Design Optimun Beam

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Optimum design of thin-walled composite beams for flexural–torsional

buckling problem

Xuan-Hoang Nguyen a,b, Nam-Il Kim a, Jaehong Lee a,⇑

a Department of Architectural Engineering, Sejong University, Seoul, South Koreab Center for Interdisciplinary Research in Technology, Ho Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam

a r t i c l e i n f o

Article history:

Available online 2 July 2015

Keywords:

Thin-walled beams

Laminated composites

Flexural–torsional buckling

Optimum design

Genetic algorithm

a b s t r a c t

The objective of this research is to present formulation and solution methodology for optimum design of

thin-walled composite beams. The geometric parameters and the fiber orientation of beams are treated

as design variables simultaneously. The objective function of optimization problem is to maximize the

critical flexural–torsional buckling loads of axially loaded beams which are calculated by a

displacement-based one-dimensional finite element model. The analysis of beam is based on the classical

laminated beam theory and applied for arbitrary laminate stacking sequence configuration. A micro

genetic algorithm (micro-GA) is employed as a tool for obtaining optimal solutions. It offers faster con-

vergence to the optimal results with smaller number of populations than the conventional GA. Several

types of lay-up schemes as well as different beam lengths and boundary conditions are investigated in

optimization problems of I-section composite beams. Obtained numerical results show more sensitivity

of geometric parameters on the critical flexural–torsional buckling loads than that of fiber angle.

2015 Elsevier Ltd. All rights reserved.

1. Introduction

Composite materials have been increasingly used in a variety of

structural fields such as architectural, civil, mechanical, and aero-

nautical engineering applications over the past few decades. The

most apparent advantages of composite materials in comparison

to other conventional materials are their high strength-to-weight

and stiffness-to-weight ratios. Furthermore, the ability to adapt

to design requirements of strength and stiffness is also cited when

it comes to composite materials. Another major advantage of com-

posites is tailorability which enables the optimization processes to

be applied in not only structural shape but materials itself as well.

Thin-walled beams are widely used in various type of structural

components due to its high axial and flexural stiffnesses with a low

weight of material. However, these thin-walled beams might be

subjected to an axial force when used in above applications and

are very susceptible to flexural–torsional buckling. Therefore, the

accurate prediction of their stability limit state is of fundamental

importance in the design of composite structures.

Up to present, various thin-walled composite beam theories

have been developed by many authors. Bauld and Tzeng [1] intro-

duced the theory for bending and twisting of open cross-section

thin-walled composite beam which was extended from the

Vlasov’s theory of isotropic materials [6,16]. A simplified theory

for thin-walled composite beams was studied by Wu and Sun

[18] in which the effects of warping and transverse shear deforma-

tion were considered. Some studies on the buckling responses of

thin-walled composite beams have been done [9,10,14,8].

Furthermore, many attempts have been made to optimize the

design of thin-walled beams. Zyczkowski [19] presented an essen-

tial review on the development of optimization of thin-walled

beams in which the stability was considered. Szymcazak [15]

optimized the weight design of thin-walled beams whose natural

frequency of torsional vibration was given. Morton and Webber [12]

described a procedure for obtaining the minimum cross-sectional

area of composite I-beam considering structural failure, local buck-

ling and displacement. Design variable of material architecture

such as the fiber orientation and the fiber volume were employed

in the investigation of Davalos et al. [4] for transversely loaded

composite I-beams. Walker [17] presented a study dealing with

the multiobjective optimization design of uniaxially loaded lami-

nated I-beams maximizing combination of crippling, buckling load,

and post-buckling stiffness. Magnucki and Monczak [11] intro-

duced variational and parametrical shaping of the cross-section

in order to search for the optimum shape of thin-walled beams.

Savic et al. [13] employed the fiber orientation as design variable

in the optimization of laminated composite I-section beams which

aimed at maximizing the bending and axial stiffnesses. Cardoso

http://dx.doi.org/10.1016/j.compstruct.2015.06.036

0263-8223/ 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.

E-mail addresses: [email protected] (X.-H. Nguyen), kni8501@gmail.

com (N.-I. Kim), [email protected] (J. Lee).

Composite Structures 132 (2015) 1065–1074

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and Valido [2] provided a sensitivity analysis of optimal design of

thin-walled composite beams in which cross-sections were taken

into account.

