Influence of stiffeners on a variable stiffness laminate optimisation Master of Science Thesis Faculty of Aerospace Structures and Computational Mechanics R.P.J. Laan I NFLUENCE OF STIFFENERS ON A VARIABLE STIFFNESS LAMINATE OPTIMISATION M ASTER OF S CIENCE T HESIS by R.P.J. Laan in partial fulfillment of the requirements for the degree of Master of Science in Aerospace Engineering at the Delft University of Technology, to be defended publicly on Monday March 21, 2016 at 2:00 PM. Thesis committee:: Dr. M. M. Abdalla, Dr. S. R. Turteltaub, Dr. S. Teixeira De Freitas, Ir. D. M. J. Peeters, TU Delft TU Delft TU Delft TU Delft This thesis is confidential and cannot be made public until March 21, 2016. An electronic version of this thesis is available at http://repository.tudelft.nl/. Copyright © Aerospace Structures & Computational Mechanics All rights reserved. D ELFT U NIVERSITY OF T ECHNOLOGY D EPARTMENT OF A EROSPACE S TRUCTURES & C OMPUTATIONAL M ECHANICS The undersigned hereby certify that they have read and recommend to the Faculty of Aerospace Engineering for acceptance a thesis entitled I NFLUENCE OF STIFFENERS ON A VARIABLE STIFFNESS LAMINATE OPTIMISATION by R.P.J.L AAN in partial fulfillment of the requirements for the degree of M ASTER OF S CIENCE Dated: March 21, 2016 Committee chairman: dr. M. M. Abdalla Committee members: dr. S. R. Turteltaub dr. S. Teixeira De Freitas ir. D. M. J. Peeters Acknowledgements In this thesis I would like to thank a lot of people who supported me during my study and this thesis. First, I would like to thank Mr. Mostafa Abdalla for giving me this project and for his support during the project. Special thanks go to Mr. Daniël Peeters who helped me by guiding me through the entire project. Without his support and time he invested in me, the thesis would have been a lot harder to complete. Thanks for answering all the questions I had, at literally every time of the day. I would like to thank my family and friends who mentally supported me when times were difficult. They made my study a lot easier. I would particular like to thank my parents Jan Laan and Margriet Laan-Dam. They supported me in every decision I made. Last but not least, I would like to thank the Thesis committee for grading my work. iv Summary Engineers are still looking for ways to improve the performance of materials used in aircraft. A major step was going from metal parts to laminates in order to save weight significantly. Those laminates existed of fibres with a constant orientation, and became laminates with changing fibre directions. Variable stiffness laminates (VSL) are used for its possibility of exploiting the anisotropic properties of composites. In order to improve the buckling load of a panel, VSL are used. Those laminates steer the loads through the laminate. This is beneficial for maximising the buckling load of a panel. An optimiser is built by the Delft University of Technology to optimise VSL by changing its fibre orientations throughout the entire panel. A manufacturing constraint is added to the optimiser to prevent the fibre paths to have a turn radius of smaller than 0.333[m]. Plate designs with fibres paths having a smaller turn radius than this constraint, are assumed to be unmanufacturable. The software optimises fibre paths of VSL against buckling and keeps the stiffness of the entire plate equal or higher than the quasi isotropic design. Since the aircraft of nowadays are stiffener dominated, an extra functionality is added to the optimiser. The optimiser is further developed to be capable of optimising stiffened plates. Stiffeners are taken into account during the optimisation. Extra functionalities such as integrated stiffeners are introduced. These are stiffeners sharing layers with the plate. The extended optimiser is verified by comparing the internal loads from five cases with the values calculated with ABAQUS. After having the optimiser verified, it is used to calculate results for twelve different cases. Those cases range from square unstiffened plates to rectangular plates with two stiffeners. The results show the influence of stiffeners on the optimised panel with changed fibre paths. Those cases are compared to the plate designs with the stiffeners excluded from optimisation. Performance increases of up to 174% are obtained for balanced symmetric laminates. The results show clearly that when stiffeners are added to a panel, the area in-between the stiffeners can be treated as a simply supported plate. Integrated stiffeners also show a buckling load increase of 113% with respect to the quasi isotropic case. The fibres of the buckling optimised plates show a relating behaviour. Fibre orientations going towards [45,-45] are present in buckling critical areas. Near stiffeners and near edges, [0,0] areas are present in order to introduce the load through the structure. A relation is found between having more [0,0] orientations in the inner plies of the laminate and significantly more [45,-45] orientations at the outer plies of the laminate. The outer plies are more critical for buckling, which requires more fibres being close to a [45,-45] in order to higher the buckling load. The inner areas are less vulnerable for buckling, which creates the opportunity of creating stiffness in the inside of the laminate. v Contents Acknowledgements iv Summary v List of Figures viii List of Tables xi List of Abbreviations xiii Introduction 1 Optimal plate design 1.1 Variable stiffness laminates 1.2 Manufacturing constraint . 1.3 Integrated stiffeners . . . . 1.4 Optimisation . . . . . . . . 1.4.1 Objective function . 1.4.2 Ranking weights . . 1.4.3 State variables . . . 1.4.4 Design variables . . 1.4.5 Constraints . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 3 3 4 4 4 5 5 5 2 The initial optimiser 2.1 About the software . . . . . . . . 2.2 Inputs and outputs . . . . . . . . 2.2.1 Option inputs . . . . . . . 2.2.2 Expected results . . . . . 2.3 Optimiser application . . . . . . 2.3.1 Mesh generation . . . . . 2.3.2 Model creation . . . . . . 2.3.3 Symmetry matrix . . . . . 2.3.4 Properties . . . . . . . . . 2.3.5 Response approximations 2.3.6 Sensitivities . . . . . . . . 2.3.7 Stopping criteria . . . . . 2.4 Developing the optimiser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 6 7 8 8 8 9 10 10 12 13 13 13 3 Stiffened panel 3.1 Single stiffener . . . . . . . . 3.1.1 Stiffener options . . . 3.1.2 Addition to the model 3.1.3 Optimisation . . . . . 3.2 Two or multiple stiffeners . . 3.3 Stiffener integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 14 14 15 16 16 17 . . . . . . . . . . . . vi CONTENTS 3.3.1 3.3.2 CONTENTS Fibre orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Verification 4.1 Input values . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Model constraints . . . . . . . . . . . . . . . . . . . . . . 4.3 Model verification . . . . . . . . . . . . . . . . . . . . . 4.3.1 Unstiffened plate . . . . . . . . . . . . . . . . . . 4.3.2 Single stiffened plate (center located) . . . . . . . 4.3.3 Single stiffened plate (off-center located) . . . . . 4.3.4 Double stiffened plate (Symmetrically located) . 4.3.5 Double stiffened plate (Asymmetrically located) 4.3.6 Crippling . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 5 Optimisation results 5.1 Initial values . . . . . . . . . . . . . . . . . 5.2 Results . . . . . . . . . . . . . . . . . . . . . 5.2.1 Unstiffened plate . . . . . . . . . . . 5.2.2 Single stiffened plate (center) . . . . 5.2.3 Single stiffened plate (off-center) . . 5.2.4 Double stiffened plate (symmetric) . 5.2.5 Double stiffened plate (asymmetric) 5.2.6 Rectangular plates . . . . . . . . . . 5.2.7 Shared layers . . . . . . . . . . . . . 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 18 . . . . . . . . . . 19 21 21 21 22 24 25 26 27 28 28 . . . . . . . . . . 29 29 30 30 31 32 33 33 34 38 40 Conclusion & recommendations 42 Bibliography 43 Appendices 45 A Laminate properties A.1 Engineering constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Material Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Lamination Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 46 46 46 B ABD-matrix 47 C Frobenius inner product 48 D FEM 49 E Verification plots E.1 Unstiffened plate . . . . . . . . . . E.2 Single stiffened plate (Center) . . . E.3 Single stiffened plate (Off-center) . E.4 Double stiffened plate (Symmetric) 51 51 54 57 60 . . . . . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS CONTENTS E.5 Double stiffened plate (Asymmetric) . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 63 List of Figures 1.1 1.2 Schematic drawing of a laminate variable stiffness laminate . . . . . . . . . . . . . . . A laminate with a stiffener integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Schematic drawing of a balanced laminate layer with steered fibres (left), and the fibre directions per node (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic drawing of a Mesh with 17x17 nodes (left) and 11x11 nodes plus element symmetry in two directions (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Software Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive (Counter Clockwise) element definition . . . . . . . . . . . . . . . . . . . . . . Standard constraint on the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The distances between the center of the laminate and the ply borders[19] . . . . . . . Placing the fibres under an angle with respect to the loading[19] . . . . . . . . . . . . 2.2 2.3 2.4 2.5 2.6 2.7 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 A schematic drawing of a plate with stiffener model . . . . . . . . . . . . . The model definition with a Main part and Sub parts . . . . . . . . . . . . Element definition, ’226 - 242 - 227’ . . . . . . . . . . . . . . . . . . . . . . A basic model with multiple stiffeners . . . . . . . . . . . . . . . . . . . . . A visualisation of an integrated stiffener . . . . . . . . . . . . . . . . . . . . A schematic drawing of two integrated stiffeners showing one layer with the fibre directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7 8 9 9 10 11 . . . . . . . . . . as . . 14 15 16 16 17 The 4 verification cases with 6 models in total . . . . . . . . . . . . . . . . . . . . . . . Nxx of the Optimiser (Left) and Nxx of ABAQUS (Right) . . . . . . . . . . . . . . . . Nyy of the Optimiser (Left) and Nyy of ABAQUS (Right) . . . . . . . . . . . . . . . . Absolute error between both models, unstiffened plate Nxx (Left) Nyy (Right) . . . . . Relative error between both models, unstiffened plate Nxx (Left) Nyy (Right) . . . . . The absolute error between both models, single stiffened plate center Nxx (Left) Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The relative error between both models, single stiffened plate center Nxx (Left) Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute error between both models, single stiffened plate off-center Nxx (Left) Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The relative error between both models, single stiffened plate off-center Nxx (Left) Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute error between both models, double symmetric stiffened plate Nxx (Left) Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The relative error between both models, double symmetric stiffened plate Nxx (Left) Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absolute error between both models, double asymmetric stiffened plate Nxx (Left) Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The relative error between both models, double asymmetric stiffened plate Nxx (Left) Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 22 22 23 23 ix . . . . . . . . . . . . . . . . . . . . arrows . . . . 2 4 17 24 24 25 25 26 26 27 27 LIST OF FIGURES LIST OF FIGURES 4.14 Buckling of model one at Pcr = 5.6e05[N ] with a stiffener height of 0.05[m] (left) Crippling of the stiffener from model two at Pcr = 2.79e05[N ] with a stiffener height of 0.15[m] (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 Fibre direction per node (Left) fibre paths of a laminate layer (Right) . . . . . . . . . Optimised fibre paths of an unstiffened plate. From the top layer (Left up) to the laminate center(Right below) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimised Fibre paths of a center stiffened plate. From the top layer (Left up) to the laminate center (Right below). The red line shows the stiffener location . . . . . . . . Optimised fibre paths of an off-center stiffened plate. From the top layer (left up) to the laminate center laminate (right below). The red line shows the stiffener location . Optimised fibre paths of a symmetric double stiffened plate. From the top layer (Left up) to the laminate center (Right below). The red line shows the stiffener location . . Optimised fibre paths of an asymmetric double stiffened plate. From the top layer (left up) to the laminate center (right below). The red line shows the stiffener location . . . Unstiffened panel (left) Stiffened panel (right) both 0.5[m] by 1[m] . . . . . . . . . . . Optimised fibre paths of an unstiffened rectangular plate of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right) . . . . . . . . . . . . . . . . . . . . . . Optimised fibre paths of a double stiffened rectangular plate with 4 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red line shows the location of the stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimised fibre paths of a double stiffened rectangular plate with 8 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red line shows the location of the stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimised fibre paths of a double stiffened rectangular plate with 12 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red line shows the location of the stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimised fibre paths of an double stiffened rectangular plate with 16 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red line shows the location of the stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimised fibre paths of an