Scientific Research and Essays Vol. 5(9), pp. 897-910, 4 May 2010 Available online at http://www.academicjournals.org/SRE ISSN 1992-2248 © 2010 Academic Journals Full Length Research Paper

Determination of thickness and stiffener locations for optimization of critical buckling load of stiffened plates

Nildem Tay�i

Department of Civil Engineering, University of Gaziantep, 27310 Gaziantep, Turkey. E-mail: taysi@gantep.edu.tr.

Tel: 90 342 3172422. Fax: 90 342 3601107.

Accepted 7 April, 2010

In this paper, buckling optimization of stiffened plates under uniform edge compression is considered. The locations of stiffeners are chosen as design variables and effects over critical buckling loads are observed. For this purpose, two types of conventional stiffened plates which are used in aerospace industry are investigated. The loaded sides of plates are simply supported and in order to reflect the other possible conditions the remaining sides considered free, clamped or simply supported. A finite strip method is used to evaluate buckling loads of plates. Sequential Quadratic Programming algorithm is used to optimize the design variables. Results are presented to show the influence of size optimizations and also stiffener locations. A parametric study is carried out in order to investigate the effect of stiffeners locations on the buckling parameters. The proposed results for the investigated samples can be used to develop an improved design for stiffened plates. Key words: Buckling analysis, structural optimization, finite strip, stiffened plates.

INTRODUCTION Stiffeners provide improvement to load carrying capacity of structures. The benefit of stiffening of a structure lies in achieving lightweight and robust design of the structure. For this purpose they have wide use in structural engineering domain. Specially, stiffened plates are used in critical and sensitive structures such as in aircrafts, ship hulls and box girders in which safety and a perfect design is crucial. Buckling is the one of the most complex phenomenon that is inevitable for heavily axially loaded stiffened plate structures. For this purpose, it is necessary to carry a deep interest and investigation about their responses under expected loads to design such structures safely.

In structural engineering, it is one of the first priorities to save weight, without loss of any strength in the used of structural elements against subjected loads. The approach of the uses of stiffeners to improve structural response is simple, but the practical stiffened plate design is a complex task. Due to involving many design variables, a complete understanding of response of such structures is not fully figured out. Therefore, a robust optimization algorithm which is integrating analysis, shape definition, sensitivity and optimization is necessary to obtain maximum efficiency from stiffened plates. The

efficiency of structural optimization algorithm is based on the computational time required in the process. Since most of the structural optimization methods are iterative, the number of structural analyses required to complete the optimum solution is large. To reduce the computational efforts of the efficient and inexpensive structural analysis method should be used.

Extensive work has been performed to present the expressions for critical buckling loads of flat plates under different load conditions. The analytical solutions for buckling are presented by Timoshenko and Gere (1936); Chajes (1974). The stability of simply supported rectangular plates under patch compression using Ritz’s energy method was studied by Liu and Pavlovic´ (2007). Both single and double Fourier series were used as deflection series to compute the values for buckling coefficients. But no theoretical solutions exist for more complicated cases such as stiffened plates. Therefore, numerical solutions are often preferred to analyze such complicated responses. Sheikh et al. (2002) investigated stability of stiffened steel plates under uniaxial compression and bending using finite element method. The parameters investigated where; the transverse slenderness of the plate, the slenderness of the web and

898 Sci. Res. Essays flange of the stiffener, the ratio of torsional slenderness of the stiffener to the transverse slenderness of the plate, and the stiffener to- plate area ratio. Kumar and Mukhopadhyay (1999) presented a stiffened plate element for stability analysis of laminated stiffened plates that the basic plate element was a combination of Allman's plane stress triangular element and a discrete thick plate bending element and includes transverse shear effects.

Brubak et al. (2007) presented an approximate semi-analytical computational model for plates with arbitrarily oriented stiffeners and subjected to in-plane loading. Their estimation of the buckling strength is made using the von Mises’ yield criterion. Peng et al. (2006) presented a mesh-free Galerkin method for the free vibration and stability analyses of stiffened plates via the first order shear deformable theory. Vörös (2009) presented the application of the new stiffener element with seven degrees of freedom per node and subsequent application in determining frequencies, mode shapes and buckling loads of different stiffened plates. The development of the stiffener is based on a general beam theory and includes the constraint torsional warping effect and the second order terms of finite rotations. Riks (2000) investigated implementation of the finite strip (FS) method that extends the scope of the determination of the post-buckling stiffness of stiffened panels for wing structures.