The existing literature reveals that, even though a significant

amount of research has been conducted on the optimization anal-

ysis of thin-walled beams, there still has been no study reported of

the optimum design of thin-walled composite beams for stability

problem by considering the geometric parameters and the fiber

orientation as design variables simultaneously. The combination

of two or more different types of design variables would offer

higher flexibility of choosing input data which results in better

optimal solution expected.

In this study, geometric parameters and fiber orientation of

I-section composite beams are employed simultaneously as design

variables for the optimization problems in which the flexural–

torsional critical buckling loads of axially loaded beams are

maximized. A micro genetic algorithm (micro-GA) is utilized as a

tool to find the optimal solutions of problems. Some adjustments

on micro-GA parameters offer lower population to be chosen

initially and faster convergence solutions are obtained.

The outline of this paper is as follows: The brief presentation of

the kinematics and analysis steps of thin-walled composite beams

is described in Section 2. Section 3 focuses on the optimization def-

initions and procedures for thin-walled composite beams. Some

parametric studies and optimization problems are demonstrated

in Section 4. In Section 5, some conclusions are reported.

2. Thin-walled composite beams

The analysis is based on the classical laminated beam theory by

Lee and Kim [9] investigating the flexural–torsional buckling

behavior of thin-walled composite beams. A brief summary of

the kinematics and analysis steps involved is going to be described

below.

2.1. Kinematics

Assuming that cross-section is rigid with respect to in-plane

deformation, the displacement components of the arbitrary point

on the thin-walled cross-section can be written as follows:

U ð x; y; z Þ ¼ uð xÞ yv 0ð xÞ zw0ð xÞ x/

0ð xÞ ð1aÞ

V ð x; y; z Þ ¼ v ð xÞ z /ð xÞ ð1bÞ

W ð x; y; z Þ ¼ wð xÞ þ y/ð xÞ ð1cÞ

where u; v , and w are the beam displacements in the x; y, and z

direction, respectively, / is the angle of twist, and x is the warping

function. The longitudinal strain of thin-walled beam is defined as

follows:

e x ¼ e0 x þ z j y þ yj z þ xjx ð2Þ

where

0 x ¼ u0 ð3aÞ

j y ¼ w00 ð3bÞ

j z ¼ v 00 ð3cÞ

jx ¼ /00 ð3dÞ

in which 0 x ; j y; j z , and jx are the axial strain, the biaxial curva-

tures in the y and z direction, and the warping curvature,

respectively.

2.2. Variational formulation

The total potential energy of system in buckled shape is

expressed as follows:

P ¼ U þ V ð4Þ

where the strain energy U is expressed as

U ¼ 1

2

Z v

r xe x þ r xyc xy þ r xz c xz

dv ð5Þ

where r x; r xy, and r xz are the axial and shear stresses, respectively.

In this study, the shear strains c xy and c xz are generated from puretorsion action which can be expressed as follows:

ca xy ¼ z z að Þj xs ð6aÞ

c3 xz ¼ yj xs ð6bÞ

where superscript ‘a’ (a = 1, 2) and ‘3’ denote the top, bottom

flanges and the web, respectively; z a is the location of

mid-surface of each flange from the shear center; j xs is the twisting

curvature defined by

j xs ¼ 2/0 ð7Þ

The potential energy V due to the in-plane stress can also be

expressed as

V ¼ 1

2

Z v

r0 x V

02 þ W 02

dv ð8Þ

where r0 x is the constant in-plane axial stress. The variation of the

strain energy is calculated by substituting Eqs. (2) and (6) into Eq.

(5) as

dU ¼

Z l0

N xde0 x þ M ydj y þ M z dj z þ M xdjx þ M t dj xs

dx ð9Þ

where N x is the axial force; M y and M z are the bending moments

about the y and z axes, respectively; M x is the warping moment;

M t is the twisting moment by pure torsion defined by

M t ¼Z A

ra xyð z z aÞ r xz y

h idA ð10Þ

By substituting Eq. (1) into Eq. (8), the variation of the potential

energy is stated as

dV ¼

Z l0

r0 xbkt k v

0dv 0 þ wdw0 þ

b2k

12þ t 2k12

þ z 2a

!/d/

0

" #dx ð11Þ

where the subscript k varies from 1 to 3, and repeated indices imply

summation; bk and t k denote the width and the thickness of flanges

and web, respectively, as shown in Fig. 1. The principle of total

potential energy is applied as

dP ¼ d U þ V ð Þ ¼ 0 ð12Þ

By introducing the relationship r0 x ¼ P 0= A and substituting Eqs.

