The paper deals with design optimization of sandwich structure made of laminate outer layers
and PUR foam core. The thickness of outer layers with the known fibre orientation angle of
individual laminae, referred to as the thickness variable, will be used as design variable. The
optimization problem with displacement constraint will be formulated to minimize the weight of
sandwich with laminate outer layers. The design is optimized using continuous design variable

Kormaníková, Eva

Transactions of the VŠB - Technical University of Ostrava Civil Engineering Series

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97 Sborník vědeckých prací Vysoké školy báňské - Technické univerzity Ostrava řada stavební, ročník 15, číslo 2, rok 2015 článek č. 12 Eva KORMANÍKOVÁ1 DESIGN OPTIMIZATION OF SANDWICH PANELS Abstract The paper deals with design optimization of sandwich structure made of laminate outer layers and PUR foam core. The thickness of outer layers with the known fibre orientation angle of individual laminae, referred to as the thickness variable, will be used as design variable. The optimization problem with displacement constraint will be formulated to minimize the weight of sandwich with laminate outer layers. The design is optimized using continuous design variable.  Keywords Optimization, sandwich panel, maximum displacement criterion, thickness design variable, weight objective function.  1 INTRODUCTION The optimization of a composite plate is important analysis for design of structures ranging from aircrafts to civil engineering structures.  The design optimization problem of current interest is the minimization of the weight function for a sandwich composite plate. This is a design optimization problem which optimizes the thickness of the sandwich layers to give the minimum weight. Of greater interest to current study are the works on the design optimization of composite sandwich plates where the thickness of outer layers and the core are taken as the design variables.   2 SANDWICH THEORY The typical sandwich structure compounds of three layers.  The outer layers are made of a material that has high strength (fiber reinforced laminates), which can transfer axial forces and bending moments, while the core is made of lightweight materials such as foam, alder wood etc. The material used in sandwich core must be resistant to compression and capable of transmitting shear. The thin cover sheets, i.e. the layers 1 and 3, have the thicknesses h1 for the lower skin and h3 for the upper skin. The thickness of the core is h2 (Fig. 1). In a general case, h1 does not have to be equal to h3, but in the most important practical case of symmetric sandwiches h1 = h3. Most sandwich structures can be modeled and analyzed using the shear deformation theory for laminate plates [1-3].  For the resultants N and M the integration is carried out over the sheets only and for the transverse shear force over the core. The constitutive equations for a sandwich are written in the hypermatrix form  γκεA000DC0BAVMNs0  (1)                                                                                                                                                                    1  Doc. Ing. Eva Kormaníková, PhD., Department of Structural Mechanics, Institute of Structural Engineering, Faculty of Civil Engineering, Technical University of Kosice, Vysokoskolska 4, 042 00 Kosice, Slovak Republic, phone: (+421) 55 602 4168, e-mail: eva.kormanikova@tuke.sk. 98 where N, M, V  are the vectors of normal forces, bending moments and transverse shear forces, respectively and γκε ,,0 are the vectors of  mid-plane strains, curvatures and transverse shear strains, respectively. The stiffness coefficients are calculated as  ,)3()1( ijijij AAA         )1()3()2(21 ijijij AAhB  , (2)  )3()1(ijijij CCC  ,       )1()3()2(21 ijijij CChD  , (3)  ;)2(hEA sijsij   i,j = 4,5 (4) where sijE  is the transverse shear modulus of the core.  3 SIZING OPTIMIZATION  The optimization process is applied to the approximate problem represented by the polynomial approximation. The coefficients of the polynomial function are determined by the least squares regression.  For regression analysis the singular value decomposition is used. When the objective function and constraints are approximated and their gradients with respect to the design variables are calculated based on chosen approximation, it is possible to solve the approximate optimization problem.  