an improved method for evolutionary structural optimisation against buckling

An improved method for evolutionary structural optimisationagainst buckling

J.H. Rong a, Y.M. Xie b,*, X.Y. Yang b

a Aircraft Structure Strength Research Institute, AVIC, P.O. Box 86, Xi'an 710065, People's Republic of Chinab Faculty of Engineering and Science, School of the Built Environment, Victoria University of Technology, P.O. Box 14428, Melbourne

City MC, VIC. 8001, Australia

Received 7 July 1999; accepted 1 May 2000

Abstract

In this paper, an improved method for evolutionary structural optimisation against buckling is proposed for

maximising the critical buckling load of a structure of constant weight. First, based on the formulations of derivatives

for eigenvalues, the sensitivity numbers of the ®rst eigenvalue or the ®rst multiple eigenvalues (for closely spaced and

repeated eigenvalues) are derived by performing a variation operation. In order to e�ectively increase the buckling load

factor, a set of optimum criteria for closely-spaced eigenvalues and repeated eigenvalues are established, based on the

sensitivity numbers of the ®rst multiple eigenvalues. Several examples are provided to demonstrate the validity and

e�ectiveness of the proposed method. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Recently, a simple new approach called evolutionary

structural optimisation (ESO) to structure optimisation

has been proposed by Xie and Steven [1,2]. It is based on

the concept of slowly removing ine�cient materials from

the structure or gradually shifting materials from the

strongest part of the structure to the weakest part until

the structure evolves to a desired optimum. Compared

to other structural optimisation methods, such as the

homogenisation method [3,4] and density function

method [5], the ESO method is attractive due to its

simplicity and e�ectiveness. In recent years, the ESO

method has been demonstrated to be capable of solving

many problems of size, shape and topology optimum

designs for static and dynamic problems [2,6,7]. At the

same time, Refs. [8±10] applied the ESO method to

optimum designs against buckling. Optimum design

against buckling can be ®nding the minimum weight

design of a structure that satis®es the prescribed buck-

ling load constraint. Alternatively, it can be maximising

the critical buckling load of the structure while keeping

its weight or volume constant. For the convenience of

comparing the e�ciency of di�erent designs, the latter

approach is generally used.

Despite the extensive research on evolutionary

structural optimisation against buckling, problems arise

when there exist repeated eigenvalues or closely-spaced

eigenvalues. First, the eigenvectors associated with the

repeated eigenvalues are not unique. Instead, they can

be any linear combination of the eigenvectors associated

with the same repeated eigenvalue. Therefore, the ei-

genvalue sensitivities for the repeated eigenvalues cannot

be determined uniquely using the formulas developed by

Fox and Kapoor [11]. And even if the derivatives of the

repeated eigenvalues corresponding to di�erent modes

are determined, these derivatives must be considered

during the optimisation process in order to increase the

buckling load factor. Second, when the ®rst eigenvalue

becomes close to the subsequent eigenvalues, there will

be serious interference between the ®rst and the subse-

quent modes. If only the ®rst eigenvalue is considered

Computers and Structures 79 (2001) 253±263

www.elsevier.com/locate/compstruc

* Corresponding author. Tel.: +61-3-9688-4787; fax: +61-3-

9688-4139.

E-mail address: [email protected] (Y.M. Xie).

0045-7949/01/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S0 04 5 -7 94 9 (00 )0 0 14 5 -0

and the e�ect of the subsequent modes is ignored, the

®rst two or multiple buckling modes may swap with

each other as a result of structural modi®cations dur-

ing the iterations thus good results cannot obtained.

In the case of closely spaced eigenvalues, Refs. [2,8]

adopted a simple optimum criterion that makes use of

an arithmetic mean of all closely-spaced eigenvalues to

calculate the sensitivity numbers (here, this method is

called the mean method). While the mean method can

generally reach convergent optimum results, it restricts

the range of optimum domains thus it cannot be ensured

that the buckling load factor is increased at each itera-

tive step. In this paper, an improved method for evolu-

tionary structural optimisation against buckling is

proposed for maximising the critical buckling load of a

structure of constant weight. First, based on the for-

mulations of derivatives for eigenvalues, the sensitivity

numbers of the ®rst eigenvalue or the ®rst multiple ei-

genvalues in the case of closely spaced and repeated

eigenvalues are derived by performing a variation op-

eration. Then, in order to e�ectively raise the buckling

load factor, a set of optimum criteria for closely-spaced

eigenvalues and repeated eigenvalues are established,

based on the sensitivity numbers of the ®rst multiple

eigenvalues. Three examples are provided to demon-

strate the validity and e�ectiveness of the proposed

method.

2. The problem statement

The linear buckling behaviour of a structure is gov-

erned by the following general eigenvalue problem:

��K� � ki�Kg�� uif g � 0f g; �1�

where �K� is the global sti�ness matrix; �Kg�, the global

stress matrix or geometric sti�ness matrix; ki, the ith

eigenvalue and uif g, the corresponding eigenvector. The

eigenvalues in Eq. (1) are used to scale the applied

loading to give buckling loads. The most critical buck-

ling load is equal to the loading multiplied by the ®rst

eigenvalue kl, and kl is referred to as the buckling load

factor.

The objective of buckling optimisation considered in

this paper is to increase the fundamental eigenvalue kl so

that the buckling load is maximised, by changing the

cross-sectional areas of the structural elements.

3. The basic formulations of eigenderivatives

If Eq. (1) has distinct eigenvalues, the eigenvalue

sensitivity was derived as follows [10,11]:

oki

od� ÿ

uif gT o�K�od � ki

o�Kg�od

� �uif g

uif gT�Kg� uif g: �2�

In the case of repeated eigenvalues, Eq. (2) in gen-

eral does not produce a unique solution due to the

non-unique nature of the eigenvectors associated with

repeated eigenvalues. To overcome this problem, the

following eigensystem is formed and solved [12]:

U� �T o�K�od

�� ki

o�Kg�od

�U� � � oki

odI� ��

aif g � 0f g; �3�

where U� � consists of the original orthogonal eigenvec-

tors that are associated with the same repeated eigen-

value ki, and I� � is the identity matrix with a dimension

corresponding to the multiplicity (r) of the repeated ei-

genvalues.

Eq. (3) is apparently an eigensystem with the re-

peated eigenvalue sensitivity oki=od being its eigenvalue

and aif g the corresponding eigenvector. If the eigen-

values of this smaller eigensystem are distinct, the

unique eigenvector aif g can be used to determine the

unique eigenvectors that is associated with the repeated

eigenvalues by the following linear combination of ei-

genvectors:

f�uig � �U� ai

n o; �4�

where f�uig stands for the unique mode shape that cor-

responds to the repeated eigenvalue ki.

After determining the unique eigenvectors for the

repeated eigenvalues, Eq. (2) can be used again to cal-

culate the eigenvalue sensitivities for those repeated ei-

genvalues, although the solutions have already been

found by solving the eigensystem of Eq. (3).

4. Sensitivity numbers and optimum criteria for bucking

load

The sensitivity analysis is used to identify the best

locations for structural modi®cations. Suppose there is a

small change in the cross-sectional area of the eth ele-

ment and assume that the eigenvector uif g is approx-

imately the same before and after such a small change

[8±10], based on the basic formulations of eigenderiva-

tives, the sensitivity numbers of the eigenvalues due to

such a change can be derived by performing a variation

operation. Generally speaking, there are four cases of

eigenderivatives in buckling optimisation, namely, dis-

tinct eigenvalues, closely-spaced eigenvalues, repeated

eigenvalues and combinations of closely spaced and re-

peated eigenvalues. According to these four cases, a set

of criteria determining elements whose cross-sectional

areas are increased and those whose cross-sectional ar-

eas are reduced will be established. The interference

254 J.H. Rong et al. / Computers and Structures 79 (2001) 253±263

between the ®rst and the subsequent eigenvalues or be-

tween repeated eigenvalues or between all these eigen-

values including closely spaced and repeated eigenvalues

are re¯ected in those criteria. We start with the simplest

case of distinct eigenvalues and establish the sensitivity

number calculation and optimum criterion.