(9) and (11) into Eq. (12), the weak form is stated as

Z l0

N xdu0 M z dv

00 M ydw00 M xd/

00 þ 2M t d/0

þ P 0 v

0dv þ w0dw þ I 0 A /

0d/

0

dx ¼ 0 ð13Þ

where I 0 is the polar moment of inertia of cross-section.

2.3. Governing equations

From the study by Lee and Kim [9], the constitutive equations of

the thin-walled composite beam are of the form

1066 X.-H. Nguyen et al. / Composite Structures 132 (2015) 1065–1074



N x

M y

M z

M x

M t

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

E 11 E 12 E 13 0 E 15

E 22 0 E 24 E 25

E 33 0 E 35

E 44 0

sym: E 55

26666664

37777775

e0 xj yj z jx

j xs

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð14Þ

where E ij are the stiffness components of thin-walled composite

beam and detailed expressions can be found in the paper of [9].

The governing equations and the natural boundary conditions

can be derived by integrating the derivatives of the varied quanti-

ties by parts and collecting the coefficients of du; dv ; dw and d/ as

follows:

N 0 x ¼ 0 ð15aÞ

M 00 z þ P 0v

00 ¼ 0 ð15bÞ

M 00 y þ P 0w00 ¼ 0 ð15cÞ

M 00x þ 2M 0t þ P 0 I 0 A /

00 ¼ 0 ð15dÞ

and

du : N x ¼ N 0 x ð16aÞ

dv : M 0 z ¼ M

00 z ð16bÞ

dv 0

: M z ¼ M 0 z ð16cÞ

dw : M 0 y ¼ M 00 y ð16dÞ

dw0 : M y ¼ M 0 y ð16eÞ

d/ : M 0x þ 2M t ¼ M

00x ð16f Þ

d/0: M x ¼ M 0x ð16gÞ

where N 0 x ; M 00 z ; M 0 z ; M 00 y ; M 0 y ; M 00x, and M 0x are the prescribed val-

ues. The explicit forms of governing equations can be obtained by

substituting the constitutive equations into Eq. (15) as follows:

E 11u00 E 12w

000 E 13v 000 þ 2E 15/

00 ¼ 0 ð17aÞ

E 13u000 E 33v

iv þ 2E 35/000 þ P 0v

00 ¼ 0 ð17bÞ

E 12u000 E 22w

iv E 24/iv þ 2E 25/

000 þ P 0w00 ¼ 0 ð17cÞ

2E 15u000 2E 35v

000 2E 25w000 E 24w

iv E 44/iv þ 4E 55/

00 þ P 0 I 0 A /

00 ¼ 0

ð17dÞ

2.4. Finite element model

The finite element model including the effects of restrained

warping and non-symmetric lamination scheme is presented. Inorder to accurately express the element deformation, pertinent

shape functions are necessary. In this study, the one-dimensional

Lagrange interpolation function W i for the axial displacement and

the Hermite cubic polynomials wi for the transverse displacements

and the twisting angle are adopted to interpolate displacement

parameters. This beam element has two nodes and seven nodal

degrees of freedom. As a result, the element displacement param-

eters can be interpolated with respect to the nodal displacements

as follows:

u ¼Xni¼1

uiWi ð18aÞ

v ¼ Xn

i¼1

v iwi

ð18bÞ

Fig. 1. Geometry of thin-walled beam.

Fig. 2. The flowchart of a micro-GA cycle in optimization problems.

X.-H. Nguyen et al. / Composite Structures 132 (2015) 1065–1074 1067

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w ¼Xni¼1

wiwi ð18cÞ

/ ¼Xni¼1

/iwi ð18dÞ

By substituting Eq. (18) into the weak statement in Eq. (13), the

finite element model of a typical element can be expressed as the

standard eigenvalue problem.