double stiffened square plate with 16 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left top) to the laminate center (Right bottom). The red and green lines show the location of the stiffeners. Green also means that the plies go into the stiffener and the fibre orientations at that location will be the fibre orientation of the stiffener layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimised fibre paths of an double stiffened rectangular plate with 16 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red and green lines show the location of the stiffeners. Green also means that the plies go into the stiffener and the fibre orientations at that location will be the fibre orientation of the stiffener layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 A drawing of master and slave nodes with their adjacent nodes (Left) and a drawing of the forces going from adjacent nodes to the master and slave nodes . . . . . . . . . . . E.1 The displacements of the unstiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . . E.2 The displacements of the unstiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . . x 28 29 30 31 32 33 34 34 35 35 36 37 37 38 39 50 51 51 LIST OF FIGURES LIST OF FIGURES E.3 The section forces of the unstiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . E.4 The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . . . E.5 The absolute error of the unstiffened plate section forces . . . . . . . . . . . . . . . . . E.6 The relative error of the unstiffened plate section forces . . . . . . . . . . . . . . . . . E.7 The displacements of the center stiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . . E.8 The displacements of the center stiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . . E.9 The section forces of the center stiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . E.10 The section forces of the center stiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . E.11 The absolute error of the center stiffened plate section forces . . . . . . . . . . . . . . E.12 The relative error of the center stiffened plate section forces . . . . . . . . . . . . . . . E.13 The displacements of the off-center stiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . E.14 The displacements of the off-center stiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . . E.15 The section forces of the off-center stiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . E.16 The section forces of the off-center stiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . E.17 The absolute error of the off-center stiffened plate section forces . . . . . . . . . . . . . E.18 The relative error of the off-center stiffened plate section forces . . . . . . . . . . . . . E.19 The displacements of the unstiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . . E.20 The displacements of the unstiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . . E.21 The section forces of the unstiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . E.22 The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . . . E.23 The absolute arror of the unstiffened plate section forces . . . . . . . . . . . . . . . . . E.24 The relative arror of the unstiffened plate section forces . . . . . . . . . . . . . . . . . E.25 The displacements of the unstiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . . E.26 The displacements of the double asymmetric stiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . E.27 The section forces of the unstiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . E.28 The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . . . E.29 The absolute arror of the unstiffened plate section forces . . . . . . . . . . . . . . . . . E.30 The relative arror of the unstiffened plate section forces . . . . . . . . . . . . . . . . . xi 52 52 53 53 54 54 55 55 56 56 57 57 58 58 59 59 60 60 61 61 62 62 63 63 64 64 65 65 List of Tables 2.1 Material properties used throughout the entire Thesis . . . . . . . . . . . . . . . . . . 7 4.1 4.2 The used material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ply orientations of the plate and stiffener(s) . . . . . . . . . . . . . . . . . . . . . . . . 21 21 5.1 5.2 Initial ply orientations of the plate and stiffener(s) . . . . . . . . . . . . . . . . . . . . The improvements of each square design compared to: 1. Stiffeners excluded from optimisation and added afterwards 2. The QI design . . . . . . . . . . . . . . . . . . . The improvements of each rectangular design compared to: 1. Stiffeners excluded from optimisation and added afterwards 2. The QI design . . . . . . . . . . . . . . . . . . . 30 5.3 xii 40 41 List of Abbreviations AFP CTS econn FEA g H MPC QI VSL Automated Fibre Placement Continuous Tow Steering Element connectivity matrix Finite Element Analysis Gradient Hessian Multi Point Contraints Quasi Isotropic Variable Stiffness Laminates xiii Introduction Designing a light weight aircraft is still one of the design objectives of aircraft engineers. The first composite-dominated planes were the B-787 and the A400M. Those were optimised by replacing metal parts by laminates. The further development of composite structures has become a trending topic. Although much progress has already been made[2], many areas are still not studied regarding composite optimisation. Many different ways of optimising composite laminates are done[3][4]. A significant weight decrease could be achieved when the fibres are having variable directions throughout the entire laminate [5]. Using those Variable Stiffness Laminates (VSL) increases the freedom in design and gives control to fine-tune materials such that local design requirements can be met [6]. The Delft Technical University developed an optimiser to optimise the fibre paths of a variable stiffness laminate. The optimiser is developed to optimise unstiffened plates against buckling. Since aircraft of nowadays are stiffener dominated, the capability of adding stiffeners to the optimisation cycle will be implemented. It is expected that when the stiffeners are also taken into account during the optimisation of the plate, its design is improved even more. This report will discuss the ’Influence of stiffeners on a variable stiffness laminate optimisation’. In order to study this influence, the existing developed optimiser will be extended. A stiffener will be added to the input of the optimiser. A new way of defining the properties will be created. Without touching the optimisation algorithm, the optimiser will be capable of showing the results of a plate optimised against buckling under influence of a stiffener. Chapter 1 discusses VSL, manufacturing constraints, and integrated stiffeners in order to make the reader familiar with those concepts. Chapter 2 discusses the initial state of the optimiser used for the research to the influence of stiffeners on variable stiffness optimisations. Chapter 3 shows the steps taken in order to implement the functionality of stiffener addition to the optimiser. The verification will be done in chapter 4. After the verification the results will be shown in chapter 5 and a conclusion will be drawn. In the end, the overall conclusion and recommendations will be discussed. 1 1 Optimal plate design The use of laminate materials is a very trending topic in the industry of aircraft design. Variable stiffness laminates (VSL) were discovered as a result from optimising laminate material. This is discussed in the first section. The Delft technical university developed an optimiser in order to optimise VSL by changing fibre paths. To keep VSL designs practical, a manufacturing constraint is introduced. This will be discussed in section two. The third section discusses another way to improve aircraft designs by integrating stiffeners. The fourth section discusses the general optimisation problem applied on designing a VSL. 1.1 Variable stiffness laminates Variable stiffness materials are, as the name already reveals, materials that do not have the same stiffness throughout the entire material. Variable stiffness materials can be controlled or uncontrolled. An example of an uncontrolled variable stiffness material is wood. Wood can be from the same tree which does not guarantee that the piece of material has the same properties throughout the entire wooden structure. Usually those stiffness changes are not immense. VSL are a type of controlled variable stiffness material. Controlled variable stiffness materials can be made of fibres and resin. The reason why VSL are used is the possibility of exploiting the anisotropic properties of composite materials to a larger extent than was previously possible[7]. The variations of fibre directions can be used to give beneficial load and stiffness distribution patterns. For buckling of composite panels it is proven that VSL can be very effective[8]. An example of a VSL can be found in figure 1.1, which is a VSL plate with a hole optimised against buckling. It can be noticed that the different areas of the laminate contain fibres in different directions ranging from -5 to -89 degrees. The buckling load has increased by 126% compared to a quasi isotropic panel [5]. Fig. 1.1: Schematic drawing of a laminate variable stiffness laminate 2 CHAPTER 1. OPTIMAL PLATE DESIGN 1.2 1.2. MANUFACTURING CONSTRAINT Manufacturing constraint Although the laminate is optimised against buckling (figure 1.1), it can be concluded that the design is not really practical because of its orientation jumps from one area to another. Those borders between two orientation areas are extremely critical for crack growth. It is thereby strongly recommended to use continuous fibres in variable stiffness laminates. A disadvantage in this case is the maximum angle the continuous fibre can make on a certain distance. So it is impossible to go from -89 degrees to -67 degrees on such a small distance as has been shown in the figure. Using continuous fibres for the already shown example would therefore not be possible. So when optimisers are optimising a VSL, its manufacturability needs to be taken into account[9]. Described by Peeters (et al.)[10], a local and global steering constraint is required for designing a continuous fibre laminate. The optimiser is thereby bounded by the constraint of obeying a minimum radius of curvature for the fibre paths. The Automated Fibre Placement (AFP) technique described by Kim (et. al)[11], makes it possible to lay fibres in any direction of choice. Inefficiencies like tow gaps and tow overlap can be prevented using Continuous Tow Steering(CTS). With both methods, the fibres need to be continuous[12]. As long as the fibres are continuous and stay within the steering constraint, it will be accepted as manufacturable design. The maximum steering angle can be represented by ζU . The following constraint can be set up: ζ 2 − ζU2 ≤ 0 (1.1) The squared form is taken because the angles can be negative as well. Changing this to an optimisation problem, together with equation 1.1 it will become: Minimize Such that max (f1 , f2 , ..., fn ) fn+1 , ..., fm ≤ 0 ζ 2 − ζU2 ≤ 0 Where f1 to fn are representing the structural responses that are optimised and fn+1 to fm are constraints. 1.3 Integrated stiffeners Laminate structures do not only exist of flat plates and basic stiffeners. According to Renton (et al.), in 2023 most composite structures are expected to be fully integrated with minimal assembly parts. The same goes for using stiffeners in aircraft. Those stiffeners will be integrated where possible. In this case, stiffeners could be integrating with the plate by using shared layers between the plate and the stiffener. Such an integrated stiffener is shown in figure 1.2. The gray layers are the layers which only continue in the plate, the yellow and pink layers are shared between the plate and stiffener, and the blue layers are only from the stiffener. The advantages of stiffener integration are[13]: • Integrated parts do no have to be assembled which means a reduce of assembly time/costs • The structure is stronger with respect to the cases of assembling a stiffener with bolts One of the main disadvantages is that the design is harder to fabricate which means that it costs more money. So a trade-off must be made between short term costs and long term costs. Weight optimisation applies to a integrated structure and less maintenance is required. Although as has been said, the production costs are rising with integrated stiffeners. 3 1.4. OPTIMISATION CHAPTER 1. OPTIMAL PLATE DESIGN Fig. 1.2: A laminate with a stiffener integration 1.4 Optimisation It needs to be clear that during the thesis, the optimisation algorithm stays untouched. But in order to understand the implementation of adding a stiffener to the optimiser, this section is written. In this thesis a plate will be optimised by changing the fibre paths in order to change the stiffness of the plate. The manufacturability is assured by adding the manufacturing constraint. Those inputs will be explained in this section. Optimisation can be described as "finding an optimal solution by changing the parameters". It has to be taken into account that there are requirements which needs to be satisfied. Those requirements can exist of "finding a maximum or minimum", and it can consist of constraints which limit the design space of the solution. This can for example be that a certain parameter is not allowed to be bigger than another certain parameter. In general an optimisation problem is defined to be a mathematical problem[14]. The design in a mathematical form is shown in equation 1.2. This equation is used to compare different designs in order to rank them[15, 16, 17]. hj = n X wi · fi (x, y) (1.2) i=1 Where: fi (x, y) wi x y 1.4.1 Is Is Is Is called the objective function the weight given for one of the ranking criteria. a vector of all state variables (which are constant inputs) a vector of all the changeable design variables (change during optimization) Objective function The outcome of the objective function (f1 (x, y)) represents the score of the variables for a certain design aspect. The objective function gives a score for how the design variables (x) together with the state variables (y) are suitable for this design. The objective functions of the optimiser discussed in this thesis is the buckling load for the first two buckling modes. The critical load for buckling is maximised during the optimisation. 