A lightweight floor system based on sandwich panels: characterization and development

Universidade do Minho Escola de Engenharia Pier Giovanni Benzo A lightweight floor system based on sandwich panels: characterization and development june 2024 UMinho | 2024 Pier Giovanni Benzo A lightweight floor system based on sandwich panels:characterization and development Pier Giovanni Benzo A lightweight floor system based on sandwich panels: characterization and development Philosophy Doctorate Thesis Civil Engineering Work conducted under supervision of: Professor Doctor José Manuel de Sena Cruz Doctor João Miguel Pereira Universidade do Minho Escola de Engenharia june 2024 ii DIREITOS DE AUTOR E CONDIÇÕES DE UTILIZAÇÃO DO TRABALHO POR TERCEIROS Este é um trabalho académico que pode ser utilizado por terceiros desde que respeitadas as regras e boas práticas internacionalmente aceites, no que concerne aos direitos de autor e direitos conexos. Assim, o presente trabalho pode ser utilizado nos termos previstos na licença abaixo indicada. Caso o utilizador necessite de permissão para poder fazer um uso do trabalho em condições não previstas no licenciamento indicado, deverá contactar o autor, através do RepositóriUM da Universidade do Minho. Licença concedida aos utilizadores deste trabalho Atribuição CC BY https://creativecommons.org/licenses/by/4.0/ iii ACKNOWLEDGEMENTS It was a long journey, at times difficult and challenging, but, nevertheless, a very rewarding experience I will always cherish. I couldn't have achieved it without the help and support of my supervisors, Professor José Sena-Cruz and Doctor João Pereira. I am grateful for their patience and enthusiasm, even for the smallest tasks. I would also like to thank Professor Paulo Lourenço for giving me the opportunity to prove myself. I would like to express my gratitude to all the people who have collaborated with me during my research work. I am grateful for the assistance and support of José Machado from Ferpainel, S.A., Professor Zlatan Denchev from the Department of Polymer Engineering, Professor Andrea Zille from the Department of Textile Engineering, and the technicians at the Laboratory of Civil Engineering of the University of Minho. I am immensely grateful to my friends and project colleagues Aloys, Guilherme, and Marco, who provided invaluable help in the laboratory and kept my spirits up during coffee breaks. Besides the people who directly contributed to my work, there are others who have been with me forever and some who joined along the way and helped me grow as a human. The biggest thanks goes to my partner, Ola. She is brilliant, loving, and caring, and none of this would have been possible without her (it is not a co-dependency statement, I could have done it on my own, but with her, it is 1000 times better). Another mega thank goes to Ali. There are no words to express the love and gratitude I have for him. A huge thank you also goes to Lorenzo, my best friend in life who never fails to “make some laugh” with me as he likes to phrase it. A big thank you goes to Pratik for being annoyingly smart and for challenging and supporting me no matter what. I am grateful to everyone in the multicultural academic community of the Department of Civil Engineering at the University of Minho. I would like to thank Ravi for his kind and wise advice. Thanks also to Chiara for pulling me and motivating me towards the end of the journey. Finally, I am grateful for the love and support of my parents and brothers. I want to thank my Mum and Dad for believing in me and providing me with everything they could from the very beginning. And I want to express my gratitude to Alessandro and Raffaele for always making me feel like time has never passed whenever I come back home. The financial support provided by the LightSlab R&D Project (POCI-01-0247-FEDER - 033865), funded by the Agência Nacional de Inovação, and by the Portuguese Foundation of Science and Technology (FCT) under the PhD grant 2020.08319, is gratefully acknowledged. iv STATEMENT OF INTEGRITY I hereby declare having conducted this academic work with integrity. I confirm that I have not used plagiarism or any form of undue use of information or falsification of results along the process leading to its elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the University of Minho. v RESUMO O setor da construção deve continuamente transformar-se de modo a mitigar os riscos associados às alterações climáticas. É crucial promover abordagens de projeto e construção que minimizem as emissões de CO2. As construções modulares e a reabilitação de edifícios existentes podem reduzir o consumo de matérias-primas e de energia, bem como a produção de resíduos. Neste contexto, sistemas de pisos leves são essenciais para tais práticas. Esta tese apresenta o desenvolvimento de um sistema de piso leve baseado em painéis sanduíche e seus sistemas de ligação que contribuíram para o resultado do projeto de investigação LightSlab. O sistema de pisos destina-se a construções modulares e reabilitação de pisos degradados em edifícios de alvenaria. A unidade modular pré-fabricada do sistema de piso é constituída por um painel sanduíche reforçado com nervuras de aço formado a frio e preenchido com espuma de poliuretano. Este trabalho de investigação visa proporcionar uma melhor compreensão do comportamento mecânico de tais membros estruturais compósitos. O painel sanduíche foi projetado utilizando ferramentas analíticas para garantir sua segurança estrutural, bem como isolamentos térmicos e acústicos adequados. Um processo de otimização foi conduzido usando algoritmos genéticos para minimizar seu peso, custo e impacto ambiental, ao mesmo tempo em que atende aos padrões estruturais e de construção. A otimização foi validada e estudos foram realizados para examinar como os principais parâmetros de projeto afetam seu desempenho. O estudo experimental analisa as propriedades mecânicas dos materiais constituintes dos painéis sanduíche e avalia o seu comportamento à flexão estático e dinâmico. Adicionalmente, os sistemas de conexão são avaliados experimentalmente. As juntas de painéis sanduíche usam chapas fixadas nas faces inferiores dos painéis adjacentes com parafusos autoperfurantes. A ligação com a estrutura vertical é materializada por cantoneiras de aço aparafusadas e/ou coladas adesivamente ao painel sanduíche. Os resultados do programa experimental são usados para verificar ferramentas analíticas e calibrar modelos numéricos com o software ABAQUS. Simulações numéricas identificam os principais aspetos físicos que regem o comportamento do painel sanduíche. Com base nos modelos analíticos validados, são desenvolvidas ferramentas de projeto simplificadas para diversas configurações de suporte e carga para promover a utilização das novas soluções de piso. Palavras-chave: aço formado a frio; algoritmo genético; espuma PUR; painel sanduíche; modelagem FE. vi ABSTRACT The building sector must continuously transform to mitigate risks associated with climate change. Modular construction and rehabilitating of existing buildings can reduce raw material consumption, energy usage, waste production and minimize CO2 emissions. Lightweight floor systems are essential for such practices. This thesis presents the development of a lightweight floor system based on sandwich panels and its connection systems which contributed to the outcome of LightSlab R&D Project. The floor system is intended for weight-sensitive applications like modular constructions and rehabilitation of degraded floors in masonry buildings. The prefabricated modular unit of the floor system is made of a cold-formed steel web core sandwich panel infilled with polyurethane foam. This work aims to provide a better understanding of the mechanical behaviour of such composite structural members. The sandwich panel was designed using analytical tools to ensure its structural safety as well as adequate thermal and acoustic insulations. An optimization process was conducted using genetic algorithms to minimize its weight, cost, and environmental impact while meeting structural and building standards. The optimization was validated, and studies were performed to examine how key design parameters affect its performance. The experimental study analyses the mechanical properties of the constituent materials of the sandwich panels and assesses their static and dynamic flexural behaviour. Additionally, an experimental program is carried out to evaluate the connection systems. To ensure interlocking between adjoining panels, a tongue and groove geometry was implemented on the webs of the panel. The adjacent panels are fastened together using cover plates that are attached to the bottom face sheets with self-drilling screws. As for the connection with vertical structures, the sandwich panels are bolted and/or adhesively bonded to steel angles that are connected to the walls. Experimental program outcomes are used to verify analytical tools and calibrate numerical models with ABAQUS software. Numerical simulations identify key mechanical aspects governing sandwich panel behaviour. Based on the validated analytical models, simplified design tools are developed for various support and load configurations to promote the use of the new floor solutions. Keywords: sandwich panel; cold-formed steel; polyurethane foam; design and optimization; analytical and FE simulations. vii TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................................................................................................. iii RESUMO ........................................................................................................................................ v ABSTRACT .................................................................................................................................... vi TABLE OF CONTENTS .................................................................................................................. vii SYMBOLS ..................................................................................................................................... xi ACRONYMS ................................................................................................................................. xiii TABLE OF FIGURES ..................................................................................................................... xiv TABLE OF TABLES ....................................................................................................................... xxv 1. INTRODUCTION ........................................................................................................................ 1 1.1. RESEARCH OBJECTIVES AND METHODOLOGY ....................................................................... 3 1.2. OUTLINE OF THE THESIS ....................................................................................................... 5 2. LIGHTWEIGHT STRUCTURES AND MATERIAL EFFICIENCY .......................................................... 9 2.1. SANDWICH PANELS ............................................................................................................. 13 2.1.1. Mechanics of sandwich panels ...................................................................................... 14 2.1.2. Structural sandwich panels ............................................................................................ 15 2.2. CFS STRUCTURES ............................................................................................................... 19 2.2.1. Lightweight CFS floor system ......................................................................................... 20 2.2.2. PUR foam infilled CFS profiles ....................................................................................... 21 2.2.3. CFS built-up members ................................................................................................... 22 2.3. OPTIMIZATION SEARCH METHODS ...................................................................................... 24 2.3.1. Genetic algorithms ........................................................................................................ 25 2.3.2. Applications in structural design .................................................................................... 27 2.4. CONCLUDING REMARKS ...................................................................................................... 31 xiv TABLE OF FIGURES Figure 2.1. Lightweight structural members: (a) sandwich panel; (b) CFS-lipped channel section. ....... 10 Figure 2.2. Current and potential applications of composite sandwich panels: (a) temporary shelter [18]; (b) bridge decks [20]. ...................................................................................................................... 10 Figure 2.3. Current and potential applications of CFS sandwich panels and profiles: (a) Esso headquarters in Leatherhead (UK) [25]; (b) CFS framing of a modular unit [24]. .................................................... 11 Figure 2.4. Degraded timber floor in an existing building in Porto (PT). .............................................. 12 Figure 2.5. Composite sandwich panel with partially removed foam to show the pin reinforcement [46]. ........................................................................................................................................................ 16 Figure 2.6. Four-point bending test of the hybrid core steel sandwich panel [57]. ............................... 17 Figure 2.7. Structural sandwich panels: (a) web-reinforced sandwich panel subjected to distributed load test [59]; (b) inclined longitudinal web sandwich panel failure in the four-point bending test [61]. ..... 18 Figure 2.8. Four-point bending test of the CFS beams and wooden boards floor system [71]. ............. 20 Figure 2.9. Dimensions of the infilled C-section [74]. All units in [mm]. .............................................. 22 Figure 2.10. Cut-out piece of the PUR foam infilled CFS profile with delamination between the foam and the steel [76]. ................................................................................................................................... 22 Figure 2.11. Comparison of actual and predicted strength of built-up box beam [83]. ........................ 24 Figure 2.12. The procedure of a generic GA. ..................................................................................... 26 Figure 2.13. Optimal solutions for different span lengths: (a) cost of the solutions; (b) plot of the penalty function [30]. ................................................................................................................................... 28 Figure 2.14. Objective function values for each sandwich panel architecture [13]. ............................. 30 Figure 3.1. Manufacturing process of steel insulating sandwich panels: (a) uncoiling CFS sheet; (b) roll forming CFS sheet; (c) PUR foam’s monomers injection; (d) PUR foam curing in heating chamber; (e) cutting of the sandwich panel. ........................................................................................................... 34 xv Figure 3.2. Layout of the sandwich panels: (a) SP1; (b) SP2; (c) SP3. ................................................ 35 Figure 3.3. Structural model adopted for the design of the sandwich panel. ....................................... 38 Figure 3.4. Typical CFS cross-sections: (a) C-section; (b) Z-section; (c) hat section; (d) roof panel; (e) curtain wall panel. ............................................................................................................................ 40 Figure 3.5. Compressive stress distribution and buckled cross section according to the EMW: (a) simply supported plate; (b) Von Karmán plate. ............................................................................................ 41 Figure 3.6. Flow chart of the calculation of the effective properties of CFS cross-sections considering the effect of local instabilities. ................................................................................................................. 43 Figure 3.7. Typical forms of stiffeners for CFS members and sheets: (a) edge stiffeners; (b) web intermediate stiffener; (c) flange intermediate stiffeners. ................................................................... 44 Figure 3.8. Equivalent structural system of an intermediate stiffener in CFS members. ...................... 44 Figure 3.9. Flow chart of the calculation of the effective properties of CFS cross-sections considering the effect of distortional instabilities. ....................................................................................................... 45 Figure 3.10. Simplified cross-section of the sandwich panel: (a) built-up box section; (b) equivalent hollow rectangular section. .......................................................................................................................... 46 Figure 3.11. Cross-section of the residential building floor considered for the thermal transmittance estimation. ....................................................................................................................................... 47 Figure 3.12. Airborne sound insulation (𝑅) of the sandwich panel, the ISO 717-1 (2013) [124] reference curve, and the translated reference curve. ......................................................................................... 50 Figure 3.13. Fire resistance test setup and instrumentation: (a) lateral view; (b) top view. All units in [mm]. ........................................................................................................................................................ 51 Figure 3.14. Thermal response of the sandwich panel. ...................................................................... 52 Figure 3.15. SP1 cross-section layout. All units in [mm]. Note: the thickness of the face sheets and webs are not to scale................................................................................................................................. 54 Figure 3.16. SP2 cross-section layout. All units in [mm]. Note: the thickness of the face sheets and webs are not to scale................................................................................................................................. 54 xvi Figure 3.17. SP3 cross-section layout. All units in [mm]. Note: the thickness of the face sheets and webs are not to scale................................................................................................................................. 