K kG ð Þ Df g ¼ 0f g ð19Þ

where K and G are the element stiffness and element geometric

stiffness matrices, respectively;k refers to the load parameter under

the assumption of proportional loading; D is the eigenvector of

nodal displacements corresponding to the eigenvalue

Df g ¼ u v w /f gT

ð20Þ

3. Design optimization

Composite materials offer higher strength and stiffness in

design of structures than those of isotropic materials due to thepresence of the advanced material properties. If it is

well-designed, they usually exhibit the best qualities of their com-

ponents and constituents. In addition, the fiber orientation can be

utilized to offer high capacity of composite structures.

Furthermore, for I-section thin-walled beams, the width of flanges

and the height of web could also be varied to fit the design require-

ments. By using optimization for a design of structure, engineers

can utilize material and geometric properties which result in

higher performance of structure. In case of thin-walled composite

beams, if it is designed and selected carefully, fiber angle could

offer high performance of structures in which objective factors

are optimal. In addition, the flexural–torsional buckling analysis

which mainly depends on the geometric dimensions of beam

allows more possibilities of applying optimization design with var-

ious types of design variables.

In this study, optimization problems involve maximizing the

critical flexural–torsional buckling load P cr under the constraints

of cross-sectional area A, ratio of web height to flange width d=b

and ratio of beam length to web height L=d. The fiber angle h,

web height d and flange width b are chosen to be design variables.

The optimization problems can be described as follows:

Find

h; d; b

Maximize

P cr ðh; d; bÞ

Subjected to

A 6 A

ð21aÞ

1 6d

b ð21bÞ

10 6L

d6 100 ð21cÞ

where A

is the upper bound value of cross-sectional area of beamswhich should not be violated by the optimal solutions.

Numerous methods are available for solving optimization prob-

lems. Basically, these methods can be categorized into two main

types which are gradient-based approach and global optimization

algorithms. The former approach works effectively for convex opti-

mization functions in continuous domain. On the other hand, the

latter one is suitable for solving non-convex functions with multi-

ple local and global optima.

Two subcategory in the global optimization algorithms are

deterministic and stochastic approaches [13]. On one hand, the

deterministic-based optimization algorithms generally guarantee

that, within a finite number of iterations, the global optimum solu-

tion can be found. In order to obtain the optimal solution using

deterministic-based approach, detailed knowledge of involved

parameters and properties of optimization problem in term of

design variables is necessary. Consequently, the complex optimiza-

tion problems with mix of discrete and continuous variables which

usually produce complicated and unpredictable trends of objective

function will be challenges for this kind of approach. On the other

hand, for the stochastic-based approach, it is not sure that the glo-

bal optimum solution can be obtained after finite steps. However,

thanks to the flexibility of searching algorithms, the stochastic

approach can be applied on most of practical optimum design

problems whose design variables are in uniformly discrete or

mix of discrete and continuous forms.

In this study, a micro genetic algorithm (micro-GA) which is

typical method of global optimization based on the stochastic

approach is employed as a tool solving proposed optimization

problems. The ideas of micro-GA are inspired by some results of

Goldberg [7]. A major advantage of the micro-GA over the regular

genetic algorithm is that it offers faster convergence results can be

obtained even a smaller number of population used [5,3]. This

improvement results in significant reduction in computational

time cost which is critical limitation of regular GA due to the eval-

uation process of fitness function for large population.

Furthermore, the micro-GA performs elitism to generate initial

population and reinitialization process which maintain the pres-

ence of the best individual of previous iteration in the next one

which means the fluctuation phenomenon in objective conver-

gence history can be avoided. The flowchart, which shows how

the micro-GA works in solving optimization problems of buckling

loads for the thin-walled composite beam, is presented in Fig. 2.

Table 1

Buckling loads of beams (N).

Lay-up S-S beam C–F beam

Kim et al. [8] This study Kim et al. [8] This study

Analytical solutions ABAQUS Analytical solutions ABAQUS

½016 1438.8 1437.5 1438.8 5755.2 5720.0 5755.2

½15= 154s 1300.0 1299.1 1300.0 5199.8 5174.0 5199.7

½30= 304s 965.2 965.1 965.3 3861.0 3848.0 3861.0

½45= 454s 668.2 668.3 668.2 2672.7 2665.0 2672.7

½60= 604s 528.7 528.8 528.7 2114.7 2119.0 2114.8

½75= 754s 487.1 487.1 487.1 1948.3 1950.0 1948.3

½0=904s 964.4 963.9 959.3 3857.8 3848.0 3837.3

½0= 45=90=452s 832.2 832.0 813.8 3328.8 3315.0 3255.3




In order to apply the micro-GA procedure, the previously

defined optimization problems need to be transferred from con-

strained optimization problems to unconstrained ones. As a conse-

quence, the newly defined optimization problems can be expressed

by maximizing the G function which posed as follows:

G ¼ P cr ½c1ð A

AÞ2

þ c2ð1 d=bÞ2

þ c3ðb L=dÞ2

ð22Þ

where c1; c2 and c3 are the penalty parameters corresponding toeach of constraints shown in Eqs. (21a)–(21c), b denotes the upper

bound or lower bound constraint of L=d and G represents the com-

bination of objective functions and penalty functions. It should be

noted that the penalty parameters are set to be zero if its corre-

sponding constraint is not violated.

4. Numerical examples

In order to illustrate the accuracy and validity of this study, the

critical buckling loads are calculated and compared with previous

published results for various stacking sequences and boundary

conditions. After that, parametric studies and optimization proce-

dures for the thin-walled composite beams are conducted in order

to investigate the influence of flange widths, web height, and

Fig. 3. S-S beam with L=d ¼ 5 for Case 1.


Fig. 5. C–F beam with L=d ¼ 5 for Case 2.

Fig. 6. C–F beam with L=d ¼ 50 for Case 2.



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length as well as fiber angle on the critical buckling load. From the

convergence test, the entire length of beams is modeled using the

eight finite beam elements in subsequent examples.

4.1. Verification

In this example, the critical buckling loads of composite beams,

as shown in Fig. 1, subjected to an axial force acting at the centroid

are evaluated for simply supported (S-S) and clamped-free (C–F)

boundary conditions. The material of beams used is the glass–

epoxy and its material properties are as follows: E 1 ¼ 53:78,

E 2 ¼ E 3 ¼ 17:93, G12 ¼ G13 ¼ 8:96, G23 ¼ 3:45 GPa, m12 ¼ m13 ¼ 0:25;

m23 ¼ 0:34. The subscripts ‘1’ and ‘2’, ‘3’ correspond to directions

parallel and perpendicular to fiber, respectively. All constituent

flanges and web are assumed to be symmetrically laminated with

respect to its mid-plane. The flange widths and the web height are

b1 = b2 = d = 50 mm, and the total thicknesses of flanges and web

are assumed to be t 1 = t 2 = t 3 = 2.08 mm. Also 16 layers with equal

thickness are considered in two flanges and web. For S-S beam

with L = 4 m and C–F beam with L = 1 m, the critical coupled

buckling loads by this study are presented and compared with

the analytical solutions from the exact stiffness matrix method

and the finite element results from the nine-node shell elements

(S9R5) of ABAQUS by Kim et al. [8] in Table 1. It can be found from

Table 1 that the results from this study are in an excellent agree-

ment with the analytical solutions and the ABAQUS’s results for

the whole range of lay-ups and boundary conditions under

consideration.

4.2. Parametric studies

The parametric study is performed for the critical buckling

loads of composite beams with various boundary conditions.

Variations of the fiber angle with respect to the length of beam

and the ratio of height to width on the critical buckling loads are

investigated. It should be noted that, in this parametric study,

the lateral displacement of beam is assumed to be restrained inorder to avoid lateral buckling. Thus, the buckling modes may be

flexural, torsional, or flexural–torsional coupled modes. Typical

graphite-epoxy material is used and its properties are as follows:

E 1 ¼ 15E 2; G12 ¼ G13 ¼ 0:5E 2; m12 ¼ 0:25. Four investigations

whose lay-up schemes are of ½h= h4s will be conducted as

follows:

Case 1: The width of flanges b varies and the height of web d

is fixed for S-S beam.

Case 2: The width of flanges b varies and the height of web d

is fixed for C–F beam.

Case 3: The height of web d varies and the width of flanges b

is fixed for S-S beam.

Case 4: The height of web d varies and the width of flanges b

is fixed for C–F beam.

For convenience, the following dimensionless buckling loads

are introduced for each cases: P cr = P cr t 21=E 2d4

for Cases 1 and 2,

and P cr = P cr t 21=E 2b4

for Cases 3 and 4.