The literature survey shows that, the finite element method is the most powerful and most preferred method in structural analysis domain and has the capability of solving all types of complex geometries, loading and boundary conditions. On the other hand, it requires plenty number of elements as input data and creates large matrix equations to solve. In this regard, these requirements increase the computational time seriously. In this point of view, an alternative method that requires less number of equations, less computational time and easy to control input and output data is desired. The FS method has proven to be an inexpensive and useful tool in analysis of prismatic structures. Structures which are simply supported on diaphragms at two opposite edges with the remaining edges arbitrarily restrained, and where the cross section does not change between the simply supported ends can be analyzed accurately and inexpensively using the FS method in cases where a full finite element analysis could be considered extravagant.

The other important component of structural optimization is optimization technique which is used. To obtain the effective, reliable and efficient optimum solutions, suitable optimization method should be used. The selection of the optimization method should be based on type of objective function, design variables, constraints and also number of design variables used in optimization problem definition. Studies on structural optimization of stiffened plates are summarized below. Iuspa and Ruocco (2008) presented the topological

optimal design of isotropic/orthotropic thin structures including stiffened panels via genetic algorithms. They developed a modified FS method used to analyze parametric structures arranged in form of plates or stiffened panels with almost arbitrary cross section shapes. Todoroki and Sekishiro (2008) proposed a fractal branch and bound method for optimizing the stacking sequences to maximize the buckling load of blade-stiffened panels with strength constraints. Bedair (1997) presented the influence of stiffener location on the stability of stiffened plates under combined compression and bending. He idealized the structure as assembled plate and beam elements rigidly connected at their junctions. Various researches have been carried out to optimize the response of plates. Özakça (1993) investigated SSO of prismatic folded plates under buckling load consideration. He used mathematical programming methods.

The literature survey shows that, shape and size optimization of stiffened plates for improving the critical buckling load was studied using various analysis and optimization methods. In these studies, effects of one or two parameters on buckling load were investigated. On the other hand, the stiffened panels are complex thin walled structure. In their design, so many parameters and interaction between parameters must be considered in order to obtain safe, economical, robust and reliable solutions. As a consequence, in the present study, it is aim to carry out a comprehensive study on stability of stiffened plates. The objective of this study is to carry out a deep interest and investigation about critical buckling load of stiffened plates. Structural optimization and a parametric study are carried out in order to investigate the effect of stiffeners location, number of stiffeners, types of design variable, boundary conditions and plate dimensions on the buckling parameters. The proposed results can be used to develop an improved design for stiffened plates.

In this study, buckling analyses are carried out using Fortran code which was developed by Özakça (1993). Theory and implementation of FS method for buckling analyses are given in Özakça et al. (2006). The buckling loads will be determined using cubic, C(0) continuity Mindlin-Reissner FSs. The code uses Sequential Quadratic Programming (SQP) to carry out structural optimization process. STRUCTURAL SHAPE OPTIMIZATION SSO can be defined as the activity of achieving the best (optimum) structural configuration to fulfill a particular structural task. In order to do this, the design must satisfy certain constraints, e.g. material failure and buckling must not occur anywhere within configuration. The objective varies depending on problem types and desired functions of problem. Critical buckling load capacity of stiffened