55 Figure 3.18. Encoding procedure of the sandwich panel: (a) phenotype; (b) genotype. ....................... 57 Figure 3.19. Implemented algorithm (OSPANEL) to optimise the design of the sandwich panel. ......... 62 Figure 3.20. Crossover operator of the algorithm OSPANEL. .............................................................. 63 Figure 3.21. Weight of the lightest sandwich panel found by OSPANEL: (a) different runs varying the value of NFT0,i; (b) zoom on the first 300 generations. ............................................................................. 64 Figure 3.22. Sensitivity analysis on the adaptive penalty function parameters: (a) 𝑁𝐹𝑇0,𝑖; (b) 𝜆. ...... 65 Figure 3.23. Comparison of the performance of the modified GA, including static penalty function, and OSPANEL. ........................................................................................................................................ 66 Figure 3.24. Cross-section and characteristics of the optimal solutions: (a) S_1 and S_3 solution; (b) S_2 solution. ........................................................................................................................................... 67 Figure 3.25. Search history of S_1 and S_3: (a) normalized constraints value; (b) zoom at the converged iteration of the normalized constraints value; (c) normalized design variable value. Note: Lines with triangle symbols are constraints that must be equal or less than 1, whereas lines with square symbols are constraints that must be equal or greater than 1. .............................................................................. 68 Figure 3.26. Search history of S_2: normalized values of the constraints; (b) normalized values of the design variables. ............................................................................................................................... 69 Figure 3.27. Parametric study on the number of webs: (a) weight of the optimal solutions; (b) cost of the optimal solutions. ............................................................................................................................. 70 Figure 3.28. Cross-section and characteristics of the optimal solutions obtained from the parametric study: (a) S_1 and S_3 solution; (b) S_2 solution. ............................................................................. 71 Figure 3.29. Values of the constraints of the optimal solutions: (a) 𝑅Md,Q; (b) 𝑅Vd,Q; (c) 𝛿max; (d) 𝑈c. .................................................................................................................................................. 72 Figure 3.30. Values of the design variables of the optimal solutions: (a) 𝑡tot; (b) 𝑡s; (c) 𝑏f. ............... 73 Figure 3.31. Most sound insulating sandwich panel retrieved by OSPANEL. ....................................... 74 xvii Figure 3.32. Prototype S_2: (a) Cross-section schematic; (b) bottom face sheet-to-web spot weld connection; (c) manufactured panels. All units in [mm]. .................................................................... 76 Figure 3.33. Intermediate prototype: (a) cross-section schematic; (b) assembly of bottom face sheet and webs; (b) failed bracing system attempt between webs; (c) manufactured panels; (d) misalignment between the top face sheet and the web. All units in [mm]. ............................................................... 77 Figure 3.34. Final prototype layout: (a) CFS components cross-section; (b) sandwich panel cross-section. All units in [mm]. .............................................................................................................................. 78 Figure 3.35. Final prototype: (a) lipped C-section on the conveyor belt; (b) manufactured panel. ........ 79 Figure 4.1. CFS tensile coupon test: (a) nominal dimensions of the specimen; (b) test setup. All units in [mm]. ........................................................................................................................................... 83 Figure 4.2. Procedure to obtain static material properties: (a) proportional limit and 0.2% proof strength; (b) dynamic stress and static stress-strain curves. ............................................................................. 85 Figure 4.3. Uniaxial coupon tensile test result: (a) stress-strain curves; (b) typical failure mode. ......... 86 Figure 4.4. Specimen preparation for zinc coating thickness measurements: (a) CFS sheet cut out from the full-scale tested sandwich panel; (b) specimen cast in epoxy resin. .............................................. 88 Figure 4.5. Optical microscope image from one of the samples. ........................................................ 88 Figure 4.6. Mechanical characterisation of the PUR foam: (a) flatwise compressive test setup; (b) flatwise tensile test setup. ............................................................................................................................. 92 Figure 4.7. Flatwise compressive test: stress-strain curves. ............................................................... 93 Figure 4.8. Flatwise tensile test: stress-strain curves. ......................................................................... 95 Figure 4.9. Area of the PUR foam specimen tracked by the DIC equipment with the corresponding layers. ........................................................................................................................................................ 96 Figure 4.10. DIC measurements of FC-40 under compression: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial compressive strain at the linear stage, at the onset of yielding and after yielding. All units in [mε]. ......................................................................... 97 xviii Figure 4.11. DIC measurements of FT-40 under tension: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial tensile strain at the linear stage and at fracture. All units in [mε]. ................................................................................................................................................ 98 Figure 4.12. DIC measurements of FC-50 under compression: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial compressive strain at the linear stage, at the onset of yielding and after yielding. All units in [mε]. ......................................................................... 99 Figure 4.13. DIC measurements of FT-50 under tension: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial tensile strain at the linear stage, at the onset of the hardening stage and fracture. All units in [mε]. ............................................................................................... 100 Figure 4.14. DIC measurements of FC-60 under compression: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial compressive strain at the linear stage, at the onset of yielding and after yielding. All units in [mε]. ....................................................................... 101 Figure 4.15. DIC measurements of FT-60 under tension: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial tensile strain at the linear stage, at the onset of the hardening stage and fracture. All units in [mε]. ............................................................................................... 102 Figure 4.16. Shear specimens’ defects: voids along one of the interfaces. ....................................... 103 Figure 4.17. Test fixture for the assessment of PUR foam shear properties. ..................................... 104 Figure 4.18. Shear test experimental setup. .................................................................................... 104 Figure 4.19. Stress-strain curves from the shear tests on PUR foam. ............................................... 105 Figure 4.20. Failure modes identified in the shear tests: (a) debonding failure at the interface PUR foamto-face sheet interface; (b) cohesive shear failure of the PUR foam................................................... 105 Figure 4.21. TGA sample preparation: (a) TL, ML, and BL of the cubic PUR foam specimen; (b) location of TGA sample extraction; (c) TGA sample in the pan of the TA Instruments Q600 SDT. ................... 108 Figure 4.22. TGA and DTGA curves of the top (TL), middle (ML), and bottom layer (BL) of the 40 kg/m3 PUR foam. ...................................................................................................................................... 109 Figure 4.23. TGA and DTGA curves of the top (TL), middle (ML), and bottom layer (BL) of the 50 kg/m3 PUR foam. ...................................................................................................................................... 109 xix Figure 4.24. TGA and DTGA curves of the top (TL), middle (ML), and bottom layer (BL) of the 60 kg/m3 PUR foam. ...................................................................................................................................... 110 Figure 4.25. Single lap shear test: (a) overall test setup; (b) adhesive joint and LVDTs. .................... 112 Figure 4.26. Load-displacement curves of the single lap shear test on the PUR adhesive. ................ 113 Figure 4.27. Observed failure modes in the single lap shear test: (a) SLST-02; (b) SLST-04. ............. 113 Figure 4.28. Sandwich panels specimens upon delivery: (a) face sheet defects; (b) milling machine treatment; (c) cohesive crack due to milling operation; (d) grinding wheel treatment. ....................... 114 Figure 4.29. Edgewise compressive test: a) overall setup and DIC equipment; b) specimen. ............ 115 Figure 4.30. Edgewise compressive test load-displacement curves: a) EC-1.0 set; b) EC-1.5 set. ...... 116 Figure 4.31. Edgewise compressive test DIC results for EC-1.0 sample: a) load-displacement curve; b) deformation history. .................................................................................................................... 117 Figure 4.32. Wrinkling half wavelength estimation using DIC: a) EC-1.0-4; b) EC-1.0-5. All units in [mm]. ...................................................................................................................................................... 117 Figure 4.33. Edgewise compressive test results for EC-1.5 sample: a) load-displacement curve; b) deformation history. ........................................................................................................................ 118 Figure 4.34. Location of the monitored points by the DIC in the edgewise compressive test. All units in [mm]. ............................................................................................................................................. 119 Figure 4.35. Comparison between LVDTs and DIC measurements for specimen EC-1.0-4. ............... 120 Figure 4.36. Comparison between LVDTs and DIC relative displacement between top and bottom crosssection for set EC-1.0. ................................................................................................................... 120 Figure 5.1. Small-scale four-point bending test setup. ...................................................................... 127 Figure 5.2. Schematic representation of the small-scale four-point bending test: (a) lateral view of the joint panels; (b) A-A section of the single panel; (c) A-A section of the joint panels. All units in [mm]. ........ 127 Figure 5.3. Small-scale four-point bending test of single sandwich panels: (a) load-displacement curve; (b) load-longitudinal strain curve. .................................................................................................... 128 xx Figure 5.4. Failure progress of single sandwich panel SP1: (a) onset of top face sheet wrinkling; (b) PUR foam crushing under loaded sections and end bearing failure of the web; (c) web local buckling under the loaded section. ............................................................................................................................... 129 Figure 5.5. Small-scale four-point bending test of joint sandwich panels: (a) load-displacement curve; (b) load-longitudinal strain curve. .......................................................................................................... 131 Figure 5.6. Progress failure of set of joint sandwich panels DP2: (a) face sheet wrinkling and end bearing failure of the web; (b) buckling of the web under the loaded section and end bearing failure of the web. ...................................................................................................................................................... 132 Figure 5.7. Design method for the estimation of the sandwich panel’s deflection: (a) M1; (b) M2. .... 133 Figure 5.8. Three-point variable span bending test setup. ................................................................ 137 Figure 5.9. Three-point variable span bending test instrumentation: (a) lateral view; (b) AA section. All units in [mm]. ................................................................................................................................ 137 Figure 5.10. 𝐿2−𝛿/𝐹𝐿 axes plot of the results: (a) 4.5 kNm bending moment test; (b) 3.0 kNm bending moment test; (c) 1.5 kNm bending moment test. .............................................................. 139 Figure 5.11. Load-longitudinal strain curve of Panel01 for a span length of 4.0 m and an applied bending moment of 4.5 kNm. .................................................................................................................... 142 Figure 5.12. Load-longitudinal strain curve of Panel03 for a span length of 4.0 m and an applied bending moment of 4.5 kNm. .................................................................................................................... 142 Figure 5.13. Flexural dynamic test setup: (a) overall view; (b) accelerometers. ................................. 143 Figure 5.14. Flexural dynamic test instrumentation: (a) lateral view; (b) section A-A. All units in [mm]. ...................................................................................................................................................... 143 Figure 5.15. Typical power spectral density curves of the sandwich panels. ..................................... 144 Figure 5.16. Typical shapes obtained from Artemis Modal (Version 6.0): a) first mode; b) second mode. ...................................................................................................................................................... 144 Figure 5.17. Four-point bending test setup. ..................................................................................... 146 Figure 5.18. Four-point bending test instrumentation: (a) lateral view; (b) AA section. All units in [mm]. ...................................................................................................................................................... 147 xxi Figure 5.19. Load-displacement curves of the sandwich panels. ...................................................... 147 Figure 5.20. Typical load-longitudinal strain curve of the sandwich panels. ....................................... 148 Figure 5.21. Evolution of failure mode of Panel01: (a) load-midspan displacement curve; (b) failure progression. ................................................................................................................................... 149 Figure 5.22. Progressive failure of the sandwich panel: (a) interest zone at ultimate load; (b) local buckling of the top plain C-section with outward deformation of the flanges; (c) interest zone at the second load drop; (d) local buckling of the webs and cohesive PUR foam crack near the top face sheet. ............. 150 Figure 6.1. Tongue and groove joint system: (a) bridge deck FRP sandwich panel [205]; (b) panel-to-panel connection with soft sealant in steel cladding wall sandwich panels [22]. ......................................... 156 Figure 6.2. Z-shape adhesive joint in GFRP sandwich floor panels [207]. ......................................... 157 Figure 6.3. Cover plate joint for panel-to-panel connection in steel web-core sandwich panel [209]. .. 158 Figure 6.4. Side lap joint in steel roof cladding sandwich panels [22]. .............................................. 158 Figure 6.5. Panel-to-panel connection systems: (a) SB; (b) SBT; (c) SW. All units in [mm]. ............... 159 Figure 6.6. Structural model adopted for the design of the panel-to-panel connection. ...................... 160 Figure 6.7. Detail of the cross-section for the panel-to-panel connection design: (a) longitudinal direction; (b) transverse direction. .................................................................................................................. 160 Figure 6.8. Panel-to-panel connection assembly process: (a) panel fastened using ratchet tie down straps; (b) drilling of the screws through the cover plate and sandwich panel. ............................................. 162 Figure 6.9. Transverse direction bending test setup. ........................................................................ 163 Figure 6.10. Transverse direction bending test instrumentation. ...................................................... 163 Figure 6.11. Load-displacement curves of the flexural test in the transverse direction. ...................... 164 Figure 6.12. Bearing failure of the screw hole in specimens SB-1 and SB-2. .................................... 164 Figure 6.13. Cohesive shear failure of the PUR foam in specimen SB-3. .......................................... 165 Figure 6.14. Tilting and pull-out of the screw in specimens SBT-1 and SBT-2. .................................. 165 Figure 6.15. Tensile cohesive failure of the PUR foam near the face sheet-to-core interface. ............. 166 xxii Figure 6.16. Test setup of the flexural test in the longitudinal direction. ........................................... 168 Figure 6.17. Test instrumentation of the flexural test in the longitudinal direction: (a) lateral view; (b) AA section. All units in [mm]. .............................................................................................................. 168 Figure 6.18. Load-displacement curves of flexural test in the longitudinal direction. .......................... 