1.4.2 Ranking weights The weights (wi ) of a function to rank the designs (hj ) are determined by the user of the optimization algorithm. In this thesis the weights are only given to the buckling objective. If multiple objectives would have been chosen, multiple weights should be given according to the importance of a given objective. 4 CHAPTER 1. OPTIMAL PLATE DESIGN 1.4.3 1.4. OPTIMISATION State variables The state variables (x) can be easily described as variables that initialize the environment. Those variables are all fixed during the optimisation. Those state variables are called such because they determine the state of the system before changes are applied. Of course those variables are allowed to be changed, but that means that the optimization process must be done again because what is optimal in one environment does not have to be optimal in another. The state variables in the optimiser are the material properties of the fibres, design dimensions such as plate width, applied loads, and the boundary conditions. When the stiffeners are added, all those input variables will be added to the state variables as well. They will not be changing during the optimisation but only be influencing the end result. 1.4.4 Design variables An optimisation problem exists of multiple types of design aspects. As mentioned above, the problem contains design variables (y). The numerical values of those variables will be used to get to the optimal solution. Those values are changed in order to find a better solution. In this optimisation problem, the design variables are the fibre angles. By changing the fibre angles, the design will score higher or lower on the objective function. The design variables define the design space which exist of all possible combinations of the design variables. Depending on the type of constraints, the design space is preferably inside the limitations set by the constraints. When strict limitations are set, the design is not allowed to get behind those limitations, but the user can also apply a ranking penalty when a constraint is not met. 1.4.5 Constraints Constraints are representing the limitations of the design. When a potential solution does not fall into the limited design space, two options can be chosen as a user of the optimization algorithm. The first is that every design which falls outside the design space surrounded by the constraints, is invalid so the ranking score for this design is equal to zero. The second option is to give this design such a penalty that potential designs which do not fall behind this constraint, have an advantage regarding to its position in the ranking sequence. Constraints exists in two different forms. Equality constraints and inequality constraints. The equality constraints requires a function or variable to be equal to another variable, function or constant. Mostly written as (=). The inequality constraints requires a function or variable to be unequal to another variable, function or constant. This can be in three ways, those are often represented by the mathematical symbols of unequal (6=), bigger than (>) and smaller than (<). Of course combinations can be made as well such as, equal to or bigger than (≥) and equal to or smaller than (≤). In some cases, those formulae are written separately in an equality constraint and an inequality constraint. The already discussed manufacturing constraint is a combination of both. Another constraint in this thesis is the optimised panel design needs to be at least as stiff as a QI design. So: Edesign ≥ EQI 5 (1.3) 2 The initial optimiser This chapter will describe the initial state of the optimiser. Discussing the modules of the optimiser will give a clear insight in the software. Some aspects must be known first in order to understand the further development of the optimiser which is going to be discussed in chapter 3. To start with section 2.1, the purpose of the software and its history will be discussed. Section 2.2 will go into a little more depth regarding the possibilities of the software and explains the inputs and expected outputs. After that, the different steps will be shown in a schematic flow chart and each of the important modules will be explained in the sub-sections 2.3.1-2.3.6. 2.1 About the software It is difficult to retrieve the exact start date when the first parts of code were written, but since some modules are dated from 2008 it can be assumed the first part are written at least 7 years ago. This software package exists of multiple modules which will be discussed in Section 2.3. It is developed on the TU Delft in order to create an optimisation tool for optimising VSL against buckling by changing the angles of the fibres in the different layers. Fig. 2.1: Schematic drawing of a balanced laminate layer with steered fibres (left), and the fibre directions per node (right) The software is capable of changing the angle of the fibres inside the layer on multiple locations. In other words, the same layer can contain fibres with multiple directions throughout the length of the composite. An example is given in figure 2.1a. As can be seen, the fibres are steered throughout the laminate. In order to determine those angle changes, the laminate is divided in elements and nodes. On every node the angle of the fibres can be different. In figure 2.1b, the different fibre angles per node are visualised with arrows. In between nodes, the average angle change is taken in order to meet the angle of the fibres which are required on the next node. The continuity of the fibres and the maximum steering angle must be taken into account since the fibres can not turn an infinite angle in an infinite small distance [10]. The multiple possibilities of the software will be explained in the next sections. 2.2 Inputs and outputs This section discusses the multiple options that the user has in order to get the wanted outputs. As has been discussed in chapter 1, state variables and design variables need to be defined. Those belong to the options of the user. 6 CHAPTER 2. THE INITIAL OPTIMISER 2.2.1 2.2. INPUTS AND OUTPUTS Option inputs At first, the user is deciding on what exactly needs to be optimised. Things like how accurate the final answer must be, and which limitations the design has, are put in the software. The accuracy of the calculations is related to the density of the mesh. Together with the symmetry properties of the elements, a mesh is generated. In figure 2.2a, an example can be found for a dens mesh of 17 by 17 nodes. Figure 2.2b shows a wider mesh and with two symmetry axis for the elements. Every node contains a vector of all the layer orientations on that location. Throughout the optimisation, the angles of the different layers change but the nodes and elements stay on the same place. Only the properties of those elements and nodes are changing during the optimisation. 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 1 -1 1 0.8 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 0.6 1 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 Fig. 2.2: Schematic drawing of a Mesh with 17x17 nodes (left) and 11x11 nodes plus element symmetry in two directions (right) The next choice for the user is about the number of layers the design will have. Each of these layers get an initial angle which is constant along the entire layer at the start of the optimisation. This means that all the nodes are having the same layer orientations at the start. After the first optimisation iteration, this will already be changed. At every node, the fibre angles will be changing towards an optimal design. The user specifies which layers are allowed to change in order to give the optimiser its design space. Other things determined by the user are the design options like having a balanced laminate, the lengths in x-axis and in y-axis, and the steering constraints. It has to be kept in mind that the laminates designed with the optimiser will always be symmetric. The final things determined by the user are the parameters for the fibre material and the thickness of each layer. The standard properties of the material can be found in table 2.1. Those properties are used throughout the entire thesis. All verification steps are done with those values. Table 2.1: Material properties used throughout the entire Thesis Material: E1 E2 G12 ν12 t AS4D/TC3500 1.54e11 [Pa] 1.08e10 [Pa] 4.02e9 [Pa] 0.317[-] 3e−4 [m] 7 2.3. OPTIMISER APPLICATION 2.2.2 CHAPTER 2. THE INITIAL OPTIMISER Expected results The expected output of the optimisation software is an optimised laminate with the number of layers kept the same. The angles of the fibres are optimised on every node and the angle differences between the nodes do not exceed the manufacturing constraint. The visual output for one layer has already been shown in figure 2.1. The output can be different for every layer. Since it is an optimisation process which does not cover all possible solutions, it is not assured that the best solution is found but an optimal solution. If more information is wanted about optimisation processes and algorithms, the reader is referred to the literature study[18]. The next section will discuss how to get from the inputs to the outputs. 2.3 Optimiser application This section will mainly be used to show and describe the working of the optimiser. A flow-chart is shown in order to make it easier for the reader to understand the process that the software is dealing with (see figure 2.3). As can be seen, multiple steps of the optimiser are visualised in the flow chart. Starting with the user inputs, and ending with the saved data containing the results. It can be noticed there is a feedback loop going from the stopping criteria to the property determination. This is a part of the optimisation phase. For every improved design, the optimiser needs to calculate the new properties, response and sensitivities. Those new values are used for the new optimisation iteration. 2.3.1 Mesh generation The mesh generation is done by dividing the square plate into triangles. The defined number of nodes and symmetry properties are determining the layout of the mesh. All lengths of the elements on each axis are equally divided. All those nodes are stored in a coordinate matrix and an element connectivity matrix. This element connectivity matrix contains all data about which nodes belong to which element. The elements must be saved such that the orientation of the element does not change. Fig. 2.3: Software Flowchart All elements are stored in a positive (counter clockwise) reference system. This is very important for determining the properties of a specific element. As can be seen in figure 2.4, element 1 is the triangular element containing the three nodes shown in the figure. The element is oriented in a counter clockwise direction, which means that the element is made of ’Node 1’-’Node 17’-’Node 2’ (in that exact order). 8 CHAPTER 2. THE INITIAL OPTIMISER 2.3. OPTIMISER APPLICATION Fig. 2.4: Positive (Counter Clockwise) element definition 2.3.2 Model creation After the generation of the mesh, the model is created. The model exists of a design, a load case and the constraints. Since the design is a square/rectangle, the design has four sides to constrain. In figure 2.5 the standard model is shown. As can be seen, the loaded edge, the edge along the X-axis and the one on the other side of the plate (parallel to the X axis at y = 0, y = L) are simply supported. In other words, at those edges: z=0 (2.1) The side parallel to the Y axis is constraint in all spatial directions. So x = y = z = 0. Fig. 2.5: Standard constraint on the model The load will be exerted on the side which is parallel to the Y axis. A constant strain load will be put on the entire edge using Multi Point Constraints (MPC). Those multi point constraints are also used for the other edge constraints. A MPC uses one master node and multiple slave nodes. The slave nodes are having the exact same displacement in x and y direction as the master node is having. Since the z direction is already constraint, all spatial directions of the slaves are equal to those of the master node. In this way, the load will be equally spread at all times during the optimisation. 