4.1. Sensitivity numbers and optimum criterion for distinct

and non-closely-spaced eigenvalues

The equation for distinct eigenvalues derivative is as

follows [2,8]:

Dki � ÿuif gT��DK� � ki�DKg�� uif g

uif gT�Kg� uif g: �5�

The change in the global sti�ness matrix �K� is equal to

the change in the element sti�ness matrix of the eth el-

ement, which can be easily calculated. However, since

�Kg� depends on the current stress distribution in the

structure and the cross-sectional change in the eth ele-

ment a�ects the stress in its surrounding elements, one

cannot assume that �Kg� is equal to the change in the

element stress matrix of the eth element only. The cal-

culation of �Kg� is generally much involved. Fortunately,

�Kg� is equal to zero if the axial or membrane stress

resultant remains constant before and after the cross-

sectional change in the elements. Such is the situation of

all statically determinate structures as the cross-sectional

changes do not a�ect the axial forces in structural

members. For a statically indeterminate structure, �Kg�can be negligible if at each iteration step the cross-

sectional modi®cations are so small that they do not

cause signi®cant changes in the axial or membrane stress

resultants. When �Kg� is ignored, Eq. (5) is reduced to

Dki � ÿ uif gT�DK� uif guif gT�Kg� uif g

� ÿ uei

� T�DKe� uei

� uif gT�Kg� uif g

; �6�

where uei

� is the eigenvector of the eth element and �Ke�

is the sti�ness matrix of the eth element. If the eigen-

vector is normalised with respect to �Kg�, Eq. (6) is fur-

ther simpli®ed to

Dki � ÿ uei

� T�DKe� uei

� : �7�

From the above equation, we de®ne the sensitivity

number for buckling load as follows:

iae � ÿ uei

� TDKe� � ue

i

� : �8�

The calculation of the above sensitivity number only

involves small matrices of individual elements. This

sensitivity number is used to measure the e�ect of

changing the cross-sectional area of the eth element on

the buckling load factor. If the structure is statically

determinate and the cross-sectional change at each it-

eration step is small, the sensitivity number gives a very

accurate estimation of the change in the buckling load

factor. Even for a statically indeterminate frame this

sensitivity is reasonably accurate if the cross-sectional

variation in the frame only results in slight changes in

the axial stress resultants. For a plate structure, this

sensitivity number only works in the case of gradually

varying the thickness of plate elements. If an element is

removed at once which may cause signi®cant changes in

the membrane or axial stress resultants in its surround-

ing elements, Eqs. (6) and (7) will be invalid and the

sensitivity number de®ned in Eq. (8) will be incorrect

too. Therefore, Eq. (8) should not be used for buckling

optimisation involving element removal. In this paper,

the buckling optimisation is carried out by changing the

cross-sectional area only.

In the case of an increase in the cross-sectional area

A, we have

DKe� � � DKe� �� Ke�A� � DA�� ÿ Ke�A�� ; �9�

and in the case of a reduction

DKe� � � DKe� �ÿ � Ke�A� ÿ DA�� ÿ Ke�A�� : �10�

Hence, to estimate the e�ect of cross-sectional changes

on the buckling load factor, two sensitivity numbers

need to be calculated for each element, one for area

increase

1a�e � ÿ ue1

� TDKe� �� ue

1

� ; �11�

and the other for area reduction

1aÿe � ÿ ue1

� TDKe� �ÿ ue

1

� : �12�

From the above de®nition of the sensitivity number, it is

clear that to raise the buckling load factor it will be most

e�ective to increase the cross-sectional areas of elements

with the highest 1a�e values and reduce those with the

highest 1aÿe values.

4.2. Sensitivity numbers and optimum criterion for distinct

and closely-spaced eigenvalues

4.2.1. Two closely-spaced eigenvalue case

During the process of buckling optimisation, it is

often observed that while the ®rst eigenvalue is in-

creasing, the subsequent eigenvalues are decreasing and

gradually the ®rst two or more eigenvalues become very

close to each other. This will cause serious interference

between the ®rst and the subsequent buckling modes.

Therefore the e�ect of all the participating eigenvectors

needed to be included. If, for example, the distance be-

tween k1 and k2 is within a certain limit, say 5%, and

the distance between k1 and k3 is greater than that

limit, we assume that the structure has become bi-modal.