169 Figure 6.19. Failure and specimen inspection after the flexural test in the longitudinal direction: (a) failure mode at one of the loaded cross-sections; (b) screws appearance after failure; (c) screw holes appearance after failure. .................................................................................................................................... 169 Figure 6.20. Displacement evolution for the three monitored cross-section of Panels A and B: (a) specimen DP01; (b) specimen DP02; (c) schematic of lateral view; (d) schematic of cross-section. All units in [mm]. ...................................................................................................................................................... 170 Figure 6.21. Load-longitudinal strain curves of the flexural test in the longitudinal direction. ............. 171 Figure 6.22. Panel-to-wall connection layouts: (a) 1L; (b) 2L; (c) 2LA. .............................................. 174 Figure 6.23. Panel-to-wall connection bending test instrumentation: (a) lateral view; (b) section AA; (c) detail B. All units in [mm]. ............................................................................................................. 175 Figure 6.24. Panel-to-wall connection four-point bending test. .......................................................... 176 Figure 6.25. Panel-to-wall connection bending test: (a) overall setup; (b) 1L; (c) 2L; (d) 2LA. ............ 176 Figure 6.26. Estimation of the flexural stiffness of the connection systems: (a) structural model of the panel-to-wall flexural test; (b) actions on the HEB450 profiles; (c) contribution to the overall deflection. ...................................................................................................................................................... 177 Figure 6.27. Analytical models used to predict the flexural stiffness of the connection systems: (a) simply supported beam; (b) fixed beam. .................................................................................................... 178 Figure 6.28. Load-displacement curves of the flexural test on the panel-to-wall systems. .................. 180 Figure 6.29. Wrinkling of the top face sheet of specimen 1L-02. ...................................................... 180 Figure 6.30. Failure progression and inspection after the flexural test of connection system 1L: (a) tilting of the bolts at the end of the test; (b) undamaged bolt holes after disassembly. ............................... 181 Figure 6.31. Local buckling at both loaded sections of specimen 2L-02. .......................................... 181 xxiii Figure 6.32. Inspection after the flexural test of connection system 2L: (a) rotation at the support; (b) bearing failure of the bolt holes after disassembly. .......................................................................... 182 Figure 6.33. Support conditions of specimen 2LA-01: (a) adhesive excess in the bottom steel angle; (b) adhesive spill in the end cross-section of the sandwich panel........................................................... 183 Figure 6.34. Support conditions of specimen 2LA-02: (a) bottom steel angle; (b) gap between the sandwich panel and the support column. ........................................................................................ 183 Figure 6.35.Failure progression of connection system 2LA: (a) crack at the adhesive-to-top face sheet interface; (b) crack at the bottom face sheet-to-adhesive interface. ................................................... 184 Figure 6.36. Debond between the top plain C-section and bottom lipped C-section of the sandwich panel. ...................................................................................................................................................... 184 Figure 7.1. Engineering stress-strain curves and predicted true stress-strain curves. ........................ 189 Figure 7.2. Tensile coupon test: mesh, loading, and support conditions. .......................................... 190 Figure 7.3. Numerical results of tensile coupon test simulations: (a) stress-strain curves; (b) deformed shape of the FE model before failure; (c) coupon specimen before failure. ....................................... 191 Figure 7.4. Crushable foam constitutive model: (a) predicted yield surface; (b) engineering stress-strain curve and predicted true stress-plastic strain curve. ......................................................................... 193 Figure 7.5. Flatwise compressive test: mesh, loading, and support conditions.................................. 194 Figure 7.6. Comparison between experimental and numerical results: (a) flatwise compressive test; (b) flatwise tensile test. ........................................................................................................................ 195 Figure 7.7. Edgewise compressive test numerical model: (a) EC-1.0 set; (b) EC-1.5 set. Notes: HC – horizontal constraint; RBC – rigid body constraint; RP – reference point; SC – symmetry constraint. 197 Figure 7.8. Mesh convergence study of the numerical model of specimens EC-1.0. ......................... 198 Figure 7.9. Buckling eigenvalues and mode shapes: a) 1st mode of EC-1.0 set; b) 1st mode of EC-1.5 set; c) 2nd mode of EC-1.5 set. .............................................................................................................. 199 A lightweight floor system based on sandwich panel: characterization and development 2 Regarding lightweight floor systems, there have been several proposed solutions that mostly involve materials such as fibre-reinforced polymer (FRP) and cold-formed steel (CFS). FRP materials are increasingly used as face sheets for sandwich panels in bridge deck and roof systems. Sandwich panels consist of two thin and stiff face sheets that are separated by a low-density core material [5]. FRP sandwich panels combine structural and building physics functions due to their high strength-to-weight ratio and low thermal conductivity. Additionally, cold-formed steel profiles are also known for their structural efficiency. They can be combined with filling materials to create sandwich structures or dry board, which can serve as the subfloor of the floor system. In this thesis, a lightweight floor system based on sandwich panels is developed and characterised within the scope of the R&D Project LightSlab – Development of innovative solutions of slab sandwich panels . The project is a joint effort of University of Minho's Department of Civil Engineering (DEC) and Department of Polymer Engineering (DEP), and Ferpainel, S.A. Each partner in the R&D project is responsible for specific tasks. The development of the adhesive connection between the components of the sandwich panels falls under the responsibility of the DEP team. Ferpainel is a manufacturing company specialising in the production of steel insulating sandwich panels for wall and roof cladding systems, as well as cold storage. They are responsible for fabricating the prototypes and making the necessary modifications to their continuous production line to accommodate the new sandwich panel. The scope of this thesis concerns the work carried out by the DEC team, which includes selecting the materials, optimising, and characterising the structural layout of the sandwich panel, and designing the panel-to-panel and panel-to-wall connection systems of the modular floor. The sandwich panel's design needed to meet the requirements of both building codes and the manufacturing process of Ferpainel the industrial partner of the R&D project. The sandwich panel will be a prefabricated modular unit manufactured in a continuous process. When working with prefabricated components, structural efficiency plays a significant role in determining the cost and environmental impact of the final product. The aim of the sandwich panel’s design is to create a structural member that is lightweight, cost-effective and has a low environmental impact. To achieve these goals, optimisation procedures based on genetic algorithms are used to explore optimal solutions based on different objectives. In addition, the modular floor system needs to be easily transported and assembled, especially in constrained spaces such as the rehabilitation of existing 1. INTRODUCTION 3 degraded floors. The system should also be lightweight enough for easy transportation from the factory to the construction site in the context of modular construction. 1.1. RESEARCH OBJECTIVES AND METHODOLOGY The research in this thesis is intended to demonstrate the integration of optimisation procedures in the design of civil engineering structures, as well as in-depth understanding of the main mechanical feature governing the behaviour of the proposed residential floor system. Furthermore, it intends to provide a database of test results that can be used to establish design tools for structural elements such as builtup and foam filled CFS profiles. These elements are not currently covered by existing building standards, which reduces the material efficiency of lightweight floor systems. The outcomes of this research provide also useful information for the inclusion of essential aspects in the numerical modelling of composite structures and the interaction between the different constituent materials. The specific objectives of this thesis were set as follows: 1. Identification of the most suitable materials and architectural layout of the sandwich panel; 2. Design of the sandwich panel and optimisation with regards to weight, cost and environmental footprint by means of genetic algorithms; 3. Mechanical characterisation of the constituent materials of the sandwich panel; 4. Development of the connection systems between sandwich panels and between those and the vertical load-bearing structure; 5. Evaluation of the mechanical performance of the sandwich panel, jointed sandwich panels, and the connection systems with the vertical load-bearing structures; 6. Development of numerical simulations to identify the key aspects governing the sandwich panel behaviour, validate analytical models, and assist to the development of simplified design tools; 7. Development of simplified analytical tools and recommendations for the design of the modular floor system compliant with the requirements of the building codes. The first objective consists of identifying the building code requirements for residential floors and selecting a range of materials for the sandwich panel's core and face sheet that can meet these requirements. The requirements pertain to the structural safety, thermal insulation, acoustic performance, and fire resistance of the floor system. A lightweight floor system based on sandwich panel: characterization and development 4 For the second objective, the available design standards in the context of sandwich panels are reviewed, and a suitable design methodology for the developed prototype is selected. After a preliminary trial-anderror optimization, an optimization procedure based on genetic algorithms is developed. The goal of the optimization is to minimize weight, cost and environmental footprint of the sandwich panel while ensuring that the minimum requirements concerning structural safety, thermal and acoustic performance are met. In the context of prefabricated structural elements, the optimization of the material efficiency of the system plays a crucial role. The algorithm is also intended to be made versatile and accessible to non-experts in order to promote structural optimization tools among practitioners. The third objective of the project involves conducting an experimental campaign on the constituent materials of the sandwich panel. The aim is to obtain their relevant mechanical characteristics. Additionally, tests are carried out to assess the physical properties of the core materials, such as thermogravimetric analysis. Microscope examinations of the coating layer of the face sheet material are also conducted. These tests provide additional information that helps to understand the influence of the manufacturing process of the core material. Furthermore, the results provide the coating layer design parameter to be included in the development of the simplified analytical tools. The fourth objective is divided into two main categories, namely, the development of panel-to-panel connection and panel-to-wall connection. A review of current practices is carried out to develop such connection systems, and the most suitable solution is selected based on the fulfilment of code requirements and the facilitation of assembly and maintenance operations. The fifth objective of the study is to perform flexural tests on small-scale and full-scale sandwich panels in threeand four-point bending configurations. The purpose of testing the single sandwich panels is to evaluate their performance at the serviceability limit state (SLS) and ultimate limit state (ULS), as well as to identify the failure mechanisms. The test results also provided the opportunity to validate the design methodology used to achieve the second objective. In addition, the results of the flexural tests provided a database to calibrate numerical models to gain further insight into the mechanical behaviour of the sandwich panel. Small-scale and full-scale jointed sandwich panels allowed for the study of the connection behaviour along the transverse and longitudinal directions, respectively. The panel-to-wall connection involved tests on full-scale sandwich panels that simulated real-service life conditions. These tests provided information on the failure mechanisms and stress distribution among the sandwich panel and the connection components. 1. INTRODUCTION 5 The sixth objective of this project is to develop finite element models that can replicate the behaviour of constituent materials, small-scale steel sandwich panels with PUR foam core, and the dynamic and static structural response of the sandwich floor panels. By comparing the experimental and analytical results with the models, insight into the failure progression and stress state within the different sandwich panel components is provided. Furthermore, this comparison assesses the reliability of simplifying assumptions made during the preliminary design stage. The seventh objective is a summary of the knowledge obtained from the analytical design, experimental campaign, and numerical modelling, concerning the behaviour of a sandwich panel. Using the validated results of the design procedure, a simplified analytical model is proposed as a design guideline for the sandwich panel floor system. 1.2. OUTLINE OF THE THESIS The thesis is organized in eight chapters. A brief summary of each chapter is presented in the next paragraphs. In the current chapter, the motivation and context of the thesis are presented. As previously mentioned, the thesis has been developed within the scope of the LightSlab R&D project, which is briefly introduced. Additionally, the research objectives and the research methodology adopted for this thesis are discussed. The literature review in Chapter 2 includes a discussion of lightweight structural systems based on sandwich panels as well as CFS structural members. The mechanical behaviour of sandwich panels is briefly introduced, and the challenges of improving the performance of the core while maintaining a reduced weight are discussed. Regarding CFS profiles, the current research trend is to enhance their resistance against instability phenomena. Furthermore, the accuracy of current analytical tools available for predicting CFS structural systems, considering their interaction with possible filling and subfloor components, is discussed. Finally, the application of a genetic algorithm procedure for optimizing the structural design of civil engineering structures is presented, with a focus on including constraints in the formulation of the optimization problem. Three sandwich panel solutions based on the combination of different core, webs and face sheets materials are presented in Chapter 3. Among these solutions, the one that is based on CFS face sheets and webs and polyurethane (PUR) foam core is selected for the optimization process due to its low cost and Ferpainel’s expertise in manufacturing steel insulating sandwich panels. The preliminary structural, A lightweight floor system based on sandwich panel: characterization and development 6 thermal, and acoustic designs of the selected solution are described in detail. Additionally, the effective width method for designing cold-formed steel members is discussed, and the proposed design methodology used in this thesis is presented along with its underlying assumptions. The thesis also illustrates the development and validation of the original optimization program based on genetic algorithms. The optimal solutions in terms of weight, cost, and environmental footprint are presented along with parametric studies on the influence of the most relevant design parameters. Finally, the most economical solution is adopted for the production of the sandwich panel prototypes. The problems faced during different stages of the manufacturing process of the prototypes and their respective solutions are discussed. The mechanical characterisation of the constituent materials of the sandwich panel is presented in Chapter 4. Tensile tests are conducted on CFS coupon specimens extracted from coils and later from the face sheets and webs of sandwich panels. A literature review is presented on how the manufacturing process affects the microstructure and mechanical properties of PUR foam. The PUR foam core material is evaluated under different loading conditions, namely flatwise compressive and tensile tests, as well as shear tests. Moreover, thermogravimetric analyses are carried out to obtain information on the physical properties of the foam. Regarding the PUR-based adhesive developed by the DEP team for joining the components of the sandwich panels, its mechanical performance is characterised by means of tensile tests on dogbone specimens and single lap shear tests. Edgewise compressive tests are carried out on small-scale sandwich panels without webs to preliminarily assess the quality of the composite structure. The results of the edgewise compressive test are also compared to the analytical prediction of the axial stiffness and ultimate load of the sandwich panel. In Chapter 5, the four-point bending test results conducted on a small-scale prototype of the final sandwich panel are presented. The design methodology employed in Chapter 3 is validated using these results. The full-scale sandwich panel's dynamic and static behaviour is investigated through flexural dynamic tests, threeand