9 2.3. OPTIMISER APPLICATION 2.3.3 CHAPTER 2. THE INITIAL OPTIMISER Symmetry matrix A symmetry matrix is a matrix which relates symmetric model parts. For a plate having symmetric loadings, symmetric properties and symmetric boundary conditions, all the elements on both sides of the symmetry axis can be related to each other. Using symmetry properties, only half of the plate needs to be calculated. Knowing this, the computational effort will be reduced significantly. Only the response for half of the plate is determined and its symmetric counterpart will have exactly the same response. So for example, the plate of figure 2.5 could be treated as such. The symmetry line could be drawn at Y = 0.5Ly (where Ly is the length of the plate in Y-direction). Then only the computations should be done on one side of the plate. The symmetry matrix creates a relation between symmetric parts of the model. Using half of the computational effort, the entire plate is analysed. The symmetry matrix is created in order to lower the computational effort before it goes into the finite element analysis. The symmetry matrix is created on a spatial point of view. The nodes are compared and processed searching for symmetry. The optimiser assumes the plate to be entirely symmetric with a symmetric load case. So the system assumes the output to be symmetric as well. Although this might save computational effort, it has to be noted that the assumption is not entirely correct! For example the laminate is symmetric regarding to the layer orientation (so the upper layers and lower layers are symmetric), and the laminate is balanced. At first sight, one would think that the load case is entirely symmetric so the response approximation would be as well. This is not true! As can be seen in figure 2.6, the distance of every layer to the middle of the laminate is different. This means that if a balanced symmetric laminate is made; for example [10, −10, 45, −45]S , the 10 laminate has a larger distance to the laminate center which requires this layer to carry slightly more load. So the plate will twist a little due to the bending phenomenon. So using the advantage of requiring less computational effort, has its price in accuracy. In this case, the accuracy is only affected by a very small amount. Fig. 2.6: The distances between the center of the laminate and the ply borders[19] 2.3.4 Properties The properties of the laminate are determined for every single node of the mesh. Every node can have different values due to different fibre angles per node. The response approximations will be done on the elements which will be discussed later. The A and D matrices are calculated for the membrane and bending properties using classical laminate theories[19]. Transforming elastic constants to engineering 10 CHAPTER 2. THE INITIAL OPTIMISER 2.3. OPTIMISER APPLICATION constants is part of the laminate theories: Qxx = Ex 1 − νxy · νyx (2.2) Qyy = Ey 1 − νxy · νyx (2.3) Where ν is the Poisson’s ratio, the Qyy is the stiffness of fibres in the direction of the force, and Qxx is the stiffness perpendicular to the force. The two other engineering constants are: Qxy = νxy Ey νyx Ex = 1 − νxy · νyx 1 − νxy · νyx (2.4) Qss = Gxy (2.5) These equations are all made for the case that the load is in the 1-direction. But for fibres, the force can also be under an angle (see figure 2.7). To go from those engineering constants to the right angle, Fig. 2.7: Placing the fibres under an angle with respect to the loading[19] the next equations are used (1 is the direction of the applied force): Q11 = m4 Qxx + n4 Qyy + 2m2 n2 Qxy + 4m2 n2 Qss (2.6) Where n = sin (θ) and m = cos (θ). The θ is the angle between the direction of the force and the direction of the fibre. The other 5 equations can be found in the Appendix A.1. In order to simplify the optimisation, the engineering constants are written as Material Invariants in a linking Matrix: U1 = (3Qxx + 3Qyy + 2Qxy + 4Qss ) 8 U1 U2 U4 0 0 0 U = U1 −U2 0 0 U5 0 0 0 U2 2 0 U2 2 0 h U3 0 −U3 0 0 U3 U3 0 0 −U3 −U3 0 V = 1 cos (2θn,l ) sin (2θn,l ) cos (4θn,l ) sin (4θn,l ) 11 (2.7) (2.8) iT (2.9) 2.3. OPTIMISER APPLICATION CHAPTER 2. THE INITIAL OPTIMISER Where θn,l is the node number and layer number respectively. The remaining U terms can be found in Appendix A.2. Calculating A and D matrices per node in the standard way: A = 2 · [U ][V ] (2.10) 2 +z ·z 2 zi+1 i i+1 + zi · [U ][V ] (2.11) 3 In order to predict the property changes in A and D for a changing angle at a node, the derivatives with respect to the angles are calculated as well: D =2· dA d[V ] = 2 · [U ] dθ dθ (2.12) z 2 + zi · zi+1 + zi2 dD d[V ] = 2 · i+1 · [U ] dθ 3 dθ (2.13) d2 A d2 [V ] = 2 · [U ] dθ2 dθ2 (2.14) 2 +z ·z 2 zi+1 d2 D d2 [V ] i i+1 + zi = 2 · · [U ] (2.15) dθ2 3 dθ2 are the distances between the ply heights and the center of the laminate (see figure Where zi and zi+1 2.6). In case of constant fibre paths per layer, all those calculations were made for each layer and not for each node. But since every node can now have different fibre angles, all those properties are calculated for every single node. After the properties of each node are determined, it goes to the response approximations module. 2.3.5 Response approximations The FEA takes a lot of computational effort. In order to save computational effort, approximations are made for the response. Those approximations are using less computational effort. With those approximations, a well defined direction for the optimiser can be computed in a very short time. If for every possible design the FEA needs to be done, it will take a lot more time. The general form of an approximated response looks like[10]: F (1) ≈ X φm : A−1 + φb : D−1 + ψm : A + ψb : D + C (2.16) n Where A and D are the in-plane and out-of-plane stiffness matrices (found in Appendix B), ’:’ is called the Frobenius inner product (Appenddix C), the φ and ψ are linear approximation terms calculated from sensitivities[20], and m and b denotes the membrane and bending part respectively. This function runs over all the nodes n. In order to perform the FEA, the global membrane/bending stiffness matrix must be known first. This matrix is determined using the A matrix, the D matrix, the model coordinates, and the model element data. The slave and master nodes along with the MPC play also a big role again (discussed in section 2.3.2). The force is divided along the edge of the plate. The MPC requires the entire edge to have the same spatial displacements. The entire calculation of the FEM can be found in Appendix D. 12 CHAPTER 2. THE INITIAL OPTIMISER 2.3.6 2.4. DEVELOPING THE OPTIMISER Sensitivities The sensitivity values are determined per node. Those values are used to indicate how large the effect will be on the properties of the laminate if the specific node undergoes some changes[20]. In this way the optimiser will be more efficient. The sensitivity of a node can be determined by calculating the second level approximation. The general form of the second level approximation is[10]: (1) F (2) ≈ F0 + g · ∆θ + ∆θT · H · ∆θ (2.17) Where g and H are the gradient and Hessian respectively. The gradient can be defined as: ∂F ∂θi ∂F ∂A ∂F ∂D = + · · ∂A ∂θi ∂D ∂θi ∂A ∂D = φm · + φb ∂θi ∂θi (2.18) ∂2F ∂θi ∂θj ∂ = ·g ∂θ ∂2A ∂2D ∂φm ∂A ∂φb ∂D = φm · + φb + · + ∂θi ∂θj ∂θi ∂θj ∂θj ∂θi ∂θj ∂θi (2.19) gi = The Hessian is calculated as followed: H= In order to be sure the Hessian is convex, the blue part of the equation is left out. Since the equation is already an approximation, and the function and gradient are equal at the approximation point, it is allowed to omit the blue part of the Hessian equation. This will give: H= 2.3.7 ∂φm ∂A ∂φb ∂D · + ∂θj ∂θi ∂θj ∂θi (2.20) Stopping criteria The optimiser is trying to optimise until one of the stopping criteria is met. Two stopping criteria are active for the optimiser. The first stopping criterion is: after at least 5 iterations, the new solution is not improved more than 0.1% (with respect to the critical load). The second stopping criterion is: a maximum of 100 iterations is reached. In both cases, the optimisation process will be ended. In every iteration, the process starts to determine the new properties and sensitivities again. It continues until at least one of the stopping criteria is met. 2.4 Developing the optimiser The optimiser is capable of optimising a VSL by changing the fibre paths in the laminate. Further developing the optimiser will make the software capable of optimising a VSL under influence of stiffeners. This chapter has shown the initial state of the optimiser. The next chapter will go into further detail of implementing one or multiple stiffeners to the optimiser. The knowledge gained in this chapter is used to further develop the optimiser with those additional features. 13 3 Stiffened panel As has been mentioned earlier, aircraft structures contain many stiffeners. Optimising composite structures with taking stiffeners into account as well, will result in even better optimised structures. The optimiser will be extended with the capability of adding a stiffer to the design, and take this into account during optimisation. This chapter discusses this new capability. Section one discusses how the implementation of a single stiffener on a plate in the software works. After having shown this implementation of a single stiffener to the optimiser, the second section will show the addition of multiple stiffeners to the optimiser. The third section discusses one step further; the implementation of the integrated stiffener functionality. 3.1 Single stiffener The first step is to design the optimiser for adding a single stiffener. Three topics regarding the implementation of a stiffener to the optimiser are discussed. The first topic is about the stiffener options. The second topic is how to add the stiffener to the model. The third topic is about how it is implemented in the optimisation. 3.1.1 Stiffener options The stiffener options are defined by the user. Starting with the number of stiffeners which need to be added. If this number is larger than zero, the remaining options will be defined. If the option of zero stiffeners will be chosen, the optimiser will work exactly the same as it would have when the optimiser was untouched. Which results in an optimised plate design without stiffeners. Many of the defined options of the unstiffened plate, are needed to be defined for the stiffener as well. The stiffener can be balanced, but is always symmetric. The spatial options are the dimensions of the stiffener, and the location of the stiffener. The mesh density in the z-direction is free to change, but the mesh density in the x-direction, is fixed to the mesh density of the plate. The stiffener will always be placed on a row of nodes of the plate (see figure 3.1). Invalid locations will be round off to the nearest row of nodes. The remaining options are the symmetry properties of the elements which are similar to the options of the plate. The dimensions are taken such that the length of the stiffener is equal to the width of the plate. This prevents having loose hanging parts of the stiffener. The height Fig. 3.1: A schematic drawing of a plate with stiffener model is taken to be 10% of the stiffener length (this can be changed by the user). So in the case of figure 3.1, the stiffener length is 0.5[m] in x-direction, zero in y-direction and 0.05[m] in z-direction. The material properties, ply thickness and volume fraction of the stiffener are defined to be equal to the plate. They are chosen by default. 14 CHAPTER 3. STIFFENED PANEL 3.1.2 3.1. SINGLE STIFFENER Addition to the model As has been discussed in section 2.3.2, a model of the plate is created. The stiffener will be added to this model. The plate is defined to be the main part, and the stiffener is going to be a sub-part. Together they are the model. This is similar to an object oriented environment where a class is created for the separate parts[21]. The plate is the main object and the stiffener is its sub-object. While every individual part is an instance of the class ’Model’, the stiffener could not ’survive’ without its main part: the plate. A visualisation of how the Main-Sub-model works can be found in figure 3.2. It shows that the stiffeners are a sub-part of the plate. They are all together the entire model. The reason for why this set up has been chosen is straight forward. The stiffener is in fact the same as an unstiffened plate. The optimiser treats stiffeners as separate plates now. Defining the properties and sensitivities for the stiffener goes similar to the ones defined for the plate. Reusing existing functions in the software saves the optimiser computational effort during optimisation and storage space since a plate concept is already known to the optimiser. The only differences between a stiffener and a plate are the boundary conditions on the edges. The small edges of the stiffener are having the same boundary conditions as the plate: simply supported. The stiffener edge touching the plate is clamped to the plate. The upper edge is free to move. Fig. 3.2: The model definition with a Main part and Sub parts The mesh of the stiffener is also build up of elements. Similar to the plate elements, the properties for the stiffener elements are calculated as well. In order to do FEA on the model, the elements of the plate and the elements of the stiffener must be treated as one model. The entire model with its elements must be merged to one part. The nodes of the stiffener must integrate with the nodes of the plate at the same location. Those merged nodes are called "connection nodes". A second model is created with all the elements of the stiffener and plate. Those elements are then connected via the connection nodes. This new model is called the "Fused model". In the fused model, all duplicates of nodes with the same coordinates are taken out. This model can now be used for the FEA using the properties gained from the initial model with main and sub parts. So the difference between the initial model and the fused model is that in the fused model it is merged to one piece while in the initial model there is still a difference between a plate and a stiffener. As has been discussed for the plate as well, the element definitions are very important. The plate elements discussed in section 2.3.1 are oriented counter clockwise. The stiffener elements need be 15 3.2. TWO OR MULTIPLE STIFFENERS CHAPTER 3. STIFFENED PANEL defined in the same way. In figure 3.3 it is shown that three nodes of an element are numbered. Since the Y -axis is going left, the X-axis is going right, and the Z-axis is going up, the counter clockwise orientation of the nodes are in the sequence of "226-242-227". Fig. 3.3: Element definition, ’226 - 242 - 227’ 3.1.3 Optimisation The default case of optimising a panel with a stiffener, is adding a stiffener with a predefined stacking sequence. This stacking sequence remains unchanged during the entire optimisation. Depending on its location, size and layers, the stiffener influences the buckling load significantly. The stiffness of the stiffener is directly put in the optimisation as being a support for the plate. For the buckling modes, the fused model (with the stiffener included) is analysed for its eigenvalues. The fused model can be analysed for buckling of the plate and for crippling of the stiffener. Since the plate and stiffener are merged to one model, for the optimiser there is no difference between both. 3.2 Two or multiple stiffeners In the case of two or multiple stiffeners, the model can be simply extended with extra stiffeners. The working of this process is similar to the single stiffener. All