plates can be increased to very high values by using properly dimensioned stiffened plate elements. This may be achieved by the use of structural shape optimization (SSO) procedures in which the shape and thickness of the structure are varied to achieve a specific objective satisfying certain constraints. Such procedures are iterative and involve several re-analyses before an optimum solution can be achieved. SSO tools can be developed by the efficient integration of structural shape definition procedures, automatic mesh generation, structural analysis, sensitivity analysis and mathematical programming methods. The algorithm which was successfully used for optimal design of axisymmetric and prismatic shell structures (Hinton and his co-workers, 1993) is improved in other studies. The basic algorithm for SSO based on the following algorithm: (i) Problem definition: Consider the case of the SSO of a panel structure in which we wish to maximize the critical buckling load subject to the constraints that the total volume of the panel should remain constant and first ten buckling loads should be greater than critical buckling load. Other types of constraints such as bounds on the design variables must also be introduced. (ii) Shape definition: The shape of the panel cross-section is defined in some convenient form that allows us to examine the sensitivities of the design to small changes in shape. Here, we describe the geometry of the plate cross-section using parametric cubic spline segments with the coordinates specified at certain key points. The thickness distribution may also be defined using cubic splines with thickness values specified at the key points. (iii) Create finite strip model: The next step is to generate a mesh of suitable FSs. Here, we use an unstructured mesh generator with mesh density specified at some key points and then interpolated through the segments appropriately. In order to ensure the accuracy of the FS model, it is necessary to make sure the refinement does not occur during the analysis in each of the optimization iteration. This means that, the strip size distribution (mesh density) remains unchanged during redesign. As the structural shape changes during the optimization process, the re-meshing is based on predetermined mesh density of each iteration, with normal FS analysis; we must also define the boundary conditions and material properties. (iv) Finite strip analysis: Next we carry out a FS analysis and in the present work, the structure is modeled using cubic, variable thickness, Mindlin-Reissner, C(0) FSs described in Özakça et al. (2006). Critical buckling loads are calculated for each panel. (v) Sensitivity analysis: The sensitivities of the buckling loads and volume of the current design are then evaluated. These design sensitivities are generally nonlinear implicit functions of the design variables and therefore difficult and expensive to calculate. Haftka and Adelman (1989) have given an excellent survey paper on

Taysi 899 the sensitivity analysis of static, transient, and eigen value problems. The numerical accuracy of sensitivity analysis affects the search directions that are used in optimization algorithms. In this study, we use the semi analytical method to calculate sensitivities. (vi) Optimize parameters: Using the objective and constraint functions and their derivatives, the sequential quadratic programming (SQP) optimization algorithm is employed to optimize the parameters or design variables. No effort has been made to study the mathematical programming methods used for structural optimization procedures and the SQP algorithm is used here essentially as a ‘black box’. The new set of values will result in a modified design. Furthermore, the constraints must be satisfied if the new design was deemed and acceptable. If the convergence criteria for optimization algorithm are satisfied, then the optimum solution has been found and the solution process is terminated. (vii) Update optimization model: After optimization, it is necessary to update the geometric model, that is, the coordinates and/or thicknesses of the primary design variables in structural optimization. This is the only part of the original input data which has to be updated with each optimization iteration, if no convergence has been achieved, the new geometry is sent to the mesh generator which automatically generates a new analysis model and the whole process is repeated from step 2. STRUCTURAL OPTIMIZATION OF STIFFENED PLATE The buckling optimization of two types of conventional stiffened plates used by aerospace industry which are named as profile A and profile B is considered. To initiate the optimization, the initial geometry of the structure must be generated. Figure 1 shows Baseline Design (BL) of the investigated of two types of stiffened plates; profile A and profile B. Width and length of plates and height of stiffeners will be constant during optimization procedure. The profiles A and B have constant volume constraints of 691480 - 578280 mm3, respectively.

The FS analysis method is used in structural optimization algorithm. FS method can analyze structures which are simply supported on diaphragms at two opposite edges with the remaining edges arbitrarily restrained, and the cross section does not change between the simply supported ends. For the sake of convenience, we adopt the following notation to describe the boundary conditions of the plates optimized in this study. A/B/C/D, which implies (boundary conditions on side y = 0)/ (boundary conditions on side x = a)/ (boundary conditions on side y = b)/ (boundary conditions on side x = 0), Figure 2 shows these boundary conditions on stiffened plate with 3 stiffeners. A simply supported edge is represented by S, a clamped edge by C and a free edge is represented by F. The stiffened plates are optimized by following boundary conditions:

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(a) Profile A (b) Profile B Figure 1. Investigated stiffened plates (All dimensions in mm).

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Figure 2. Boundary conditions.

(a) S/F/S/F supports, (b) S/S/S/S supports, (c) S/C/S/C supports. The loaded sides of plates are simply supported. The plates are loaded under uniformly distributed compre-ssion force in stiffeners direction. Force applied as a

uniformly distributed compressive load over the plate and stiffeners. Note that uniform compressive load is redistributed according to the changes in element dimensions during optimization process. The stiffened plates are made of aluminum alloy which is the preferred material for aerospace industry. The following material properties are used: Poisson’s ratio 33.0=ν and