four-point bending tests. The analytical predictions are compared with the experimental results, considering different assumptions on the level of contribution of the core to the overall mechanical performance of the sandwich panel. Additionally, the influence of assuming the gross and effective properties of the CFS cross-section is also assessed. The development and characterisation of the panel-to-panel and panel-to-wall connection systems are presented in Chapter 6. After reviewing the literature on the currently used panel-to-panel joints, the analytical design of the proposed connection system is described. Three different connection systems 1. INTRODUCTION 7 were tested using three-point bending on joint small-scale sandwich panels in the transverse direction. Considering factors such as the ease of assembly of the joint and mechanical performance, the connection system based on a cover plate fastened by means of self-drilling screws to the bottom face sheet of the sandwich panel is adopted for full-scale testing. Four-point bending tests are carried out on joint full-scale sandwich panel to assess the mechanical behaviour of the connection in the longitudinal direction. In the second stage, the topic of the connection between floor systems and the existing vertical structure is reviewed. The proposed connection systems are illustrated, and their performance is investigated by means of four-point bending tests. The numerical modelling of the constituent materials and dynamic and static flexural behaviour of the sandwich panel is presented in Chapter 7. The material models for the CFS face sheet and PUR foam core have been validated against the results obtained from the mechanical characterization campaign. The strategies used to replicate the mechanical behaviour of the full-scale sandwich panel are discussed, highlighting the key physical features that should be included to obtain reliable results. Finally, based on a comparison of the analytical and numerical models against experimental results, a simplified design tool is proposed. This tool is based on the most conservative prediction of the sandwich panels' performance. The conclusions drawn from the work developed in this thesis are summarised in Chapter 8. Potential solutions for addressing the issues that arose during the development of sandwich floor panels are outlined, and suggestions are provided for future research work that involves both experimental and numerical analyses. A lightweight floor system based on sandwich panel: characterization and development 8 9 CHAPTER 2 2. LIGHTWEIGHT STRUCTURES AND MATERIAL EFFICIENCY The slenderness of the structural composing elements and the lightness of the look have always been a concern for the design of structures. In the past, when technology was not yet advanced enough, perspective helped disguise squat and heavy elements necessary for the stability of the structure, such as in the Parthenon [8]. As the master constructors accumulated experience, they pushed the limit of natural materials in search of highly slender structures. This effort is still vivid in the architecture of Gothic cathedrals [9]. Iron, steel, and reinforced concrete emerged as new building materials capable of outperforming stone, wood, and bricks after the Industrial Revolution [10]. Steel bridges and skyscrapers are nowadays some of the most iconic structures of the human metropolis. Currently, the search for efficient and lightweight building materials is led not only by appeasing the human eye. The sustainability challenge and rapidly progressing climate change must be addressed by the construction sector [11]. As technological advancements progress, new combinations of materials are made possible. New optimisation approaches based on machine learning allow us to improve the design of structural systems in terms of cost, mass, and environmental footprint [12], [13]. The advantages of lightweight structures include, among others, reduced raw material consumption, ease of transportation and installation, decreased manufacturing costs and construction time [14]. In the following paragraph, a brief description of structural systems recently developed for weight-sensitive applications in civil engineering is presented. Further detail on each system will be provided in the next sections. Structural systems based on sandwich panels, cold-formed steel (CFS) profiles or a combination thereof have been employed and studied in weight-sensitive applications in civil engineering in recent years. Sandwich panels are made of two thin face sheets separated, usually, by a low-density core material (see A lightweight floor system based on sandwich panel: characterization and development 10 Figure 2.1a). The face sheets are generally composed of stiff and high-strength materials whereas the core is relatively light and provide adequate stiffness in the direction normal to the faces of the panel [15]. (a) (b) Figure 2.1. Lightweight structural members: (a) sandwich panel; (b) CFS-lipped channel section. CFS profiles are produced by bending flat sheets of metal at room temperature into shapes that will provide a higher load-bearing capacity than the original flat sheets (see Figure 2.1b) [16]. Sandwich panels and CFS profiles have been used in weight-sensitive applications such as temporary shelters, bridge decks, façade cladding, roof systems, modular constructions, and the rehabilitation of degraded floors in existing buildings ([17], [18]). The corresponding benefits of these lightweight structural systems for each application are summarised hereafter. In the context of sandwich panels, fibre-reinforced polymer (FRP) face sheets provide strength properties similar to those of metals, with a weight approximately four times lighter than metallic materials [15]. Lightness is a key factor when developing a temporary house for disaster-hit areas (see Figure 2.3a) since accessibility to roads is usually limited [19]. Thus, such composite solutions show potential to be employed as modular systems in temporary houses [18]. (a) (b) Figure 2.2. Current and potential applications of composite sandwich panels: (a) temporary shelter [18]; (b) bridge decks [20]. Bridge deck solutions based on composite sandwich panels provide longer lifespans eliminating structural integrity issues related to corrosion [21]. Furthermore, lightweight panels allow for rapid construction time and minimum traffic disruption when replacing old concrete deck slabs (see Figure 2.3b) [20]. Flat or 2. LIGHTWEIGHT STRUCTURES AND MATERIAL EFFICIENCY 11 lightly profiled metal sheets with low-density polymeric foam cores possess a high strength-to-weight ratio, durable thermal insulation, and can be rapidly erected and installed even in extreme weather conditions, and economically mass-produced [22]. Metal sandwich panels are already a mature product in several industries, including the industrial partner of the LightSlab R&D Project, and are commonly used to clad low-rise industrial and commercial buildings (see Figure 2.3a). Nevertheless, their use has been restricted to non-structural or secondary structural elements [17]. CFS C-section profiles generally constitute the framing of the load-bearing unit of modular constructions (see Figure 2.3b) [23]. The CFS framing may be combined with sandwich wall panels, though further studies are required in view of their application in residential buildings [4]. The use of lightweight materials for modular construction is necessary due to the limit imposed by the unit transportation. Off-site manufacturing of the units offers several advantages such as reduced construction schedule, improved quality, and reduced resource wastage [24]. (a) (b) Figure 2.3. Current and potential applications of CFS sandwich panels and profiles: (a) Esso headquarters in Leatherhead (UK) [25]; (b) CFS framing of a modular unit [24]. Finally, sandwich paneland CFS-based structural systems show potential to be exploited as lightweight floor systems for the rehabilitation of existing buildings. Degraded timber floors are commonly found in several types of masonry buildings across Europe (see Figure 2.3) [26]. A lightweight floor system based on sandwich panel: characterization and development 18 In research, it has been found that the effectiveness of the web in reinforcing structures is similar to increasing the density of the core material. A study showed that a sandwich panel with a 3% core crosssection occupied by longitudinal webs had equivalent flexural properties to a similar panel with twice the core material density [59] (see Figure 2.7a). High-density core sandwich panels are considered a solution that underutilizes the composite face sheet, as its longitudinal strain is found to be far within the ultimate strain at the peak of the flexural test [60]. Similar conclusions were drawn in another study where a GFRP sandwich beam with inclined longitudinal webs, conferring it a trapezoidal cross-section shape, with lowdensity PUR foam core was compared to an unreinforced GFRP beam with high-density PUR foam core [61]. The flexural test showed that the web-reinforced beam was more than twice as stiff as the unreinforced beam (see Figure 2.7b). Additionally, in the web-reinforced beam, the shear deflection was estimated to be around 1%, demonstrating that the shear stresses are almost completely carried by the webs. (a) (b) Figure 2.7. Structural sandwich panels: (a) web-reinforced sandwich panel subjected to distributed load test [59]; (b) inclined longitudinal web sandwich panel failure in the four-point bending test [61]. The use of low-density core materials is required to achieve the thermal insulation properties required by building codes, making the web-core system an advantageous option for metal sandwich panels. In a study of roofing system applications, a box beam composed of CFS face sheets and C-section profiles acting as webs was investigated [62]. Three and four-point bending tests were performed to assess the resistance of the metal webs to bearing failure and shear buckling. Analytical models estimate that the PUR foam contributes to almost 80% of the web bearing failure resistance at the supports. Another study focused on the flexural resistance of steel web-core sandwich panels. The web core system included trapezoidal sheeting welded to the face sheets. In flexural tests, the panel tended to fail due to instability phenomena such as wrinkling and local instability of the webs at midspan [63]. 2. LIGHTWEIGHT STRUCTURES AND MATERIAL EFFICIENCY 19 There are various methods being used to improve the mechanical behaviour of sandwich panels in order to make them suitable as primary structural elements. Among these methods, the web-core system has shown to be the most effective. However, it adds significant weight which can be a hindrance in replacing degraded timber floors in existing buildings. For composite face sheet sandwich panels, it is possible to enhance the mechanical properties of the polymeric core material, as both materials have high thermal performance. However, sandwich panels with high-density core materials tend to underutilise the composite material and fail in a brittle manner due to debonding between the core and the face sheet. To address this, low-density foam core materials combined with longitudinal webs are an attractive solution to reinforced steel face sheet sandwich panels, particularly for thermal insulation performance. Steel web-core sandwich panels are also ductile, which is a desirable feature for primary structural elements. 2.2. CFS STRUCTURES CFS members have been extensively researched in recent years, and their use as primary structural elements has consequently increased [64]. The main advantages with respect to hot-rolled steel elements are the following ones: i) wider possibilities of cross-section shapes which leads to more efficient material usage [65]; and ii) ease of transportation, installation, and connection [66]. Nevertheless, CFS profiles are sensitive to local and global instabilities due to the thinness of the plate elements. Therefore, different strategies have been developed to improve the resistance to instability phenomena which can be grouped into three categories: i) lightweight floor system consisting of CFS components, self-drilling screws, and a dry board topping layer; ii) CFS profiles infilled with PUR foam; iii) CFS built-up members. In the following sections, a review of the various strategies used to reduce the slenderness of the plate elements which compose the cross-section of CFS profiles is presented. Despite the different methods employed, the literature highlights common challenges. There is a general lack of understanding of the physical features of the structural system, such as the behaviour of the interfaces between different materials or the degree of interaction provided by connection systems (be it adhesive or mechanical). Due to this lack of understanding, analytical tools have so far failed to incorporate these aspects into design standards. Simplified analytical models which exclude such features have been proposed with varying degrees of success. A lightweight floor system based on sandwich panel: characterization and development 20 2.2.1. Lightweight CFS floor system Lightweight floor systems are based on the combination of CFS profiles and a topping layer which may consists of dry boards of different materials as well as hot-rolled steel plates [67], [68]. Nevertheless the current design codes on CFS members do not include any provision on the degree of composite action to be considered between the boards and the CFS profiles. The assessment of the degree of composite action has been addressed by comparing numerical and simplified analytical models with the results of experimental tests [69]. Analytical model generally considers a transformed T-shape cross-section where the floorboard material is converted into an equivalent area of steel material [70]. The validated numerical models may then be used to explore the influence of different parameters on the degree of composite action reached by the composite structures such as thickness of the CFS profiles, spacing and type of the shear connectors, thickness and type of dry boards, and depth of the CFS profiles [64]. In [71] various shear transfer mechanisms between CFS C-lipped profiles and wooden floorboards are investigated, namely self-drilling screws, wood adhesive and epoxy resin. Four-point bending test are carried out on the composite structures and the results are compared to the performance of bare steel sections and analytical models (see Figure 2.8). Furthermore, the tests revealed that the highest level of composite action is achieved by bonding the CFS beams and floorboards with epoxy resin. The analytical approach considers a perfect bond between the floor components and a plastic distribution of stresses. The ultimate load ratio between experimental and analytical prediction is found to be between 0.4 and 0.9 depending on the shear transfer mechanism. Regarding the estimation of the flexural stiffness, the analytical predictions show a better agreement with the experimental results with the ratio included between 0.6 and 0.85. Figure 2.8. Four-point bending test of the CFS beams and wooden boards floor system [71]. 2. LIGHTWEIGHT STRUCTURES AND MATERIAL EFFICIENCY 21 2.2.2. PUR foam infilled CFS profiles Foam-filled CFS profiles can provide significant benefits in terms of mechanical performance, particularly regarding the bending resistance [72]. The PUR foam filling provides a continuous support for the thin plate composing the CFS cross-section thus improving the buckling resistance. Furthermore, with a little increase in weight, major improvement in terms of shock absorption, acoustic and thermal insulations can be achieved [73]. Nevertheless, to optimise the design of such structures further experiments are required on the single constituents and the composite members as well as the development of adequate design methods [74]. The flexural performance of two types of CFS built-up members is compared in [75]. The first type is a bare CFS built-up member consisting of two main profiles laced together by L-profiles. The second type is the same member encased in CFS sheets and filled with PUR foam having a density of 40 kg/m3. Based on the three and four-point bending configuration, flexural tests were conducted to investigate the influence of foam filling on the bending and shear response of the beam. The results of the tests indicate that the load-carrying capacity of members filled with PUR foam is two to three times higher than the load-carrying capacity of bare CFS members in terms of flexural and shear resistance, respectively. The authors suggest that for the design of PUR foam infill CFS members, the design code for hot-rolled steel members should be used since no premature localised failure such as local or distortional buckling was observed during the experiments. Analytical predictions show that the hot-rolled steel guidelines provide a conservative estimation of the load-carrying capacity of infilled CFS members. The effects of foam stiffening on the flexural resistance of CFS lipped C-sections filled with PUR foam is investigated in [74]. The foam is of various densities and the beam specimens have different width-tothickness ratios. The performance of the infilled C-sections is compared to that of bare CFS C-sections. All the specimens show the same type of failure, which is local buckling of the compressed flange. However, the infilled profiles reach the ultimate load at higher levels of stress in the plate elements of the cross-section. Analytical predictions that take into account the stiffening effect of the foam by introducing a buckling coefficient are compared to the experimental results. The predictions are accurate up to a width-to-thickness ratio of about 150. Beyond 150, the predictions become nonconservative as the yielding stress and the buckling stress of the plates approach each other, making the elastic buckling stress less accurate. A lightweight floor system based on sandwich panel: characterization and development 22 Figure 2.9. Dimensions of the infilled C-section [74]. All units in [mm]. Methods to improve the local and distortional buckling load of CFS (cold-formed steel) members is discussed in [76]. The slenderness of the plate, which affects the buckling loads, can be improved by using stiffeners or corrugations. However, these methods primarily improve local buckling rather than distortional buckling. In this research, PUR foam is suggested as a strategy to enhance the performance of lipped C-sections. Both numerical simulations and experimental testing are carried out to assess the axial load-carrying capacity of the infilled profiles. The numerical simulations showed an improvement of approximately 50% in terms of ultimate load compared to bare CFS profiles. However, the axial compression test showed a more limited improvement of approximately 25%. The authors suggest that the reason for not achieving the same level of ultimate load in the test is due to the poor adhesion between the PUR foam and the plates of the CFS cross-section (see Figure 2.10). Figure 2.10. Cut-out piece of the PUR foam infilled CFS profile with delamination between the foam and the steel [76]. 