loading cases in order to satisfy the limits on the horizontal displacements,
∆−∆
=
′′
300
)()(
mem
c
mem
c
mem
c
mem
c
LU
1
YYSer
n
, L
G
n,n,
n
xx
and mem
c
mem
c 1 Nn ,,Λ= (8.14)
where mem
cn
L is the length of the column under consideration. The indexes (U and L)
define the position of the two–column ends.
b) Imposing the limits on the vertical nodal displacements (maximum value within a
beam) due to the unfactored imposed loading case.
=
360
)(
)(
mem
b
mem
b
mem
b
max
2
Ser
n
n
n, L
G
x
x
δ
,
mem
b
mem
b 21 N,,,n Λ= (8.15)
where mem
bn
L is the length of the beam under consideration.
The flowchart given in Figure 8.3 illustrates the design procedure to BS 5950.
Description of the program developed for the design of steel frame structures is given in
Appendix C.

B C DA
YESNO
Start
Apply notional horizontal loading case, compute horizontal nodal
displacements and determine whether the framework is sway or
non–sway using step 2
Apply loading case q Q,,, Λ21= : if the framework is sway, then
include the notional horizontal loading case
Analyse the framework, compute normal forces, shearing forces
and bending moments for each member
Design of member
mem
n =
mem
21 N,,, Λ
Evaluate the design strength )(mem
y, j,in
xp of the member
Tension
member?
Compute the effective buckling lengths according the required
approach mentioned in step 3
Determine the type of the section (slender, semi–compact,
compact or plastic) utilising Table 7 of BS 5950
Figure 8.3a. Flowchart of design procedure of structural steelwork
Check the slenderness criteria employing (8.20) – (8.6)
DCBA

NO
YES
NO
YES
B
Local capacity check Local capacity check
Lateral torsional buckling
check
Overall capacity check
Check of the serviceability criteria using (8.13) – (8.15)
Is
mem
n = mem
N ?
Is q = Q?
Compute the horizontal and vertical nodal displacements due to the
specified loading cases
Carry out the checks of shear applying (8.10) and
shear buckling using (8.12) if necessary
C
End
Figure 8.3b. (cont.) Flowchart of design procedure of structural steelwork
DA

8.3 Problem formulation and solution technique
The general formulation of the design optimization problem can be expressed by
=
=
mem
mem
memmem
1
)(Minimize
N
n
nn
LWF x
subject to: 1)(
Str
mem ≤x
q,
n,r
G , r = 1, 2, 3, 4, q = 1,2, , Q,Λ
1)(
Sle
mem ≤xn,s
G , s = 1, 2
1)(
Ser
mem ≤xn,t
G , t = 1, 2, 3
1
bs
bs
1
x
x
≤− n,n
n,n
I
I
, ss 21 N,,,n Λ= , 121 bb += N,,,n Λ (8.16)
)21( TTTT
Jj ,,,, xxxxx Λ= , J,,,j Λ21=
jji Dx ∈, and
)(
21 λ
Λ
,j
,,
,j
,
,jj dddD =
where mem
n
W is the mass per unit length of the member under consideration and is taken
from the published catalogue. )(
Str
mem x
q,
n,r
G , )(
Sle
mem xn,s
G and )(
Ser
mem xn,t
G reflect the
strength, slenderness and serviceability criteria respectively. The vector of design
variables x is divided into J sub–vectors Jx . The components of these sub–vectors take
values from a corresponding catalogue jD . In the present work, the cross–sectional
properties of the structural members, which form the design variables, are chosen from
two separate catalogues (universal beams and columns covered by BS 4).
The flowchart in Figure 8.4 demonstrates the applied solution technique.

Input data files: GA
parameters, FE model,
loading cases etc.
YES
NO
Save the feasibility checks of the design set
Randomly generate the initial population
Design set =1, 2,
o
pN,Λ
Decode binary chromosomes to integer values and
select the sections from the appropriate catalogue
according to their corresponding integer values
Evaluate the objective and penalised functions
Design set = o
pN ?
Select the best pN individuals out of
o
pN , and impose
them into the first generation of GA algorithm
Apply the design procedure illustrated in flowchart
given in Figure 8.3 to check strength, sway stability
and serviceability criteria to BS 5950
A
New design
Figure 8.4a. Flowchart of the solution technique
Start

YES
YES
NO
NO
Save the feasibility checks of the design set
Generation 1: Calculate the new penalised objective
function, then carry out crossover and mutation
Design set = 2, 3, pN,Λ
Decode binary chromosomes to integer values and
select the sections from the appropriate catalogue
according to their corresponding integer values
Evaluate the objective and penalised functions
Convergence
occurred?
Store the best individuals, and impose them into the next
generation and carry out crossover and mutation
New generation
Apply the design procedure illustrated in flowchart
given in Figure 8.3 to check strength, sway stability
and serviceability criteria to BS 5950
A
Stop
Figure 8.4b. (cont.) Flowchart of the solution technique
New designDesign set = pN ?

