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Accepted Manuscript Optimum Design of Thin-Walled Composite Beams for Flexural-Torsional Buckling Problem Xuan-Hoang Nguyen, Nam-Il Kim, Jaehong Lee PII: DOI: Reference: S0263-8223(15)00500-0 http://dx.doi.org/10.1016/j.compstruct.2015.06.036 COST 6535 To appear in: Composite Structures 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 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain 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-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 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 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 Keywords: Thin-walled beams; Laminated composites; Flexural-torsional buckling; Optimum design; Genetic algorithm Introduction Composite materials have been increasingly used in a variety of structural fields such as architectural, civil, mechanical, and aeronautical 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-toweight 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 composites 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 (1984) introduced 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 A simplified theory for thin-walled composite beams was 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 and Lee 2004, Shin et al 2007, Kim et al 2008) ∗ Correspongding author Email addresses: hoangnguyence@gmail.com (Xuan-Hoang NGUYEN), kni8501@gmail.com (Nam-Il KIM), jhlee@sejong.ac.kr (Jaehong LEE) 98 Gunja Dong, Gwangjin Gu, Seoul, 143-747, South Korea Preprint submitted to Composite Structures June 26, 2015 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 considered Szymcazak (1984) optimized the weight design of thin-walled beams whose natural frequency of torsional vibration was given Morton (1994) described a procedure for obtaining the minimum cross-sectional area of composite I-beam considering structural failure, local buckling 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 (1996) for transversely loaded composite I-beams Walker (1998) presented a study dealing with the multiobjective optimization design of uniaxially loaded laminated I-beams maximizing combination of crippling, buckling load, and post-buckling stiffness 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 stiffnesses Cardoso (2011) 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 analysis 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 Section focuses on the optimization definitions and procedures for thin-walled composite beams Some parametric studies and optimization problems are demonstrated in Section In Section 5, some conclusions are reported Thin-walled composite beams The analysis is based on the classical laminated beam theory by Lee and Kim (2001) investigating the flexuraltorsional 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(x, y, z) = u(x) − yv (x) − zw (x) − ωφ (x) (1a) V(x, y, z) = v(x) − zφ(x) (1b) W(x, y, z) = w(x) + yφ(x) (1c) where u, v, and w are the beam displacements in the x, y, and z direction, respectively, φ is the angle of twist, and ω is the warping function The longitudinal strain of thin-walled beam is defined as follows: ε x = ε0x + zκy + yκz + ωκω (2) z t1 b1 z1 y d t3 x t2 z2 L b2 Figure 1: Geometry of thin-walled beam where =u (3a) κy = −w (3b) κz = −v (3c) κω = −φ (3d) x in which x0 , κy , κz , and κω are the axial strain, the biaxial curvatures 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: Π=U+V (4) where the strain energy U is expressed as U= σ x ε x + σ xy γ xy + σ xz γ xz dv (5) v where σ x , σ xy , and σ xz are the axial and shear stresses, respectively In this study, the shear strains γ xy and γ xz are generated from pure torsion action which can be expressed as follows: γαxy = (z − zα ) κ xs γ3xz = −yκ xs (6a) (6b) where superscript ‘α’ (α=1, 2) and ‘3’ denote the top, bottom flanges and the web, respectively; zα is the location of mid-surface of each flange from the shear center; κ xs is the twisting curvature defined by κ xs = 2φ The potential energy V due to the in-plane stress can also be expressed as (7) V= σ0x V + W (8) dv v where σ0x 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 l δU = N x δε0x + My δκy + Mz δκz + Mω δκω + Mt δκ xs dx (9) where N x is the axial force; My and Mz are the bending moments about the y and z axes, respectively; Mω is the warping moment; Mt is the twisting moment by pure torsion defined by σαxy (z − zα ) − σ xz y dA Mt = (10) A By substituting Eq (1) into Eq (8), the variation of the potential energy is stated as l t2 b δV = σ0x bk tk v δv + wδw + k + k + z2α φδφ dx 12 12 (11) where the subscript k varies from to 3, and repeated indices imply summation; bk and tk 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 δΠ = δ (U + V) = By introducing the relationship stated as l σ0x (12) = P /A and substituting Eqs (9) and (11) into Eq (12), the weak form is N x δu − Mz δv − My δw − Mω δφ + 2Mt δφ + P0 v δv + w δw + I0 φ δφ A dx = (13) where I0 is the polar moment of inertia of cross-section 2.3 Governing equations From the study by Lee and Kim (2001), the constitutive equations of the thin-walled composite beam are of the form Nx E11 M y Mz = M ω M sym E12 E22 E13 E33 E24 E44 t 0 E15 εx E25 κy E35 κ z κ ω E55 κ xs (14) where Ei j are the stiffness components of thin-walled composite beam and detailed expressions can be found in the paper of Lee and Kim (2001) The governing equations and the natural boundary conditions can be derived by integrating the derivatives of the varied quantities by parts and collecting the coefficients of δu, δv, δw and δφ as follows: Nx = (15a) Mz + P v = (15b) My + P0 w = I0 Mω + 2Mt + P0 φ = A (15c) (15d) and δu : N x = N x0 δv : Mz = Mz δφ : (16a) (16b) δv : Mz = Mz0 (16c) δw : My = My0 δw : My = My0 Mω + 2Mt = Mω0 δφ : Mω = Mω0 (16d) (16e) (16f) (16g) where N x0 , Mz0 , Mz0 , My0 , My0 , Mω0 , and Mω0 are the prescribed values The explicit forms of governing equations can be obtained by substituting the constitutive equations into Eq (15) as follows: + 2E15 φ = (17a) + P0 v = (17b) + P0 w = I0 − E24 wiv − E44 φiv + 4E55 φ + P0 φ = A (17c) E11 u − E12 w E13 u E12 u 2E15 u − 2E35 v − 2E25 w − E13 v − E33 viv + 2E35 φ − E22 wiv − E24 φiv + 2E25 φ (17d) 2.4 Finite element model The finite element model including the effects of restrained warping and non-symmetric lamination scheme is presented In order to accurately express the element deformation, pertinent shape functions are necessary In this study, the one-dimensional Lagrange interpolation function Ψi for the axial displacement and the Hermite cubic polynomials ψi 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 parameters can be interpolated with respect to the nodal displacements as follows: n u= ui Ψi (18a) v i ψi (18b) wi ψi (18c) φ i ψi (18d) i=1 n v= i=1 n w= i=1 n φ= i=1 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 − λG) {∆} = {0} (19) where K and G are the element stiffness and element geometric stiffness matrices, respectively; λ refers to the load parameter under the assumption of proportional loading; ∆ is the eigenvector of nodal displacements corresponding to the eigenvalue {∆} = {u v w φ}T (20) 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 of their components 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 requirements 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 various types of design variables In this study, optimization problems involve maximizing the critical flexural-torsional buckling load Pcr 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 θ, web height d and flange width b are chosen to be design variables The optimization problems can be described as follows: Find θ, d, b Maximize Pcr (θ, d, b) Subjected to A ≤ A∗ d 1≤ b L 10 ≤ ≤ 100 d (21a) (21b) (21c) where A∗ is the upper bound value of cross-sectional area of beams which should not be violated by the optimal solutions Numerous methods are available for solving optimization problems 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 optimization functions in continuous domain On the other hand, the latter one is suitable for solving non-convex functions with multiple local and global optima 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 deterministicbased approach, detailed knowledge of involved parameters and properties of optimization problem in term of design variables is necessary Consequently, the complex optimization 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 global 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 (Goldberg 1989) 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 (Dozier et al 1994, Coello and Pulido 2001) This improvement results in significant reduction in computational time cost which is critical limitation of regular GA due to the evaluation process of fitness function for large population Furthermore, the micro-GA performs elitism