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Table 1. The lower and upper bounds of thickness design variables. Abbreviation Min (mm) Max (mm) Thickness of plate skin tskin 1.2 3.0 Thickness of stiffener tstiff 1.2 3.0

modulus of elasticity 29 /1073 mNE ×= . Based on the type of the design variables, the stiffened plates are optimized for following cases: (i) Only size optimization (TO): Design variables are thicknesses of plate and stiffener. (ii) Only shape optimization (LO): Design variables are the location of stiffeners. (iii) Size and shape optimization (LTO): Design variables are thicknesses of plate and stiffeners and location of stiffeners. The lower and upper bounds of thickness design variables which are used for both profiles A and B optimizations are shown in Table 1. The limits of shape design variables are shown in Figure 3. Note that, to maintain the symmetry, the distance from symmetry axes to right and left stiffeners are forced to equal linking. The thicknesses of plate skin (tskin) and stiffeners (tstiff) will be kept equal in baseline design. Their initial values are computed by satisfying constant cross-sectional area. Various numbers of stiffeners are also considered in order to observe the effect of number of stiffeners on critical buckling loads. In present study, numbers of stiffeners are considered as: (i) Three stiffeners plate, (ii) Four stiffener plate, and (iii) Five stiffener plate. Discussion of optimization results The FS method is inexpensive semi analytical method. Due to this advantages, very fine mesh of cubic FSs are used in analyses. The optimization is carried out for two profile types, three different boundary conditions, three different types of design variables and three stiffeners cases. The total 54 stiffened plates are optimized. The optimum critical buckling loads are presented in Tables 2 - 7 for different numbers of stiffeners and profile types. The initial and optimum values of design variables, critical buckling loads and percent improvements of profiles are investigated and the following results can be obtained: (i) For the case involving only size optimization, the improvements in the critical buckling loads are small for all boundary condition. Two thickness design variables are used in optimization. Thicknesses of skin and stiffener have minor effects on the critical bucking loads. The range of percentage improvement is between

1.04-10.79% for all stiffened plate considered. (ii) For the case involving only shape optimization, the significant improvements are obtained for all boundary conditions. The improvements in critical buckling loads are minimum of 7.52% and maximum of 73.50%. The optimization results indicate critical buckling loads which are so sensitive to location of stiffeners. (iii) The best improvement is obtained in optimizations when size and shape design variables are defined together. It means that, increasing the number of design variables in optimization process causes increased of the critical buckling loads. But higher number of design variables requires more computational time. (iv) The locations of stiffeners mainly depend on the boundary conditions. The stiffeners are placed near to the edge and the distance between the stiffeners are large at the S/F/S/F supported stiffened plates. On the other hand, the stiffeners are placed near to the center of the plate at the S/C/S/C supported stiffened plates. (v) The effect of number of stiffeners on critical buckling load is shown clearly from Tables 2 - 7. When the number of stiffeners increases, the critical buckling loads also increase in the initial and optimum solutions. (vi) In order to observe the size (dimension of plate) effect on the optimization, the optimization is repeated in profile A and B types where only the dimensions (the length and width of the stiffened plate) are different. The improvements in the critical buckling loads show similar trends for profile A and B. Parametric study A parametric study is carried out in order to investigate the effect of stiffeners locations on the buckling parameters. To observe the variation in critical buckling loads due to stiffener location, a series of analysis are done. In these analyses, skin and stiffeners thicknesses are kept constant and the positions of stiffeners are changed systematically. The parametric study is carried out for two profile types, three different boundary conditions, and three stiffeners cases. And totally, 18 stiffened plates are analyzed. The critical buckling loads are presented in Figures 4 - 9 for different numbers of stiffeners and profile types. Maximum critical buckling loads are compared with the optimization results. Discussion of parametric studies The parametric study shows that location of stiffeners has

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(a) Three stiffener

(b) Four stiffener

(c) Five stiffener

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Figure 3. Design variables of stiffened plates for various numbers of stiffeners.

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Table 2. Baseline design and optimization results of profile ‘’A” with three stiffeners. S/F/S/F S/S/S/S S/C/S/C BL TO LO LTO BL TO LO LTO BL TO LO LTO tskinn 2.237 2.434 2.237 2.434 2.237 2.398 2.237 2.365 2.237 2.434 2.237 2.370 tstiff 2.237 1.200 2.237 1.200 2.237 1.387 2.237 1.566 2.237 1.200 2.237 1.534 S1 293.33 293.33 317.47 317.44 293.33 293.33 221.69 218.11 293.33 293.33 205.22 204.20 Pcr 59.73 62.78 64.25 72.64 90.78 98.34 137.02 144.30 91.53 101.41 158.81 166.81 % imp 5.10 7.57 21.61 8.33 50.94 58.95 10.79 73.50 82.25

Table 3. Baseline design and optimization results of profile “B” with three stiffeners.