For a bi-modal structure, the ®rst two buckling modes

may swap with each other as a result of structural

J.H. Rong et al. / Computers and Structures 79 (2001) 253±263 255

modi®cation during the iterations if only the sensitivity

numbers of the ®rst eigenvalue are considered. There is

no point in trying to increase k1 only to see k2 drop its

value below previous k1 in the next step. In Ref. [2], this

case is dealt with by a simple strategy called Ômean

methodÕ, i.e. to increase the average of k1 and k2, instead

of increasing a single eigenvalue k1 only. However, it is

not ensured in the mean method that k2 is kept above

the previous k1 in the next step thus restricting the range

of optimum domains. Therefore, it is necessary to com-

prehensively consider the sensitivity numbers of the ®rst

two eigenvalues to determine the element modi®cation.

First, in order to clearly describe the idea of the

proposed method and to unify the formula for various

cases of two and multiple closely spaced eigenvalues,

some mathematics notations are de®ned as follows:

S � 1; 2; . . . ;mf g;N�1 � S; Nÿ1 � S;

�13�

N�2 � ej 2a�e�

> ÿ e e 2 Sg;Nÿ2 � ej 2aÿe

�> ÿ e e 2 S

;

�14�

where m is the number of structural ®nite elements to be

modi®ed, and 2a�e and 2aÿe are the sensitivity numbers of

the second eigenvalue of eth element for area increasing

and decreasing, respectively. N�2 and Nÿ2 are the sets of

the structural elements that are de®ned by 2a�e > ÿe, and2aÿe > ÿe, respectively. e is a small positive real number

which can be changed in each optimum step. An initial

value of e at current step is de®ned as

e0 � k2 ÿ k1

N; �15�

where N is the total number of elements in which ele-

mental sectional-areas are changed at current step. It is

set up in advance based on the requirement of keeping

structural weight constant. Then de®ne

N� � N�1 \ N�2 Nÿ � Nÿ1 \ Nÿ2 ; �16�where \ represents the intersect set operator of two sets.

After determining N� and Nÿ from Eqs. (11)±(16),

we check up the sum �NR� of the numbers of elements in

N� and Nÿ. If NR < N , then de®ne

ek � bekÿ1 �k � 1; 2; 3; . . .�; �17�where b > 1. The above procedures using Eqs. (13)±(17)

will be repeated until NR is equal to or greater than N.

Now, it is clear that to increase k1 and keep k2 above

k1 in the next step, namely to raise the buckling load

factor, it will be most e�ective to increase the cross-

sectional areas of elements that belong to N� and are of

the highest 1a�e values, and reduce the cross-sectional

areas of elements that belong to Nÿ and are of the


4.2.2. Multiple closely-spaced eigenvalue case

Similar to the bi-modal case, a structure become

multi-modal when the distance between the rth eigen-

value kr and the ®rst eigenvalue k1 is within a certain

limit, say 5%, and the distance between k�r�1� and k1 is

greater than that limit. In such a case, N�2 , Nÿ2 , N�3 ,

Nÿ3 ,. . .,N�r , and Nÿr are constructed in the following form

for the second, the third,. . ., the rth eigenvalue, respec-

tively.

N�j � ej ja�e�

> ÿ ej e 2 S

Nÿj � ej jaÿe�

> ÿ ej e 2 S �j � 2; 3; . . . ; r�: �18�

Similarly, an initial value of ej is given as follows:

e0j �

kj ÿ k1

N�j � 2; 3; . . . ; r�: �19�

Then it follows:

N� � N�1 \ N�2 \ � � � \ N�r ;

Nÿ � Nÿ1 \ Nÿ2 \ � � � \ Nÿr :�20�

After determining N� and Nÿ, from Eqs. (18)±(20),

the sum of the numbers of elements in N� and Nÿ are

checked up and similar equation to Eq. (17) can be de-

®ned where applicable:

ekj � bekÿ1

j �k � 1; 2; 3; . . .�: �21�

After N� and Nÿ are determined, it is clear that to

increase k1 and keep k2; k3; . . . ; kr above the previous k1,

it will be most e�ective to increase the cross-sectional

areas of elements that belong to N� and are of the

highest 1a�e values, and reduce the cross-sectional areas

of elements that belong to Nÿ and are of the highest 1aÿevalues.

4.3. Sensitivity numbers and optimum criterion for

repeated eigenvalues

It is de®ned that

U� � � u1f g; u2f g; . . . ; urf g� �:

When the cross-sectional area of an element is changed

in the current structure, assuming that �DKg� is ignored,

the following equation can be derived from Eq. (3):

A�� oki

odI� ��

aif g � 0f g �22�

in which

A �k11 k12 . . . k1r

k21 k22 . . . k2r

..