the stiffener options must be given for every additional stiffener. Those can be different from each other. As can be seen in figure 3.4, a plate with two stiffeners is shown. This is one of the ways, the stiffeners could be placed. Fig. 3.4: A basic model with multiple stiffeners 16 CHAPTER 3. STIFFENED PANEL 3.3 3.3. STIFFENER INTEGRATION Stiffener integration The next step for extending the optimiser is developing the stiffener integration capability. As has been discussed in chapter 1, those are stiffeners with shared layers from the plate to the stiffener. Figure 3.5 shows an example of a integrated stiffener. The two brown top layers of the plate, go into the stiffener. The blue stiffener layers are having a predefined stacking sequence. First the fibre orientations of the shared layers will be discussed. Then how this is implemented in the optimiser will be explained. Fig. 3.5: A visualisation of an integrated stiffener 3.3.1 Fibre orientations Sharing layers with the plate is partially restricting the stacking sequence of the stiffener. The stiffener is restricted to have the same stacking sequence as the plate has, at the connection nodes. Only the shared layers are restricted, the blue layers from figure 3.5 are not shared so those are not restricted. Those are predefined by the user. Figure 3.6 shows a composite layer of a plate with two stiffeners which are integrated. As can be seen, the fibre orientations are visualised by arrows. The blue arrows represent the fibre orientations of the plate. They have each its own orientation along the plate. The red arrows are visualising the fibre orientations of the stiffener. As can be seen, the red arrows above each other are having the same orientation as the blue arrow below them. The plate layer with a certain orientation, goes up and those fibre orientations remain the same as the last fibre orientation of the plate at the connection node. Fig. 3.6: A schematic drawing of two integrated stiffeners showing one layer with the arrows as fibre directions 17 3.3. STIFFENER INTEGRATION 3.3.2 CHAPTER 3. STIFFENED PANEL Optimisation The optimiser will be slightly adapted due to this extra functionality. The entire stacking sequence of the stiffener is no longer constant through the optimisation. The fibre orientations of the shared layers in the stiffener change when the underlying nodes of the plate are changing (figure 3.6). This means that the influence of those connection nodes on the model response is increased. When one of the fibre orientations at the connected nodes is changed, the entire column of all stiffener nodes above it are changing as well. Therefore the sensitivities of those connection nodes will be significantly increased. This is done by adding the gradient of the above laying nodes to the gradient of the connection node. This is the same gradient as has been discussed in section 2.3.6. gi = φ m · n ∂A ∂D X ∂D ∂A + φb + + φb +φm · ∂θi ∂θi j=1 ∂θj ∂θj Where ’i’ is the plate node and ’j=1,2,...,n’ are the nodes above the plate node. 18 (3.1) 4 Verification This chapter shows the verification process of the developed optimiser with its added features. Four main cases are verified. Some of those cases have two different configurations: 1. Unstiffened plate 2. Stiffened plate with one stiffener (at two locations) 3. Stiffened plate with two stiffeners (symmetrically placed and assymmetrically placed) 4. Crippling of a single stiffened plate The first three cases will be verified by comparing the displacement and section forces from the optimiser model with the ABAQUS model using the exact same mesh and same forces. The fourth case is verified by showing that the optimiser is taking crippling into account when it is expected to occur. It will be shown that for a specific case, crippling occurs at a lower load than plate buckling. The four cases with the six models can be found in figure 4.1. The determination of the Displacement is done during the FEA, and the section forces can be calculated using: Nxx h i εxx N = Nyy = A εyy (4.1) Nxy εxy Where ε is the strain and A is the A-matrix discussed earlier. Normally the stresses are used in order to verify such models. In this case the section forces are used since the stresses are changing from layer to layer in a laminate. The section forces are taken from all the layers together. A mean error of the models is calculated with the mean error of the vector lengths: n X 1 |NA |−|NM | = · n i=1,..,n |NA | δmean q n X 1 = · n i=1,..,n NA · NTA − q i q NM · NTM NA · NTA (4.2) i Where n is the number of elements, A means ABAQUS model, and M means MATLAB model. The input values are similar for each case and are discussed first. In the second section, plots will be shown in order to make the different cases comparable. All the results of the computed displacements and sectional forces can be found in Appendix E. Starting with the unstiffened plate, its sectional forces Nxx and Nyy will be discussed. After that, the absolute error and the relative error of the same two sectional forces are discussed. Since the absolute error and relative error of the sectional forces are of most interest, only those will be discussed per design case. For more plots, the reader will be referred to the appendix where the remaining sectional forces and the displacements of each model can be found. In the end of this chapter a conclusion will be drawn about the verification. 19 CHAPTER 4. VERIFICATION Fig. 4.1: The 4 verification cases with 6 models in total 20 CHAPTER 4. VERIFICATION 4.1 4.1. INPUT VALUES Input values The mesh for each model is similar. The plate has a mesh of 21x21 nodes and the stiffener(s) have a mesh of 3x21 nodes. The only exception is the model for verifying whether the optimiser is taking crippling into account. This stiffener mesh is 7x21 nodes since this stiffener has another height than the stiffeners in the other cases. The mesh is equally spread in every model. The dimensions of the plates are 0.5 [m] by 0.5 [m]. The Stiffeners are all 0.05 [m] except for the stiffener of the crippling verification which is 0.15[m]. The material properties can be found in table 4.1. Every case in this entire thesis uses a constant thickness for plies. Table 4.1: The used material properties E1 1.54e11[Pa] E2 1.08e10[Pa] G12 4.02e9[Pa] ν12 0.317[-] tply 3e-4[m] The layups of the plate and stiffener laminates can be found in table 4.2. Those lay-ups are chosen semi-randomly, it is preferred to have at least three layer orientations at least at 15 degrees difference. So although the choice of fibre orientations is random, they were not allowed to all point in the same direction. It has to be kept in mind that in this verification process, no fibre angle optimisations are done. Those constant ply orientations are predefined and non-changing throughout the entire verification. The fibre orientations are all with respect to the X-axis. Table 4.2: Ply orientations of the plate and stiffener(s) Plate Layup [10,-10, 20,-20, 80,-80, 60,-60, 30,-30, 70,-70]S 4.2 Stiffener Layup [45,-45, 30,-30, 60,-60, 0, 0]S Model constraints Every model has simply supported edges (as has been discussed in chapter 2). A force of 1[N] is exerted along the edge at x = L. A MPC is added for this edge in order to keep the constant strain along the edge. The short edge of the stiffener is also included in the MPC. The force will also be applied on the stiffener. 4.3 Model verification The six models discussed in the beginning of this chapter are compared in this section. The verification is done by comparing the sectional forces and the displacements during the load case. Those data are visualised in plots. The section force plots and the displacements plots can be found in Appendix E. The error between the optimiser model and the Abaqus model are of more interest, so those plots are shown and discussed in this chapter as well. The displacement plots will not be discussed in this chapter. By studying the response and section forces of the six different cases, a conclusion about the additional stiffener concept can be drawn. 21 4.3. MODEL VERIFICATION 4.3.1 CHAPTER 4. VERIFICATION Unstiffened plate The first model to be verified is the initial unstiffened plate. Starting with the section forces. Figure 4.2 shows the Nxx of the optimiser and the ABAQUS calculations. Figure 4.3 shows the Nyy of the optimiser and the ABAQUS calculations. Only for the plate, the section forces are shown. Since the errors between both models is of more interest, showing the sectional forces has only be done for the unstiffened plate. For the other cases, only the errors are shown in this chapter. Fig. 4.2: Nxx of the Optimiser (Left) and Nxx of ABAQUS (Right) Fig. 4.3: Nyy of the Optimiser (Left) and Nyy of ABAQUS (Right) The error between the optimiser and the Abaqus model is shown in figure 4.4, the relative error is shown in figure 4.5. As can be notices, the relative errors for the Nxx are very small. At most of the area, it is even within the 0.1% error. At some outliers, the error reaches 8% max. For the Nyy , the relative errors are significantly higher, although the absolute errors are not immense high. Especially the corners where probably force concentrations occur are having deviating values. Most of the errors are still within the 10% range. For the Nxy (which is shown in Appendix E.1) the errors are even worse. But this is also because of the relative forces. The forces calculated for Nxy are so small, that an error of 0.01 can result in a relative error of 1500%. A good example can be a vector of [0.01, 0.01, 1] with an error of [0.01, 0.01, 0.01]. In this example the error in the first two directions 22 CHAPTER 4. VERIFICATION 4.3. MODEL VERIFICATION are 100% while the error in the third direction is only 1%. The total error is very small because of the large third component. Since the direction of interest is the direction of the force (x-direction), the errors for Nxx can mainly be used to draw a conclusion. The mean error (equation 4.2) compensates for this occurrence by dividing the absolute error by the length of every [Nxx ,Nyy ,Nxy ]-vector. The so called mean error is: δ = 0.08%. This is enormously low because of the contribution of the very low errors of the Nxx . So this unstiffened plate is verified. Fig. 4.4: Absolute error between both models, unstiffened plate Nxx (Left) Nyy (Right) Fig. 4.5: Relative error between both models, unstiffened plate Nxx (Left) Nyy (Right) 23 4.3. MODEL VERIFICATION 4.3.2 CHAPTER 4. VERIFICATION Single stiffened plate (center located) The second model is the single stiffened plate with a stiffener placed at the center of the plate. The error plots can be found in figure 4.6 for the absolute error and figure 4.7 for the relative error. The stiffener is placed at y = 0.25[m]. The lay-up is equal to the other stiffeners and is discussed in the beginning of this chapter. The same as for the unstiffened plate, the errors of the Nxx are relatively low. It can be noticed that they are higher than for the unstiffened plate. The relative errors for Nyy are also significantly higher. As has been discussed, this is the result of absolute errors in a region where very small forces are determined. According to the mean error δ = 1.64%, and the responses (which can be found in Appendix E.2) the single center stiffened plate with its stiffener in the centre is verified. Fig. 4.6: The absolute error between both models, single stiffened plate center Nxx (Left) Nyy (Right) Fig. 4.7: The relative error between both models, single stiffened plate center Nxx (Left) Nyy (Right) 24 CHAPTER 4. VERIFICATION 4.3.3 4.3. MODEL VERIFICATION Single stiffened plate (off-center located) The off-center located stiffener is placed at 0.75 of the plate length. So at y = 0.375[m]. The error plots can be found in figure 4.8 for the absolute error en figure 4.9 for the relative error. The calculated mean error δ = 1.63%, is remarkably similar to the error calculated for the single center stiffened plate. The errors for Nxx and Nyy are also similar. The same conclusion for this case can be drawn as for the single center stiffened plate. The single off-center stiffened is verified as well according to the responses (Appendix E.3) and the mean calculated error. Fig. 4.8: The absolute error between both models, single stiffened plate off-center Nxx (Left) Nyy (Right) Fig. 4.9: The relative error between both models, single stiffened plate off-center Nxx (Left) Nyy (Right) 25 4.3. MODEL VERIFICATION 4.3.4 CHAPTER 4. VERIFICATION Double stiffened plate (Symmetrically located) The double stiffened plate has stiffeners located at 0.25 and 0.75 of the plate length. So at y = 0.125[m] and y = 0.375[m] respectively. The error plots can be found in figure 4.10 for the absolute error en figure 4.11 for the relative error. As can be seen, especially on the edges, the error is relatively higher than towards the center of the panel. Again the Nxx has relatively low errors, but also the relative errors are bigger than for the single stiffened plates. The mean error is calculated to be: δ = 2.62%. The errors are still within an acceptable range. The double symmetrically stiffened plate is verified according to the mean calculated error and its responses which can be found in Appendix E.4. Fig. 4.10: The absolute error between both models, double symmetric stiffened plate Nxx (Left) Nyy (Right) Fig. 4.11: The relative error between both models, double symmetric stiffened plate Nxx (Left) Nyy (Right) 26 CHAPTER 4. VERIFICATION 4.3.5 4.3. MODEL VERIFICATION Double stiffened plate (Asymmetrically located) The asymmetric double stiffened plate has stiffeners located at 0.5 and 0.75 of the plate length. So at y = 0.25[m] and y = 0.375[m] respectively. The error plots can be found in figure 4.12 for the absolute error en figure 4.13 for the relative error. The mean error is calculated to be: δ = 2.59%. The same as has been concluded for the case of a symmetrically double stiffened plate, this case is verified as well according to the responses (Appendix E.5) and the calculated mean error. Fig. 