One of the algorithms used in the optimization module is called the Modified Feasible Direction method (MFD). The solving process is iterated until convergence is achieved. It is important to distinguish the iteration inside the approximate optimization from the loop in the overall optimization process. Convergence of MFD to the optimum is checked by criteria of maximum iterations and criteria of objective function changes.  Besides the previously mentioned criteria, the Kuhn-Tucker conditions necessary for optimality must be satisfied. Convergence or termination checks are performed at the end of each optimization loop in general optimization. The optimization process continues until either convergence or termination occurs.  4 MODELING AND SOLUTION OF SANDWICH PLATE  For the numerical solution the simply supported panel with laminate facings was used [4, 5]. Panel length is L = 3750 mm, nominal width is B = 1000 mm. Thickness of the facings is h1 = h3 and core is h2 (Fig. 1). On the panel affects uniform static wind load with intensity of 2 kPa in the bending plane. The laminate Carbon/epoxy facings are composed of eight identical thickness layers of a symmetric laminate [0/±45/90]s.  It was considered the carbon fibres in epoxy matrix, while unidirectional laminate layer has characteristics: Ef = 230 GPa; Em = 3 GPa; f  = 0.2; m = 0.3; Vf = 0.6; ρk = 1580 kg/m3.  Sandwich core, consisting of polystyrene, has material constants: EP = 16 MPa;  νP = 0.3; ρP = 150 kg/m3.  Laminate properties were determined by homogenization techniques [4, 12]. Computational program MATLAB was used to calculate the effective material properties of laminate facings. Numerical solutions were conducted through the COSMOS/M program. STAR module for solving linear static was used for calculations. There were used finite elements of the type SHELL4L. These 99 are the 4-node multi-layer quadrilateral elements with membrane and bending response and can be enter up to fifty layers.     h3       h2     h1  Laminate facings   Fig. 1: Scheme of sandwich structure Design optimization problems can be written as follows:  Optimization problem 1:    1hGF X  min   [N] 01.0101 14   h [m] 1.02 h  [m] 0  w0.0375   [m] Optimization problem 2:    2hGF X  min   [N] 2.0101 22   h [m] 001.01 h  [m] 0  w0.0375   [m] Optimization problem 3:    21,hhGF X  min   [N] 002.0105 14   h  [m] 2.0105 22   h    [m] 0  w0.0375   [m] The initial values and bounds of design variables, constraints and the objective function are shown in the Table 1 for optimization problem 1. Tab. 1: Summary of results of the optimization problem 1 Optimization parameters Initial values  Final values Tolerance  Design variable h1 [m] 0.001 5.683∙10-4 1∙10-5 Objective function G [N] 573.75 568.983 1∙10-3 Constraint w [m] 0.02378 0.0375 3.75∙10-4   1                                 2                                 3                                  4                                5                                6      0.001  0.00095  0.00091  0.00086  0.00082  0.00078  0.00073  0.00069  0.00065  0.0006  0.00056  Fig. 2: Variation of design variable h1 [m] during the optimization process 1 100 The initial values and bounds of design variables, constraints and the objective function are shown in the Table 2 for optimization problem 2. Tab. 2: Summary of results of the optimization problem 2 Optimization parameters Initial values Final values Tolerance  Design variable h2 [m] 0.1 7.755∙10-2 1∙10-5 Objective function G [N] 573.75 447.445 1∙10-3 Constraint w [m] 0.02378 0.0375 3.75∙10-4    1                                         2                                          3                                          4                                         5       0.1  0.09562  0.09125  0.08687  0.08250  0.07812  0.07375  0.06937  0.06500  0.06062  0.05625  Fig. 3: Variation of design variable h2 [m] during the optimization process 2 The initial values and bounds of design variables, constraints and the objective function are shown in the Table 3 for optimization problem 3. Tab. 3: Summary of results of the optimization problem 3 Optimization parameters Initial values Final values Tolerance  Design variable h1 [m] 0.001 0.002 1∙10-5 Design variable h2 [m] 0.158 0.05067 1∙10-5 Objective function G [N] 900 341.31 1∙10-3 Constraint w [m] 0.01092 0.0375 3.75∙10-4 0.002  0.00185  0.00171  0.00156  0.00142  0.00128  0.00113  0.00099  0.00085  0.00070  0.00056  1         2         3         4        5          6        7         8         9         10       11       12      13       