. ... ..

. ...

kr1 kr2 . . . krr

2666437775; �23�


kij � uei

� TDKe� � ue

j

n o�i � 1; 2; . . . ; r; j � 1; 2; . . . ; r�;

�24�where I� � is a r � r identity matrix. To solve Eq. (22), the

matrix aif g�i � 1; 2; . . . ; r� can be obtained, then the

unique element eigenvector �uei

� can be calculated from

the following equation:

�uei

n o� ue

1

� ; ue

2

� ; . . . ; ue

r

� � �aif g �i � 1; 2; . . . ; r�:

�25�Substituting �ue

i

� to Eqs. (11) and (12), the sensitivity

numbers (1a�e ;1aÿe ;

2a�e ;2aÿe ; . . . ; ra�e and raÿe ) for the re-

peated eigenvalues can be obtained easily.

De®ne

0N�1 � S; 0 Nÿ1 � S; �26�

0N�j � ej ja�e�

P 0 e 2 S;

0Nÿj � ej jaÿe�

P 0 e 2 S �j � 2; 3; . . . ; r�; �27�

0N� � 0N�1 \ 0N�2 \ � � � \ 0N�r ;0Nÿ � 0Nÿ1 \ 0Nÿ2 \ � � � \ 0Nÿr ;

�28�

0N � 0N� [ 0Nÿ; �29�

eN�j � ej ja�e�

> ÿ e e 2 S;

eNÿj � ej jaÿe�

> ÿ e e 2 S �j � 2; 3; . . . ; r�; �30�

eN� � 0N�1 \ eN�2 \ � � � \ eN�r ;eNÿ � 0Nÿ1 \ eNÿ2 \ � � � \ eNÿr ;

�31�

eN � eN� [ eNÿ; �32�where [ represents the sum set operator of two sets.

In order to have the ®rst eigenvalue increased,

the sectional areas of elements whose values of 2a�e ;3a�e ; . . . ; and ra�e ; or 2aÿe ;

3aÿe ; . . . ; and raÿe all are

greater than zero, and are of the highest 1a�e values or

the highest 1aÿe values, should be changed. In the general

case, it cannot be ensured that 0N� and 0Nÿ all are not

an empty set. When one of 0N� and 0Nÿ is an empty set,

we can have eN� and eNÿ to be a non-empty set by

setting up a small positive parameter e (referring to

Section 4.2), then change the sectional areas of elements

belonging to 0N or eN , and being of the highest 1a�evalues, or the highest 1aÿe values. When eN instead of 0Nis used in the optimisation process, the reduction in the

®rst eigenvalue can be the smallest even if an increase

cannot be ensured. Moreover, for many engineering

structural optimisations, the occurrence of repeated ei-

genvalues is much fewer compared to the total iterations

and thus a temporary eigenvalue decrease at one or two

steps will not a�ect the solution process as a whole.

4.4. Optimum criterion for the combination of closely

spaced and repeated eigenvalues

When the repeated eigenvalues are not the ®rst ei-

genvalue but are very close to the ®rst eigenvalue, we can

refer to Sections 4.2 and 4.3 and their sensitivity num-

bers are calculated by using Eqs. (11), (12) and (22)±(25).

For the determination of modi®ed elements, each re-

peated eigenvalue is respectively treated as a closely

spaced eigenvalue in relation to the ®rst eigenvalue. In

doing so, it may happen that one or two eigenvalues

become repeated with respect to the ®rst eigenvalue.

This case is solved by combining the optimum criteria

presented in Sections 4.2 and 4.3. ~N� and ~Nÿ can be

de®ned and obtained in such a combination. They serve

as the same purpose as N� and Nÿ in the case of closely

spaced eigenvalues, and 0N� and 0Nÿ (or eN� and eNÿ)

in the case of repeated eigenvalues.

5. Evolutionary procedure for buckling optimisation

Similar to Ref. [2], the evolution procedure for

buckling optimisation of columns, frames and plates is

outlined as follows:

Step 1: Discretise the structure using a ®ne ®nite-

element mesh.

Step 2: Solve the eigenvalue problem (1).

Step 3: Calculate the sensitivity numbers according

to each case:

Case 1: distinct eigenvalues: calculate 1a�e and 1aÿefor each element.

