kameshki_2001_optimum design of nonlinear steel frames with semi-rigid connections

7/27/2019 Kameshki_2001_Optimum Design of Nonlinear Steel Frames With Semi-rigid Connections

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Optimum design of nonlinear steel frames with

semi-rigid connections using a genetic algorithm

E.S. Kameshki *, M.P. Saka

Department of Civil and Architectural Engineering, University of Bahrain, P.O. Box 32038, Isa Town, Bahrain

Received 8 October 1999; accepted 22 February 2001

Abstract

The realistic modeling of beam-to-column connections plays an important role in the analysis and design of steel

frames. A genetic algorithm based optimum design method is presented for nonlinear multistorey steel frames with

semi-rigid connections. The design algorithm obtains a frame with the least weight by selecting appropriate sections

from a standard set of steel sections such as wide ange sections of AISC or universal sections of British standard. The

algorithm accounts for the serviceability and strength constraints as specied in BS5950. A nonlinear empirical model is

used to include the momentrotation relation of beam-to-column connections. Furthermore, the PD eect is also

accounted for in the analysis and design of the multistorey frame. The eective length factors for columns which are

exibly connected to beams are obtained from the solution of the nonlinear interaction equation. A number of frames

with end plate without column stieners are designed to demonstrate the eciency of the algorithm. 2001 Civil-

Comp Ltd. and Elsevier Science Ltd. All rights reserved.

Keywords: Structural optimization; Genetic algorithm; Semi-rigid connections; Nonlinear analysis; Steel design; Unbraced frame

1. Introduction

The structural response of a steel frame is closely

related with the behavior of its beam-to-column con-

nections. The realistic modeling of a steel frame, there-

fore, requires the use of realistic connection modeling, if

an accurate response of the frame is desired to be ob-

tained. It is common engineering practice to assumeeither pinned or a fully rigid connections between beams

and columns. Experiments, however have shown that

the actual behavior lies somewhere between these two

idealized models which makes them exible. Further-

more, experiments have also shown that when a moment

is applied to a exible connection, the relation of relative

beam column rotation is nonlinear. Momentrotation

curves of various types of connections are shown in Fig.

1 [1]. The rotational distortion of the connections aects

the displacements of the frame and brings about redis-

tribution of moments between columns and beams.

Thus, in the analysis and design of steel frames beam-

to-column connections should be modeled as semi-rigid

connections.

The exibility of a connection is dependent on the

geometric parameters of the elements used in the con-

nection such as bolt size and dimensions of end plate orangle sections. Extensive research works, experimental

as well as numerical have been carried out to establish

momentrotation relationship to be used for predicting

the actual behavior of the exible connections [26]. As

a result of these studies several mathematical expressions

have been proposed which vary from a simple linear

model to polynomial and power models. These relation-

ships are used in the modeling of the steel frame con-

nections and they provide a fairly accurate prediction of

frame response. Although the problem of analysis of

steel frames with semi-rigid connections has drawn great

attention, it is not possible to state the same for the

design of such frames. In spite of the fact that design

Computers and Structures 79 (2001) 15931604

www.elsevier.com/locate/compstruc

*Corresponding author.

0045-7949/01/$ - see front matter 2001 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.

PII: S0 0 4 5 -7 9 4 9 (0 1 )0 0 0 3 5 -9


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codes such as AISC-LRFD [7] and BS5950 [8] allows

the designer to consider partially restrained connections

in the design of steel frames, no specic guidelines are

given for the design in these codes. In some of the recent

research works the design problem of steel frameswith semi-rigid connections is addressed [912]. In all

these works, mathematical programming techniques

are used to formulate and obtain the solution of the

optimum design problem. Due to the discrete nature

of the programming problem, the solution techniques

available in mathematical programming are complex,

require the computation of sensitivities of design con-

straints and they are not very ecient particularly for

large size structures. Furthermore, in all the research

works drift constraints were not considered in the design

problem.