Figs. 3–6 show the variation of the critical buckling loads of

beams with L=d ¼ 5 and L=d ¼ 50 with respect to the fiber angle

change for Cases 1 and 2. It can be observed from Figs. 3–6 that

the critical buckling load decreases as the value of d=b increases

for different type of boundary conditions and the ratio of L=d.

Besides, the critical buckling loads are minimum at the fiber angle

of 90. On the other hand, the fiber angle at which the maximum

buckling load occurs depends on the boundary condition and the

values of L=d and d=b. The variation of the buckling loads with

L=b ¼ 60 and L=b ¼ 120 are plotted through Figs. 7–10 for CasesFig. 8. S-S beam with L=d ¼ 120 for Case 3.

Fig. 9. C–F beam with L=b ¼ 60 for Case 4.

Fig. 10. C–F beam with L=b ¼ 120 for Case 4.




3 and 4. From Figs. 7–10, it is observed that unlike for Cases 1 and

2, the buckling load does not decrease with increase of d=b through

the whole range of fiber angle. Thus, it can be realized from para-

metric studies that the maximum buckling loads of thin-walled

composite beams corresponding to fiber angle change are difficult

to predict, especially when flange widths b and web height d are

simultaneously changed. This observation motivates us to study

on the optimization of critical buckling load for the thin-walledcomposite beams which are essential for the practical design of

compressed structural elements.

4.3. Optimal designs

In this Section, couples of optimization problem for the

thin-walled composite beams are presented. A FORTRAN-based

computer program has been developed to integrate subroutines

of buckling analysis of thin-walled composite beams and the micro

genetic algorithm which is employed to be an optimization tool.

Input parameters of the optimization problem are prescribed and

the lower and upper bounds of design variables as well as con-

straints of optimization problem are provided. For sufficient runs

of genetic algorithm, the parameters such as population size, max-

imum generation, crossover rate, and penalty parameters need to

be selected carefully. The material and geometric properties,

bounds of design variables and input parameters of genetic algo-

rithm are presented in Tables 2–4, respectively. It can be found

from Table 3, there are 58 and 19 possibilities for the design vari-

able type of width (or height) and fiber angle which result in the

chromosome lengths storing for each type are of 6 and 5, respec-

tively. As previous parametric studies, the lateral displacement of

beam is constrained to avoid lateral buckling.

Two types of boundary conditions such as S-S and C–F ones are

considered with arbitrary values of beam length. Couples of lay-up

schemes of ½h1= h14s; ½h1= h24s, and ½h1= h1=h2= h22s are

introduced in the optimization problems. Table 5 shows optimiza-

tion results for S-S beams where design variables are h1; h2; b, and

d. For each lay-up scheme, the different values of beam length

which are L = 1 m, L = 2 m, and L = 5 m are considered. In order to

illustrate effectiveness of the proposed optimization methodology,

a regular design which satisfies all optimization constraints in Eqs.

(21a) and (21b), should be provided. Case 4 in Table 5 demon-

strates an assumed regular design whose fiber angles are all 0 uni-

directional, the flange width and the web height are 25 mm and

100 mm, respectively. Table 6 consists of two cases where the

same set of fiber angles from 45 to 90 are employed. The only

difference is that all possible fiber angle should be presented in

the solution which is composed a quasi-isotropic stacking

sequence in the first case. The second case, however, does notask for the presence of all type of fiber angles which means each

lamina is free to select its fiber orientation from the set of four pos-

sibilities of 45; 0; 45, or 90.

As can be seen in Tables 5 and 6, all cases of lay-up schemes

with different L produce the optimal values of critical buckling

loads which are greater than the solutions obtained from the

assumed regular design. These results clearly demonstrate effec-

tiveness of the proposed optimization procedure and its possible

application for the practically optimal design of thin-walled com-

posite beams. Furthermore, in the most of cases, the

Table 2

Material and geometric properties of thin-walled composite beams used in opti-

mization problems.

Parameter Value

E 1 15 E 2E 2 1.0 GPa

G12 0.5 E 2G23 0.8 E 2m12 0.25

t 1; t 2 4 mm

t 3 4 mm

Ply thickness 0.25 mm

A 600mm2

Table 3

Design variables in optimization problems.