8.4 Benchmark examples
Having introduced the design procedure according to BS 5950, formulated the problem
and the solution technique, the process of optimization is now carried out.
Three representative frameworks are demonstrated here to illustrate the
effectiveness and benefits of the developed GA technique as well as investigating the
effect of the employed approach for determining the effective buckling lengths on the
optimum design attained. The sectional members are chosen from BS4 as described in
Section 7.2.1.
In the present work, it is assumed that o
pN and pN are 1000 and 60 respectively.
One–point crossover is applied. Probability of crossover cP and mutation mP are 70 %
and 1 % respectively. The elite ratio rE is 30 %. The technique described in Section 6.2
is utilised where the simple "exact" penalty function employed is
Minimize =
violated.sconstraintofany0
satisfiedsconstraintall)(C
)(
,
,F-
F
x
x (8.17)
The convergence criteria and termination conditions detailed in Section 5.6.3.7 are
utilised where av
C = 0.001, cu
C = 0.001 and 200max
=gen .
8.4.1 Example 1: Two–bay two–storey framework
The optimum design of the two–bay two–storey framework shown in Figure 8.5 is
investigated. The loading cases described in Section 7.3.2 were considered. The
optimization process was carried out when the number of design variables representing
the framework members is 4 and 6 respectively. The linking of design variables are the
same as those described in Section 7.2.2. The three approaches described in Section 8.2
for the determination of the effective length were also applied.

The problem was run utilising the solution parameters described in Section 8.4.
When 4 design variables representing the framework members are taken into account,
the optimization process was carried out using 10 runs for each approach mentioned in
step 3 of Section 8.2. The optimization process was automatically terminated when one
of the termination conditions was satisfied. The solutions are listed in Table 8.1 while
the corresponding design variables of the optimum solution are given in Table 8.2.
Table 8.1. The solutions for the two–bay two–storey framework (4 design variables)
Weight (kg)
Run
First approach
(code)
Second approach
(FE)
Third approach
(conservative)
1 8640 7910 8870
2 8430 8010 8490
3 8690 7950 8630
4 8730 8360 8690
5 8630 7910 8630
6 8550 8110 8490
7 8430 8010 8750
8 8490 7910 8590
9 8750 8150 8870
10 8450 8110 8630
Average weight 8579 8043 8664
Minimum weight 8430 7910 8490
109
87
6
5
4
3
2
1
10.00 m 10.00 m
5.00 m
5.00 m
Figure 8.5. Two–bay two–storey framework

Table 8.2. The optimum solution for the two–bay two–storey framework (4 design
variables)
Cross sections
Design
variable
Member
No. First approach
(code)
Second approach
(FE)
Third approach
(conservative)
1 1, 2, 5, 6 356 × 368 × 177 UC 356 × 368 × 129 UC 356 × 368 × 153 UC
2 3, 4 356 × 368 × 177 UC 356 × 368 × 129 UC 356 × 368 × 153 UC
3 7, 8 457 × 191 × 74 UB 610 × 229 × 101 UB 610 × 229 × 113 UB
4 9, 10 533 × 210 × 82 UB 610 × 229 × 101 UB 533 × 210 × 82 UB
Weight (kg) 8430 7910 8490
The convergence characteristics of the weight of the framework were then
examined during the optimization process. Figure 8.6 shows the changes of the best
framework design with number of generations performed.
Figure 8.6. Two–bay two–storey framework (4 design variables):
best design versus generation number
Similarly, the minimum weight design of the same framework under the same
loading cases is investigated when 6 design variables representing the framework
members are considered. The solutions obtained are listed in Table 8.3 while the
7000
8000
9000
10000
11000
12000
0 10 20 30 40 50 60 70
First approach (code)
Second approach (FE)
Third approach (conservative)
Generation number
Bestdesign(kg)

