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Please cite this article as: Nguyen, X-H., Kim, N-I., Lee, J., Optimum Design of Thin-Walled Composite Beams for Flexural-Torsional Buckling Problem, Composite Structures (2015), doi: http://dx.doi.org/10.1016/j.compstruct 2015.06.036

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Optimum Design of Thin-Walled Composite Beams for Flexural-Torsional

Buckling Problem

Xuan-Hoang NGUYEN1, Nam-Il KIM1, Jaehong LEE1,∗

Department of Architectural Engineering, Sejong University, Seoul, South Korea

Abstract

The objective of this research is to present formulation and solution methodology for optimum design of thin-walledcomposite beams The geometric parameters and the fiber orientation of beams are treated as design variables simul-taneously 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 Theanalysis of beam is based on the classical laminated beam theory and applied for arbitrary laminate stacking sequenceconfiguration A micro genetic algorithm (micro-GA) is employed as a tool for obtaining optimal solutions It offersfaster convergence 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 lems of I-section composite beams Obtained numerical results show more sensitivity of geometric parameters on thecritical flexural-torsional buckling loads than that of fiber angle

prob-Keywords: Thin-walled beams; Laminated composites; Flexural-torsional buckling; Optimum design; Geneticalgorithm

1 Introduction

Composite materials have been increasingly used in a variety of structural fields such as architectural, civil, chanical, and aeronautical engineering applications over the past few decades The most apparent advantages ofcomposite 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 itcomes to composite materials Another major advantage of composites is tailorability which enables the optimizationprocesses to be applied in not only structural shape but materials itself as well

me-Thin-walled beams are widely used in various type of structural components due to its high axial and flexuralstiffnesses with a low weight of material However, these thin-walled beams might be subjected to an axial forcewhen used in above applications and are very susceptible to flexural-torsional buckling Therefore, the accurateprediction 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 andTzeng (1984) introduced the theory for bending and twisting of open cross-section thin-walled composite beam whichwas extended from the Vlasov’s theory of isotropic materials A simplified theory for thin-walled composite beamswas studied by Wu and Sun (1992) in which the effects of warping and transverse shear deformation were considered.Some studies on the buckling responses of thin-walled composite beams have been done (Lee and Kim 2001, Lee andLee 2004, Shin et al 2007, Kim et al 2008)

Furthermore, many attempts have been made to optimize the design of thin-walled beams Zyczkowski (1992)presented an essential review on the development of optimization of thin-walled beams in which the stability was con-sidered Szymcazak (1984) optimized the weight design of thin-walled beams whose natural frequency of torsionalvibration was given Morton (1994) described a procedure for obtaining the minimum cross-sectional area of com-posite I-beam considering structural failure, local buckling and displacement Design variable of material architecturesuch as the fiber orientation and the fiber volume were employed in the investigation of Davalos et al (1996) fortransversely loaded composite I-beams Walker (1998) presented a study dealing with the multiobjective optimizationdesign of uniaxially loaded laminated I-beams maximizing combination of crippling, buckling load, and post-bucklingstiffness Magnucki and Monczak (2000) introduced variational and parametrical shaping of the cross-section in order

to search for the optimum shape of thin-walled beams Savic (2001) employed the fiber orientation as design variable

in the optimization of laminated composite I-section beams which aimed at maximizing the bending and axial stinesses Cardoso (2011) provided a sensitivity analysis of optimal design of thin-walled composite beams in whichcross-sections were taken into account

ff-The existing literature reveals that, even though a significant amount of research has been conducted on the mization analysis of thin-walled beams, there still has been no study reported of the optimum design of thin-walledcomposite beams for stability problem by considering the geometric parameters and the fiber orientation as designvariables simultaneously The combination of two or more different types of design variables would offer higherflexibility of choosing input data which results in better optimal solution expected

opti-In this study, geometric parameters and fiber orientation of I-section composite beams are employed neously as design variables for the optimization problems in which the flexural-torsional critical buckling loads ofaxially loaded beams are maximized A micro genetic algorithm (micro-GA) is utilized as a tool to find the optimalsolutions of problems Some adjustments on micro-GA parameters offer lower population to be chosen initially andfaster convergence solutions are obtained

simulta-The outline of this paper is as follows: simulta-The brief presentation of the kinematics and analysis steps of thin-walledcomposite beams is described in Section 2 Section 3 focuses on the optimization definitions and procedures forthin-walled composite beams Some parametric studies and optimization problems are demonstrated in Section 4 InSection 5, some conclusions are reported

2 Thin-walled composite beams

The analysis is based on the classical laminated beam theory by Lee and Kim (2001) investigating the torsional buckling behavior of thin-walled composite beams A brief summary of the kinematics and analysis stepsinvolved is going to be described below

V=1

2 vσ0 x

12+ t

2 k

12+ z2 α

The governing equations and the natural boundary conditions can be derived by integrating the derivatives of thevaried quantities by parts and collecting the coefficients of δu, δv, δw and δφ as follows:

ωare the prescribed values The explicit forms of governing equations can

be obtained by substituting the constitutive equations into Eq (15) as follows:

E11u00− E12w000− E13v000+ 2E15φ00= 0 (17a)

E13u000− E33viv+ 2E35φ000+ P0

E12u000− E22wiv− E24φiv+ 2E25φ000+ P0w00= 0 (17c)2E15u000− 2E35v000− 2E25w000− E24wiv− E44φiv+ 4E55φ00+ P0I0