S/F/S/F S/S/S/S S/C/S/C BL TO LO LTO BL TO LO LTO BL TO LO LTO tskin 2.177 2.326 2.177 2.322 2.177 2.273 2.177 2.235 2.177 2.342 2.177 2.211 tstiff 2.177 1.439 2.177 1.456 2.177 1.699 2.177 1.884 2.177 .1359 2.177 2.003 S1 245.33 245.33 264.71 267.00 245.33 245.33 185.68 184.32 245.33 245.33 171.17 169.90 Pcr 66.08 70.41 73.35 77.36 100.45 104.72 155.31 159.01 101.28 110.11 175.58 180.93 % impr. 6.55 11.00 17.07 4.25 54.61 58.30 8.72 73.36 78.64

Table 4. Baseline design and optimization results of profile “A” with four stiffeners.

S/F/S/F S/S/S/S S/C/S/C BL TO LO LTO BL TO LO LTO BL TO LO LTO tskin 2.123 2.325 2.123 2.308 2.123 2.267 2.123 2.220 2.123 2.300 2.123 2.227 tstiff 2.123 1.342 2.123 1.396 2.123 1.556 2.123 1.742 2.123 1.539 2.123 1.715 S1 110.00 110.00 117.04 116.54 110.00 110.00 88.32 87.92 110.00 110.00 82.70 82.56 S2 330.00 330.00 348.66 351.03 330.00 330.000 265.02 262.78 330.00 330.00 248.17 243.64 Pcr 97.56 107.54 110.24 118.91 135.66 144.65 191.57 194.46 136.35 148.08 214.63 219.12 % impr. 10.23 12.99 21.88 6.63 41.21 43.34 8.60 57.41 60.70

Table 5. Baseline design and optimization results of profile “B” with four stiffeners.

S/F/S/F S/S/S/S S/C/S/C BL TO LO LTO BL TO LO LTO BL TO LO LTO tskin 2.059 2.173 2.059 2.146 2.059 2.097 2.059 2.014 2.059 2.160 2.059 2.005 tstiff 2.059 1.643 2.059 1.740 2.059 1.922 2.059 2.225 2.059 1.689 2.059 2.257 S1 92.00 92.00 98.19 99.42 92.00 92.00 65.80 75.37 92.00 92.00 62.65 69.14 S2 276.00 276.00 291.81 295.12 276.00 276.00 214.01 223.46 276.00 276.00 202.56 209.10 Pcr 106.87 112.09 121.06 124.03 148.54 150.35 200.51 206.01 149.35 155.65 225.68 231.21 % impr. 4.88 13.28 16.06 1.22 34.99 38.69 4.22 51.11 54.81 a huge impact on critical buckling loads. The results of parametric studies show good agreement with the optimum solutions presented in Table 2 - 7. The parametric studies and optimization results verify the necessity of the stiffeners in plate structures for improvement of the buckling load.

CONCLUSION In this paper, structural optimization and parametric study are carried out to observe the effect of various parameters on critical buckling load. These parameters are stiffeners location, number of stiffeners, types of

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Figure 4. Buckling load of profile “A” stiffened plate with three stiffeners.

Table 6. Baseline design and optimization results of profile “A” with five stiffeners. S/F/S/F S/S/S/S S/C/S/C BL TO LO LTO BL TO LO LTO BL TO LO LTO tskin 2.021 2.195 2.021 2.173 2.021 2.146 2.021 2.106 2.021 2.1784 2.021 1.843 tstiff 2.021 1474 2.021 1.542 2.021 1.628 2.021 1.752 2.021 1.525 2.021 2.580 S1 176.00 176.00 184.48 182.32 176.00 176.00 145.06 145.63 176.00 176.00 135.98 141.02 S2 352.00 352.00 367.90 368.75 352.00 352.00 295.15 291.26 352.00 352.00 276.02 282.31 Pcr 136.55 148.32 155.42 163.72 178.30 185.65 232.75 234.87 178.88 188.44 252.93 255.23 %impr. 8.62 13.82 19.90 4.12 30.54 31.73 5.34 41.40 42.68

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Figure 5. Buckling load of profile “B” stiffened plate with three stiffeners.