corresponding design variables of the best solution of each approach are also given in
Table 8.4. The convergence history of the best designs are also displayed in Figure 8.7.
Table 8.3. The solutions for the two–bay two–storey framework (6 design variables)
Weight (kg)
Run
First approach
(code)
Second approach
(FE)
Third approach
(conservative)
1 8490 7955 8700
2 8650 8015 8560
3 8600 8090 8495
4 8415 7870 8495
5 8430 7975 8570
6 8630 8030 8730
7 8600 8160 8630
8 8430 7870 8510
9 8550 8115 8495
10 8415 8100 8740
Average weight 8521 8018 8592.5
Table 8.4. The optimum solution for the two–bay two–storey framework (6 design
variables)
Cross sections
Design
variable
Member
No. First approach
(code)
Second approach
(FE)
Third approach
(conservative)
1 1, 5 356 × 368 × 153 UC 356 × 368 × 177 UC 356 × 368 × 153 UC
2 2, 6 254 ×254 × 73 UC 356 × 368 × 129 UC 356 × 368 × 153 UC
3 3 356 × 368 × 153 UC 356 × 368 × 177 UC 356 × 368 × 202 UC
4 4 203 × 203 × 86 UC 356 × 368 × 129 UC 356 × 368 × 153 UC
5 7, 8 610 × 229 × 101 UB 533 × 210 × 82 UB 533 × 210 × 82 UB
6 9, 10 762 × 267 × 147 UB 533 × 210 × 82 UB 610 × 229 × 101 UB
Weight (kg) 8415 7870 8495

Figure 8.7. Two–bay two–storey framework–6 design variables:
From Tables 8.1 and 8.3, it can be observed that there is more than one solution
available, and the difference in weight between them is small. This could be of benefit
in using an automated design procedure that allows the designer to choose the
appropriate solution depending on the availability of the sections provided by
manufacturer. Moreover, applying design optimization allows the designer to achieve
better solutions when utilising more accurate methods for evaluating the effective
buckling lengths.
It is of interest also to compare the design variables of two solutions having the
same value of the objective function. This could add a new perspective to the
advantages of using automated design. In the first solution presented in Table 8.5, it can
be observed that the cross sections corresponding to the design variables representing
the columns are identical. The design variables corresponding to columns (1, 3 and 5)
are also the same in the second solution. This indicates that it may be economical to use
7000
8000
9000
10000
11000
12000
0 10 20 30 40 50 60 70 80
Generation number
Bestdesign(kg)

the developed algorithm to decide the optimum grouping of the members in a
framework.
Table 8.5. Comparison between the design variables of two solutions having the same
value of the objective function
Cross sections
Design
variable
Member
No. First solution Second solution
1 1, 5 356 × 368 × 177 UC 356 × 368 × 177 UC
2 2,6 356 × 368 × 177 UC 203 × 203 × 46 UC
3 3 356 × 368 × 177 UC 356 × 368 × 177 UC
4 4 356 × 368 × 177 UC 203 × 203 × 71 UC
5 7,8 457 × 191 × 74 UB 610 × 229 × 101 UB
6 9, 10 533 × 210 × 82 UB 762 × 267 × 147 UB
Weight (kg) 8430 8430
8.4.2 Example 2: Five–bay five–storey framework
The next example to study is the five–bay five–storey framework shown in Figure 8.8.
The loading cases described in Section 7.3.3 are taken into account.
Figure 8.8. Five–bay five–storey framework
2P
16 21
5554535251
50
17
49484746
45
19
18
44434241
4039383736
3534333231
26
27
28
29
3025
24
23
22
2015
14
13
12
11
10
9
8
7
6
5
4
3
2
1
2P
2P
2
P
2P
2P
2P
2P
2P
4P 4P 4P 4P
4P 4P 4P 4P
4P4P4P 4P
4P4P4
P
0.01P
0.01P
0.01P
0.01P
0.01P
4P
2P 2P2P PP
3.00 m
3.00 m
3.00 m
3.00 m
3.00 m
5.00 m5.00 m5.00 m5.00 m5.00 m

The design optimization process was carried out using different numbers of design
variables representing the framework members. Here, 8 and 10 design variables were
considered. Figures 8.9 and 8.10 show the linking of 8 and 10 design variables
respectively. The three approaches described in Section 8.2 for the determination of the
effective lengths were applied (see Toropov et. al., 1999).
10 10 10 99
10 10 10 9
91010109
91010109
7888
9
7
4
4
4
4
4 4
44
5
5
66
5
55
5
6
5
5
6
1
1
2
2
33
2
1
2
1
Figure 8.10. Five–bay five–storey framework showing the arrangement
of 10 design variables
Figure 8.9. Five–bay five–storey framework showing the arrangement
of 8 design variables
8 8 8 88
8 8 8 8
88888
88888
7777
8
7
4
4
4
4
4 4
44
5
5
66
5
55
5
6
5
5
6
1
1
2
2
33
2
1
2
1