14       15       16       17  Fig 4: Variation of design variables h1 [m] during optimization process 3 101  0.158  0.14787  0.13774  0.1276  0.11747  0.10734  0.09720  0.08707  0.07694  0.06680  0.05067  1         2         3         4        5          6        7         8         9         10       11       12      13       14       15       16       17  Fig. 5: Variation of design variables h2 [m] during optimization process 3    Fig. 6: Contour plot of deflections w before and after the optimization process 3   Fig. 7: Contour plot of stresses xz  at the bottom of core layer before and after the optimization process 3  6 CONCLUSIONS The first order shear laminate theory was used by the FEM analysis of the problem [5-11]. The problem was formulated as a minimum weight of simply supported rectangular sandwich plate subject to deflection constraint in the middle of the plate. Design variable were thicknesses h1 and h2 of sandwich layers. The optimal problem was solved using SLP and MFD method with maximum 70 iterations in each own optimization loop. In the Figs. 2-5 there are depicted variations of design variables during the optimization processes 1, 2 and 3, respectively. Initial and final values of optimization processes 1, 2 and 3 are shown in the Tables 1, 2 and 3, respectively. Contour plot of 102 deflections w and stresses xz at the bottom of the core layer before and after optimization process 3 are illustrated in the Figs. 6 and 7, respectively. The most general optimization procedure is optimization process 3 with two design variables, where designer can optimize whole thickness of sandwich panel within to take into account both constraints for thicknesses h1 and h2. There was not taken into account a hygrothermal effect of environment. Only static analysis under mechanical loading was performed. ACKNOWLEDGMENT This work was supported by the Scientific Grant Agency of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences under Project VEGA 1/0477/15. REFERENCES [1] GÜRDAL, Z., HAFTKA, R.T. and HAJELA, P. Design and Optimization of Laminated Composite   Materials, J. Wiley & Sons, 1999. [2] BARBERO, E. J. Finite element analysis of composite materials, CRC Press, USA, ISBN-13: 978-1-4200-5433-0, 2007. [3] MIHALIKOVA, M. [et al.] Influence of strain rate on automotive steel sheet breaking, Chemické listy. Vol. 105, no. 17 (2011), pp. 836-837. - ISSN 0009-2770. [4] SYKORA, J. SEJNOHA and M. SEJNOHA, J. Homogenization of coupled heat and moisture transport in masonry structures including interfaces, Applied Mathematics and Computation, 219 (13), pp. 7275-7285, 2013. [5] HARABINOVA, S. and PANULINOVA, E. Properties of Aggregates of Steel-Making Slag, GeoConference on Energy and Clean Technologies: conference proceedings, Albena, Bulgaria – Sofia, Volume 2, 2014, pp. 199-202. [6] JENDZELOVSKY N. Analysis of the 3D state of stress of a concrete beam, Advanced Materials Research, Volume 969, pp. 45-50, 2014.  [7] KREJSA, M., JANAS, P., YILMAZ, I., MARSCHALKO, M. and BOUCHAL, T. The use of the direct optimized probabilistic calculation method in design of bolt reinforcement for underground and mining workings, The Scientific World Journal, Volume 2013, Article number 267593, 2013. [8] ZMINDAK, M., PELAGIC, Z. and Bvoc, M. Analysis of high velocity impact on composite structures, Applied Mechanics and Materials, Volume 617, pp. 104-109, 2014.  [9] KRALIK, J. Optimal design of npp containment protection against fuel container drop, Advanced Materials Research, Volume 688, pp. 213-221, 2013. [10] MELCER, J. and LAJCAKOVA, G. Comparison of finite element and classical computing models of reinforcement pavement, Advanced Materials Research, Vol. 969, pp. 85-88, 2014.  [11] KOLÁŘ, V. and NĚMEC, I.: The Efficient Finite Element Analysis of Rectangular and Skew Laminated Plates. International Journal for Nurnerical Methods in Engineering. Vol. 7, No. 3, 1973. IJNMBH 7(3) 241-412. [12] SEJNOHA, M. AND ZEMAN, J. Micromechanical modeling of imperfect textile composites, (2008)  International Journal of Engineering Science, 46 (6), pp. 513-526. Reviewers: Doc. Ing. Jozef Dický, PhD., Department of Structural Mechanics, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Slovakia. Doc. Ing. Ivan Němec, CSc., Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Czech Republic. 

Design optimization of sandwich panels

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Design optimization of sandwich panels

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