2.2.3. CFS built-up members CFS profiles are also combined with each other to improve the resistance of common open C and Zsections to lateral torsional buckling [77]. Common solutions involve joining together back-to-back CFS- 2. LIGHTWEIGHT STRUCTURES AND MATERIAL EFFICIENCY 23 lipped C-sections to obtain a doubly symmetric open cross-section or assembling one CFS-lipped Csection with a plain C-section to obtain a box beam [78]. Self-drilling screws are used to connect the CFS members. Nevertheless, CFS design codes [79], [80] do not provide methods for the design of built-up CFS cross-sections, namely the degree to which different members work as a monolithic element [81]. Several researchers have proposed different design methods. According to [82], the flexural capacity of a built-up box section subjected to concentric loading may be obtained by application of the superposition method. The sum of the capacity of the individual members yields the strength of the built-up member. A numerical model was created to replicate the results of flexural tests on a box beam composed of a CFS-lipped C-section and a track section fastened by self-drilling screws. The numerical model was validated and used to conduct a parametric study, investigating the impact of various factors such as steel yield strength, width-to-thickness ratio, screw spacing, and location of load application. The study concluded that the superposition method [79] is only reliable for concentric loading, and design guidelines are more accurate for height-to-thickness ratios greater than approximately 220 and screw spacings less than 150 mm. Different analytical methods for predicting the flexural resistance of regular and lipped CFS C-sections and built-up box beams composed of the latter profiles are compared against experimental and numerical results in [83]. The EWM described in [79] accurately predicts the flexural strength of lipped C-sections but not plain C-sections. The superposition method for predicting the flexural strength of built-up box beams leads to slight overestimations and requires a correction factor of 0.9. The Chinese Specification for thin-walled steel buildings (2011) [84] proposes an equivalent box beam method which considers the built-up section as an equivalent thin-walled rectangular hollow section. The overlapping flanges are converted into a single stiffened element through the equivalent built-up box beam, with the thickness of a single flange. This assumption is deemed reliable for built-up box sections with a width-to-thickness (𝑤/𝑡) ratio not greater than 100 (see Figure 2.11). In reference to predicting deflection at the serviceability limit state (SLS), [66] has compared the results of flexural tests on back-to-back built-up C-sections with the analytical predictions based on the EWM as described in [80]. Different methods have been proposed, taking varying assumptions into account: i) variation of the modulus of elasticity along the beam; ii) variation of the effective properties of the beam based on the stress level along the beam; iii) variation of the previous two along the beam. The comparison with experimental results shows that Young's modulus has a negligible influence on deflection prediction. However, the effective properties of the cross-section have a significant impact on analytical A lightweight floor system based on sandwich panel: characterization and development 24 estimations. Assuming that the gross-cross-section properties remain constant throughout the beam's length leads to an underestimation of deflection by approximately 8%. To account for the variation of effective properties along the beam, the interpolation method described in [80] is a simplified yet reliable method. It slightly overestimates the vertical deflection by about 4%. Figure 2.11. Comparison of actual and predicted strength of built-up box beam [83]. Based on the literature review of lightweight structural systems based on CFS profiles the following conclusions may be drawn: i) PUR foams with densities not greater than 40 kg/m3 are commonly used as an infill material to improve the resistance of single CFS profiles and built-up CFS cross-section; ii) analytical models for considering the mechanical contribution of the PUR foam on the overall behaviour of the CFS infilled profiles lack experimental and numerical validation; iii) building codes design guidelines appeared to be nonconservative for built-up box beam and even for single CFS profiles; iv) assembly processes of CFS profiles-based structural system are not yet automated; v) connection methods mostly involve self-drilling screws and laser welding; vi) adhesive connections between CFS elements have not yet been explored. 2.3. OPTIMIZATION SEARCH METHODS The mathematical definition of an optimisation problem for the objective function 𝑓(𝒙) is the following: given a variable vector 𝒙 in the solution space 𝑿, find the vector 𝒙∗ that minimises (or maximises) the objective function so that min𝑓(𝒙)=𝑓(𝒙∗). Furthermore, the solution space 𝑿 is generally restricted 2. LIGHTWEIGHT STRUCTURES AND MATERIAL EFFICIENCY 25 by one or multiple constraints. The constraints may be expressed as inequalities in the form of 𝑔i(𝒙)≤0 with 𝑖=1,…,𝑚 and/or as equalities such as ℎj(𝒙)=0 with 𝑗=𝑚,…,𝑝 [85]. In civil engineering, the challenge often involves finding the optimal combination of design variables from all feasible designs for a structural element. This solution should minimize construction costs while also meeting strength and serviceability requirements [86]. Traditional search methods for optimisation problems, such us calculus-based, enumerative, and random, lack robustness [87]. When searching for the optimal solution, calculus-based methods explore the surrounding area of the initial search point. However, they rely on continuity and derivative existence in equations, which are not always present in real-world problems with a vast multimodal (multi-peaks) search space full of discontinuities. On the other hand, enumerative methods involve assessing the objective function value at each point in the space, but this approach can be inefficient for extensive search spaces. Metaheuristic search methods have received increasing popularity among researchers in the solution of optimisation problems [12]. Such procedures sample a subset of the solution space otherwise too large to be fully investigated. Random choices are used by these procedures to guide a highly exploitative search of the solution space [87]. Genetic algorithms are an example of such a search procedure. When compared to other search methods, the main advantages of these methods can be summarised as follows: (i) they can handle various types of variables such as continuous, discrete, integer, and/or categorical; (ii) they do not require any extra information apart from the value of the objective function itself; and (iii) being a population-based approach, they are less likely to be trapped in local optima [88]. In the next section, an overview of genetic algorithms is presented, as well as a description of the main genetic operators. 2.3.1. Genetic algorithms A genetic algorithm (GA) is an optimisation and search procedure based on the Darwinian principle of the survival of the fittest. An initial population 𝑃1 of 𝑁 individuals (chromosomes), which represents the potential solutions to the optimisation problem, is randomly initialised. The individuals are evaluated using a predefined fitness (objective) function, and the best solutions are given higher chances to reproduce, whereas the weakest may not reproduce at all. New individuals (offspring) are produced by recombination (crossover) of the parent’s chromosomes. Sometimes random new features may be introduced through mutation in the offspring population 𝑄t. The solutions in 𝑄t are evaluated and, according to different selection procedure, they are merged with solutions from 𝑃t to form the population 𝑃t+1. Improvement A lightweight floor system based on sandwich panel: characterization and development 26 in the population results from the repeated selection of the best-performing individuals and the consequent elimination of low-performers [89]. A generic GA procedure is illustrated in Figure 2.12. Figure 2.12. The procedure of a generic GA. Although all GAs follow the steps shown in Figure 2.12, various techniques have been employed to customise their application for specific purposes. When dealing with multi-objective optimisation problems, GAs can be categorised into two primary groups. In the first category, the objective functions are merged into a single composite function. This is typically done by assigning weights to each objective and creating a linear combination that incorporates all of them. Another approach consists in retrieving a Pareto solution set. The Pareto solution set includes solutions that are not dominated by any other solutions, meaning that no other solutions are superior in all attributes [90]. Implementing the former approach is advantageous as it is a simple extension of the single objective GA. On the other hand, Paretobased approaches offer a set of solutions equally distributed in the performance space, which allows a possible decision-maker to perform trade-off studies and select the most suitable solution. Another issue that has been addressed concerns the transformation of the objective function, a measurement of the performance of the solution, into the probability of reproduction of that solution [91]. The direct use of a solution's performance level to determine its mating probability can result in premature convergence at the start of the run, as it may favour “super individuals” [92]. For the same reason, the procedure is predicted to slow down at the end of the search when the difference between the individuals’ 2. LIGHTWEIGHT STRUCTURES AND MATERIAL EFFICIENCY 27 performance is negligible as the algorithm approaches convergence. To overcome these issues, two types of fitness assignment have been developed: i) scaling, which consists in calculating the fitness by means of a non-linear function of the objective functions; and ii) ranking which orders the population according to the objective function and then assigns a fitness value based on the absolute position of the individual in the population [89]. Finally, it is necessary to devise a way to incorporate constraints in GAs, as they are unconstrained search methods [29]. One of the most common strategies is the introduction of a penalty function [93]. The penalty function involves adding a specific value to the objective function, depending on the number of constraint violations in a given solution. Penalty functions may be divided into three main categories: i) static; ii) dynamic; and iii) adaptive. Static penalty functions add a constant penalty to infeasible solutions, i.e. solutions that do not satisfy the constraint of the optimisation problem. Dynamic penalty functions generally increase the severity of the penalty with an increasing number of generations in order to force the final solution to be feasible. Finally, adaptive penalty functions will increase the severity of the penalty based on the ongoing success of the search for the global optimum. The increase in the penalty depends on the feasibility of the best solutions found so far in the evolutionary process. The choice of the penalty functions is particularly important since the global optimum often lies at the boundary of the feasible solution space [94]. 2.3.2. Applications in structural design Over the last few decades, numerous studies have been conducted on optimization techniques [63]. However, the utilization of such methods has been limited in current engineering practices, often relying on trial-and-error procedures [26]. This may be due to various factors, as highlighted in [64], which recommends that researchers collaborate with engineering firms to develop practical design optimization software. Additionally, much of the literature on optimization algorithms focuses on their mathematical aspects and their success in solving rather trivial examples [95]. Therefore, the following paragraphs will focus on studies related to the development of civil engineering problem-oriented optimisation methods, to support the creation of a benchmark library of existing and new heuristic search procedures, as suggested in [96]. The creation of such a repository may contribute to fostering the use of optimisation methods among practitioners. In [30] the objective is to find the most economical design for a sandwich panel consisting of steel face sheets and a PUR foam core. The optimal sandwich panel should also be able to cover a wide range of A lightweight floor system based on sandwich panel: characterization and development 34 pressed on top of the expanding PUR foam; v) the sandwich panel is heated up to 40 °C to speed up the curing of the PUR foam; vi) the continuous panel is cut to the required lengths by a flying saw. Based on the current manufacturing process of Ferpainel, in the R&D proposal, it was established that all the different panel architectures had to include metal face sheets. This is due to the fact that changing the face sheet material would imply modifications to the production line of Ferpainel not achievable during the timeline of the project. (a) (b) (c) (d) (e) Figure 3.1. Manufacturing process of steel insulating sandwich panels: (a) uncoiling CFS sheet; (b) roll forming CFS sheet; (c) PUR foam’s monomers injection; (d) PUR foam curing in heating chamber; (e) cutting of the sandwich panel. As discussed in the literature review in Chapter 2, sandwich panels with low-density cores do not currently meet the necessary requirements for primary structural applications in civil engineering. Thus, the sandwich panel developed in this research implements a web-core system to enhance shear stiffness 3. DESIGN, OPTIMISATION, AND MANUFACTURING 35 while maintaining a reduced weight. For that purpose, core reinforcing elements made of steel and GFRP, and core materials made of PUR foam and balsa wood are the chosen options based on information gathered in the literature review. Low-density PUR foams have low thermal conductivity, which can lead to significant weight reduction. On the other hand, balsa wood has higher mechanical properties that can provide improved shear stiffness, reducing the number of webs (core reinforcing elements) necessary to achieve the required structural performance. A schematic of various panel architectures and their constituent materials which are considered as possible modules of the sandwich panel floor system is illustrated in Figure 3.2. Detailed information about the dimensions of the cross-sections and the characteristics of the sandwich panels obtained from the trial-and-error optimization procedure are described in Section 3.5. Solution SP1 uses CFS face sheets, PUR foam core, and longitudinal webs in the shape of Cand Z-section profiles. SP2 involves CFS face sheets, GFRP webs, and PUR foam core. Finally, SP3 is made up of a balsa wood core enclosed by CFS face sheets and C-section outer webs. (a) (b) (c) Figure 3.2. Layout of the sandwich panels: (a) SP1; (b) SP2; (c) SP3. A lightweight floor system based on sandwich panel: characterization and development 36 3.1. STRUCTURAL AND TECHNICAL REQUIREMENTS The sandwich floor panels shall meet the standards for building floors, which include structural, thermal, acoustic, and fire performance requirements. However, this thesis only focuses on the structural design of the sandwich panel and does not cover its thermal and acoustic behaviour, as well as fire performance. These aspects are important for the overall performance of the sandwich panels and should be addressed in future studies. Nonetheless, a preliminary study of the thermal, acoustic, and fire resistance performances was also conducted. The structural analysis and design process adhered to the standards set by Eurocodes, the relevant transposed Portuguese norm, and the European Recommendations for Sandwich Panels that were applicable at the time of execution of this task of the research project. The following guidelines and standards were taken into consideration during the process: • Eurocode 0 – Basis of structural design (2002) [100]; • Eurocode 1 – Actions on structures – Part 1-1: General actions – Densities, self-weight, imposed loads for buildings (2002) [101]; • Eurocode 3 – Design of steel structures – Part 1-3: General rules – Supplementary rules for coldformed members and sheetin (2006) [80]; • Eurocode 3 – Design of steel structures – Part 1-5: Plated structural elements (2006) [102]; • “Eurocódigo 3 – Projeto de estruturas de aço – Parte 1-1: Regras gerais e regras para edifícios” (2010) [103]; • CNR – DT 205/2007, Guide for the Design and Construction of Structures made of FRP Pultruded Elements (2007) [104]; • CEN WG 4, Fibre Reinforced Polymer Structures – Scientific and Technical Report (2014) [105]. In the preliminary thermal design of the sandwich panel, the thermal transmittance limit value was taken into consideration as per the Portuguese code for building energy efficiency [106]. Such limits are set to ensure good thermal performance of the building or its autonomous part with regard to energy consumption, technical system efficiency, and condensation reduction. The design was carried out for the most rigorous climatic zone, which has a maximum thermal transmittance limit of 0.30 W/m2K. The primary application of the sandwich floor panel is for the rehabilitation of existing buildings, which does not envisage any specific acoustic requirement. However, to ensure a minimal level of sound insulation for its use in new constructions, it is desirable to consider the acoustic performance. The 3. DESIGN, OPTIMISATION, AND MANUFACTURING 37 Portuguese code on the acoustic requirements for buildings [107] specifies limit values for airborne and impact sound insulation. Lightweight building elements have inherent difficulties in achieving the required sound insulation, as sound insulation is directly proportional to the mass of the building element. Therefore, reduced values of acoustic requirements were considered, which are 60% of those established by the Portuguese standards. For the sandwich panel design, a minimum value of 50 dB was considered for airborne sound insulation (𝐷nT,w). The airborne sound insulation value is a measure of the amount of sound pressure reduction offered by the panel separating two rooms. For impact sound insulation (𝐿nT,w), a maximum value of 60 dB was considered as the threshold sound pressure level in the room. Impact sound insulation is expressed as the sound pressure level registered in a room when one of the enveloping elements is excited by impact. The ability of a building component to resist fire is measured in terms of its resistance to collapse or excessive deflection (R), resistance to flame and hot gases penetration while maintaining structural integrity (E), and insulation offered to the unexposed face to prevent ignition of material in contact with it (I). These values are represented by a number which indicates the time duration for which the criteria must be satisfied. The Portuguese fire safety code [108] specifies the values that are applicable based on the risk category of the building. For floors and roofs of residential buildings belonging to the lowest risk category, a minimum resistance of 30 minutes (REI 30) should be provided. 