In the present study, a genetic algorithm based op-timum design method is developed for unbraced multi-

storey steel frames with semi-rigid beam-to-column

connections. Lateral displacements in such frames are

much more than those of rigid frames due to joint

exibility. The geometric nonlinearity due to PD eect

is taken into account in the frame analysis. The design

algorithm obtains the frame with the least weight by

selecting appropriate sections for the beams and col-

umns of the frame from the standard set of available

sections such as universal section of BS standard or wide

ange section of AISC. The serviceability and combined

strength constraints are implemented in the design al-

gorithm as they are described in BS5950.

2. Discrete optimum design of unbraced steel frames with

partially restrained connections

The design of unbraced steel frames necessitates the

selection of steel sections for its columns and beams

from a standard steel section tables such that the framesatises the serviceability and strength requirements

specied by the code of practice while the economy is

observed in the overall or material cost of the frame.

Further more, most of the present design codes such as

AISC and BS5950 allow the designer to carry out a

simple design, rigid design and semi-rigid design de-

pending upon the assumptions made for the beam-

to-column connections. While the rigid joint assumption

implies that full slope continuity exists between the ad-

joining members, the simple joint assumption on the

other hand implies that the beams behave as simply

supported members. In reality, experimental studieshave shown that all connections exhibit semi-rigid de-

formation behavior which fall between fully rigid and

ideally pinned connections. Partially restrained connec-

tions aect the moment distribution in the beams and

columns as well as the increase of frame drift. Hence,

designing steel frames without taking into account

the eect of joint exibility may lead to uneconomical

frames.

2.1. Design problem

The discrete optimum design problem of unbracedsteel frame where the minimum weight is taken as the

objective and the constraints are implemented from

BS5950 [8] has the following form:

Minimize Wngr1

mrtrs1

ls 1a

Subjected to

dj dj1=hj 6 dju j 1; . . . ; ns 1bdj 6 dju i 1; . . . rd 1c

Fk

Agkpy Mxk

Mcxk6 1 k 1; . . . ; nc 1d

or

Fk

Agkpck mMxk

Mbk6 1 k 1; . . . ; nc 1e

Mxl6Mcxl l 1; . . . ; nb 1f

Bbfr6Bcfr r 1; . . . ; nd 1g

Dut6Dlt t nby 2; . . . ; nj 1h

Fig. 1. Connections momentrotation curves.

1594 E.S. Kameshki, M.P. Saka / Computers and Structures 79 (2001) 15931604


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where Eq. (1a) denes the weight of frame. mr is the unit

weight of steel section to be adopted for group r from

the standard steel section table. tr is the total number of

members in group r and ng is the total number of groups

in the frame. ls is the length of member s.

Eq. (1b) represents the interstorey drift limitationof the multistorey frame. dj and dj1 are the lateral de-ections of two adjacent storey levels and hj is the storey

height. ns is the total number of storeys in the frame.

BS5950 limits the horizontal deection of columns due

to unfactored imposed and wind loads to height of

column/300 in each storey of a building with more than

one storey. Eq. (1c) denes the displacement restrictions

that may be required to be included other than drift

constraints such as deections in beams. BS5950 limits

such deections under the unfactored imposed loads to

span/360, if they carry plaster or other brittle nish. rdis

the total number of such restricted displacements in theframe.

Eqs. (1d) and (1e) dene the local capacity and overall

buckling checks for beam columns. These expressions are

given in clause 4.8.3 of BS5950 which covers the design of

compression members with moments. Eq. (1d) insures

that at the points of greatest bending moment and axial

load, yielding or local buckling do not take place. Fk and

Mxk in this equation are the ultimate axial force and the

ultimate bending moment about the major axis at the

critical region of member k. Agk is the gross cross sec-

tional area and py is the design strength of the steel grade

used for member k. Mcxk is the moment capacity of the

member about the major axis. BS5950 carries out theoverall buckling check of a beam-column by using either

the simplied or more exact approach. Eq. (1e) repre-

sents the simplied approach where m is the equivalent

uniform moment factor given in Table 18 of the code.