to generate initial population and reinitialization process which maintain the presence of the best individual of previous iteration in the next one which means the fluctuation phenomenon in objective convergence 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 In order to apply the micro-GA procedure, the previously defined optimization problems need to be transferred from constrained optimization problems to unconstrained ones As a consequence, the newly defined optimization problems can be expressed by maximizing the G function which posed as follows: G = Pcr − [γ1 (A∗ − A)2 + γ2 (1 − d/b)2 + γ3 (β − L/d)2 ] (22) where γ1 , γ2 and γ3 are the penalty parameters corresponding to each of constraints shown in Eq (21a) to (21c), β denotes the upper bound or lower bound constraint of L/d and G represents the combination of objective functions and penalty functions It should be noted that the penalty parameters are set to be zero if its corresponding constraint is not violated BEGIN Initialize GA parameters Population Initialization Individual selection Decode variable's chromosomes Assemble structures Analyze structures Next generation Objective and Fitness evaluation Yes Convergence condition No Select best individual END Yes Elitism selection Elitism No Rank individual with its chromosomes Crossover operation Mutation operation New population Figure 2: The flowchart of a micro-GA cycle in optimization problems 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= m and C-F beam with L= 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 It can be found from Table 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) Lay-up [00 ]16 [150 / − 150 ]4s [300 / − 300 ]4s [450 / − 450 ]4s [600 / − 600 ]4s [750 / − 750 ]4s [00 /900 ]4s [00 / − 450 /900 /450 ]2s S-S beam Kim et al (2008) Analytical solutions ABAQUS 1438.8 1300.0 965.2 668.2 528.7 487.1 964.4 832.2 1437.5 1299.1 965.1 668.3 528.8 487.1 963.9 832.0 This study 1438.8 1300.0 965.3 668.2 528.7 487.1 959.3 813.8 C-F beam Kim et al (2008) Analytical solutions ABAQUS 5755.2 5199.8 3861.0 2672.7 2114.7 1948.3 3857.8 3328.8 5720.0 5174.0 3848.0 2665.0 2119.0 1950.0 3848.0 3315.0 This study 5755.2 5199.7 3861.0 2672.7 2114.8 1948.3 3837.3 3255.3 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 in order 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: 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∗cr = Pcr t12 /E2 d4 for Cases and 2, and P∗cr = Pcr t12 /E2 b4 for Cases and Figs to show the variation of the critical buckling loads of beams with L/d = and L/d = 50 with respect to the fiber angle change for Cases and It can be observed from Figs to 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 = 60 and L/b = 120 are plotted through Figs to 10 for Cases and From Figs to 10, it is observed that unlike for Cases 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 parametric 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-walled composite beams which are essential for the practical design of compressed structural elements d /b d /b d /b d /b d /b -5 P c r* ( x ) = = = = = 1 0 θ( d e g ) Figure 3: S-S beam with L/d = for Case 4.3 Optimal Designs In this Section, couples of optimization problem for the thin-walled composite beams are presented A FORTRANbased 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 constraints of optimization problem are provided For sufficient runs of genetic algorithm, the parameters such as population size, maximum 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 algorithm are presented in Tables to 4, respectively It can be found from Table 3, there are 58 and 19 possibilities for the design variable type of width (or height) and fiber angle which result in the chromosome lengths storing for each type are of and 5, respectively 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 [θ1 / − θ1 ]4s , [θ1 / − θ2 ]4s , and [θ1 / − θ1 /θ2 / − θ2 ]2s are introduced in the optimization problems Table shows optimization results for S-S beams where design variables are θ1 , θ2 , 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 d /b d /b d /b d /b d /b = = = = 1 P c r* (x -6 ) = 0 θ( d e g ) Figure 4: S-S beam with L/d = 50 for Case d /b d /b d /b d /b d /b = = = = -6 ) = P c r* (x 0 θ( d e g ) Figure 5: C-F beam with L/d = for Case 10 d /b d /b d /b d /b d /b = = = = -7 ) = P c r* (x 0 θ( d e g ) Figure 6: C-F beam with L/d = 50 for Case d /b = d /b = d /b = -5 ) P c