2.4 Finite element model

The finite element model including the effects of restrained warping and non-symmetric lamination scheme is sented In order to accurately express the element deformation, pertinent shape functions are necessary In this study,the one-dimensional Lagrange interpolation functionΨifor the axial displacement and the Hermite cubic polynomials

pre-ψifor the transverse displacements and the twisting angle are adopted to interpolate displacement parameters Thisbeam element has two nodes and seven nodal degrees of freedom As a result, the element displacement parameterscan be interpolated with respect to the nodal displacements as follows:

3 Design optimization

Composite materials offer higher strength and stiffness in design of structures than those of isotropic materials due

to the presence of the advanced material properties If it is well-designed, they usually exhibit the best qualities oftheir components and constituents In addition, the fiber orientation can be utilized to offer high capacity of compositestructures Furthermore, for I-section thin-walled beams, the width of flanges and the height of web could also bevaried to fit the design requirements By using optimization for a design of structure, engineers can utilize materialand 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 factorsare 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 various types of design variables

In this study, optimization problems involve maximizing the critical flexural-torsional buckling load Pcrunder theconstraints of cross-sectional area A, ratio of web height to flange width d/b and ratio of beam length to web heightL/d The fiber angle θ, web height d and flange width b are chosen to be design variables The optimization problemscan be described as follows:

Find

θ, d, bMaximize

Two subcategory in the global optimization algorithms are deterministic and stochastic approaches (Savic et al.2001) On one hand, the deterministic-based optimization algorithms generally guarantee that, within a finite number

of iterations, the global optimum solution can be found In order to obtain the optimal solution using based approach, detailed knowledge of involved parameters and properties of optimization problem in term of designvariables is necessary Consequently, the complex optimization problems with mix of discrete and continuous vari-ables which usually produce complicated and unpredictable trends of objective function will be challenges for thiskind of approach On the other hand, for the stochastic-based approach, it is not sure that the global optimum solutioncan be obtained after finite steps However, thanks to the flexibility of searching algorithms, the stochastic approachcan be applied on most of practical optimum design problems whose design variables are in uniformly discrete or mix

deterministic-of discrete and continuous forms

In this study, a micro genetic algorithm (micro-GA) which is typical method of global optimization based onthe stochastic approach is employed as a tool solving proposed optimization problems The ideas of micro-GAare inspired by some results of Goldberg (Goldberg 1989) A major advantage of the micro-GA over the regulargenetic algorithm is that it offers faster convergence results can be obtained even a smaller number of population used(Dozier et al 1994, Coello and Pulido 2001) This improvement results in significant reduction in computational timecost which is critical limitation of regular GA due to the evaluation process of fitness function for large population

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G= Pcr− [γ1(A∗− A)2+ γ2(1 − d/b)2+ γ3(β − L/d)2] (22)where γ1, γ2and γ3 are the penalty parameters corresponding to each of constraints shown in Eq (21a) to (21c), βdenotes the upper bound or lower bound constraint of L/d and G represents the combination of objective functionsand penalty functions It should be noted that the penalty parameters are set to be zero if its corresponding constraint

Objective and Fitness evaluation

Convergence condition Select best individual

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 procedures for the thin-walled composite beams are conducted in order to investigate the influence

of flange widths, web height, and length as well as fiber angle on the critical buckling load From the convergence

test, the entire length of beams is modelled 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: E1= 53.78 GPa, E2= E3= 17.93

GPa, G12 = G13= 8.96 GPa, G23= 3.45 GPa, ν12 = ν13= 0.25, ν23 = 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 t1=t2=t3= 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 (2008) in

Table 1 It can be found from Table 1 that the results from this study are in an excellent agreement with the analytical

solutions and the ABAQUS’s results for the whole range of lay-ups and boundary conditions under consideration

Table 1: Buckling loads of beams (N)

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 in order to avoid lateral buckling Thus, the buckling modes may be flexural,

tor-sional, or flexural-torsional coupled modes Typical graphite-epoxy material is used and its properties are as follows:

E1 = 15E2, G12 = G13 = 0.5E2, ν12 = 0.25 Four investigations whose lay-up schemes are of [θ/ − θ]4s 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∗

Figs 3 to 6 show the variation of the critical buckling loads of beams with L/d= 5 and L/d = 50 with respect tothe fiber angle change for Cases 1 and 2 It can be observed from Figs 3 to 6 that the critical buckling load decreases asthe value of d/b increases for different type of boundary conditions and the ratio of L/d Besides, the critical bucklingloads are minimum at the fiber angle of 90◦ On the other hand, the fiber angle at which the maximum buckling loadoccurs depends on the boundary condition and the values of L/d and d/b The variation of the buckling loads with

L/b = 60 and L/b = 120 are plotted through Figs 7 to 10 for Cases 3 and 4 From Figs 7 to 10, it is observed thatunlike for Cases 1 and 2, the buckling load does not decrease with increase of d/b through the whole range of fiberangle Thus, it can be realized from parametric studies that the maximum buckling loads of thin-walled compositebeams corresponding to fiber angle change are difficult to predict, especially when flange widths b and web height dare simultaneously changed This observation motivates us to study on the optimization of critical buckling load forthe thin-walled composite beams which are essential for the practical design of compressed structural elements

FORTRAN-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 [θ1/ − θ1]4s, [θ1/ − θ2]4s, and [θ1/ − θ1/θ2/ − θ2]2sare introduced in the optimizationproblems Table 5 shows optimization results for S-S beams where design variables are θ1, θ2, b, and d For eachlay-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

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