Mbk is the buckling resistance moment capacity of mem-

ber k about its major axis computed as explained in

clause 4.3.7. pck is the compression strength of member k

which is obtained from the solution of the quadratic

PerryRobertson equation given in Appendix c.1 of

BS5950. It is apparent that computation of compression

strength of a compression member requires its eective

length. The computation of the eective length of acompression member connected to beams with semi-rigid

connections is automated and included in the algorithm

developed. nc in Eq. (1d) represents total number of

compression members in the frame.

The constraint (1f) represents the moment capac-

ity check for the laterally supported beams. Design of

members in bending is given in clause 4.2 of BS5950. It is

assumed in this study that slabs in the steel building

provide sucient lateral restraint for the beams. Mxl in

Eq. (1f) is the ultimate bending moment in member

l determined at the critical region. Mcxl is the moment

capacity of member l which is computed as explained in

clause 4.2.5 and 4.2.6 of the code.

The constraint (1g) is required to insure that the steel

section selected for the column has wider ange width

than the steel section adopted for the beam. Bbfr and Bcfrare the ange widths of the steel sections used for beam

and column respectively. This requirement is imposed at

every joint in the frame where a beam and column meet.nj is the total number of joints in the frame except

supports.

The last constraint (1h) is included to insure that the

steel sections selected for upper oor columns are not

wider in depth than that of lower oor columns. nby is

the number of bays in the frame. Dut and Dlt are the

depths of the steel section adopted for the upper and

lower oor columns.

2.1.1. Section classication

It is worthwhile mentioning that BS5950 necessitates

the determination of the classication of the cross sec-

tion of the steel sections selected for the frame members

prior to computation of their load capacity of a member

depending upon whether its cross section is plastic or

compact or semi-compact or slender.

2.1.2. Eective column-length factor

The eective length factor k for the columns in

an unbraced steel frame with semi-rigid connections is

determined from the following interaction equation [7,

24]

GHAGH

B p=k 2 366 GHA GHB

p=ktan p=k 2

where GHA and GH

B are modied relative stiness factors at

Ath and Bth ends of column and given as:

GHA

AEIL

c

A aufEIL

b

; GHB

BEIL

c

B aufEIL

b

3

where subscript b and c denote beam and column re-

spectively. auf is a coecient which represents the con-

nection condition. It is equal to 1 for rigid connections

and computed from the following expression for semi-

rigid beam end connections.

auf 1

2EIbLbK

0R 4

where

R 1

4EIbLbKA

1

4EIb

LbKB

EIb

Lb

24

KAKB5

in which KA and KB are the rotational stiness of the

semi-rigid connections at the rst and the second ends

of the beam. Ib and Lb are the moment of inertia and

E.S. Kameshki, M.P. Saka / Computers and Structures 79 (2001) 15931604 1595


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the length of the beam. K is the smaller of KA and KB.The solution of the nonlinear Eq. (2) for k results

in the eective length factor k for the column in con-

sideration.

2.2. Solution by genetic algorithm

The solution of the optimum design problem given

from Eqs. (1a)(1f) requires the selection of appropriate

steel sections from a standard list such that the weight of

the frame becomes minimum while the constraints are

satised. This turns the design problem into a discrete

programming problem. The solution techniques avail-

able in mathematical programming for obtaining the

solution of such problems are somewhat cumbersome.

On the other hand, the genetic algorithms which are

recent addition to optimization techniques are easy to

apply and provide eective solutions to discrete opti-mum design problems.

Genetic algorithms are developed by applying the

principle of survival of the ttest into a numerical search

method. They are used as a function optimizers, par-

ticularly when the variables have discrete values. They

achieve this by rst selecting an initial population where

each individual is constructed by bringing together the

total number of variables respectively in a binary or

other coded form. These individuals are called articial

chromosomes and they have a nite length string. The

binary code for each design variable represents the se-

quence number of this variable in the discrete set.