First, the optimization process was run using 8 design variables representing the
framework members. The solutions over 5 runs are given in Table 8.6. The design
variables corresponding to the optimum design of the three approachs are listed in Table
8.7.
Table 8.6. The solutions for the five–bay five–storey framework (8 design variables)
Weight (kg)
Run
First approach
(code)
Second approach
(FE)
Third approach
(conservative)
1 15455 14675 16101
2 15385 14851 15926
3 15465 14390 15991
4 15321 14935 15973
5 15367 14725 16299
Average weight 15398.6 14715.2 16058
Table 8.7. The optimum solution for the five–bay five–storey framework (8 design
variables)
Cross sections
Design
variable First approach
(code)
Second approach
(FE)
Third approach
(conservative)
1 356 × 368 × 153 UC 305 × 305 × 118 UC 305 × 305 × 118 UC
2 356 × 368 × 129 UC 305 × 305 × 118 UC 305 × 305 × 118 UC
3 356 × 368 × 129 UC 305 × 305 × 97 UC 254 × 254 × 89 UC
4 356 × 368 × 129 UC 356 × 368 × 129 UC 356 × 368 × 129 UC
5 254 × 254 × 107 UC 305 × 305 × 97 UC 305 × 305 × 137 UC
6 203 × 203 × 52 UC 203 × 203 × 71 UC 254 × 254 × 73 UC
7 406 × 140 × 39 UB 305 × 102 × 28 UB 254 × 102 × 28 UB
8 406 × 140 × 39 UB 305 × 165 × 40 UB 406 × 140 × 46 UB
Weight (kg) 15321 14390 15926

It is of interest to note that the optimizer is able to obtain more than one suitable
solution for each approach, and the difference in the weight between them is little. This
can be concluded when comparing the average value of the solutions with each solution
separately. Using the more accurate approach for determining the effective buckling
length may results in achieving better solutions.
During the optimization process, the solutions are monitored to examine their
convergence history. Then, the graphical representation of changes of the best design
with the number of generations performed achieved to reach the optimum design is
shown in Figure 8.11. It is worth observing that the solution convergence is achieved in
90 generations using a population size of only 70.
Figure 8.11. Five–bay five–storey framework (8 design variables):
Second, the problem was similarly analysed when utilising 10 design variables
representing the framework members. The solutions obtained are given in Table 8.8
while the design variables corresponding to the optimum design of each approach are
listed in Table 8.9.
12000
14000
16000
18000
20000
22000
24000
0 10 20 30 40 50 60 70 80 90
Generation number
Bestdesign(kg)

Table 8.8. The solutions for the five–bay five–storey framework (10 design variables)
Weight (kg)
Run
First approach
(code)
Second approach
(FE)
Third approach
(conservative)
1 15391 14723 16309
2 15571 14461 16239
3 15371 14195 16941
4 15753 14809 15819
5 15679 14455 16469
Average weight 15553 14528.6 16355.4
Table 8.9. The optimum solution for the five–bay five–storey framework (10 design
variables)
Cross sections
Design
(code)
Second approach
(FE)
Third approach
(conservative)
1 305 × 305 × 97 UC 305 × 305 × 137 UC 305 × 305 × 137 UC
2 305 × 305 × 97 UC 305 × 305 × 137 UC 305 × 305 × 97 UC
3 254 × 254 × 107 UC 203 × 203 × 52 UC 254 × 254 × 89 UC
4 356 × 368 × 129 UC 356 × 368 × 129 UC 356 × 368 × 129 UC
5 254 × 254 × 107 UC 254 × 254 × 73 UC 356 × 368 × 129 UC
6 203 × 203 × 46 UC 203 × 203 × 46 UC 203 × 203 × 60 UC
7 533 × 210 × 92 UB 533 × 210 × 92 UB 356 × 171 × 51 UB
8 254 × 146 × 31 UB 254 × 102 × 25 UB 254 × 146 × 37 UB
9 356 × 171 × 51 UB 356 × 127 × 39 UB 406 × 178 × 54 UB
10 406 × 140 × 46 UB 406 × 140 × 39 UB 406 × 140 × 39 UB
Weight (kg) 15371 14195 15819
Figure 8.12 demonstrates the convergence history of the optimum designs during
the optimization process. It can be observed that the convergence has been achieved in
80 generations due to the termination conditions described in Section 8.4.