4.12: The absolute error between both models, double asymmetric stiffened plate Nxx (Left) Nyy (Right) Fig. 4.13: The relative error between both models, double asymmetric stiffened plate Nxx (Left) Nyy (Right) 27 4.4. CONCLUSION 4.3.6 CHAPTER 4. VERIFICATION Crippling The crippling verification is only meant to verify the fact that the optimiser software should account for crippling. The method chosen is creating two models with both almost identical dimensions. Both models are created in the optimiser. No ABAQUS model is required to verify the crippling capabilities of the optimiser. Both models have a plate of 0.5[m] by 0.5[m] and a stiffener placed in the centre of the plate at y = 0.25[m]. The height of the stiffener is the only difference between the two models. The first model has a stiffener height of 0.05[m] and the second model has a stiffener height of 0.15[m]. The stiffener from the second case, has an increased height to intentionally let the stiffener cripple earlier than the plate will buckle. So in order to verify the crippling functionality of the optimiser, the second model should buckle later because of the increased stiffness of the plate plus stiffener. But it should cripple even earlier than it should buckle. Using eigenvalue analysis on the stiffness matrices, the first has a critical load of Pcr = 5.6e + 05 and the second case has a critical load of Pcr = 2.79e + 05. From those two values it can already be concluded that the crippling is occurring since a stiffer stiffener is added to the second case but the critical load has decreased. After having the displacements plotted in figure 4.14, it can be concluded that those expectations were right. The displacements are shown larger than they really are, in order to visualise the crippling of the model in the figure. Fig. 4.14: Buckling of model one at Pcr = 5.6e05[N ] with a stiffener height of 0.05[m] (left) Crippling of the stiffener from model two at Pcr = 2.79e05[N ] with a stiffener height of 0.15[m] (Right) 4.4 Conclusion This chapter shows that each of the cases is verified. It can be concluded that those models have all some outliers in the different section forces but the mean calculated errors are within a range of 3%. The relatively high errors for Nyy and Nxy can be assumed normal because those sectional force values are really small as well. This makes a small absolute error, relatively big while this is not the case. The mean errors are compensating for this effect. Together with analysing the relative values of the biggest sectional force component (which is in those cases Nxx ), and the responses (which can be found in Appendix E) a valid conclusion can be drawn. All the errors of the main sectional force (Nxx ) stay within the 10% to -10% range while most of the errors are even within the 1% to -1% range. For crippling it is shown that the optimiser takes this into account. With the stiffener of an 0.05[m] height it clearly shows that the plate is buckling. With the case of a stiffer height of 0.15[m], the stiffener is crippling before the entire model buckles. Its critical load is lower in the second case since the stiffener starts to cripple earlier. So therefore, the optimiser is verified for adding one or multiple stiffeners to the plate. 28 5 Optimisation results After verification of the renewed optimiser, the optimiser is ready to obtain results. This chapter will discuss the results of the optimiser. Five main cases will be studied and discussed. Those cases are listed: 1. Unstiffened panel 2. Single stiffened panel (centered and off-centered) 3. Double stiffened panel (symmetric and asymmatric) 4. Double stiffened panel with changing stiffener layers 5. Integrated stiffeners (square and rectangular plate) The unstiffened case is used to show the influence of stiffeners by comparing the initial unstiffened case with the cases with stiffeners. The case with double stiffeners has multiple variations. The number of plies varies between the 0 and 16 plies in order to show the influence of adding and removing stiffener plies. Two cases for the integrated stiffeners will be studied. Integrated stiffeners mean that plies from the plate go into the stiffener as has been discussed in chapter 1.3. This will all be discussed in this chapter. The results can be shown in two ways (see figure 5.1); 1. the fibre directions at the nodes are shown with an arrow 2. the fibre paths through the laminate layer. The second way of showing the results is chosen because those are easier to imagine for the reader. It has to be kept in mind that all the results are showing two layers which are balanced as can be seen in the figure. All the designs are optimised for the first two buckling modes and being as stiff as the Quasi Isotropic configuration. In this case Quasi Isotropic is determined by putting all lamination parameters equal to zero. Fig. 5.1: Fibre direction per node (Left) fibre paths of a laminate layer (Right) 5.1 Initial values The same model constraints are valid as for the verification phase. The mesh is different from the verification phase and will be noted for every case. The local steering constraint is 333[mm], which means that the radius of curvature of the fibres needs to be at least 333 [mm]. The initial guess for ply orientations can be found in table 5.1. The initial values such as dimensions and stiffener locations 29 5.2. RESULTS CHAPTER 5. OPTIMISATION RESULTS will be specified for each case in the next section. All the stiffeners are the same size: 0.5[m] by 0.05[m]. The mesh density does change from case to case and will be mentioned per case as well. For the stiffener(s) there are always 3 nodes in the direction perpendicular to the plate and the number of nodes in the length direction are always equal to the number of plate nodes in the same direction. Table 5.1: Initial ply orientations of the plate and stiffener(s) Plate Layup [30, −30, 30,-30, 30,-30, 30,-30, 30,-30, 30,-30]S 5.2 Stiffener Layup [45,-45, 30,-30, 60,-60, 0, 0]S Results This section shows the result of optimised plates coming out of the optimiser. Since every laminated plate in this chapter is symmetric and balanced, only six pictures with layups are shown. Each laminate contains 24 plies. Every picture is showing a balanced layer (two plies at once), and the symmetric counterpart of the laminate is now shown since those layers are exactly the same. Only the top 6 balanced layers are shown and in the order of the top layer first (left up), then going to the right for the second layer and ending with the most inside balanced layer on the right bottom of each figure. Next to the layups, a small 3D visualisation of the model is shown in order to show the locations of the stiffeners. For every case, the buckling load will be given in order to draw conclusions on the increase or decrease of performance of the specified case. 5.2.1 Unstiffened plate The unstiffened plate contains 19x19 equally spread nodes on an area of 0.5[m] by 0.5[m]. The results can be found in figure 5.2. As can be seen in the figure, the layers do look similar but are not identical. It can be noticed that every layer has a large area of the layer which goes towards [45,-45].It is known from buckling analysis that [45,-45] layers are wanted in a design optimised against buckling[22]. Fig. 5.2: Optimised fibre paths of an unstiffened plate. From the top layer (Left up) to the laminate center(Right below) Since the outer plies have more influence on the buckling mode, the outer layers are more optimised 30 CHAPTER 5. OPTIMISATION RESULTS 5.2. RESULTS towards a [45,-45] laminate than the more inner layers. As can be seen, the inner layers have areas going towards [45,-45] as well but have more fibre directions going towards [0,0] since this is optimal for creating stiffness which is an optimising requirement. As has been discussed, the plate must be at least as stiff as the quasi isotropic plate. The buckling load of the quasi isotropic plate is: Pcr = 1.51e05[N ]. The buckling load of this optimise plate is: Pcr = 2.36e05[N ]. As can be noticed, the critical load has increased by 56.3% with respect to the QI case. This unstiffened plate will be taken as standard result in order to compare the other results with. The influence of stiffeners on this optimisation process will be discussed with respect to this standard case. 5.2.2 Single stiffened plate (center) The single stiffened plate with a stiffener located at the center of the plate is optimised for a mesh with 17x17 nodes on an area of 0.5[m] by 0.5[m]. The results can be found in figure 5.3. The red line in the figure shows the location of the stiffener. As can be seen, the first two plies are even closer to a [45,-45] orientation compared to the unstiffened plate (counting the plies from top of the laminate to the center of the laminate). The fourth, fifth and sixth ply are clearly showing the influence of the stiffener. Towards the location of the stiffener, the fibres are going towards 0 degrees (parallel to the stiffener) in order to introduce the direct loads. The areas between the stiffener and the outside edges are treated more or less as separate plates with each area optimised against buckling. The plies closer to the laminate center contain more fibre orientations closer towards [0,0]. Stiffness is mainly created in the center of the laminate and the [45,-45] degrees fibres are more wanted on the outside of the laminate. This is because the outside is more sensitive for buckling than the center of the laminate. Fig. 5.3: Optimised Fibre paths of a center stiffened plate. From the top layer (Left up) to the laminate center (Right below). The red line shows the stiffener location The single center stiffened plate has a critical load of Pcr = 7.9773e05[N ]. It would be useless to compare an unstiffened plate with a stiffened plate. Adding the exact same stiffener to the optimised unstiffened plate would give a good insight in the optimised performance. This critical load is: Pcr = 6.5413e05[N ]. This is an increase of 22.0%. In fact that is the difference between adding the stiffener to the optimisation procedure or only added after the optimisation has been done. The QI model has a Pcr = 6.3244e05[N ], which means an increase of 26.1%. 31 5.2. RESULTS 5.2.3 CHAPTER 5. OPTIMISATION RESULTS Single stiffened plate (off-center) The single stiffened plate with a stiffener on a quarter length of the plate is optimised for a mesh of 17x17 nodes on an area of 0.5[m] by 0.5[m]. The results can be found in figure 5.4. Those plies show clearly the influence of the stiffener regarding the separation of two areas. The stiffener is preventing the plate from buckling in the area where the stiffener is placed. This automatically creates two areas with non-identical properties. The larger area is still critical for buckling modes. This can be concluded from the results as well, the larger area consists of more [45,-45] area than the smaller area above the stiffener which contains more fibre orientations going towards [0,0]. Comparing the fourth, fifth and sixth layer (the three layers closest to the laminate center) with the same layers of the center stiffened plate, the larger area of the off-center stiffened plate shows more [45,-45]. This probably results from the fact that the buckling mode of the smaller area above the stiffener is far from critical regarding the buckling modes which makes it more optimal to design this area to create stiffness. This can be concluded from the plots as well. Fig. 5.4: Optimised fibre paths of an off-center stiffened plate. From the top layer (left up) to the laminate center laminate (right below). The red line shows the stiffener location The critical load for this single off-center stiffened plate is: Pcr = 7.5919e05[N ]. Adding the same stiffener at the unstiffened plate gives: Pcr = 4.8112e05[N ]. So, adding the stiffener to the optimising procedure gives an increase of 57.8% in the critical load. Compared to the critical load of the QI case having a Pcr = 3.7320e05[N ], which is an increase of 103%. 32 CHAPTER 5. OPTIMISATION RESULTS 5.2.4 5.2. RESULTS Double stiffened plate (symmetric) The double symmetric stiffened plate is optimised for a mesh of 17x17 nodes on an area of 0.5[m] by 0.5[m]. The results can be found in figure 5.5. It can be seen that also in this double stiffened case, the outer layer (layer one) is dominated by fibre going towards [45,-45] orientations. The remaining layers contain fibres which are going towards a [45,-45] orientation in the middle area, and going towards a [0,0] orientation on the outer two areas. As could already been seen in the figure of the single off-centred stiffened plate, the small areas between the edges and the stiffeners contain many fibres with an orientation going towards a [0,0] orientation because those areas are less sensitive for buckling modes. The results of this double symmetric stiffened plate will be used to compare to a larger panel discussed in section 5.2.6. Fig. 5.5: Optimised fibre paths of a symmetric double stiffened plate. From the top layer (Left up) to the laminate center (Right below). The red line shows the stiffener location The critical load of this double stiffened plate is: Pcr = 1.7666e06[N ]. Adding the stiffeners after the optimisation gives an critical load of: Pcr = 1.5804e06[N ]. That is an increase of 11.8%. The critical load for the QI case is: Pcr = 8.6987e05[N ]. This is an increase of 103%. 5.2.5 Double stiffened plate (asymmetric) The double asymmetric stiffened plate is optimised for a mesh of 17x17 nodes on an area of 0.5[m] by 0.5[m]. The results can be found in figure 5.6. As could be expected, the result of this double asymmetric stiffened plate is a combination of the single stiffened plates. The first layer (counting from top layer towards the laminate center) contain areas mainly orientated as [45,-45] and the second to the sixth layer the fibre orientations are more going to [0,0] orientations in the two upper areas in order to create stiffness. The smaller areas are again less vulnerable for buckling. The bigger area is mainly optimised against buckling. The critical load for this double asymmetric stiffened panel is: Pcr = 1.6765e06[N ]. Adding those stiffeners to the unstiffened plate gives: Pcr = 7.6551e05[N ]. This is an increase of 119%. The QI case has a critical load of Pcr = 7.8407e05[N ], which is an increase of 114%. 