A genetic algorithm initiates a search for nding theoptimum in a discrete space by rst selecting a number

of individuals randomly and collecting them together

to constitute the initial population. It then makes use

of four operators to generate a new population. These

operators are selection, mating, crossover and mutation.

A detailed explanation of these operators is given in

Refs. [1318]. Among these, the crossover operator is

probably the one which plays an important role in the

production of the new generation. There are several

types of crossover operators such as single point, two

point, multipoint, uniform and variable to variable

crossover. It is shown that two or three point crossoverperforms much better among the multipoint crossover

techniques [17]. In the detailed study carried out in Ref.

[19] on the evaluation of crossover techniques it was

shown that direct design variable exchange produced the

best solutions in the test problems considered.

The genetic algorithm works on a population of in-

dividuals instead of single solution. The population is

obtained at the beginning of the computations by col-

lecting the individuals randomly. Those individuals in

the population who are t are then selected for mating.

This selection is carried out according to a tness cri-

teria. In order to establish a tness criterion, it is nec-

essary to transform the constrained design problem of

(1) into an unconstrained one. This is achieved by using

a penalty function. There are dierent types of penalty

functions used in conjunction with genetic algorithm

such as linear double segment, linear multiple segment

and quadratic penalty functions [15,16]. In this study the

transformation is based on the violation of normalizedconstraints as suggested in Ref. [20]. The normalized

form of the design constraints given in Eqs. (1a)(1h) are

expressed as follows:

gi di= diu 16 0 i 1; . . . ; rd 6a

gj dj dj1= hjdiu 16 0 j 1; . . . ; ns 6b

gk FkAgkpy

Mxk

Mcxk

16 0 k 1; . . . ; nc 6c

or

gk FkAgkpck

mMxk

Mbk

16 0 k 1; . . . ; nc 6d

gl Mxl=Mcxl 16 0 l 1; . . . ; nb 6e

gr Bbfr=Bcfr 16 0 r 1; . . . ; nj 6f

gt Dut=Dlt 16 0 t nby 2; . . . ; nj 6g

The unconstrained function P is then constructed

P W 12

Cmt1

mt

37

where W is the objective function given in Eq. (1a), C

is a constant to be selected depending on the prob-

lem and mt is a violation constant computed as in the

following:

if gt > 0 then mt gtif gt6 0 then mt gt

8

where t varies from 1 to m which is the total number

of constraints. The expression for tness is selected as

Ft Pmax Pmin Pt 9where Ft is the tness of the individual t, Pmax and Pminare the maximum and minimum values of the uncon-

strained function of (7) for the entire population. Pt is

the value of the same function for the individual t only.

The tness factor for each individual is then calculated

as Ft=Fav; where Fav is the mean tness of the entirepopulation. Individuals are then selected according to

their tness factor, coupled randomly. The crossover

operator is applied and o-springs are produced to ob-

tain a new population. The details of selection, crossover



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and mutation are explained in Ref. [18] and are not re-

peated here.

3. Analysis of steel frames with semi-rigid connections

A genetic algorithm moves from one generation to

another until either a certain individual dominates the

population or a predetermined number of generations is

reached. It is apparent that during the computation of

the tness factor for each individual the evaluation of

constraints is required. This in turn necessitates the

analysis of the frame in order to update its structural

response under the external loads. In the analysis of steel

frames with semi-rigid connections, the eect of con-

nection exibility is modeled by attaching rotational

springs with stiness moduli Ka and Kb to the rst and

second ends of a member as shown in Fig. 2.The nonlinear stiness matrix of member iwith semi-

rigid restraints at the ends in global coordinates has the

following form:

S

a ...

b d ...

c1 e1 f1..

.

a b c1 ... ab d e1 ..