Case 2: distinct and closely-spaced eigenvalues:

calculate ia�e �i � 1; 2; . . . ; r� and iaÿe �i � 1; 2;. . . ; r� for each element.

Case 3: repeated eigenvalues: calculate ia�e �i �1; 2; . . . ; r� and iaÿe �i � 1; 2; . . . ; r� for each ele-

ment.

Case 4: combination of closely-spaced and re-

peated eigenvalues: calculate the sensitivity num-

ber of closely spaced and repeated eigenvalues for

each element at the same time.

Step 4: According to each case, determine the ele-

ments whose cross-sectional areas are to be increased,

and those whose cross-sectional areas are to be re-

duced. The requirements on structural symmetry

and loading conditions are considered in the element

modi®cation.

Case 1: Increase the cross-sectional areas of the el-

ements that have the highest 1a�e values, and de-

crease the cross-sectional areas of the same

number of elements that have the highest 1aÿe val-

ues.



ements that belong to N� (Eqs. (16) and (20)) and

have the highest 1a�e values, and decrease the

cross-sectional areas of the same number of ele-

ments that belong to Nÿ and have the highest1aÿe values.


ements that belong to 0N� or eN� (Eq. (28) or Eq.

(31)) and have the highest 1a�e values, and decrease

the cross-sectional areas of the same number of el-

ements that belong to 0Nÿ or eNÿ and have the



ements that belong to ~N� and have the highest 1a�evalues, and decrease the cross-sectional areas of

the same number of elements that belong to ~Nÿ

and have the highest 1aÿe values.

Step 5: Repeat steps 2 to 4 until the increase in the

®rst eigenvalue becomes very slight for a consecutive,

say four or ®ve iterations.

During the evolution, the cross-sectional area is al-

lowed to vary in small steps. For beam elements of

rectangular cross-sections, either the breadth or the

depth can be changed. For plate elements, the thickness

can be changed. The change in the element sti�ness

matrix �DKe� can be easily calculated for the above ®nite-

element types. Meanwhile, information on eigenvalues

and eigenvectors required for the sensitivity number

calculation is readily available from the ®nite element

analysis. In the above procedures, the number of ele-

ments subjected to cross-sectional changes and the

step size of the change at each iteration need to be pre-

scribed.

6. Examples

In order to demonstrate the validity and e�ectiveness

of the proposed method, two simple frames and a box

structure displaying closely spaced or repeated eigen-

values during buckling optimisation are considered. In

these three examples, the initial design is of uniform

cross-section, and the YoungÕs modulus E � 200 GPa,

PoissonÕs ratio m � 0:3 and mass density q � 2700 kg/m3

are assumed.

6.1. Three-member portal frame

A three-member pin based frame, which was analy-

sed in Refs. [2,13] is considered for closely-spaced

buckling eigenvalues. The frame structural model and

the loading are shown in Fig. 1. All the members are of

circular cross-sections and of equal length of 1 m. Initial

uniform radius square r2 is 20 mm2, and it is allowed to

vary to the maximum 40 mm2 and to the minimum

5 mm2 in steps of 1 mm2. Each member is divided into

10 elements of equal length. A modifying ratio 24% and

closely-spaced eigenvalue parameter ec � 4% are used.

For the frame with a uniform cross-section, the ®rst

buckling mode is anti-symmetric with sway and the

second buckling mode is symmetric with closely-spaced

eigenvalues from the outset.

The optimum radius square r2 for each element is

shown in the column chart Fig. 2. The buckling load is

1.2514 times that of the uniform frame, in comparison to

1.125 times by only considering the single mode, and

1.2474 by using the mean method [8]. The evolutionary

histories of the ®rst two eigenvalues using the mean

method and the proposed method are given in Figs. 3

and 4, respectively. Fig. 5 compares the evolutionary

histories of the ®rst eigenvalue using these two methods.

It is seen that although the optimum factors obtained by

the two methods di�er only by 0.32%, fewer iterations

are involved in the proposed method.

The above problem is analysed with di�erent values

of closely-spaced model parameter ec � 2%, 4% and

4.5%, as well as di�erent modifying ratios c � 10%, 15%

and 24%. While the iteration histories of eigenvalues

vary slightly in intermediate designs, no di�erence is

observed in the ®nal design.

