DESIGN OPTIMIZATION OF 2D STEEL FRAME · PDF fileDesign Optimization of 2D Steel Frame...

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


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;


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)


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


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)


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.


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


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


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.


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


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 ?


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.


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


252

Table 8.2. The optimum solution for the two–bay two–storey framework (4 design variables)

Cross sections Design

var iable Member

No. First approach (code)



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



Third approach (conservat ive)

Generation number

Bes

t des

ign

(kg)


253

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




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)


var iable Member

No. First approach (code)



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


254

Figure 8.7. Two–bay two–storey framework–6 design variables: best design versus generation number

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

Bes

t des

ign

(kg)


255

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


var iable 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 215554535251

5017

49484746

45

19

1844434241

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

2P

2P

2P

2P

2P

2P

4P 4P 4P 4P

4P 4P 4P 4P

4P4P4P 4P

4P4P4P

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


256

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

910 10 10 9

910 10 10 9

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


257

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




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)


var iable First approach (code)



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


258

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): best design versus generation number

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

Bes

t des

ign

(kg)


259

Table 8.8. The solutions for the five–bay five–storey framework (10 design variables)

Weight (kg) Run




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)





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.


260

Figure 8.12. Five–bay five–storey framework (10 design variables):

best design versus generation number

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

Bes

t des

ign

(kg)


261

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

2P 2P 4P 4P 4P

2P 2P 4P 4P 4P

2P 2P 4P 4P 4P

2P 2P 4P 4P 4P

2P 2P 4P 4P 4P

2P 2P 4P 4P 4P

2P 2P 4P 4P 4P

2P 2P 4P 4P 4P

2P 2P 4P 4P 4P

P P 2P 2P 2P

4.00 m 4.00 m 4.00 m 4.00 m

16.00 m

32.0

0 m

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


262

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.


263

3 6 6 6 3

P14

P23 P5 P11 P17

P24 P6 P12 P18

P23 P5 P11 P17

P24 P6 P12 P18

P23 P5 P11 P17

P24 P6 P12 P18

P23 P5 P11 P17

P24 P6 P12 P18

P23 P5 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

P26 P2 P8 P14 P20

P27 P3 P9 P15 P21

P26 P2 P8 P14 P20

P27 P3 P9 P15 P21

P26 P2 P8 P14 P20

P27 P3 P9 P15 P21

P26 P2 P8 P14 P20

P27 P3 P9 P15 P21

P26 P2 P8 P20

P2P1 P7 P13 P19

Figure 8.14. Four–bay ten–storey framework

H10

H9

H8

H7

H6

H5

H4

H3

H2

H1


264

Table 8.10. Loads applied on the four–bay ten–storey framework (in kN)

Loading case Load 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


265

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




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





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


266

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

Bes

t des

ign

(kg)


267

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)( ffffY,X, nj,in

LxLLL en

en

x ) and the equivalent uniform factor

)(mem xq

nm are user defined. The default value for each is unity. At this stage, it is

worth noting that )(mem xq

nm 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.


268

To validate the applied FORTRAN code, the problem described in Section 8.3

should be first run when )(mem xq

nm 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

nm 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 sections Design var iable

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

nm is 1.

Therefore, the cross sections of beams, obtained when applying technique 2, have more


269

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 runFif th run

Bes

t des

ign

(kg)

Generation number

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


271

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


272

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.

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