Figure 8.12. Five–bay five–storey framework (10 design variables):
8.4.3 Example 3: Four–bay ten–storey framework
The final example is the four–bay ten–storey framework shown in Figure 8.13. In this
figure, the loading pattern for the stability analysis and member numbering are shown
where 01.0=α . The problem formulated in Section 8.4.1 utilising 8 design variables
representing the framework members are considered and the linking is given in Figure
8.13. It is assumed that the spacing between successive frameworks is 6.00 m. The
framework will be used for offices and computer equipment purposes. The following
eight loading cases were considered.
1. The beams are subjected to the vertical loads LL.DL.P 6141v
+= .
2. The beams are subjected to the vertical loads LL.DL.P 6141v
+= , and the left hand
side of the framework is subjected to the notional horizontal loads.
12000
14000
16000
18000
20000
22000
24000
0 10 20 30 40 50 60 70 80
Generation number
Bestdesign(kg)

87 88 89 90
83 84 85 86
79 80 81 82
75 76 77 78
71 72 73 74
67 68 69 70
63 64 65 66
59 60 61 62
55 56 57 58
51 52 53 54
41
50
49
48
47
46
45
44
42
43
31
40
39
38
37
36
35
34
32
33
21
30
29
28
27
26
25
24
22
23
11
20
19
18
17
16
15
14
12
13
1
10
9
8
7
6
5
4
2
3
2P2P 4P 4P 4P
2P2P 4P 4P 4P
2P2P 4P 4P 4P
2P2P 4P 4P 4P
2P2P 4P 4P 4P
2P2P 4P 4P 4P
2P2P 4P 4P 4P
2P2P 4P 4P 4P
2P2P 4P 4P 4P
PP 2P 2P 2P
4.00 m4.00 m4.00 m 4.00 m
16.00 m
32.00m
5.00 m
3.00 m
3.00 m
3.00 m
3.00 m
3.00 m
3.00 m
3.00 m
3.00 m
3.00 m
αP
αP
αP
αP
αP
αP
αP
αP
αP
αP
Figure 8.13. Four–bay ten–storey framework: dimensions, member numbering
and loading pattern for the stability analysis

3. The beams of the first bay (counting from the left) are exposed to the vertical loads
LL.DL.P 6141v
+= while the rest of the beams are subjected to the vertical loads
DL.P 41v
= .
4. The beams of the first two bays (counting from the left) are subjected to the vertical
loads LL.DL.P 6141v
+= while the rest of the beams are subjected to the vertical
loads DL.P 41v
= .
5. LL.DL.P 6141v
+= and DL.P 41v
= are distributed in a staggered way. This
means that the loads applied to the top left storey are LL.DL.P 6141v
+= while the
adjacent beams either in the same storey level or the storey beneath carry vertical
loads DL.P 41v
= .
6. The beams are subjected to vertical loads LL.DL.P 2121v
+= and the left hand side
of the framework is subjected to the factored wind loads WL.P 21h
= .
7. The beams are subjected to the vertical loads LL.P 01v
= and the left hand side of
the framework is subjected to unfactored wind loads WL.P 01h
= . This loading
pattern is considered to check horizontal displacements at the nodes.
8. The beams are subjected to vertical loads LL.P 01v
= . This loading pattern is taken
into account to check vertical displacements at nodes.
Figure 8.14 shows a loading pattern in which the values of the nodal loads of each
loading case, stated above, can be identified from Table 8.10.

36663
P14
P23P5 P11 P17
P24P6 P12 P18
P23P5 P11 P17
P24P6 P12 P18
P23P5 P11 P17
P24P6 P12 P18
P23P5 P11 P17
P24P6 P12 P18
P23P5 P11 P17
P2P4 P10 P16
8 8 8 8
8 8 8 8
8 8 8 8
8 8 8 8
8 8 8 8
8 8 8 8
8 8 8 8
8 8 8 8
8 8 8 8
7 7 7 7
1
3
3
2
2
2
2
1
1
4
6
6
5
5
5
5
4
4
4
6
6
5
5
5
5
4
4
4
6
6
5
5
5
5
4
4
1
3
3
2
2
2
2
1
1
P26P2 P8 P14 P20
P27P3 P9 P15 P21
P26P2 P8 P14 P20
P27P3 P9 P15 P21
P26P2 P8 P14 P20
P27P3 P9 P15 P21
P26P2 P8 P14 P20
P27P3 P9 P15 P21
P26P2 P8 P20
P2P1 P7 P13 P19
Figure 8.14. Four–bay ten–storey framework
H10
H9
H8
H7
H6
H5
H4
H3
H2
H1