Parameter Lower

bound

Upper

bound

Interval No. of

possibilities

No. of

genes

b 15 mm 300 mm 5 mm 58 6

d 15 mm 300 mm 5 mm 58 6

hð1;2Þ 0 90 5 19 5

Table 4

GA parameter for a typical run of optimization problem of 5 m-long S-S beams with

½h1= h14s lamination.

Parameter Value

Population size 50

Max. generation 100

c1 108

c2 108

c3 108

Crossover rate 0.5

Table 5

Optimization results for S-S beams with design variables of h

1; h

2 ; b

, and d

.

Case Lay-up L (m) Optimization results

h1 h2 b (mm) d (mm) P cr (N) d=b L=d

1 ½h1= h14s 1.00 30 – 50 50 2.296E+04 1.00 20.00

2.00 30 – 40 70 1.126E+04 1.75 28.60

5.00 25 - 15 120 4.204E+03 8.00 41.70

2 ½h1= h24s 1.00 35 30 50 50 2.369E+04 1.00 20.00

2.00 15 30 45 60 1.600E+04 1.33 33.30

5.00 30 15 15 120 4.433E+03 8.00 41.70

3 ½h1= h1=h2= h22s 1.00 40 25 50 50 2.383E+04 1.00 20.00

2.00 40 15 40 70 1.193E+04 1.75 28.60

5.00 35 5 15 120 4.437E+03 8.00 41.70

4 ½016 1.00 25 100 3.936E+03 4.00 10.00

2.00 25 100 1.836E+03 4.00 20.00

5.00 25 100 1.249E+03 4.00 50.00

Assumed regular design for the comparison with optimal results.


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½h1= h1=h2= h22s lay-up offers the best optimal solutions due to

its highest flexibility of choosing stacking sequence comparing to

other lay-up schemes.

Figs. 11–15 describe the optimal solutions presented in Tables 5

and 6. In each graph, the relation of the optimal critical buckling

load and the length of beam are plotted featuring the shape of

cross-section. The same relations of the assumed regular designs

are also printed for comparison purpose. Similarly, Tables 7 and

8 present optimization results for the C–F beam problem. The

solutions show the same trends in comparison with the S-S beam

problem in which the optimal critical buckling load increases as

the beam length decreases. From two cases of boundary condi-

tions, we can observe that even though the lay-up scheme is

changed, the design variables of flange width b and web height d

maintain same value corresponding to length of beam L. This

means that the values of b; d; L, in other word d=b and L=d but

not the fiber angle are critical factors which highly influence the

optimal critical buckling load.

Table 6

Optimization results for S-S beams with design variables of b; d and some specific fiber angles.

Case Fiber angles L (m) Optimization results

Lay-up b (mm) d (mm) P cr (N) d=b L=d

1 f45; 0; 45; 90g2s 1.00 ½45=45=0=902s 50 50 1.828E+04 1.00 20.0

2.00 ½45= 45=0=902s 35 85 9.055E+03 2.29 25.0

5.00 ½0=90= 45=452s 15 120 2.523E+03 8.00 41.7

2 f45; 0; 45; 90g2s 1.00 ½45= 45=0=02s 50 50 2.257E+04 1.00 20.02.00 ½45=45=0=02s 40 70 1.128E+04 1.75 28.6

5.00 ½45=0=45=02s 15 120 3.921E+03 8.00 41.7

All angles have to be presented in the optimal stacking sequence. All angles are not required to be presented in the optimal stacking sequence.

Fig. 11. Optimization results for S-S beams with lay-up of ½h1= h14s.

Fig. 12. Optimization results for S-S beams with lay-up of ½h1= h24s.

Fig. 13. Optimization results for S-S beams with lay-up of ½h1= h1=h2= h22s.

Fig. 14. Optimization results for S-S beams with a set of fiber angles of

f45; 0; 45 ; 90g2s , require all angles to be presented.

Fig. 15. Optimization results for S-S beams with a set of fiber angles of f45; 0; 45 ; 90g2s , not require all angles to be presented.