.

b d

c2 e2 f2 ...

c2 e2 g

PTTTTTTTTTTTTTR

QUUUUUUUUUUUUUS

10

in which

a EAL

cos2 a 12EIL3

fx1 /5 sin2 a

b EAL

12EI

L3 fx1 /5

cos a sin a

d

EA

L sin2 a

12EI

L3 f

x1 /

5 cos2 a

c1 6EIL2

fx2 /2 sin a

c2 6EIL2

fx3 /2 sin a

e1 6EIL2

fx2 /2 cos a

e2 6EIL2

fx3 /2 cos a

f1 4EIL

fx4 /3

f2

2EI

L fx5

/3

g 4EIL

fx6 /4

11

where Eis the modulus of elasticity, A, I, L and a are the

area, the moment of inertia, the length and the direction

cosine of the member respectively. The eects of the

exible connections are included in the stiness matrix

by modifying the stiness terms of rigid frame member

through the use of fx1, fx2, fx3, fx4, fx5, and fx6. These

coecients are given in [22] as:

fx1

Ka

Kb

Ka

Kb=KK

fx2 Ka Kb 2=KKfx3 Kb Ka 2=KKfx4 Ka Kb 3=KKfx5 Ka Kb=KKfx6 Kb Ka 3=KK

KK KaKb 4Ka Kb 12

12

where Ka and Kb are the stiness moduli of the exible

connections at the ends of the member. The eect of

axial forces on the deformed shape of the member are

included in the stiness matrix by using stability func-

tions of/2, /3, /4, and /5. These functions are derivedFig. 2. Semi-rigid plane frame member. (a) End forces and end

displacements and (b) end rotations.



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in Ref. [21] and used in the optimum design of unbraced

rigid frames [23]. They have the following form:

/1 b cotb; /2 b2=3 3/1/3 3/2 /1=4; /4 3/2 /1=2/5 /1/2

13

where

b 0:5p qp ; q Fl=Pcr Fll2= p2EI in which Fl is the axial force in the member, and Pcr is the

Euler critical load for a pin-ended member of the same

length and stiness of the member.

3.1. Determination of Ka and Kb

The stiness moduli Ka and Kb of the exible con-nections are determined by considering nonlinear con-

nection behavior. There are several mathematical

models to describe the Mhr relationship obtained by

tting a curve to experimental data [6]. Among these the

polynomial model proposed by Frye and Morris [3] is

adopted in this study due to its easy implementation.

This model is represented by an odd power polynomial

of the form

hr c1 KM 1 c2 KM 3 c3 KM 5 14

where K is the standardization constant which dependsupon connection type and geometry; and c1; c2 and c3are the curve tting constants. The values of these

constants are given in Ref. [6] for dierent types of

connections. The rotational stiness Ka and Kb of the

springs at the ends of the member are calculated as a

tangent stiness using the nonlinear standardized func-

tion given in Eq. (14). This is simply achieved by rst

computing the exibility of the connection as dhr=dM.The stiness of the connection is then obtained as a re-

ciprocal of the exibility calculated for a certain value of

a moment, if the connection is loaded [6]. The stiness of

the connection is taken as its initial stiness, if the

connection is unloaded as shown in Fig. 3.

3.2. PD eect

Analysis of steel frames with semi-rigid connections

yields an increase in lateral displacements. This in turn

makes it necessary to consider the eect of axial forces in

the structural response of the frame. The nonlinear

stiness matrix which accounts for this eect through

the use of stability functions is shown in Eqs. (10), (11)

and (13). The algorithm utilized to account for PD

eects is given in detail in Refs. [21,23] and it has the

following steps:

1. Assume the axial forces in members to be zero ini-

tially.

2. Setup the overall stiness matrix, analyze the frame

under the external loads, obtain joint displacements

and member end forces.

3. Use the axial forces found for the members, calculate

the corresponding stability functions.