Table 8.10. Loads applied on the four–bay ten–storey framework (in kN)
Loading caseLoad
symbol 1 2 3 4 5 6 7 8
P1 45.0 45.0 45.0 45.0 45.0 35.0 15.0 15.0
P2 90.0 90.0 90.0 90.0 45.0 65.0 40.0 40.0
P3 90.0 90.0 90.0 90.0 90.0 65.0 40.0 40.0
P4 90.0 90.0 90.0 90.0 90.0 70.0 30.0 30.0
P5 180.0 180.0 180.0 180.0 60.0 130.0 80.0 80.0
P6 180.0 180.0 180.0 180.0 180.0 130.0 80.0 80.0
P7 90.0 90.0 70.0 90.0 70.0 70.0 30.0 30.0
P8 180.0 180.0 120.0 180.0 120.0 130.0 80.0 80.0
P9 180.0 180.0 120.0 180.0 120.0 130.0 80.0 80.0
P10 90.0 90.0 45.0 90.0 45.0 70.0 30.0 30.0
P11 180.0 180.0 60.0 180.0 180.0 130.0 80.0 80.0
P12 180.0 180.0 60.0 180.0 60.0 130.0 80.0 80.0
P13 90.0 90.0 60.0 70.0 70.0 70.0 30.0 30.0
P14 180.0 180.0 60.0 120.0 120.0 130.0 80.0 80.0
P15 180.0 180.0 60.0 120.0 120.0 130.0 80.0 80.0
P16 90.0 90.0 60.0 45.0 90.0 70.0 30.0 30.0
P17 180.0 180.0 60.0 60.0 60.0 130.0 80.0 80.0
P18 180.0 180.0 60.0 60.0 60.0 130.0 80.0 80.0
P19 90.0 90.0 60.0 60.0 70.0 70.0 30.0 30.0
P20 180.0 180.0 60.0 60.0 120.0 130.0 80.0 80.0
P21 180.0 180.0 60.0 60.0 120.0 130.0 80.0 80.0
P22 90.0 90.0 60.0 60.0 45.0 70.0 30.0 30.0
P23 180.0 180.0 60.0 60.0 180.0 130.0 80.0 80.0
P24 180.0 180.0 60.0 60.0 180.0 130.0 80.0 80.0
P25 45.0 45.0 25.0 25.0 70.0 35.0 30.0 30.0
P26 90.0 90.0 45.0 45.0 90.0 65.0 80.0 80.0
P27 90.0 90.0 45.0 45.0 45.0 65.0 80.0 80.0
H1 0.0 3.5 0.0 0.0 0.0 11.0 9.2 0.0
H2 0.0 7.0 0.0 0.0 0.0 16.7 14.0 0.0
H3 0.0 7.0 0.0 0.0 0.0 15.6 13.0 0.0
H4 0.0 7.0 0.0 0.0 0.0 14.5 12.0 0.0
H5 0.0 7.0 0.0 0.0 0.0 13.4 11.2 0.0
H6 0.0 7.0 0.0 0.0 0.0 12.2 10.2 0.0
H7 0.0 7.0 0.0 0.0 0.0 11.1 9.25 0.0
H8 0.0 7.0 0.0 0.0 0.0 10.0 8.35 0.0
H9 0.0 7.0 0.0 0.0 0.0 8.9 7.5 0.0
H10 0.0 7.0 0.0 0.0 0.0 7.5 6.25 0.0