Table 7. Baseline design and optimization results of profile “B” with five stiffeners. S/F/S/F S/S/S/S S/C/S/C BL TO LO LTO BL TO LO LTO BL TO LO LTO tskin 1.955 2.108 1.955 1.957 1.955 1.905 1.955 1.600 1.955 1.976 1.955 1.663 tstiff 1.955 1.794 1.955 1.948 1.955 2.103 1.955 2.999 1.955 1.894 1.955 2.813 S1 147.20 147.20 157.36 155.12 147.20 147.20 100.79 127.35 147.20 147.20 90.78 118.15 S2 294.40 294.40 309.74 308.51 294.40 294.40 230.82 255.83 294.40 294.40 220.83 240.11 Pcr 147.94 151.283 168.02 168.44 181.83 193.32 224.98 245.52 195.09 197.12 247.91 278.36

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Figure 6. Buckling load of profile “A” stiffened plate with four stiffeners.

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Figure 7. Buckling load of profile”B” stiffened plate with four stiffeners.

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Figure 8. Buckling load of profile “A” stiffened plate with five stiffeners.

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Figure 9. Buckling load of profile “B” stiffened plate with five stiffeners.

910 Sci. Res. Essays design variable, boundary conditions and plate dimension. From the present study the following conclusions can be drawn: (i) Increase in the critical buckling load is small when only size optimization is done. However, shape optimization gives better results. The best improvement is obtained in optimizations when size and shape design variables are defined together. (ii) The locations of stiffeners mainly depend on the boundary conditions. The stiffeners are placed near to edge at the S/F/S/F boundary condition and away from the edge at the S/C/S/C boundary condition. As expected S/C/S/C boundary condition gives higher critical buckling loads. It means that, when the plate is well supported critical buckling load increases. (iii) The main increase in the critical buckling load is obtained when the large numbers of stiffeners are used. (iv) The parametric study and optimization study show that, the critical buckling loads are very sensitive to the location of stiffener. REFERENCES Bedair OK (1997). Influence of stiffener location on the stability of

stiffened plates under compression and in-plane bending. Int. J. Mech. Sci. 39: 33-49.

Brubak L, Hellesland J, Steen E (2007). Semi-analytical buckling strength analysis of plates with arbitrary stiffener arrangments. J. Constr. Steel Res. 63: 532-543.

Chajes A (1974) Principles of structural stability theory. Englewood Cliffs N.J. Prentice-Hall.

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sensitivity Analysis. Structural Optimization 1: 137-151. Iuspa L, Ruocco E (2008). Optimum topological design of simply

supported composite stiffened panles via genetic algorithm. Computers and Structures 86: 1 718-1737.

Kumar SYV, Mukhopadhyay M (1999). A new finite element for buckling analysis of laminated stiffened plates. Composite Structures 46: 321-331.

Liu YG, Pavlovic MN (2007). Elastic stability of flat rectangular plates under patch compression. Int. J. Mech. Sci. 49: 970-982.

Özakça M (1993). Analysis and Optimal Design of Structures with Adaptivity. Ph. D. thesis, Department of Civil Engineering University College of Swansea, U.K.

Özakça M, Murphy A, Veen SVD (2006). Buckling and post-buckling of sub-stiffened or locally tailored aluminum panels. 25th International Congress of the Aeronautical Sciences.

Peng LX, Liew KM, Kitipornchai S (2006). Buckling and free vibration of stiffened plates using the FSDT mesh-free method. J. Sound Vibration 289: 421-449.

Riks E (2000). Buckling and post-buckling analysis of stiffened panels in wing box structures. Int. J. Solids Struct. 37: 6795-6824.

Sheikh IA, Grodin GY, Elwi AE (2002). Stiffened steel plates under uniaxial compression. J. Constr. Steel Res. 58: 1061-1080.

Timoshenko S, Gere J (1936). Theory of Elastic Stability. McGraw-Hill, New York.

Todoroki A, Sekishiro M (2008). Stacking sequence optimization to maximize the buckling load of blade-stiffened panels with strength constraints using the iterative fractal branch and bound method. Composites Part B: Engineering 39: 842-850.

Vörös MG (2009). Buckling and free vibration analysis of stiffened panels. Thin Walled Structures 47: 382-390.