3.2. STRUCTURAL DESIGN All the sandwich panels were designed according to the requirements of the Eurocode 0 (2002) [100] and Eurocode 1 (2002) [101]. A detailed description of all the design procedures, including design equations and safety factor coefficient, may be found in [109] and [110]. In this section, the overall design approach is presented, and specific details are provided only for the design of the CFS components of the sandwich panel since they are present in all the proposed solutions. Furthermore, the influence on the sandwich panel final cross-section related to challenges faced during the prototype manufacturing process and the connection system between sandwich panels are also discussed. The adopted structural model is a simply supported beam subjected to a uniformly distributed load (see Figure 3.3). This represents the worst-case scenario for design as it leads to high stresses and mid-span deflections. Furthermore, this assumption fails to consider the impact of the connections between panels and walls, which could prove advantageous in terms of decreasing the vertical deflection. The span length A lightweight floor system based on sandwich panel: characterization and development 38 is 5.0 m, which is considerable in comparison to the average span length found in old masonry buildings across Europe, approximately 4.0 m, according to the literature on sandwich panels for civil engineering applications [111] [36]. The uniformly distributed load (𝑞) considers the self-weight of the sandwich panel, an additional permanent load of 1.5 kN/m2 to account for the presence of floor finishing, partition walls and suspended ceiling, and the load due to variable action (imposed load) on residential buildings, which is set equal to 2.0 kN/m2 according to Eurocode 1 (2002) [101]. Figure 3.3. Structural model adopted for the design of the sandwich panel. The self-weight of the non-structural members envisages the possibility of using additional systems such as insulation layers, floating floors, and drop-down ceilings. A thick layer of PUR foam in sandwich panels generally ensures good thermal insulation. Nevertheless, the presence of metal webs inside the core represents thermal bridges which may make necessary the addition of adequate insulation layers. Floating floors and drop-down ceilings will probably be required due to the poor acoustic insulation of sandwich panels. Indeed, acoustic insulation properties rely on the mass of the structural element, which in the case of sandwich panels, is generally reduced. The plenum space between the sandwich panels and the drop-down ceiling may accommodate sprinklers and detectors, which will improve the fire resistance of the sandwich floor panel. Nevertheless, the thermal, acoustic and fire resistance performances of the sandwich floor panel will not be discussed in detail as they are out of the scope of this thesis. The general design approach at the ultimate limit state (ULS) consists in estimating the design crosssection resistance to bending moment (𝑀Rd) and shear (𝑉Rd). Substituting 𝑀Rd and 𝑉Rd in the equations of the bending moment and shear diagrams of the simply supported beam and solving for the load carrying capacity of the sandwich panel yields: 𝑅Md=8∙𝑀Rd 𝐿2 (3.1) 𝑅Vd=2∙𝑉Rd 𝐿 (3.2) where 𝐿 is the span length, 𝑅Md is the load carrying capacity of the panel with respect to bending moment, and 𝑅Vd is the load carrying capacity of the panel with respect to shear. The structural safety 3. DESIGN, OPTIMISATION, AND MANUFACTURING 39 verifications at the ULS are considered satisfied if the load carrying capacity of the panel is equal or greater than the design load calculated according to the fundamental load combination of Eurocode 0 (2002) [100] as stated in Equations (3.3) and (3.4): 𝑅Md≥𝛾G∙𝐺k+𝛾Q∙𝑄k (3.3) 𝑅Vd≥𝛾G∙𝐺k+𝛾Q∙𝑄k (3.4) where 𝐺k represent the sum of the self-weight of the sandwich panel and the additional 1.5 kN/m2 of permanent loads, 𝑄k represent the imposed load for residential buildings, 𝛾G and 𝛾Q are the partial safety factors for permanent and live loads and they are set equal to 1.35 and 1.5, respectively [100]. For what concerns the serviceability limit state (SLS), the sandwich panel's stiffness is designed to fulfil the requirements imposed on the vertical deflection of generic floors in the national annex of the transposed Portuguese version of the Eurocode 3 Part 1-1 [103]. The total vertical deflection (𝛿max) is the sum of the deflections due to the permanent actions (𝛿1) and due to the variable action for residential buildings (𝛿2). The limits on the above-mentioned vertical deflections are given by the following equations: 𝛿max≤𝐿 250 (3.5) 𝛿2≤𝐿 300 (3.6) Regarding the vibrations, the Portuguese National Annex [103] considers the requirements fulfilled if the natural frequency of the element is higher than 3 Hz. The estimation of the natural frequency may be exempted if the total vertical deflection estimated for the characteristic load combination, given in Equation (3.7), is less than 28 mm. Such a requirement is already fulfilled if Equation (3.5) is satisfied, given that 𝐿/500 for a span length of 5.0 m result in a limit of 20 mm. It should be noted that for the assessment of the SLS, the loads should be combined according to the following equation [100]: 𝐺k+𝜓0∙𝑄k (3.7) where 𝜓0 is the factor for the characteristic value of the variable action for residential areas set equal to 0.7 [100]. A lightweight floor system based on sandwich panel: characterization and development 40 3.2.1. Effective width method As previously stated, CFS members may be a preferable option compared to the common hot-rolled steel members. A wide range of shapes may be obtained for CFS members' cross-section by cold forming operations, and therefore favourable strength-to-weight ratio can be achieved economically. Furthermore, CFS members are ideal for prefabrication and mass production, they are fast and easy to erect and install, economical to transport and handle, and they are recyclable. All these advantages can provide significant cost saving in construction [112]. Usual shapes for CFS members are C-section, Z-section, hat sections, as well as profiled sheets for roof and wall panels as illustrated in Figure 3.4. (a) (b) (c) (d) (e) Figure 3.4. Typical CFS cross-sections: (a) C-section; (b) Z-section; (c) hat section; (d) roof panel; (e) curtain wall panel. The verification process for these profiles is similar to class 4 hot-rolled steel profiles as outlined in Eurocode 3 Part 1-1 (2005) [113]. The effective properties of the profiles are determined using methods described in Eurocode 3 Part 1-3 (2006) [80] and 1-5 (2006) [102]. Eurocode 3 Part 1-3 (2006) [80] specifically addresses the verification of cold-formed steel profiles, while Eurocode 3 Part 1-5 (2006) [102] covers both hot-rolled and cold-formed steel profiles of class 4 and provides guidance on determining their effective properties. The approach proposed in the Eurocode assumes that each element of the cross-section of a cold-formed steel member (flanges, webs, edge stiffeners) may be considered as a thin flat plate supported on its lateral edges by their contiguous part. These plates may buckle at a stress level generally lower than the yielding stress of the material, or the stress required to trigger global buckling. The calculation of the elastic critical buckling load of a plate represents a first assessment of the behaviour of thin plates. Solving the differential equation of the vertical displacement of a thin plate simply supported on all four sides and 3. DESIGN, OPTIMISATION, AND MANUFACTURING 41 subjected to a uniform compressive state of stress in one direction yields the elastic critical buckling of the plate (𝜎cr) according to Equation (3.8): 𝜎cr=𝑘σ∙𝜋2∙𝐸 12(1−𝜈2)∙(𝑡𝑏)2 (3.8) where 𝑏 is the width of the plate, 𝐸 is the Young’s modulus of the plate material, 𝑘σ is the plate buckling coefficient depending on the stress ratio and the boundary conditions, 𝑡 is the thickness of the plate, and 𝜈 is the Poisson’s ratio of the material. However, plates, unlike columns, may sustain greater loads than the predicted buckling load. This is due to the redistribution of the compressive stresses once the plate buckles as a result of the membrane tensile and shear stresses developed in the direction perpendicular to loading. The method to include the postbuckling reserve in the analytical estimation of the ultimate strength of thin plates was first proposed by von Karmán [114]. The method, known as EWM, assumes that when the plate reaches collapse a non-uniform distribution of stresses arises as illustrated in Figure 3.5a. The stresses are maximum at the edges of the plate and may be approximated to a uniform distribution of stresses (𝜎av). (a) (b) Figure 3.5. Compressive stress distribution and buckled cross section according to the EMW: (a) simply supported plate; (b) Von Karmán plate. A lightweight floor system based on sandwich panel: characterization and development 42 The effective width is the width of an equivalent plate which fails when the compressive stresses reach the yielding strength of the material (𝜎y). Thus, the effective width (𝑏eff) may be expressed as: 𝜎av∙𝑏=𝜎y∙𝑏eff (3.9) Imposing the equivalence between the critical load of the plate reaching collapse, given by Equation (3.8), and the critical load of the effective width, and solving for 𝑏eff yields: 𝑏eff=𝑏∙√𝜎cr 𝜎y (3.10) Substituting Equation (3.9) into Equation (3.10) yields the ultimate stress of the plate considering the postbuckling reserve: 𝜎av=√𝜎cr∙𝜎y (3.11) The Eurocode approach involves the estimation of the slenderness of the plate (𝜆p) to establish the effective width of the plate element by combining Equations (3.8) and (3.10): 𝜆p=𝑏 𝑏eff=√𝜎y 𝜎cr=𝑏𝑡∙√12(1−𝜈2) 𝜋2∙𝐸∙𝑘σ∙𝜎y (3.12) A further condition on 𝜆p is prescribed by the Eurocode to determine the reduction factor of the crosssection due to local instabilities (𝜌). The correction coefficient is based on studies of postbuckling strength of plates conducted by Winter [115]. They include the effect of out-of-plane imperfection, and residual stresses on real cold-formed steel members. When a CFS members is subjected to a compressive state of stress due to axial loading, bending moments or shear, in order to evaluate the ultimate strength, it is necessary to estimate the effective widths of all the plate elements composing the cross-section. The iterative procedure proposed in Eurocode 3 Part 15 (2006) [102] to account for the effect of local buckling is illustrated in Figure 3.6. First, the stress distribution (𝜓) in the plate elements of the cross-section is calculated according to the gross cross-section characteristics. The reduction factor for the area of each plate element is determined according to the slenderness of the plate which in turn depends on the buckling coefficient of the plate. The effective characteristics of the cross-section are estimated. The stress distribution is calculated again according to the effective cross-section characteristics until convergence is reached. In the scope of this 3. DESIGN, OPTIMISATION, AND MANUFACTURING 43 thesis, the location of the neutral axis (𝑦G) was considered the convergence criterion. Convergence was assumed when the difference between the neutral axis location between two consecutive iterations was less than 10-3 mm. Figure 3.6. Flow chart of the calculation of the effective properties of CFS cross-sections considering the effect of local instabilities. 3.2.2. Effect of distortional instabilities CFS members are often augmented with stiffeners which further complicate the estimation of the ultimate strength [116]. Stiffener may consist of folds or bends located at the extremities or in the middle of the plate element (see Figure 3.7). The instability mode associated with the stiffener is defined as distortional buckling. It generally involves the movement of the line junction between the flange and the stiffener without a rigid body rotation or translation of the whole cross-section [16]. This is opposed to local buckling modes where typically outof-plane deformation of the web or flange elements occur without movement of the line junction between those elements and those elements and the eventual stiffener. A lightweight floor system based on sandwich panel: characterization and development 50 𝐿n=155−30log(𝑚´)+10log(𝑇s)+10log(𝜎)+log(𝑓 𝑓ref) (3.25) 𝐿n=43+30log(𝑓)−𝑅 (3.26) where 𝑇s is the structural reverberation time. In EN ISO 12354-2 (2017) two alternative methods are proposed for the estimation of 𝐿n as reported in Equations (3.25) and (3.26). Therefore, the average value obtained from the two expressions is considered. Both the values of 𝑅 and 𝐿n are plotted over an interval of frequencies ranging from 100 Hz to 3500 Hz. To simplify the design, a single number quantity of the sound insulation values can be estimated according to the methods proposed in the EN ISO 717-1 (2013) [124] and ISO 717-2 (2013) [125]. The method consists in translating a reference curve until the difference with the estimated curve is less than 32 dB (see Figure 3.12). Figure 3.12. Airborne sound insulation (𝑅) of the sandwich panel, the ISO 717-1 (2013) [124] reference curve, and the translated reference curve. The value of the shifted reference curves at 500 Hz represents the single number quantity sought, namely the weighted sound reduction index (𝑅w) and the weighted normalized impact sound pressure level (𝐿n,w). 𝑅w and 𝐿n,w are then converted into the weighted standardized level difference (𝐷nT,w) and the weighted standardized impact sound pressure level (𝐿nt,w) according to the following Equations: 𝐷nT,w=𝑅w+10log(0.32∙𝑉 𝑆s) (3.27) 𝐿nT,w=𝐿n,w−10log(0.32∙𝑉 𝑆s) (3.28) 3. DESIGN, OPTIMISATION, AND MANUFACTURING 51 where 𝑉 is the volume of the laboratory receiving room and 𝑆s is the surface of the sandwich floor panels separating the receiving room from the source room. 3.5. FIRE RESISTANCE ASSESSMENT Modelling the thermo-mechanical behaviour of sandwich panels in fire is a challenging task because many thermal, physical, and mechanical processes occur simultaneously, and they all depend on various factors, such as temperature, heat flux, and the duration of the fire [126]. Therefore, research studies involve conducting fire tests to quantify the fire resistance of the sandwich panels and to gather the necessary input material thermal properties for developing numerical and analytical models [127], [128]. Although the analysis of the sandwich panel's fire resistance behaviour is out of the scope of this thesis, the results of the fire resistance test (see Figure 3.13) carried out at the laboratory of the Spanish Association for the Promotion of Research and Fire Safety Technology (AFITI) in collaboration with Kolumbarium Consulting SL are presented below. The experimental campaign's outcome will serve as a reference database for future studies on the sandwich panel's fire resistance performance. (a) (b) Figure 3.13. Fire resistance test setup and instrumentation: (a) lateral view; (b) top view. All units in [mm]. A lightweight floor system based on sandwich panel: characterization and development 52 The specimen analysed in this experimental campaign is the final prototype described in Section 3.8.2 of this chapter. The sandwich panel has a length of 2000 mm, a total thickness of 170 mm, and a width of 500 mm. The experiment was carried out in accordance with EN1363-1 (2020) [129] using a four-point bending loading configuration over a clear span length of 1800 mm and a shear span of 600 mm. To apply a total dead load of 10 kN, steel blocks were placed on the unexposed face sheet, while the panel was subjected to the ISO 834-1 (1999) [130] fire curve. Six type K thermocouples (TCK) monitored the temperature of the unexposed face sheet (see Figure 3.13). The temperature measured over the area of the top face sheet of the sandwich panels presented similar qualitative development. Therefore, the evolution of the mean temperature over time on the unexposed face sheet is presented in Figure 3.14. Figure 3.14. Thermal response of the sandwich panel. The temperatures on the top face sheet of the panel present a very small increase until approximately 18 min, when a steep increase is registered. The higher heating rate is most likely due to the degradation of the upper part of the core. Two fire safety criteria were assessed based on the test results, namely integrity (E) and insulation (I). However, since there was no measurement available for the vertical deflection, it was not possible to evaluate the load bearing capacity (R). For what concerns the (E) criterion, the sandwich panel ensured its separating function for 21 min when sustained flames could be observed at one of the ends of the specimen. Regarding criterion (I), the sandwich panel registered an increase of more than 140 K after 3. DESIGN, OPTIMISATION, AND MANUFACTURING 53 20 min at the location of Thermocouple 3. Thus, the panel did not comply with the minimum requirements imposed by the Portuguese fire safety code [108]. The panel showed poor resistance to fire, which highlights the importance of installing fire protective systems in view of its application in residential buildings. However, the use of suspended calcium silicate board as a passive fire protection system has proven to be an effective solution to increase the fire endurance of sandwich panels by 20 to 35 minutes [131], making it feasible for the sandwich panel to meet the minimum fire safety requirements. 