Figs. 16 and 17 show the effectiveness of the micro-GA over the

regular-GA in term of the number of generation and population

size. The two graphs are generated from the cases of 5 m long

S-S beams whose optimal solutions are printed in Table 5. As can

be seen in Figs. 16 and 17, by using the micro-GA with population

of 50, the optimal critical buckling loads are obtained just after 25

and 13 iterations for cases of ½h1= h14s and ½h1= h24s, respec-

tively. However, with the same or even larger amount of

population and number of generations, the solutions by regular

GA are still worse than those by micro-GA. It is found in these

investigation that in order to get convergence solutions which

are identical to those of micro-GA solution, one should use the reg-

ular GA with the number of population of 800 and 1800 for the

cases of ½h1= h14s and ½h1= h24s, respectively. Furthermore,

while the regular GA experiences some kind of fluctuation of objec-

tive function in the process of optimization, the micro-GA presents

Table 7

Optimization results for C–F beams with design variables of h1; h2 ; b, and d .

Case Lay-up L (m) Optimization results

h1 h2 b (mm) d (mm) P cr (N) d=b L=d

1 ½h1= h14s 1.00 30 – 40 70 1.126E+04 1.75 14.29

2.00 20 – 30 90 5.099E+03 3.00 22.22

5.00 0 – 30 90 1.084E+03 3.00 55.56

2 ½h1= h24s 1.00 35 20 40 70 1.132E+04 1.75 14.292.00 30 5 30 90 5.363E+03 3.00 22.22

5.00 0 0 30 90 1.084E+03 3.00 55.56

3 ½h1= h1=h2= h22s 1.00 40 15 40 70 1.192E+04 1.75 14.29

2.00 30 0 30 90 5.414E+03 3.00 22.22

5.00 0 0 30 90 1.084E+03 3.00 55.56

4 ½016 1.00 25 100 1.836E+03 4.00 10.00

2.00 25 100 1.312E+03 4.00 20.00

5.00 25 100 1.050E+03 4.00 50.00

Assumed regular design for the comparison with optimal results.

Table 8

Optimization results for C–F beams with design variables of d ; b and some specific fiber angles.

Case Fiber angles L (m) Optimization results

Lay-up b (mm) d (mm) P cr (N) d=b L=d

1 f45; 0; 45; 90g2s 1.00 ½45=45=0=02s 35 80 9.054E+03 2.29 12.50

2.00 ½0= 45=45=902s 30 90 2.852E+03 3.00 22.22

5.00 ½0= 45=45=902s 30 90 4.563E+02 3.00 55.56

2 f45; 0; 45; 90g2s 1.00 ½45= 45=0=0 2s 40 70 1.128E+04 1.75 14.29

2.00 ½45=0=0=02s 30 90 5.010E+03 3.00 22.22

5.00 ½0=0=0=0 2s 30 90 1.084E+03 3.00 55.56

All angles have to be presented in the optimal stacking sequence. All angles are not required to be presented in the optimal stacking sequence.

Fig. 16. Optimization convergence history of ½h1= h14s lay-up problem: the

micro-GA versus the regular-GA.

Fig. 17. Optimization convergence history of ½h1= h24s lay-up problem: the

micro-GA versus the regular-GA.


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a stable growth. This is due to the elitism of selection process in

micro-GA in which the best individual of previous generation is

always guaranteed to be appeared in next iteration.

5. Concluding remarks

This paper presented the formulation and the methodology for

the optimum design of thin-walled composite beams. The para-metric studies show that the effects of fiber angle and

cross-section geometry on the critical buckling load are varied

for the different boundary condition and length of beam. In some

cases, the increase of d=b is followed by the decrease of critical

buckling load through the range of fiber angle and the variation

of d=b produces diverse trends of critical buckling load with

respect to fiber angle change. In addition, formulation and investi-

gation of optimization problems of thin-walled composite beams

have been presented by maximizing the flexural–torsional buck-

ling load. The fiber angle and the cross-section geometry are

employed as design variables simultaneously. It reveals that the

optimization result heavily depends on the ratios of L=d and d=b

but less sensitive to the variation of the fiber angle. The

micro-GA has been applied to find the optimal solutions.Moreover, the optimal solutions and convergence rates of the

micro-GA are apparently better than those of the regular GA. The

micro-GA also eliminates the fluctuation of objective function phe-

nomenon which usually appears in regular GA due to the elitism of

population selection process. The micro-GA enables a possibility to

use just a small number of initial populations to obtain an appro-

priate solution of optimization problems.

Acknowledgments

This research was supported by National Research Foundation

of Korea (NRF) funded by the Ministry of Education, Science and

Technology through 2015R1A2A1A01007535. The support is grate-

fully acknowledged.

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