6.2. Three-member space frame example

A space frame with three beams pinned at the base

and clamped at the apex is considered for the optimi-

sation of structures with repeated eigenvalues against

buckling. The frame model and the loading condition

are shown in Fig 6. All the members are of circular

cross-sections and of equal length of 1 m. Initial uni-

Fig. 1. Structural model and loading case of the three-member

portal frame.


form r2 is 20 mm2, and it is allowed to vary to the

maximum 40 mm2 and to the minimum 5 mm2 in steps

of 1 mm2. Each member is divided into 10 elements of

equal length. c � 24% and ec � 4% are used. This is a

triple symmetric structure and the ®rst three eigenvalues

coincide for the uniform design and remain coincided

Fig. 2. Ratio of radius squares of beam cross-sections at optimum point to corresponding initial uniform values.

Fig. 3. Optimisation histories of the ®rst two eigenvalues for the three-member frame by using the mean method.

Fig. 4. Optimisation histories of the ®rst two eigenvalues for the three-member frame by using the proposed method.


throughout the optimisation process. The optimum

beam sectional parameters obtained by using the pro-

posed method are given in Fig. 7. The ratios of the ®nal

to initial uniform radius of the cross-section are dis-

played in this ®gure for one member, as it is identical for

all members. The optimum buckling load is 1.275 times

that of the uniform frame and it is achieved after 10

iterations. The iteration histories of the ®rst three ei-

genvalues (namely three repeated eigenvalues) are given

in Fig. 8.

6.3. Box frame

In the box frame shown in Fig. 9, all the members are

of rectangular cross-sections with a constant breadth

b � 40 mm and an initial uniform depth d � 40 mm.

The horizontal members at the top and bottom are di-

vided into 12 elements of equal length, and diagonal and

vertical members in the middle are divided into three

elements of equal length. Numbering of beam elements

is shown in the model.

Buckling optimisation of this frame was considered

to be one of the most di�cult examples in the literature.

It was studied in Ref. [10] using the mean method of

ESO. In applying the method proposed in this paper, the

initial value of the design variable, beam depth d is al-

lowed to change without upper limit and to the mini-

mum depth of 1 mm in steps of 1 mm. c � 24% and

ec � 4% are used. The optimum depth ratio is shown

Fig. 10. The evolutionary histories of the ®rst ®ve ei-

genvalues using the proposed method and the mean

method are shown in Figs. 11 and 12, respectively. It is

observed that while all cases of closely spaced and re-

peated eigenvalues occur during optimisation process,

the ®rst eigenvalue is kept increasing at non-repeated

eigenvalue points by using the proposed method. It also

takes fewer iterations than the mean method. Fig. 13

shows the results of the ®rst eigenvalues in these two

methods. The buckling load factor is 1.9214, compared

to 1.8678 with a di�erence of 2.9%.

It is seen that members of the optimum design have

segmented cross sections which are not manufacturally

appealing. One solution to this can be some smoothing

techniques using interpolation functions so that the

structural member displays a smoother outer shape. This

point for ESO method is presented in detail in Ref. [2].

Fig. 5. Optimisation histories of the ®rst eigenvalue for the three-member frame by using the proposed method and the mean method.

Fig. 6. Optimum model and loading case of the three-member

space frame.


Fig. 9. Finite-element model and loading case of the box frame (allowable minimum depth � 1 mm).

Fig. 8. Optimisation histories of the ®rst eigenvalue for the three-member space frame by using the proposed method.

Fig. 7. Optimum result of section radius squares of beam elements for the three-member space frame.


Fig. 10. Optimum result of beam section depths for the box frame.

Fig. 11. Optimisation method histories of the ®rst ®ve eigenvalues for the box frame by using the mean method.

Fig. 12. Optimisation histories of the ®rst ®ve eigenvalues for the box frame by using the proposed method.


7. Conclusion

An improved approach to optimising the structures

against buckling is proposed and illustrated with ex-

amples. The results demonstrate that the proposed

method is valid and e�ective and is suitable for various

complex cases of practical structures. The method can

obtain better optimum design for structures against

buckling than the mean method. It can be readily im-

plemented in any of the existing ®nite-element codes.

References

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Fig. 13. Optimisation histories of the ®rst eigenvalue for the box frame by using the mean method and the proposed method.


an improved method for evolutionary structural optimisation against buckling

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