3.6. SANDWICH PANEL SOLUTIONS Three different sandwich panel solutions were considered by combining different face sheets and core materials. As stated previously, the different solutions are based on the web-core system. This architectural layout was deemed the most appropriate for the new flooring system. It is cost-efficient as it allows the use of economic and low-strength core materials that are currently available in the market and does not complicate the manufacturing process of sandwich panels [132]. In the preliminary design stage, mechanical connections were envisaged between the webs and the face sheets of all the solutions. However, in a later stage, due to a requirement made necessary by the manufacturing process, adhesive bonding was selected as a preferable option. A tailored polyurethane-based adhesive was developed by the team of the Department of Polymer Engineering of the University of Minho. Further information on the subject will be provided in Section 4.3 of this thesis. The different solutions are illustrated in Figure 3.15, Figure 3.16, and Figure 3.17. Their cross-section is the result of a preliminary optimisation process by trial-and-error method. The optimisation parameters were the total thickness of the panel and the number of reinforcing elements in the core. The target of the optimisation was to retrieve the architecture with the lowest cost in fulfilment of the structural safety requirements at the ULS and SLS. Solution SP1 includes CFS face sheets and webs and a PUR foam core (see Figure 3.15). The outer webs enclosing the core are constituted by C-section profiles, whereas the inner webs are composed by Z-section profiles. The design considered both the face sheets and the webs composed of continuous hotdip zinc-coated carbon steel sheets with a yielding and ultimate strength of 220 MPa and 300 MPa, respectively. The contribution of the PUR foam to the mechanical resistance was neglected as a conservative assumption. The governing design factor was the resistance of the cross-section to bending moment. A lightweight floor system based on sandwich panel: characterization and development 54 Figure 3.15. SP1 cross-section layout. All units in [mm]. Note: the thickness of the face sheets and webs are not to scale. Solution SP2 includes CFS face sheets, GFRP webs and a PUR foam core (see Figure 3.16). The face sheets possess the same characteristics as those considered for solution SP1. The GFRP webs were selected from the commercial Cand I-profiles manufactured by the company Fiberline Composites A/S. The laminate architecture is symmetrical and balanced. The internal and external laminas are made up of unidirectional glass fibre roving and bi-directional fabric mats, respectively. The design considered the mechanical properties attributed to pultruded profiles of class E23 according to EN 13706-3 (2002) [133]. The contribution of the PUR foam was neglected as per the previous solution design. The structural safety verifications were carried out according to the Italian Guide for the Design and Construction of Structures made of FRP Pultruded Elements (2007) [105]. The design was carried out for the most stressed reinforcing element, namely the GFRP I-profile. The resisting cross-section included the effective part of the CFS face sheets calculated according to the Eurocode 3 Part 1-3 (2006) [80] and Part 1-5 (2006) [102]. Ultimately, the design was governed by the deflection limit at the SLS. Figure 3.16. SP2 cross-section layout. All units in [mm]. Note: the thickness of the face sheets and webs are not to scale. Solution SP3 includes CFS face sheets and webs and a balsa wood core (see Figure 3.17). The face sheets and webs have the same characteristics as those in the previous solutions. The balsa wood core consists of veneer layers with the grain oriented in the direction perpendicular to the face sheets of the panel. Average properties of balsa wood core and sandwich panel failure modes considered in the design were based on the guidelines reported in the CEN WG 4 Scientifical and Technical Report (2014) [104]. 3. DESIGN, OPTIMISATION, AND MANUFACTURING 55 The area of the compressed CFS face sheet was considered to be fully effective. However, the wrinkling stress accounting for the contribution of the balsa wood core was considered instead of the steel yield strength. As a conservative assumption, the shear action was considered to be carried only by the CFS web elements. Thus, their verification was carried out according to the Eurocode 3 Part 1-3 (2006) [80] and Part 1-5 (2006) [102]. The driving factor of the design was the shear resistance of the CFS webs. Figure 3.17. SP3 cross-section layout. All units in [mm]. Note: the thickness of the face sheets and webs are not to scale. The outcome of the design process, including the imposed load carrying capacity with respect to bending moment (𝑅Md,Q) and shear (𝑅Vd,Q), as well as the deflection caused by permanent and variable loads (𝛿max), weight, and cost, are summarised in Table 3.1. The weight and cost of the sandwich panels are based on the nominal dimensions, mass density, and unit cost of the materials used. The unit costs are estimated based on the prices provided by the supplier of the sandwich panel manufacturer. Table 3.1. Structural performance, weight, and cost of solutions SP1, SP2, and SP3. Solution 𝑹𝐌𝐝,𝐐 [kN/m2] 𝑹𝐕𝐝,𝐐 [kN/m2] 𝜹𝐦𝐚𝐱 [mm] Weight [kg/m2] Cost [€/m2] SP1 3.1 8.4 13 33.4 32 SP2 3.4 8.6 13 30.6 93 SP3 4.2 3.4 19 36.2 128 The preliminary design and optimisation process yielded expected results. Solution SP2 is the lightest due to the presence of the GFRP webs, whereas solution SP1 resulted in the most economical prototype. Solution SP3 has a smaller thickness compared to SP2 and SP3, demonstrating that the inclusion of the core in the contribution to the overall structural performance leads to significant saving of materials. Due to its low cost and the experience of Ferpainel in the production of steel insulating sandwich panels, solution SP1 was selected for the next stage of the design process, namely the optimisation process by means of GA. A lightweight floor system based on sandwich panel: characterization and development 56 3.7. GENETIC ALGORITHM OPTIMISATION PROCESS In the scope of this thesis, a new GA has been developed to optimise the design of solution SP1 in terms of weight, cost, and environmental impact. A primary target of the optimisation procedure is to ensure that it is easily accessible to non-experts. In order to achieve this, the GA should be straightforward to implement, flexible, and computationally efficient. With this aim, a code was developed and implemented in C language, based on the GA described in this section. Despite the aim of approaching the solution of realistic engineering problems with GAs, it is necessary to discuss the strategies used in the evolutionary process in order to evaluate the reliability of the method. In particular, attention will be dedicated to the description and tuning of the penalty function as well as explaining its versatility and ease of implementation with the verifications of the Eurocode 3 Part 1-3 (2006) [80]. Finally, the result of the optimisation process in terms of optimal solutions will be presented. 3.7.1. Formulation of the optimisation problem GAs operators, which will be described in detail in the next section, manipulate the genetic information of the candidate solution to improve it in the next generation. Thus, the first step in the formulation of the optimisation problem is the translation of the geometric and material properties of SP1 into a chromosome-like structure. This process, known as encoding, is shown in Figure 3.18 and converts the observable features of the sandwich panel, also called its phenotype, into its corresponding genetic constitution, called the genotype. The following design variables are considered in the optimisation process of the sandwich panel: the total thickness of the sandwich panel (𝑡tot); ii) the thickness of the CFS sheet (𝑡s); iii) the width of the web flange and (𝑏f); iv) the number of reinforcing webs in the core (𝑛w); and v) the density of the PUR foam core (𝜌PUR). As mentioned earlier, when using the trial-and-error optimization process, the weight, cost, and GWP of the sandwich panel are determined based on the nominal dimensions and the mass density, unit cost, and carbon footprint of the steel, PUR foam, and adhesive (see Table 3.2). The assessment of the unit carbon footprints is based on the values found in the literature [12] and the life cycle assessment study (LCA) conducted by different hot-dip galvanized steel manufacturing companies [134]. The LCA studies considered in this work follow the cradle-to-gate approach. These studies assess the environmental effects of building materials starting from the extraction of raw materials, up to the completion of the finished product ready for shipment from the factory gate. However, they do not consider downstream activities 3. DESIGN, OPTIMISATION, AND MANUFACTURING 57 such as transportation, final use, disposal, and recycling. The impact category studied is global warming whose indicator is the GWP for a time horizon of 50 years. Furthermore, given the small amount of adhesive to be used in the panel its contribution to the overall mass and carbon footprint is neglected. (a) (b) Figure 3.18. Encoding procedure of the sandwich panel: (a) phenotype; (b) genotype. Table 3.2. Mass density, unit cost, and carbon footprint of the components of the sandwich panel. Material Mass density [kg/m3] Unit cost [€/kg] Unit carbon footprint [kgCO2-eq./kg] Steel 78.5 0.82 2.8 PUR foam 35, 40, 50, 60, 70, 80, 90, 100, 110, 120 1.79 4.14 Adhesive Neglected 9.54 Neglected The weight (𝑤𝑡), cost (𝑐𝑠𝑡), and global warming potential (𝑔𝑤𝑝) objective functions are aggregated into a single objective function (𝑓(𝒙)) according to the weighted-based genetic algorithm (WBGA) approach: 𝑓(𝒙)=∑𝑤i∙𝑓i′(𝒙) 3 i=1 (3.29) where 𝒙 is the design variable vector, 𝑤i is the weight assigned to the 𝑖−𝑡ℎ objective function, and 𝑓i′(𝒙) is the normalized value of the objective functions. Indeed, the magnitude of each objective function 𝑓i is different, thus, a min–max normalization is implemented. The min-max normalization consists in using the weight, cost, and environmental footprint of the best and worst individual at each generation to calculate the normalized values of the objective functions. The objective function's minimum value is scaled to 0, the maximum value is scaled to 1, while every other value is scaled to a decimal between 0 A lightweight floor system based on sandwich panel: characterization and development 58 and 1. The normalization process allows the comparison of objective functions of different magnitudes. Consider, for instance, the normalised value of the weight of an individual 𝑤𝑡′j: 𝑤𝑡′j=𝑤𝑡j−𝑤𝑡min 𝑤𝑡max−𝑤𝑡min (3.30) where 𝑤𝑡j is the weight of the 𝑗 individual, 𝑤𝑡max is the weight of the heaviest individual, and 𝑤𝑡min is the weight of the lightest individual. The WBGA approach aggregates the objectives, producing a single compromise solution according to the decision-maker [85]. It is a simple extension of the single objective GA and therefore, it is straightforward to implement. Taking into consideration the aforementioned aspect, the WBGA approach was deemed the most adequate to retrieve the optimal cross-section of the sandwich, which had to be ultimately manufactured and tested. Nevertheless, from a scientific standpoint and under the pressure of climate change issues related to the construction sector (or rather humankind itself), it is interesting to retrieve at least the extreme of the Pareto optimal set. Therefore, assigning different weights in Equation (3.29), different alternatives are sought, namely the lightest, the most economical and the least environmental impacting solutions. This requires the code to be run multiple times with different weights of the objective functions. However, if the decision-maker has no clear preference among the initial objectives the Paretobased approach should be used [135]. This approach has the advantage of providing a Pareto set of solutions which should be uniformly distributed and diverse over the true Pareto front in a single run. A trade-off study has to be carried out to select a compromise solution from the Pareto set. 3.7.2. Constraints and penalty function When solving an optimisation problem, the constraints can be divided into two categories: i) box condition constraints; and ii) behavioural constraints. The former group of constraints limits the range of values that the design variables can take, as detailed in Table 3.3. Table 3.3. Box conditions and mutation steps of the design variable of the sandwich panel. Design variable Symbol Unit Minimum Maximum Step Total thickness 𝑡𝑡𝑜𝑡 [mm] 50 200 1 Steel sheet thickness 𝑡s [mm] 0.5 2.0 0.1 Width of web flange 𝑏f [mm] 20 50 ∙𝑡s 1 Number of webs 𝑛w [mm] 3 4 1 PUR foam mass density 𝜌PUR [kg/m3] 35 120 1 3. DESIGN, OPTIMISATION, AND MANUFACTURING 59 The range of values for the design variables was limited for various reasons: i) 𝑡tot can range between 50 mm and 200 mm, which is comparable to those of traditional floor systems; ii) the limit for 𝜌PUR is set according to the manufacture capability of Ferpainel; iii) 𝑛w (which includes the outer webs) upper limit is related to manufacturing constraints, whereas the lower limit is based on the research of sandwich panels that indicates that typically at least two webs in the core are necessary for adequate structural behaviour [59], [61]; iv) 𝑡s and 𝑏f are within the interval permissible by Eurocode 3 Part 1-3 (2006) [80]. Regarding the behavioural constraints, they are implicit functions of 𝒙 and they follow from the structural, thermal, and acoustic requirements described in the previous sections. The behavioural constraints are incorporated in 𝑓(𝒙) employing an adaptive penalty function (𝑓p(𝒙,𝑡)) proposed in [93] given in Equation (3.31): 𝑓p(𝒙,𝑡)=𝑓(𝒙)+(𝐹feas(𝑡)−𝐹all(𝑡))∙∑ 𝑑i 𝑁𝐹𝑇i 6 i=1 (3.31) where 𝑡 is the number of generations, 𝐹all(𝑡) the unpenalized value of the best solution yet found, 𝐹feas(𝑡) the value of the best feasible solution yet found, 𝑑i the distance metric of each constraint, and 𝑁𝐹𝑇i is the near feasible threshold described later on in this section. The penalty function involves adding a penalty based on the difference between the solutions performance and the legal requirement of the corresponding performance. In the literature of GAs, this difference is known as cost to completion [136] and represents the distance of the individual from the feasible solution space (𝑑i). Regarding constraints stemming from structural engineering practice, verification is usually expressed in terms of inequality between the design resistance (𝑅d) and the design value of the relevant force (𝐹Ed). This inequality can be used to calculate 𝑑i of the penalty function adopted in the GA: 𝑑i={𝐹Ed−𝑅d if 𝑅d−𝐹Ed<0 0 if 𝑅d−𝐹Ed≥0 (3.32) The implementation of the current design practice requirements is straightforward and demonstrates the versatility of the optimisation method for structural members other than sandwich panels. The definition of 𝑑i for the constraint described in Section 3.1 is given in Table 3.4. A lightweight floor system based on sandwich panel: characterization and development 66 the search. It can be concluded that the proposed adaptive penalty function allows i) to explore a wider solution space, ii) avoid premature convergence, and iii) make the results of the GA less sensitive to the starting population. Figure 3.23. Comparison of the performance of the modified GA, including static penalty function, and OSPANEL. In Table 3.5, the parameters of the genetic operators adopted in the search of the optimal solutions presented in the next section are summarised. Table 3.5. The genetic parameters adopted in OSPANEL. Genetic parameters Symbol Default value Number of individuals in a population 𝑁 1000 Near-feasible threshold upper bound 𝑁𝐹𝑇0 5% Dynamic search parameter 𝜆 0.1 Selective pressure of exponential ranking method 𝑠 0.7 Type of selection scheme - SUS Type of crossover operator - Multi-point crossover Crossover rate - 1 Mutation rate - 0.8 Elitist strategy - TRUE Maximum number of generations 𝑡max 20000 3.7.5. Optimal solutions OSPANEL is used to retrieve the extreme of the Pareto front, namely the lightest (S_1), the most economical (S_2), and the least polluting (S_3) sandwich panel designs. A summary of the characteristics of the optimal solutions is illustrated in Figure 3.24a and Figure 3.24b. S_1 solution coincides with the 3. DESIGN, OPTIMISATION, AND MANUFACTURING 67 S_3 one which agrees with the result on the optimization study of sandwich panels [12]. This may be explained by the fact that the carbon footprint is proportional to the weight of the component materials. The steel weight makes up for 80% of the total weight of the sandwich panel, thus the GA minimizes its use. A different scenario may arise if a less conservative structural design approach would be available, and the PUR foam contribution to the resistance of the sandwich panel is considered. In such a case, a lighter solution may be obtained increasing the density of the foam to allow for the use of a thinner steel sheet with an improved buckling resistance resulting from the denser core support. However, this evolutionary strategy may not lead to a less polluting solution since the PUR foam has a greater environmental impact than steel according to Table 3.2. (a) (b) Figure 3.24. Cross-section and characteristics of the optimal solutions: (a) S_1 and S_3 solution; (b) S_2 solution. The normalized value of the constraints during the search of the S_1 and S_3 solutions is shown in Figure 3.25a and Figure 3.25b, respectively. The bending moment resistance is the active constraint [141], i.e. the requirements which is ultimately governing the design as it is the last one to be fulfilled during the A lightweight floor system based on sandwich panel: characterization and development 68 search. This depends mostly on the total thickness of the panel which is the last design variable converging to its optimal value as shown in Figure 3.25c. The shear resistance and the midspan vertical deflection are relatively less stringent as they are fulfilled earlier in the run. The value of the thermal resistance fluctuates when great variations in the total thickness are observed. This is an expected result as the thermal performance relies on the thickness of the core insulating layer which makes up most of the total thickness. (a) (b) (c) Figure 3.25. Search history of S_1 and S_3: (a) normalized constraints value; (b) zoom at the converged iteration of the normalized constraints value; (c) normalized design variable value. Note: Lines with triangle symbols are constraints that must be equal or less than 1, whereas lines with square symbols are constraints that must be equal or greater than 1. 