33 5.2. RESULTS CHAPTER 5. OPTIMISATION RESULTS Fig. 5.6: Optimised fibre paths of an asymmetric double stiffened plate. From the top layer (left up) to the laminate center (right below). The red line shows the stiffener location 5.2.6 Rectangular plates In order to study the optimising behaviour of the fibres under influence of stiffeners, the area inbetween two stiffeners is studied more close as well. The area between two stiffeners is often assumed to be a plate which is simply supported. In this section, a rectangular plate with two stiffeners is discussed. The area in-between the two stiffeners contains the same dimensions as the unstiffened square plate at the beginning of this chapter. In this way, the influence of the stiffener can be studied and the assumption of having a simply supported plate in-between the two stiffeners is discussed. A rectangular plate with 17x33 nodes and an area of 0.5[m] by 1[m] is optimised for five different cases. The only difference between the five cases is the stiffener design, having zero, four, eight, twelve, or sixteen plies. In this way, the influence of the stiffener can be studied in more detail by changing its stiffness. The stiffeners are located at 0.25 and 0.75 of the panel length. Starting with the unstiffened (zero plies) rectangular plate (see figure 5.7). Fig. 5.7: Unstiffened panel (left) Stiffened panel (right) both 0.5[m] by 1[m] 34 CHAPTER 5. OPTIMISATION RESULTS 5.2. RESULTS Unstiffened This unstiffened rectangular plate has fibres in slightly other directions than expected (see figure 5.8). The upper layer exists of an inner area with [0,0] and contains more [45,-45] areas on the sides. From this it can be concluded that due to the rectangular shape, the buckling phenomenon is less critical. In order to satisfy the constraint of having a stiff panel which is at least being as stiff as the QI panel, in all six layers the areas with [0,0] oriented fibres can be found. Going towards the center of the laminate, more [0,0] directions go to the upper and lower edge of the panel, and the directions going towards [45,-45] are going towards the center where buckling is more critical. The critical load of Fig. 5.8: Optimised fibre paths of an unstiffened rectangular plate of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right) the unstiffened rectangular plate is: Pcr = 2.3229e05[N ]. The QI rectangular plate with the same dimensions has a critical load of: Pcr = 1.1883e05[N ]. That is an increase of 95%. Double stiffened (four plies) This double stiffened rectangular plate has two stiffeners with each four plies of [45,-45,-45,45]. Comparing those six layers with the case of no stiffeners, the design contains less areas with [0,0] oriented fibres (see figure 5.9). Fig. 5.9: Optimised fibre paths of a double stiffened rectangular plate with 4 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red line shows the location of the stiffeners 35 5.2. RESULTS CHAPTER 5. OPTIMISATION RESULTS This means that the stiffeners add a rather large contribution to the stiffness of the plate which causes the layers to be optimised especially against buckling. Comparing those plies with the double stiffened square plate, it becomes clear that the stiffeners in this rectangular design are not preventing the entire panel from buckling. In this case the panel is already divided in three separate buckling areas. As the number of stiffener plies increase, it is expected that this separation of buckling areas will occur again but will be even more visible. The critical load of this rectangular plate with 2 stiffeners of both 4 plies thick is: Pcr = 7.1653e05[N ]. Adding the stiffeners after optimisation of the plate leads to a critical load of: Pcr = 4.1964e05[N ]. This is an increase of 70.7%. The QI case has a critical load of Pcr = 2.6177e05[N ], that is an increase of 174%. Double stiffened (eight plies) The results of the double stiffened rectangular plate with two stiffeners of each 8 plies thick can be found in figure 5.10. The stiffener plies exist of [45,-45,30,-30]S . It can be seen that those stiffeners have a slightly bigger influence than the one from the previous case. The areas on the plate are partly dividing again like what happened with the square plate. The middle area becomes more critical with respect to buckling. The outer areas are going towards the [0,0] design again in order to create stiffness. Fig. 5.10: Optimised fibre paths of a double stiffened rectangular plate with 8 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red line shows the location of the stiffeners The critical load of this double stiffened rectangular plate is: Pcr = 8.3771e05[N ]. Doing the optimisation for the unstiffened case and adding the stiffeners afterwards again, leads to: Pcr = 4.3368e05[N ]. This is an performance increase of 93.2%. The QI case has a critical load of Pcr = 3.5804e05[N ], that is an increase of 134%. Double stiffened (twelve plies) The double stiffened plate with two stiffeners with each having twelve plies of [45,-45,30,-30,60,-60]S is optimised. The results of the optimisation can be found in figure 5.11. The same as for the square plate, the inner area of this design contains fibres close to the [45,-45] orientation and the two outer areas contain fibres going towards a [0,0] orientation. Remarkable is that the area between the stiffeners go more and more to an unstiffened square plate when the number of plies per stiffeners is increasing. This verifies the assumption of the middle area can be treated as a simply supported 36 CHAPTER 5. OPTIMISATION RESULTS 5.2. RESULTS square plate. But only when the stiffeners are stiff enough. Stiffer stiffeners create a clear separation between the three areas of the laminate. Fig. 5.11: Optimised fibre paths of a double stiffened rectangular plate with 12 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red line shows the location of the stiffeners The critical load of this optimised rectangular plate is: Pcr = 8.5300e05[N ]. When the stiffeners would be added after optimisation: Pcr = 4.3909e05[N ]. This is an increase of 94.3%. The QI case has a critical load of Pcr = 3.7131e05[N ], that is an increase of 130%. Double stiffened (sixteen plies) The double stiffened plate with two stiffeners with each having sixteen plies of [45,-45,30,-30,60,60,0,0]S is optimised. The results of this optimisation can be found in figure 5.12. It is clear that due Fig. 5.12: Optimised fibre paths of an double stiffened rectangular plate with 16 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red line shows the location of the stiffeners to the increased stiffness of the stiffeners the panel can be divided in three areas again. The middle area comes really close to the results of an unstiffened plate with the same dimensions. As has been discussed, the middle area can be treated as a simply supported square plate when the stiffeners are stiff enough. The other smaller areas are mainly used to create stiffness for the stiffness requirement. The critical load for this stiffened rectangular plate is: Pcr = 8.8838e05[N ]. The critical load for adding the same stiffeners afterwards: Pcr = 4.5581e05[N ]. This is an increase of 94.9%. The QI case has a critical load of Pcr = 3.8242e05[N ], that is an increase of 132%. 37 5.2. RESULTS 5.2.7 CHAPTER 5. OPTIMISATION RESULTS Shared layers This section discusses the results of the integrated stiffeners functionality which has already been discussed in section 1.3. Two cases are shown, the square plate and the rectangular plate both with two fully integrated stiffeners. Those results will be compared to the constant non-integrated stiffened plates. The advantages of integrated stiffeners has already been discussed in section 1.3. In this section, fully integrated stiffeners are discussed. This means that every layer from the plate goes into the stiffener. Square plate Looking at the results of the square plate with two integrated stiffeners (see figure 5.13), it looks quite similar to the double stiffened square plate with non-shared stiffeners. In the figure it can be noticed that the first four balanced layers contain green lines. This means that the layers are going up into the stiffeners with the same fibre orientations. These are the integrated layers of the stiffener (see section 1.3 for information about integrated stiffeners). Since the layers of the stiffeners have the same fibre orientations as the plate fibres at that same location, the design has become a little different. Where the square plate without integrated stiffeners had much [0,0] fibre oriented areas, the plate with the integrated layers does not have it. Which means that integrated layers can have a significant influence on the design. The reason is that if the plate layers below the stiffeners go to [0,0] oriented fibres, the stiffeners are changing its fibre orientation such as well. The critical load for this plate is: Pcr = 1.4208e06[N ]. Comparing to the square plate with predefined stiffener layups gives: Pcr = 1.5804e06[N ]. This is a decrease in performance of 10.1%. The critical load for the QI case is: Pcr = 8.6987e05[N ]. This is an increase of 81.7%. Fig. 5.13: Optimised fibre paths of an double stiffened square plate with 16 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left top) to the laminate center (Right bottom). The red and green lines show the location of the stiffeners. Green also means that the plies go into the stiffener and the fibre orientations at that location will be the fibre orientation of the stiffener layer 38 CHAPTER 5. OPTIMISATION RESULTS 5.2. RESULTS Rectangular plate The influence of integrated stiffeners on the rectangular plate can be found in figure 5.14. In this figure the fibre orientations of the stiffeners are not fixed. The plies go from the plate into the stiffener (as has been discussed in section 1.3). The same as in the previous section, the green lines represent the layers going up from the plate into the stiffener. From the results it can be concluded that the optimised plate does not change significantly compared to the rectangular plate with 16-ply-stiffeners which are having a constant fibre orientation. From this it can be concluded that especially the thickness of the stiffener influences the design significantly instead of the fibre orientation. This could have been expected since the geometric stiffness of the stiffener provides stiffness against buckling. This prevents the plate to buckle on smaller loadings. Comparing the results of the plate with sixteen shared layers of the stiffener and the rectangular plate with sixteen constant layers, can be used to verify the extra functionality of stiffener integration. The initial optimiser discussed in chapter 2 was not able to account for integrated stiffeners. Fig. 5.14: Optimised fibre paths of an double stiffened rectangular plate with 16 plies per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red and green lines show the location of the stiffeners. Green also means that the plies go into the stiffener and the fibre orientations at that location will be the fibre orientation of the stiffener layer The critical load of the rectangular plate with integrated stiffeners is: Pcr = 8.6279e05[N ]. The critical load of the optimised rectangular plate with no integrated stiffeners (discussed in section 5.2.6) is: Pcr = 8.8838e05[N ]. This is a decrease of only 2.89%. This verifies the fact that integrated stiffeners can be used without affecting the critical loads too much. Although this percentage looks very small, it can not be used to represent the decrease of performance in each possible case. This time it is compared to a case with constant stiffeners of [45,-45, 30,-30, 60,-60, 0, 0]S , an extra study must be done to know the influence for the worst cases. This is also the reason why only a fully integrated stiffener is discussed in this thesis, from sem-integrated stiffeners with half non-integrated and half integrated no conclusions can be drawn either. The critical load of the QI plate with two stiffeners is: Pcr = 4.0563e05[N ]. This means that the rectangular design with the integrated stiffeners does still have a performance increase of 113% with respect to the QI case. 39 5.3. CONCLUSION 5.3 CHAPTER 5. OPTIMISATION RESULTS Conclusion Looking at the results of this chapter, the designs have all improved regarding the critical loads. Fibre orientations going towards [0,0] are beneficial for creating stiffness, and fibre orientations going towards [45,-45] are used to optimise the buckling critical areas. Inner plies overall contain more fibre paths in the direction of the load than the outer layers where buckling modes are more critical. This is because of the distance from those layers to the center of the laminate. The influence of the stiffeners is clearly visible in the fibre path optimisations. The stiffer the stiffeners are, the more the optimiser treats the panel as three separate areas with simply supported properties. A great example is the comparison between the middle area of the rectangular plate with stiff stiffeners, and the unstiffened square plate. It shows that the middle area can be taken as a separate plate with simply supported boundary conditions as long as the stiffeners are stiff enough. Those designs get close to each other regarding the fibre paths. The rectangular panel with shared layer stiffeners looks very similar to the panel of stiffeners with constant fibre orientations. This verifies the optimiser with the functionality of using integrated stiffeners. The integrated stiffener designs are a better than the QI designs: 81.7% for the square and 113% for the rectangular plate. From comparing a predefined non-changing stiffener with a constantly changing one, no