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
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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
VII I
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
Lmemc,Y n′∆
Umemc,Y n′∆
X ′
sNh
b1
x
n,I
maxmemb
nδ
11x
,I
1sx
,nI
1sx
,NI
X
Y
Y
Y
Y
YY
YY
YY 1bsx
+N,NI
1bsx
+N,nI
1b1x
+N,I
X
X X
X
X
X
X
X
X
maxmemb
Nδ
max
1δ
Z′
Y ′
Figure 8.2. Deformed configuration of a framework combined with coordinate systems
1h
1B bNB
Design Optimization of 2D Steel Frame Structures
240
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
)(UY mem
cx
n,′∆ and lower )(L
Y memc
xn,′
∆ horizontal nodal displacements of each column
member memcn satisfies the following condition:
1
2000
)()(
memc
memc
memc
LY
UY
≤
��
�
�
��
�
�
∆−∆′′
n
n,n,
L
xx, mem
cmemc 21 N,,,n Λ= . (8.1)
Step 3. Calculation of the effective buckling lengths effmemX, n
L and effmemY, n
L of columns
and beams. For columns, effmemc
,X nL is determined according to one of the following three
approaches:
• using the charts from BS 5950 as described in Section 2.6.2.2;
Design Optimization of 2D Steel Frame Structures
241
• 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 effmemb
X, nL of a beam equals the unrestrained length of
the compression flange that occurs on the underside of a beam (see MacGinley, 1997).
To evaluate )(effmemY, 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, )(effmemY, j,ixL
n equals to the half of the length of the member memn
L .
Step 4. Calculation of the slenderness ratios )(memX,x
nλ and )(memY, j,ix
nλ of the
member memn using
mem
mem
mem
X,
X,
X,
)()(
eff
nr
Ln
n
xx =λ , (8.2)
mem
mem
mem
Y,
Y,
Y,
)()(
eff
n
j,i
j,ir
xLx
n
n=λ (8.3)
where memX, nr and memY, n
r are the radius of gyrations of the section about X and Y axes.
Step 5. Check of the slenderness constraints Sle
memn,sG for each member using
1)(Sle
mem ≤xn,s
G , s = 1, 2 (8.4)
where 180
)()(
mem
memX,
1
Sle xx n
n,G
λ= and (8.5)
180
)()(
mem
memY,
2
Sle j,i
j,in,
xxG
nλ
= . (8.6)
Design Optimization of 2D Steel Frame Structures
242
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 memn 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 memy n,p of the member.
c) Check of the strength constraints )(Str
mem xq,
n,rG 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 ≤xq,
n,rG , 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
nF is the axial force, )(memX,
xq
nM is the moment about the major local
axis (x) at the critical region of the member under consideration, )(memy, j,ixpn
is the
design strength of the member and )(memCX j,in,xM is the moment capacity of the
Design Optimization of 2D Steel Frame Structures
243
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 xAxAnn
are equal.
For each member, the overall capacity )(Str
mem2x
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
nm is the equivalent uniform factor and is calculated as discussed in
Chapter 2 for each loading case (q). )(membx
n,M is the buckling resistance moment.
The shear capacity )(Str
mem3x
q,
n,G is computed by
)(
)()(
mem
mem
mem
Y,
Y,
3
Str
j,in
q
n
n, xP
FG
q, xx = (8.10)
where )(memY, j,inxP is the shear capacity of the member, and )(memY,
xq
nF is the critical
shear force under the specified loading case (q).
Each member should also satisfy the shear buckling constraint )(Str,
mem4x
q
n,G if
)(63)(
)(,
,
,ji
ji
jix
xt
xdε≥ . (8.11)
Hence, )(Str
mem4x
q,
n,G is computed by
)(
)()(
mem
mem
mem
cr,
Y,
4
Str
j,in
q
n
n, xV
FG
q, xx = (8.12)
Design Optimization of 2D Steel Frame Structures
244
where )(memcr, j,inxV 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
)()(
memc
memc
memc
memc
LU
1
YYSer
n, L
Gn,n,
n
xx and mem
cmemc 1 Nn ,,Λ= (8.14)
where memcn
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
)()(
memb
memb
memb
max
2
Ser
n
n
n, LG
xx
δ, mem
bmemb 21 N,,,n Λ= (8.15)
where membn
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.
Design Optimization of 2D Steel Frame Structures
245
B C DA
YES NO
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 memn = mem21 N,,, Λ
Evaluate the design strength )(memy, j,inxp 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)
D C B A
Design Optimization of 2D Steel Frame Structures
246
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 memn = memN ?
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
D A
Design Optimization of 2D Steel Frame Structures
247
8.3 Problem formulation and solution technique
The general formulation of the design optimization problem can be expressed by
�=
=mem
memmemmem
1
)(MinimizeN
nnn
LWF x
subject to: 1)(Str
mem ≤xq,
n,rG , 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
1bs
bs
1x
x ≤− n,n
n,n
I
I, ss 21 N,,,n Λ= , 121 bb += N,,,n Λ (8.16)
)21( TTTTJj ,,,, xxxxx Λ= , J,,,j Λ21=
jji Dx ∈, and
)(21 λ
Λ,j
,,,j
,,jj dddD =
where memnW is the mass per unit length of the member under consideration and is taken
from the published catalogue. )(Str
mem xq,
n,rG , )(
Slemem x
n,sG and )(
Sermem x
n,tG 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.
Design Optimization of 2D Steel Frame Structures
248
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, opN,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 = opN ?
Select the best pN individuals out of opN , 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
Design Optimization of 2D Steel Frame Structures
249
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 design Design set = pN ?
Design Optimization of 2D Steel Frame Structures
250
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 opN 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
xx (8.17)
The convergence criteria and termination conditions detailed in Section 5.6.3.7 are
utilised where avC = 0.001, cuC = 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.
Design Optimization of 2D Steel Frame Structures
251
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
876
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
Design Optimization of 2D Steel Frame Structures
252
Table 8.2. The optimum solution for the two–bay two–storey framework (4 design variables)