The problem was analysed employing the solution parameters mentioned in
Section 8.4. The optimization process was carried out using 5 runs for each approach for
determining the effective buckling lengths. The optimization process was automatically
terminated when one of the termination conditions, stated in Section 8.4, is satisfied.
The solutions achieved are listed in Table 8.11 while the corresponding design variables
of the optimum solution of each approach are given in Table 8.12.
Table 8.11. The solutions for the four–bay ten–storey framework
Weight (kg)
Run
First approach
(code)
Second approach
(FE)
Third approach
(conservative)
1 34421 30835 35125
2 34400 30649 35393
3 34424 29301 35649
4 34337 30904 34934
5 34406 30727 36992
Average weight 34397.6 30483.2 35618.6
Table 8.12. The optimum solution for the four–bay ten–storey framework
Cross sections
Design
(code)
Second approach
(FE)
Third approach
(conservative)
1 356 × 406 × 235 UC 356 × 368 × 177 UC 356 × 406 × 235 UC
2 356 × 368 × 153 UC 305 × 305 × 118 UC 356 × 368 × 153 UC
3 356 × 368 × 129 UC 203 × 203 × 71 UC 356 × 368 × 129 UC
4 356 × 406 × 235 UC 356 × 368 × 202 UC 356 × 406 × 235 UC
5 305 × 305 × 118 UC 356 × 368 × 129 UC 356 × 368 × 129 UC
6 305 × 305 × 118 UC 254 × 254 × 73 UC 356 × 368 × 129 UC
7 254 × 146 × 31 UB 305 × 102 × 33 UB 305 × 102 × 25 UB
8 457 × 152 × 52 UB 457 × 152 × 52 UB 457 × 152 × 52 UB
Weight (kg) 34337 29301 34934

It can be observed that there is little difference in the values of the solution for
each approach, listed in Table 8.11. This indicates the developed algorithm can be
successfully applied to reach a good solution. It is also interesting to note that the
column members, belonging to group 1 and 4 were grouped separately, but the same
universal column (356 × 406 × 235 UC) was adopted for both groups when using either
the first or third approach. Similarly, the cross sections, corresponding to the third, fifth
and sixth design variable of the optimum design of the third approach, are also the same.
This indicates that it may be more economical to use the developed algorithm to decide
the best grouping of the framework members.
During the optimization process, the convergence characteristics of each solution
were examined. Figure 8.15 shows the changes of the best design with the number of
generations performed to reach the optimum design.
Figure 8.15. Four–bay ten–storey framework: best design versus
generation number
It is worth noting that the optimum solutions were reached within 50 generations,
and the rest of the computations were carried out to satisfy the convergence criteria.
20000
30000
40000
50000
60000
0 10 20 30 40 50 60 70 80
Generation number
Bestdesign(kg)

8.5 Validation of the optimum design
This section shows that the values of the constraints obtained by applying the developed
FORTRAN code for the design of steel frame structures to BS 5950 are in a good
agreement with those obtained by CSC software.
Since 1975, CSC UK Ltd. (1998) has specialised in developing PC–based
software for structural engineering design. The product S–FRAME was introduced to
analyse a framework under specified loading cases, then by switching to the product S–
STEEL the framework members can be checked for compliance with BS 5950 design
criteria. Due to the innovative use of graphics, both S–FRAME and S–STEEL have a
user interface facility. The user interface facility provides the designer to visualise the
orientation of the sections of the members, coordinate system, member numbering and
the design results. The following steps can summarise the used procedure.
1) In S–FRAME, the framework geometry, member sections and loading cases are
defined. Then, the bending moments, shear forces, displacements are calculated
applying the linear analysis facility.
2) Starting to S–STEEL program. This automatically detects the framework geometry,
loading cases, bending moments, shear forces and displacements and member
sections. The design checks are then carried out. Here, the effective length factors
( memmemmemmem )(and)( ffff
Y,X, nj,in
LxLLL e
n
e
n
x ) and the equivalent uniform factor
)(mem xq
n
m are user defined. The default value for each is unity. At this stage, it is
worth noting that )(mem xq
n
m is computed in the developed FORTRAN code as given
in clause 4.3.7.6 of BS 5950 (technique 1) for each member at each loading case.
3) The design results are then visualised in a separate window as shown later.