4. Repeat the steps from 2 until the dierence between

two successive sets of axial forces is smaller than a

specic tolerance.

During these iterations the determinant of the overallstiness matrix is calculated and loss of stability is

checked. If the convergence in the axial forces is ob-

tained without loss of stability, the joint displacements

and member forces obtained in this nonlinear response

are used in the computation of tness values for this

individual. It should be pointed out that in this algo-

rithm the design load is not applied incrementally in the

nonlinear analysis. Instead it is applied immediately and

iterations are carried out at this load. It should also

be noted that during the nonlinear analysis the xed

end moments change from one iteration to another due

to rotational springs attached at the end of beams. Themodied xed end moments are calculated by taking

into account the eect of exible end connection for a

frame member which is loaded as described in Ref. [2].

4. Optimum design procedure

The optimum design algorithm developed for steel

frames with semi-rigid connections and based on genetic

algorithm consists of the following steps:

1. An initial population which consists of 50 individuals

is constructed randomly.

Fig. 3. Momentrotation behavior of semi-rigid connection.



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2. For each individual decoding is carried out and steel

sections adopted for the design variables from the

standard steel section table are identied.

3. The nonlinear analysis of the frame is carried out

under the applied loads for these sections by account-

ing for nonlinear behavior of the semi-rigid connec-tions and PD eect.

4. The joint displacements and member forces obtained

from the nonlinear analysis are used to calculate the

values of normalized constraints and unconstrained

function P from Eqs. (6a)(6g) and (7) for each indi-

vidual.

5. Fitness value and tness factor are computed for each

individual and depending on their tness factor indi-

viduals are copied into mating pool.

6. Individuals are coupled randomly and reproduction

operator is applied using two point cross sites and

the value of 0.8 for probability of crossover. Twoo-springs are generated from each couple and a

new population is obtained.

7. Mutation is applied to the new population with the

probability value of 0.001.

8. The initial population is replaced with the new popu-

lation and steps 1 to 7 are repeated until the same in-

dividual dominates 80% of the new population or

preselected number of generations is reached. The t-

test individual of all the generations represents the

best solution.

In order to insure that the best individual of each

generation is not destroyed from one generation to an-

other elitist strategy is followed in the design algorithm.

At each generation, among the individuals which satisfy

all the design constraints, the one with minimum weight

is stored and compared with the similar individual of the

next generation. If the new one is heavier than the old

one, there is then a loss of good genetic material. This

Fig. 4. End plate connection without column stieners.

Fig. 5. Three-storey, two-bay steel frame.



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situation is rectied by replacing the individual having

the lowest tness of the current generation with the

previous individual. In this way the loss of good indi-

viduals during the generations is prevented.

5. Design examples

The design algorithm presented is used to design two

unbraced steel frames with semi-rigid connections. The

modulus of elasticity was 210 kN/mm2 in both examples.

It is assumed that beam-to-column connections are

made using end plate with no column stieners as shown

in Fig. 4. The curve tting and standardization constants

of Mhr polynomial relationship shown in Eq. (14) for

end plates without column stieners are given in Ref. [6]

as:

c1 1:83 103; c2 1:04 104;c3 1:24 108

and

k d2:4g t0:4p d1:5b 15

where dg and tp are shown in Fig. 4 and db is the dia-

meter of the bolt used in the connection. In the ex-

amples considered the thickness of end plate is selected

as 12 mm. The value of dg is calculated depending upon

Table 1

Optimum design of three-storey, two-bay steel frame

Group no. Member type Steel section designation

Linear frame analysis Nonlinear frame analysis

Rigid connection Semi-rigid con-nection Rigid connection Semi-rigid con-nection

1 Column W30X90 W21X50 W24X55 W21X73

2 Column W18X55 W18X35 W21X73 W21X73

3 Column W10X33 W18X35 W18X40 W6X15

4 Column W24X68 W27X84 W21X50 W24X68

5 Column W18X55 W24X55 W16X36 W18X35

6 Column W14X34 W18X46 W12X40 W18X35

7 Beam W12X26 W18X35 W18X35 W16X26

Total weight (kg) 4217.40 4230.00 4434.00 3938.10

Top storey sway (cm) (allowable 3:66 cm) 1.32 0.39 0.96 1.20Maximum interstorey drift (cm)

(allowable 1:22 cm)0.59 0.21 0.48 0.52

Maximum interaction ratio 1.00 1.00 0.73 0.84

Fig. 6. Design history of the three-storey, two-bay steel frame.