conclusions could be drawn. Multiple designs should be studied before a valid conclusion can be drawn about the performance of integrated stiffener designs with respect to non-integrated stiffener designs. The rate of improvement of the square and rectangular plate with predefined stiffener stacking sequences, can be found in table 5.2 and table 5.3. In both tables, the first column shows the design which is optimised. The second column shows the rate of improvement with respect to the same design but the stiffeners excluded from the optimiser. The third column shows the rate of improvement compared to the same design with a QI material. According to the calculated critical loads, all the non-integrated stiffeners are having a positive influence on the optimised design. Ranging from a critical load increase of 11.8% to 119% for the squared plate and 70.7% to 94.9% for the rectangular plate. This means that when the stiffeners are not included in the optimisation cycle, the design could still have been improved by the rate shown in the second column. The real rate of improvement is shown in the third column when it is compared to the QI case. It is shown that it is worth optimising with stiffeners included. Table 5.2: The improvements of each square design compared to: 1. Stiffeners excluded from optimisation and added afterwards 2. The QI design Unstiffened Plate Single Stiffened (center) Single Stiffened (off-center) Double Stiffened (symmetric) Double Stiffened (asymmetric) 40 Excl. Stiffeners 22.0% 57.8% 11.8% 119% QI 56.3% 26.1% 103% 103% 114% CHAPTER 5. OPTIMISATION RESULTS 5.3. CONCLUSION Table 5.3: The improvements of each rectangular design compared to: 1. Stiffeners excluded from optimisation and added afterwards 2. The QI design Unstiffened Stiffeners of 4 plies Stiffeners of 8 plies Stiffeners of 12 plies Stiffeners of 16 plies Excl. Stiffeners 70.7% 93.2% 94.3% 94.9% 41 QI 95% 174% 134% 130% 132% Conclusion & recommendations Variable stiffness laminates (VSL) are proven to be an excellent substitution for constant fibre laminates in aerospace structures. The optimiser developed at the Delft University of Technology is perfect for optimising VSL against buckling. The software includes a manufacturing constraint in order to model manufacturable designs. But the optimiser was short on the capability of including stiffeners in the optimisation. After further development of the optimiser, it is now capable of adding stiffeners to the optimisation. This implementation is realised by merging stiffeners to the plate with an object oriented programming approach. Using main- and sub-parts, computational effort is saved by reusing already existing calculation functionalities. The addition of stiffeners is verified by comparing displacements and sectional forces from the optimiser with the calculated values of ABAQUS. The overall mean errors, stayed under the 3%, despite of the somewhat enormous outliers in the verification process. The optimiser accounts for crippling of stiffeners which has been verified as well. The optimiser has optimised 12 simply supported models for the first two buckling modes while the models were constrained to be at least as stiff as their quasi isotropic designs. Looking at the results of the optimiser; the influence of stiffeners with a predefined stacking sequence gives a buckling load increase of 11.8% to 119%. Stiffeners separate the areas on the plate. The area in-between two stiffeners shows similar properties to a simply supported plate. The larger areas are more sensitive to buckling than the smaller areas. This results in optimal fibre paths going to a [45,-45] fibre orientation. The other areas are creating stiffness by changing its fibre orientations closer to a [0,0] design. The outer layers of a laminate are found to be more critical to buckling than the inner layers. The outer layers contain more fibres going towards a [45,-45] direction. And since the inner layers are less sensitive to buckling, stiffness is created here by orienting the fibres closer to a [0,0] orientation. The integrated stiffener designs are shown to be better than the quasi isotropic design but can be less performing compared to the stiffeners with a fixed layup. No valid conclusions can be drawn about using shared layers with respect to its performance increase. More research needs to be done to draw more conclusions about comparing integrated stiffeners with non-integrated stiffeners. Recommendations can be done regarding this thesis and for future work: • The optimiser could be optimised even more to reduce computational effort and storage space. • As already been mentioned, more study can be done to integrated stiffeners. If the stiffeners become VSL as well, a structure is expected to be optimised even further. This requires the stiffener layers to be included in the optimisation as well. • The optimiser could be developed in an entire object oriented environment. This will cost a lot of time to develop but will save a lot of time during analysis. It will also save a lot of time if other types of structures are going to be implemented. For example, plates with holes. • A further study could be done to different load cases. 42 Bibliography [1] Frontpage picture, from website: https://en.wikipedia.org/wiki/Tailored_fiber_placement [visited March 6, 2016] [2] Gurdal, Z., Haftka, R.T., Hajela, P., 1999. Design and Optimization of Laminated Composite Materials, New York: John Wiley & Sons [3] Callahan, J.K., Weeks, G.E., 1992. Optimum Design of Composite Laminates Using Genetic Algorithms. Composites Engineering, 2(3), pp. 149-160 [4] Ghiasi, H., Pasini, D., Lessard, L., 2009. 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Optimum Design of Laminated Fibre Composite Plates, International Journal for Numerical Methods in Engineering, 11(5), pp. 623-640 44 Appendices 45 A Laminate properties A.1 Engineering constants Q11 = m4 Qxx + n4 Qyy + 2m2 n2 Qxy + 4m2 n2 Qss (A.1) Q22 = n4 Qxx + m4 Qyy + 2m2 n2 Qxy + 4m2 n2 Qss (A.2) Q12 = m2 n2 Qxx + m2 n2 Qyy + m4 + n4 Qxy − 4m2 n2 Qss Q66 = m2 n2 Qxx + m2 n2 Qyy − 2m2 n2 Qxy + m2 − n2 2 (A.3) Qss (A.5) (A.6) Q16 = m3 nQxx − mn3 Qyy + mn3 − m3 n Qxy + 2 mn3 − m3 n Qss Q26 = mn3 Qxx − m3 nQyy + m3 n − mn3 Qxy + 2 m3 n − mn3 Qss A.2 Material Invariants 3Qxx + 3Qyy + 2Qxy + 4Qss 8 Qxx − Qyy U2 = 2 Qxx + Qyy − 2Qxy − 4Qss U3 = 8 Qxx + Qyy + 6Qxy − 4Qss U4 = 8 Qxx + Qyy − 2Qxy + 4Qss U5 = 8 U1 = A.3 (A.4) (A.7) (A.8) (A.9) (A.10) (A.11) Lamination Parameters A V0,1,2,3,4 B V0,1,2,3,4 1 = 2 Z1 [1, cos 2θ(z), cos 4θ(z), sin 2θ(z), sin 4θ(z)] dz (A.12) [1, cos 2θ(z), cos 4θ(z), sin 2θ(z), sin 4θ(z)] · zdz (A.13) [1, cos 2θ(z), cos 4θ(z), sin 2θ(z), sin 4θ(z)] · z 2 dz (A.14) −1 Z1 = −1 D V0,1,2,3,4 3 = 2 Z1 −1 46 B ABD-matrix V0A V1A V2A 0 0 A11 A A A V −V V 0 0 A22 1 2 0 A A A 0 −V2 V0 0 0 12 A= = h 0 0 −V2A 0 V0A A66 V3A A A16 V 0 0 0 4 2 A V3 A A26 −V4 0 0 0 2 0 V1B V2B B11 0 −V B V2B B22 1 B 0 0 −V2B h2 12 B= = 0 0 −V2B B66 4 V3B B16 V4B 0 2 B V3 B26 −V4B 0 2 0 0 0 0 0 0 0 0 (B.2) 0 0 0 0 V0D V1D V2D 0 0 D11 D D D −V1 V2 0 0 D22 V0 D D D 3 0 0 −V V 0 2 0 h 12 D D D= = 0 0 −V 0 V 2 0 D66 12 D V3 D D16 V 0 0 0 4 2 V3D D D26 0 −V4 0 0 2 (B.1) (B.3) ! ABDmatrix = 47 A B B D (B.4) C Frobenius inner product The Frobenius inner product is written as A : B. It is the component-wise inner product of the A and B matrices as though they are vectors. It can be calculated as follows: A:B= X Aij Bij = vec (A)T vec (B) i,j Where ’vec’ stands for vectorization. This can also be written in the better known form of ’traces’ of matrices: vec (A)T vec (B) = tr AT B 48 D FEM (from section 2.3.5): The general problem which needs to be solved numerically is: Km · U = F The real problem is more difficult than it looks like. It needs to be taken into account that the model has free master degrees of freedom, free degrees of freedom and slave degrees of freedom. In the software, the stiffness matrix of the free degrees of freedom is called K11 , the free master degree of freedom K22 , and the stiffness between the free and free master degrees of freedom K12 . The internal forces exerted on the free nodes is in equilibrium with the structure: K11 u1 = f1 It has to be noticed that this equation is only valid for a complete structure which only exists of free nodes. The influence of the master and slave nodes must be taken into account as well. So the real equation becomes: K11 u1 = f1 − f21 Where f21 is the internal force exerted by the adjacent nodes of the master and slave nodes (see figure D.1): f21 = K21 · u2 Writing this out gives: K11 · u1 = f1 − K12 · u2 Using Cholesky decomposition for K11 : R0 · R · u1 = f1 − K12 · u2 Where R0 is the lower triangle, and R the upper triangle. The working of the Cholesky decomposition can be found in Appendix C. Rewriting the equation results in: u1 = R\ R0 \f1 − R0 \K12 · u2 Using A = R0 \f1 and b = R0 \K12 gives: u1 = R\(A − b · u2 ) In order to write out u2 , the same principle is valid as well: K22 · u2 = f2 − f12 Where: f12 = K12 · u1 Which results in: K22 · u2 = f2 − K12 · u1 Substituting u1 : K22 · u2 = f2 − K12 · [R\(b − A · u2 )] 49 (D.1) APPENDIX D. FEM Rewriting this: f2 = K22 · u2 + K12 · [R\(b − A · u2 )] f2 = K22 · u2 + [R\(K12 · b − K12 · A · u2 )] f2 = R\(K12 · b − K12 · A · u2 + R · K22 · u2 ) R · f2 = K12 · b + (R · K22 − K12 · A) · u2 Bringing u2 to the left side: (R · K22 − K12 · A) · u2 = R · f2 − K12 · b − R · K12 (R · K22 − K12 · A) · u2 = R · f2 − K12 · b − R · K12 u2 = (R · K22 − K12 · A) \(R · f2 − K12 · b) In order to write it easier, A0 = R\K12 : u2 = K22 − A0 · A \ f2 − A0 · b (D.2) Those u1 and u2 can fill up the displacement matrix ’U ’, only the MPC-matrix must be multiplied Fig. D.1: A drawing of master and slave nodes with their adjacent nodes (Left) and a drawing of the forces going from adjacent nodes to the master and slave nodes with the u2 in order to calculate the displacements of the slave nodes as well. For calculating the force in every degree of freedom, equation D can be easily used. Where U is the displacement matrix containing all the displacements for every degree of freedom of the model. 50 E Verification plots E.1 Unstiffened plate Fig. E.1: The displacements of the unstiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom Fig. E.2: The displacements of the unstiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom 51 E.1. UNSTIFFENED PLATE APPENDIX E. VERIFICATION PLOTS Fig. E.3: The section forces of the unstiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom Fig. E.4: The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom 52 APPENDIX E. VERIFICATION PLOTS E.1. UNSTIFFENED PLATE Fig. E.6: The relative error of the unstiffened plate section forces Fig. E.5: The absolute error of the unstiffened plate section forces 53 E.2. SINGLE STIFFENED PLATE (CENTER) E.2 APPENDIX E. VERIFICATION PLOTS Single stiffened plate (Center) Fig. E.8: The displacements of the center stiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom Fig. E.7: The displacements of the center stiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom 54 APPENDIX E. VERIFICATION PLOTS E.2. SINGLE STIFFENED PLATE (CENTER) Fig. E.9: The section forces of the center stiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom Fig. E.10: The section forces of the center stiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom 55 E.2. SINGLE STIFFENED PLATE (CENTER) APPENDIX E. VERIFICATION PLOTS Fig. E.11: The absolute error of the center stiffened plate section forces Fig. E.12: The relative error of the center stiffened plate section forces 56 APPENDIX E. VERIFICATION PLOTS E.3 E.3. SINGLE STIFFENED PLATE (OFF-CENTER) Single stiffened plate (Off-center) Fig. E.13: The displacements of the off-center stiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom Fig. E.14: The displacements of the off-center stiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom 57 E.3. SINGLE STIFFENED PLATE (OFF-CENTER) Fig. E.15: The section forces of the off-center stiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom APPENDIX E. VERIFICATION PLOTS Fig. E.16: The section forces of the off-center stiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom 58 APPENDIX E. VERIFICATION PLOTS E.3. SINGLE STIFFENED PLATE (OFF-CENTER) Fig. E.17: The absolute error of the off-center stiffened plate section forces Fig. E.18: The relative error of the off-center stiffened plate section forces 59 E.4. DOUBLE STIFFENED PLATE (SYMMETRIC) E.4 APPENDIX E. VERIFICATION PLOTS Double stiffened plate (Symmetric) Fig. E.19: The displacements of the unstiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom Fig. E.20: The displacements of the unstiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom 60 APPENDIX E. VERIFICATION PLOTS E.4. DOUBLE STIFFENED PLATE (SYMMETRIC) Fig. E.21: The section forces of the unstiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom Fig. E.22: The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom 61 E.4. DOUBLE STIFFENED PLATE (SYMMETRIC) APPENDIX E. VERIFICATION PLOTS Fig. E.24: The relative arror of the unstiffened plate section forces Fig. E.23: The absolute arror of the unstiffened plate section forces 62 APPENDIX E. VERIFICATION PLOTS E.5. DOUBLE STIFFENED PLATE (ASYMMETRIC) E.5 Double stiffened plate (Asymmetric) Fig. E.25: The displacements of the unstiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom Fig. E.26: The displacements of the double asymmetric stiffened plate in all three spatial directions (ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom 63 E.5. DOUBLE STIFFENED PLATE (ASYMMETRIC) APPENDIX E. VERIFICATION PLOTS Fig. E.27: The section forces of the unstiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom Fig. E.28: The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx on top, Ny in the middle, Nz at the bottom 64 APPENDIX E. VERIFICATION PLOTS E.5. DOUBLE STIFFENED PLATE (ASYMMETRIC) Fig. E.29: The absolute arror of the unstiffened plate section forces Fig. E.30: The relative arror of the unstiffened plate section forces 65

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