To validate the applied FORTRAN code, the problem described in Section 8.3
should be first run when )(mem xq
n
m for each member equals 1. This is named as
technique 2. Then, CSC software is used to check the obtained results.
The optimum design of two–bay two–storey framework is investigated when 4
design variables representing the framework members are considered. The framework is
shown in Figure 8.5. The framework is subjected to the same loads as mentioned in
Section 7.3.2. The optimization process was carried out utilising the design procedure
discussed in Section 8.2 while the solution parameters and the convergence criteria are
considered as those given in Section 8.4. Five runs were carried out when applying the
first approach for determining the effective buckling lengths. The design variables
corresponding to the optimum solution were then tabulated in Table 8.13. It is worth
comparing the best solution obtained with that achieved in section 8.4.1 (technique 1)
when a more accurate equation for determining )(mem xq
n
m was applied. This comparison
is also presented in Table 8.13.
Table 8.13. The best solution for the two–bay two–storey framework (4 design
variables)
Cross sectionsDesign
variable
Member
No. Technique 1 Technique 2
1 1, 2, 5, 6 356 × 368 × 177 UC 305 × 305 × 118 UC
2 3, 4 356 × 368 × 177 UC 305 × 305 × 118 UC
3 7, 8 457 × 191 × 74 UB 610 × 229 × 101 UB
4 9, 10 533 × 210 × 82 UB 762 × 267 × 147 UB
Weight (kg) 8430 8500
It is known from clause 4.3.7.6 of BS 5950 that the upper limit of )(mem xq
n
m is 1.
Therefore, the cross sections of beams, obtained when applying technique 2, have more

strength than those achieved by employing technique 1. This allows the optimizer to
obtain solution (8500 kg), which has column sections (305 × 305 × 118 UC) having
strength less than those (356 × 368 × 177 UC) of technique 1.
The graphical representation of changes of the best design with the number of
generations performed for each trial is shown in Figure 8.16.
Figure 8.16. Two–bay two–storey framework: best design versus
generation number.
At this stage, the framework weight is optimized and the section of each member
is known. The optimizer is also modified to indicate whether the framework is sway or
non–sway. Here, the optimizer identifies the framework as a non–sway framework. This
is also successfully examined when using S–FRAME.
Following the three steps stated at the beginning of this section, the obtained
results are validated and the design results from S–STEEL are displayed in Figure 8.17.
8000
9000
10000
11000
12000
0 10 20 30 40 50 60 70
First run
Second run
Third run
Fourth run
Fifth run
Bestdesign(kg)
Generation number

Figure 8.17. The design results of two–bay two–storey framework (captured from S–STEEL)

In this figure, the numbering of the framework members, type of cross section of
each member and node are shown. The design checks are indicated in colour in which
the code utilisation menu gives the range for of each colour. It is worth noting that the
design results vary between 0.8 and 1.0. Among the strength constraints, the overall
buckling constraints have the largest value.
8.6 Concluding remarks
Optimization technique based on GA was applied for design optimization of steel frame
structures. Multiple loading cases were considered. The design method obtained a steel
frame structure with the least weight by selecting appropriate sections for beams and
columns from BS 4. The following concluding remarks can be made.
1) It has been proven that the developed GA approach can be successfully incorporated
in design optimization in which framework members have to be selected from the
available sections taken from BS 4 while the design satisfies the design criteria
according to BS 5950.
2) It is also worth noting that different numbers of design variables are considered for
each framework and the optimizer is able to obtain a good solution in a reasonable
number of generations. This indicates that the developed approach can be utilised by
a practising designer.
3) The optimizer is successfully linked to a finite element package for a more accurate
treatment of the determination of the effective buckling length that leads to
achieving a more economical design.
4) In the present chapter, the constraints imposed on the second moment of area of two
adjacent columns in two adjacent storey levels are chosen to reflect the designers
experience. Other constraints, such as sectional dimensions, sectional area, etc., can

also formulated. This indicates that the optimizer is able to treat different practical
constraints depending on the skills and experience of the designer.
5) It can be observed that the optimizer helps to identify the best arrangement of
grouping to obtain economical design. This illustrates that it may be economical to
use the developed algorithm to decide the optimum grouping of the members in a
framework using multi–objective functions.
6) It can also be concluded that the developed optimizer is able to obtain more than one
suitable solution, and the difference between them is small. This adds a benefit of
using an automated design that allows the designer to choose the appropriate
solution depending on the availability of the sections provided by manufacturer.
7) It is interesting to note that even some of the powerful computer software packages
available today for the design of steel frameworks such as CSC and STAAD–III
require the structural designer to input the effective buckling length factor as a
parameter. In this study, computation of the effective buckling length is automated
and included in the developed algorithm. This is achieved by employing three
different approaches as discussed in Section 8.2.
Two questions arise. The first is whether or not the developed optimizer can
obtain a solution of minimum weight design of three–dimensional steelwork. This is a
more complex problem and the formulation of the problem includes more constraints.
The bracing members, which take discrete values from BS 4848 have to be incorporated
in the design problem. The second is what difference could be achieved in the optimum
design when using either of these approaches for evaluating the effective buckling
length. These questions will be answered in the next chapter.

steel structure optimisation

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