9/12

the standard steel section adopted for the beam. During

the design process the bolt diameter to be used in the

connection is computed from the design of the connec-

tion according to clause 6.3 of BS5950 for bending

moment and shear considering grade 8.8 of ordinary

bolts. With this information, the nonlinear expression(14) can easily be used to obtain the stiness of the

exible connection for any particular value of a bending

moment.

In the design examples considered, rst British stan-

dard sections given in Ref. [25] were used. The sections

for beams were selected from the universal beam sec-

tions varying from 914 419 388 UB to 254 107 28 UB and the sections for columns were adoptedfrom the set varying from 356 406 634 UC to152 152 23 UC. It was noticed that these sets ofsections were able to provide optimum solutions for

small size steel frames, but unable to provide any solu-tion for moderate or large size frames due to their limi-

ted total number in the set, particularly when semi-rigid

connections were considered for beam-to-column con-

nections. This diculty was overcome by replacing

British standard steel sections with that of American

wide ange sections [7], which are used for both beams

and columns. This has increased the total number of

available sections from 96 to 512 providing much greater

exibility for the algorithm to reach even a lighter frame

than the case of using British sections. The steel grade of

A36 is considered for wide ange sections.

5.1. Three-storey, two-bay frame

A three-storey, two-bay frame is designed with rigid

and semi-rigid connections considering linear and non-

linear (PD eect) frame behavior. Fig. 5 shows the

frame conguration, dimensions, loading, numbering of

joints and grouping of members. The frame is taken

from [26] where it was used to demonstrate the eect of

semi-rigid connections in the analysis of steel frames.

The allowable interstorey drift was 1.219 cm and al-

lowable sway of the top storey was 3.65 cm as specied

by the code.The optimum results after 400 generations are pre-

sented in Table 1. For each case, 10 dierent designs

were performed and those reported in the table are the

lightest among 10 runs and they are obtained for the

seed value equal to six. In the case where the PD eect

is not considered in the frame analysis, a linear modeling

was used to represent the semi-rigid connection. In this

modeling, the initial stiness of the connection is utilized

throughout the analysis. The initial stiness of the beam-

to-column connection adopted in this study is close to

the rigid connection. This is why when PD eect is not

considered, minimum weights obtained for both rigid

and semi-rigid frames turn out to be almost the same. It

is apparent from Table 1 that in both cases ultimate

strength constraints but not the drift constraints were

dominant in the design problem. While the maximum

interaction ratio was equal to 1.0 in rigid and semi-rigid

Fig. 7. Ten-storey, one-bay steel frame.



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optimum frame, the drift values were below their al-

lowable limits.

The design history of generations for the optimum

nonlinear frame with semi-rigid connections given in

Table 1 is shown in Fig. 6. It is apparent that after 120

generations the minimum weight almost remains thesame. This veries that selection of the terminal genera-

tion number of 400 is reasonable.

In the case where the PD eect is considered in the

frame analysis, the semi-rigid frame is 11% lighter than

the rigid frame. In this case neither the ultimate strength

nor the drift constraints are active in the design problem.

Instead the size limitations govern the selection of sec-

tions for the frame members. However, Table 1 shows

that semi-rigid optimum frame has 25% larger drift.