r* (x 0 θ( d e g ) Figure 7: S-S beam with L/d = 60 for Case 11 d /b = d /b = d /b = -5 ) P c r* (x 0 θ( d e g ) Figure 8: S-S beam with L/d = 120 for Case d /b = d /b = d /b = -5 ) P c r* (x 0 θ( d e g ) Figure 9: C-F beam with L/b = 60 for Case 12 d /b = d /b = d /b = -5 P c r* ( x ) 0 0 θ( d e g ) Figure 10: C-F beam with L/b = 120 for Case constraints in Eqs (21a) to (21c) should be provided Case in Table demonstrates an assumed regular design whose fiber angles are all 0◦ unidirectional, the flange width and the web height are 25 mm and 100 mm, respectively Table 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 not ask 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 possibilities of −45◦ , 0◦ , 45◦ , or 90◦ As can be seen in Tables 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 effectiveness of the proposed optimization procedure and its possible application for the practically optimal design of thin-walled composite beams Furthermore, in the most of cases, the [θ1 / − θ1 /θ2 / − θ2 ]2s lay-up offers the best optimal solutions due to its highest flexibility of choosing stacking sequence comparing to other lay-up schemes Figs 11 to 15 describe the optimal solutions presented in Table and Table 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 and 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 conditions, 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 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 m long S-S beams whose optimal solutions are printed in Table 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 [θ1 / − θ1 ]4s and [θ1 / − θ2 ]4s , respectively 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 13 which are identical to those of micro-GA solution, one should use the regular GA with the number of population of 800 and 1800 for the cases of [θ1 /−θ1 ]4s and [θ1 /−θ2 ]4s , respectively Furthermore, while the regular GA experiences some kind of fluctuation of objective function in the process of optimization, the micro-GA presents 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 Table 2: Material and geometric properties of thin-walled composite beams used in optimization problems Parameter Value E1 E2 G12 G23 ν12 t1 , t2 t3 Ply thickness A∗ 15E2 1.0 GPa 0.5E2 0.8E2 0.25 mm mm 0.25 mm 600 mm2 Table 3: Design variables in optimization problems Parameter Lower bound Upper bound Interval No of possibilities No of genes b d θ(1,2) 15 mm 15 mm 0◦ 300 mm 300 mm 90◦ mm mm 5◦ 58 58 19 6 Table 4: GA parameter for a typical run of optimization problem of 5m-long S-S beams with [θ1 / − θ1 ]4s lamination Parameter Value Population size Max generation γ1 γ2 γ3 Crossover rate 50 100 108 108 108 0.5 Concluding Remarks This paper presented the formulation and the methodology for the optimum design of thin-walled composite beams The parametric 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 investigation of optimization problems of thin-walled composite beams have been presented by maximizing the flexural-torsional buckling 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 14 Table 5: Optimization results for S-S beams with design variables of θ1 , θ2 , b, and d Case Lay-up L (m) Optimization results θ1 θ2 b (mm) d (mm) Pcr (N) d/b L/d [θ1 / − θ1 ]4s 1.00 2.00 5.00 30◦ 30◦ 25◦ - 50 40 15 50 70 120 2.296E+04 1.126E+04 4.204E+03 1.00 1.75 8.00 20.00 28.60 41.70 [θ1 / − θ2 ]4s 1.00 2.00 5.00 35◦ 15◦ 30◦ 30◦ 30◦ 15◦ 50 45 15 50 60 120 2.369E+04 1.600E+04 4.433E+03 1.00 1.33 8.00 20.00 33.30 41.70 [θ1 / − θ1 /θ2 / − θ2 ]2s 1.00 2.00 5.00 40◦ 40◦ 35◦ 25◦ 15◦ 5◦ 50 40 15 50 70 120 2.383E+04 1.193E+04 4.437E+03 1.00 1.75 8.00 20.00 28.60 41.70 [00 ]†16 100 100 100 3.936E+03 1.836E+03 1.249E+03 4.00 4.00 4.00 10.00 20.00 50.00 Case Fiber angles 1.00 25 2.00 25 5.00 25 † Assumed regular design for the comparison with optimal results Table 6: Optimization results for S-S beams with design variables of b, d and some specific fiber angles {−450 , 00 , 450 , 900 }†2s L (m) 1.00 2.00 5.00 Optimization results Lay-up b (mm) d (mm) Pcr (N) d/b L/d [−450 /450 /00 /900 ]2s [450 / − 450 /00 /900 ]2s [00 /900 / − 450 /450 ]2s 50 35 15 50 85 120 1.828E+04 9.055E+03 2.523E+03 1.00 2.29 8.00 20.0 25.0 41.7 50 70 120 2.257E+04 1.128E+04 3.921E+03 1.00 1.75 8.00 20.0 28.6 41.7 1.00 [450 / − 450 /00 /00 ]2s 50 2.00 [−450 /450 /00 /00 ]2s 40 5.00 [450 /00 /450 /00 ]2s 15 † All angles have to be presented in the optimal stacking sequence †† All angles are not required to be presented in the optimal stacking sequence {−450 , 00 , 450 , 900 }†† 2s 15 Table 7: Optimization results for C-F beams with design variables of θ1 , θ2 , b, and d Case Lay-up L (m) Optimization results θ1 θ2 b (mm) d (mm) Pcr (N) d/b L/d [θ1 / − θ1 ]4s 1.00 2.00 5.00 30◦ 20◦ 0◦ - 40 30 30 70 90 90 1.126E+04 5.099E+03 1.084E+03 1.75 3.00 3.00 14.29 22.22 55.56 [θ1 / − θ2 ]4s 1.00 2.00 5.00 35◦ 30◦ 0◦ 20◦ 5◦ 0◦ 40 30 30 70 90 90 1.132E+04 5.363E+03 1.084E+03 1.75 3.00 3.00 14.29 22.22 55.56 [θ1 / − θ1 /θ2 / − θ2 ]2s 1.00 2.00 5.00 40◦ 30◦ 0◦ 15◦ 0◦ 0◦ 40 30 30 70 90 90 1.192E+04 5.414E+03 1.084E+03 1.75 3.00 3.00 14.29 22.22 55.56 [00 ]†16 100 100 100 1.836E+03 1.312E+03 1.050E+03 4.00 4.00 4.00 10.00 20.00 50.00 Case Fiber angles 1.00 25 2.00 25 5.00 25 † 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 {−450 , 00 , 450 , 900 }†2s L (m) 1.00 2.00 5.00 Optimization results Lay-up b (mm) d (mm) Pcr (N) d/b L/d [−450 /450 /00 /00 ]2s [00 / − 450 /450 /900 ]2s [00 / − 450 /45/ 900 ]2s 35 30 30 80 90 90 9.054E+03 2.852E+03 4.563E+02 2.29 3.00 3.00 12.50 22.22 55.56 70 90 90 1.128E+04 5.010E+03 1.084E+03 1.75 3.00 3.00 14.29 22.22 55.56 1.00 [450 / − 450 /00 /00 ]2s 40 2.00 [−450 /00 /00 /00 ]2s 30 5.00 [00 /00 /00 /00 ]2s 30 † All angles have to be presented in the optimal stacking sequence †† All angles are not required to be presented in the optimal stacking sequence {−450 , 00 , 450 , 900 }†† 2s 16 Optimal critical buckling load (N) 2.5x104 Optimal result ([300/-300]4s,b=50,d=50) Regular design 2.0x10 1.5x104 ([300/-300]4s,40,70) 1.0x104 0.5x104 ([250/-250]4s,15,120) ([00]16,25,100) ([00]16,25,100) ([00]16,25,100) 1.0 2.0 L (m) 5.0 Optimal critical buckling load (N) Figure 11: Optimization results for S-S beams with lay-up of [θ1 / − θ1 ]4s 2.5x104 Optimal result ([350/-300]4s,b=50,d=50) Regular design 2.0x10 0 ([15 /-30 ]4s,45,60) 1.5x104 ([300/-150]4s,15,120) 1.0x104 0.5x104 ([00]16,25,100) ([00]16,25,100) ([00]16,25,100) 1.0 2.0 L (m) 5.0 Figure 12: Optimization results for S-S beams with lay-up of [θ1 / − θ2 ]4s 17 Optimal critical buckling load (N) 2.5x104 ([400/-400/250/-250]2s,b=50,d=50) Optimal result Regular design 2.0x10 1.5x104 ([400/-400/150/-150]2s,40,70) ([350/-350/50/-50]2s,15,120) 1.0x10 0.5x104 ([00]16,25,100) ([00]16,25,100) ([00]16,25,100) 1.0 2.0 L (m) 5.0 Optimal critical buckling load (N) Figure 13: Optimization results for S-S beams with lay-up of [θ1 / − θ1 /θ2 / − θ2 ]2s 2.5x104 Optimal result Regular design 2.0x10 ([-450/450/00/900]2s,b=50,d=50) 1.5x104 ([450/-450/00/900]2s,35,85) 1.0x10 ([00/900/-450/450]2s,15,120) 0.5x104 ([0 ]16,25,100) ([00]16,25,100) ([00]16,25,100) 1.0 2.0 L (m) 5.0 Figure 14: Optimization results for S-S beams with a set of fiber angles of {−450 , 00 , 450 , 900 }2s , require all angles to be presented 18 Optimal critical buckling load (N) 2.5x104 Optimal result ([450/-450/00/00]2s,b=50,d=50) Regular design 2.0x10 1.5x104 ([-450/450/00/00]2s,40,70) ([450/00/450/00]2s,15,120) 1.0x104 0.5x104 ([00]16,25,100) ([00]16,25,100) ([00]16,25,100) 1.0 2.0 5.0 L (m) O b je c t iv e f u n c t io n ( P c r) Figure 15: Optimization results for S-S beams with a set of fiber angles of {−450 , 00 , 450 , 900 }2s , not require all angles to be presented x x 3 x x x m ic r o G A - p o p re g G A -p o p re g G A -p o p 0 0 0 G e n e r a tio n Figure 16: Optimization convergence history of [θ1 / − θ1 ]4s lay-up problem: the micro-GA versus the regular-GA 19 O b je c t iv e f u n c t io n ( P c r) x x 3 x x x m ic r o G A - p o p re g G A -p o p re g G A -p o p 0 0 0 G e n e r a tio n Figure 17: Optimization convergence history of [θ1 / − θ2 ]4s lay-up problem: the micro-GA versus the regular-GA convergence rates of the micro-GA are apparently better 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Journal of Global Optimization 1998;13:433-444 Zyczkowski M, Recent advances in optimal structural design of shells, European Journal of Mechanics - A/Solids 1992;11:5-24 21 ... 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... variation of d/b produces diverse trends of critical buckling load with respect to fiber angle change In addition, formulation and investigation of optimization problems of thin-walled composite beams. .. presentation of the kinematics and analysis steps of thin-walled composite beams is described in Section Section focuses on the optimization definitions and procedures for thin-walled composite beams