Another interesting result obtained is that conside-

ration ofPD eect in the design of rigid frame generates

5% heavier frame while it causes 7% lighter frame insemi-rigid modeling. It is clear that in the rigid frame

consideration of PD eect yields heavier frame. How-

ever the same was not observed for the semi-rigid frame.

This is due to the fact that in the optimum frame it was

size constraint, which govern the design not strength or

drift limitations. It was noticed that ve linear analyses

were required to obtain the nonlinear response of the

frame in the case of exible connection modeling.

5.2. Ten-storey, one-bay frame

The ten-storey, one-bay frame of Fig. 7 is also de-signed using the algorithm presented. The frame con-

guration dimensions, loading, joint numbering and

member grouping is shown in the gure. This frame is

designed with and without considering PD eect to-

gether with rigid and semi-rigid connection modeling.

The allowable interstorey drift was 1.22 cm while the top

storey sway was limited to 12.5 cm.The optimum frames obtained after 400 generations

for each case are shown in Table 2. The variation of

weight for nonlinear frame with semi-rigid connections

against number of generations during the optimum de-

sign process is shown in Fig. 8. Each case designed 10

times, each time using dierent seed value and the

lightest among these obtained for the seed value of 4 is

reported in the table.

In the case where PD eect is not considered semi-

rigid connection modeling results in 16% lighter frame.

In both rigid and semi-rigid case ultimate strength

constraints are dominant in the design. Intermediatedrift and top storey sway of the frame are well below

their upper limits. When PD eect is considered, semi-

rigid frame becomes 15% lighter than the rigid frame.

Once again, it is the ultimate strength constraint, which

decides the sections for frame members. While these

constraints are at their upper limits of 1 for number of

members in the frame, both drift constraints were well

below the allowable limits. It is noticed that in the case

of rigid frame, the nonlinear response is obtained after

two to three linear analysis while in the semi-rigid frame

7 to 8 linear analysis are required to reach the nonlinear

behavior of the frame. It is also interesting to notice that

consideration of PD eect in the frame analysis yields16% and 15% lighter frames in both cases.

Table 2

Optimum design of ten-storey, one-bay steel frame

Group no. Member type Steel section designation

Linear frame analysis Nonlinear frame analysis

Rigid connection Semi-rigid con-

nection

Rigid connection Semi-rigid con-

nection

1 Column W40X167 W36X182 W40X192 W36X1822 Column W36X182 W33X118 W36X182 W33X118

3 Column W36X182 W30X108 W36X280 W33X118

4 Column W30X108 W27X84 W36X182 W27X84

5 Column W21X166 W21X111 W24X104 W14X82

6 Beam W30X90 W24X68 W24X68 W33X118

7 Beam W24X68 W27X84 W30X90 W30X108

8 Beam W21X73 W27X84 W18X86 W21X93

9 Beam W21X83 W12X45 W18X55 W21X50

Total weight (kg) 28607.00 23971.00 31788.00 26963.00

Top storey sway (cm) (allowable 12:5 cm) 0.77 0.36 0.63 1.03Maximum interstorey drift (cm)

(allowable

1:22 cm)

0.24 0.11 0.21 0.44

Maximum interaction ratio 0.97 0.96 0.91 0.91



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6. Summary and conclusions

A genetic algorithm based optimum design method is

presented for nonlinear steel frames with semi-rigid

connections. Design examples are included to demon-

strate the eect of connection exibility and geometric

nonlinearity in the design of steel frames.

It is noticed from the design examples that semi-rigid

connection modeling produces lighter frames. Connec-

tion exibility aects the distribution of forces in the

frame and causes increase in the drift of the frame. This

in turn necessitates the consideration of PD eect in the

frame analysis. It required ve to eight iterations in the

design examples considered to obtain the nonlinear re-

sponse of frame which clearly indicates the signicance

of geometric nonlinearity in the design of semi-rigid steel

frames. It is also noticed that consideration ofPD eect

yields a heavier frame in the case of semi-rigid as well as

rigid frame.

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