3. DESIGN, OPTIMISATION, AND MANUFACTURING 69 Despite the large range of geometric design variables explored during the search, the sound insulation constraints show little variation. This corroborates the difficulty of improving the acoustic performance of sandwich panels. A 4 dB increase in the sound insulation can only be achieved by doubling the weight of the sandwich panel. OSPANEL identifies in the 4 webs configuration, i.e. the maximum number allowable, a preferable option both for the S_1, S_2, and S_3 solutions. This confirms the findings of previous studies on the flexural behaviour of sandwich panels [59], [61]. The webs are fundamental in providing the required bending stiffness to the structural element. It should also be noted that in the lightest solution the flanges of the webs are at the limit of the geometrical requirements of the Eurocode 3 Part 1-3 (2006) [80]. This is due to the fact that they provide a constraint for the top face sheet against buckling. The normalized values of the constraints and design variables in the search history of S_2 solution are illustrated in Figure 3.26a. The constraints and design variables search history (see Figure 3.26b) show a similar pattern to those of S_1 and S_3 solutions. In comparison to the latter, the S_2 solution presents thicker steel sheets and less wide flanges. This indicated that the price of the adhesive is relevant to the overall cost of the sandwich panel. The results of OSPANEL indicates that it is more economical to increase the flexural stiffness of the panel by thickening the steel sheets rather than adhesively joining together wider flanges and top face sheet. (a) (b) Figure 3.26. Search history of S_2: normalized values of the constraints; (b) normalized values of the design variables. Fluctuations and singularities can be observed in the plot of the design variables and the constraints in the search history of both optimal solutions (see Figure 3.25 and Figure 3.26). In the search for the A lightweight floor system based on sandwich panel: characterization and development 70 optimal solution, when a solution with a low value of the objective function, not fitting the constraints, ceases to be the best solution, two new designs might appear with very similar values of the penalized objective function. OSPANEL takes some generations to establish the best among the two competing designs which result in the fluctuations and singularities observed in the graphs. OSPanel switches between the two competing solutions until the closest to meet the constraints emerges as the best one. This may be explained by the fact that both 𝑁𝐹𝑇0,i and 𝜆 values are set to be small (the former one in terms of percentage) so that the unfeasible region of the solution space that OSPANEL is encouraged to search is slowly decreasing through the generations. Indeed, in Figure 3.21 and Figure 3.22 it is possible to observe that fluctuations and singularities tend to disappear when the above-mentioned parameters are set in order to perform a faster search and explore an unfeasible region that is rapidly decreasing. 3.7.6. Parametric studies with OSPANEL code To explore a wider solution space, the values of design variables and requirements are modified keeping the other factors constant. Firstly, the maximum number of webs in the core is progressively increased, disregarding the manufacturing constraints. The population is forced to maintain a constant number of webs during the whole search so that multiple optimal solutions with a different number of webs are retrieved. The result of the parametric study is shown in Figure 3.27. (a) (b) Figure 3.27. Parametric study on the number of webs: (a) weight of the optimal solutions; (b) cost of the optimal solutions. The S_3 solution with 8 webs is the lightest and least polluting sandwich panel (Figure 3.28). However, it is also the most expensive since a large amount of adhesive is required to bond the flanges with the 3. DESIGN, OPTIMISATION, AND MANUFACTURING 71 top face sheet. The S_2 solutions with 6 and 7 webs are both economically competitive. Furthermore, the 7 webs architecture appears to be a local optimum as the S_1, S_2, and S_3 solutions are identical. (a) (b) Figure 3.28. Cross-section and characteristics of the optimal solutions obtained from the parametric study: (a) S_1 and S_3 solution; (b) S_2 solution. In Figure 3.29 the imposed load-carrying capacities, deflection, and thermal transmittance of the S_1, S_2, and S_3 solutions are reported for each number of webs. 𝑅Md,Q for optimal solutions with less than 8 webs approaches the limit value, that is the bending moment is the active constraints for these sandwich panels. On the other hand, 𝑅Vd,Q constraint is always largely fulfilled. The assumption of a constant thickness for the whole section facilitates the effective width calculation but it seems to preclude the possibility of further improvement in the optimization of the webs, which govern the shear resistance of the panel. The vertical deflections increase with increasing number of webs. This may be explained by the fact that the optimal solutions with a greater number of webs have smaller total and steel sheet thicknesses that result in a lower lever of arm produced by the face sheets and reduced effective area, A lightweight floor system based on sandwich panel: characterization and development 72 respectively. Both these phenomena contribute to the decrease of the bending stiffness of the panel. Finally, thermal performances are illustrated in Figure 3.29d. The thermal transmittances of the S_1 and S_2 solution with more than 9 and 8 webs, respectively, are slightly above the limit. Thus, for these sandwich panel architectures the governing criterion of the design is not related to the structural performance but to the thermal one. (a) (b) (c) (d) Figure 3.29. Values of the constraints of the optimal solutions: (a) 𝑅Md,Q; (b) 𝑅Vd,Q; (c) 𝛿max; (d) 𝑈c. The design variables for the different layouts are plotted in Figure 3.30. OSPANEL reduces the cost of the panel mainly by thinning the thickness of the steel sheet and narrowing the width of the web flanges. On 3. DESIGN, OPTIMISATION, AND MANUFACTURING 73 the other hand, S_1 and S_3 are obtained by reducing the total thickness as well as the steel sheet thickness. (a) (b) (c) Figure 3.30. Values of the design variables of the optimal solutions: (a) 𝑡tot; (b) 𝑡s; (c) 𝑏f. In the second stage, the influence of the cost of the adhesive is also investigated offsetting the original price by +10% and -20%. A decrease in the price of the adhesive makes the S_1 and S_3 solution with 4 webs also the most economical solution. An increase in the price provokes instead a narrowing of the flanges in the S_2 solution, thus diminishing the surface area to be bonded. At the same time, the total thickness of the panel is increased to compensate for the loss of the moment of inertia. This suggests that for an adhesive with a unitary cost equal to or less than approximately 8.5 €/kg there is no relevant A lightweight floor system based on sandwich panel: characterization and development 74 difference in the overall price for a sandwich panel with an adhesive or mechanical connection between its face sheets and webs. Finally, the full sound insulation requirements are imposed in the adaptive penalty function to verify the possibility of using sandwich panels as separating elements of dwellings in new construction. The highest acoustic insulation is achieved when OSPANEL is set to search for the S_1 and S_3 solution. The sandwich panel fulfils 80% and 65% of the airborne and impact sound insulation requirements, respectively. OSPANEL conceive a much heavier sandwich panel compared to the previous one making use of most of the steel mass allowable by the box conditions and the highest density PUR foam (see Figure 3.31). Figure 3.31. Most sound insulating sandwich panel retrieved by OSPANEL. 3.7.7. Discussion of the optimisation results with OSPANEL code Based on the obtained solutions and the manufacturing constraints, the S_2 with 4 webs is selected as the optimal architecture for the sandwich panel to be produced in the scope of the Lightslab R&D Project. The nominal dimensions are modified to conform to more convenient and traditional sizes of floor and CFS profiles, i.e. the total thickness and the web flange width are set equal to 170 mm and 45 mm, respectively. The lightest sandwich panels coincide with the least polluting as the carbon footprint is directly proportional to the mass of the sandwich panel. The difference between the S_1 and S_2 solutions is the use of the adhesive bonding connection. The least expensive sandwich panels have a reduced width of the web flanges to reduce the cost of the adhesive bonded area. 3. DESIGN, OPTIMISATION, AND MANUFACTURING 75 The plot of the constraint functions shows that the issue of the low acoustic insulation of the sandwich panel shall be addressed in view of their application as a residential floor system. Additionally, the thickness of the face sheet and the webs shall be considered as separate design variables to improve the efficiency of the design with respect to bending moment and shear resistances. Disregarding the manufacturing constraint imposed by Ferpainel production line, optimal solutions with a number of webs greater than 4 present a total thickness ranging between 170 mm and 190 mm and a steel sheet thickness exclusive of the coatings varying between 0.7 mm and 1.0 mm. In particular, the 7 webs architecture present a good compromise between all the objective functions. On the other hand, sandwich panel architectures with more than 8 webs are not advisable both for economical and thermal issues. 3.8. MANUFACTURING PROCESS AND PANELS’ CONNECTION The manufacturing process of the optimised prototype, i.e. S_2 with the appropriate changes to fit standard measures of flooring systems and CFS profiles, consisted of three stages. The challenges encountered in the first two manufacturing attempts eventually led to the final cross-section of the sandwich panel. Furthermore, the final prototype included in its design considerations related to the panelto-panel connection system. In the next sections, a detailed description of the three manufacturing attempts is presented. 3.8.1. Manufacturing of prototype S_2 In the first manufacturing attempt made by Ferpainel, a modification was applied to the recommendations provided by the DEC team of the University of Minho. The builder team of Ferpainel opted for a spot weld connection between the bottom face sheet and the webs (see Figure 3.32). The bottom part of the sandwich panel was then manually positioned on the conveyor belt for the foaming process. The top face sheet-to-web connection was achieved by means of adhesive bonding. The DEP team of the University of Minho developed the polyurethane-based adhesive for such connection. The adhesive was prepared by the builder team at Ferpainel and applied on the top flanges of the webs as the sandwich panels advanced on the conveyor belt to the injection system of the monomer of the PUR foam. However, the bottom face sheet-to-web connection did not ensure the necessary stability of the webs during the manufacturing process. As a result, the manufactured panels highlighted rather large deviation from straightness of the A lightweight floor system based on sandwich panel: characterization and development 82 compressive tests of small scale sandwich panel with steel face sheet and PUR foam core. Different PUR foam core densities (40 kg/m3, 50 kg/m3, and 60 kg/m3) and steel sheet thicknesses (1.0 mm and 1.5 mm) were tested according to a preliminary study on the range of properties of interest for the application of sandwich panels in residential flooring systems. Thermogravimetric analyses (TGA) were also carried out to qualitatively assess the PUR foam microstructure. Indeed, information can be gathered about the main physical and chemical reactions needed to create a closed-cell microstructure by assessing foam degradation caused by temperature. Civil engineering structures, particularly residential buildings, are designed to last 50 years. Polymeric materials used in the proposed sandwich panels, such as the PUR-based adhesive and the PUR foam, are prone to creep deformation under permanent loads [148], [149]. Shear stress-induced creep in PUR foam is less relevant because the CFS webs have greater shear stiffness and absorb most stress. On the other hand, the PUR-based adhesive transfers shear stresses from the plain CFS section to the lipped CFS section. Thus, the creep deformation and the effect of temperature on the adhesive´s mechanical properties shall be assessed to ensure the integrity of the sandwich panel during its service life. However, obtaining experimental data on the creep behaviour and temperature's effect on the adhesive's mechanical properties is beyond the scope of this thesis and will be addressed in future works. Finally, CFS specimens were extracted from the full-scale tested sandwich panels to perform additional tensile coupon tests. Especially during the Covid-19 pandemic, Ferpainel purchased steel sheets from multiple suppliers to save costs due to extreme fluctuations in raw material supply costs. Therefore, verifying the mechanical properties of the steel sheet with which the final prototypes were manufactured was deemed necessary. Furthermore, zinc coating thickness measurements by microscopical examination were carried out to establish the core thickness of the metal sheets. 4.1. COLD-FORMED STEEL CFS specimens were laser cut out from the steel sheet used in the production of the steel insulating sandwich panels produced by Ferpainel and subsequently from the final manufactured prototypes. Specimens' dimensions were established based on the type of test to be carried out, i.e. tensile coupon tests and zinc coating thickness microscopical examinations. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 83 4.1.1. Tensile coupon test Tensile coupon tests were carried out according to ISO 6892-1 (2016) [150] on three sets of specimens. The first batch of specimens included two sets with a nominal thickness of 1.0 mm (TC#1-1.0) and 1.5 mm (TC#1-1.5), respectively. TC#1-1.0 and TC#1-1.5 were extracted from the steel sheet used in Ferpainel’s line of steel insulating sandwich panels. The third set of specimens (TC#2-1.0) had a nominal thickness of 1.0 mm and were extracted from the full-scale panels described in Chapter 5. Therefore, sheets with different nominal dimensions (35 cm × 40 cm and 35 cm × 8 cm) were extracted out from the sandwich panels tested in a four-point bending configuration up to failure. Regions of low bending and shear stresses in the face sheets and webs of the panels were identified as the extraction zone for the sheets by numerically simulating the four-point bending test (see Chapter 7). All the coupon specimens were extracted in the longitudinal direction of the steel sheets. The specimens were machined according to the specification of EN 10346 (2015) [151] and are illustrated in Figure 4.1a. Prior to testing, gauge marks were applied in the region of interest, and the cross-section dimensions of the specimens were measured with a digital calliper. The coupon tensile tests were carried out in a universal testing machine (UTM) with a capacity of 200 kN. The tensile load is applied by gripping the ends of the specimens with clamps. The axial strain was measured with an extensometer with a base length of 100 mm (precision of 0.01 mm) placed at the middle specimen height. The overall test setup is shown in Figure 4.1b. (a) (b) Figure 4.1. CFS tensile coupon test: (a) nominal dimensions of the specimen; (b) test setup. All units in [mm]. A lightweight floor system based on sandwich panel: characterization and development 84 The coupon specimens were tested in a quasi-static monotonic-instantaneous loading up to failure under displacement control. The displacement-based protocol followed the suggestions of [152] to eliminate the influence of the strain rate on the test results. A displacement rate of 0.003 mm/min was applied until reaching the proportional limit. The displacement rate was then increased to 0.015 mm/min and kept constant until the yielding phase was completely developed. Finally, the displacement rate is increased to 0.066 mm/min until failure to keep the test duration within a reasonable time. The loading was paused for 100 s at different critical locations to obtain the static drops due to stress relaxation. The pauses were near the 0.2% proof stress, at the beginning of the hardening region, and near the ultimate strength. As stated previously, despite the tensile coupon test being a highly standardised test procedure, errors may occur in the analysis and interpretation of the data. Therefore, to correctly estimate the properties of the CFS specimens, the influence of the loading rate had to be removed from the experimental stressstrain curves, i.e. the dynamic curves, according to the method described in [152]. The procedure to obtain the static material properties of the CFS coupons is the following (see Figure 4.2): 1. The Young’s modulus (𝐸) is calculated as the slope of the initial linear portion of the stress-strain curve in the range between 20% and 45% of the nominal proof strength (𝜎0.2); 2. A straight line is drawn from the origin of the axis with the slope of the estimated 𝐸. The proportional limit (𝜎p) is established at the point of separation of the straight line from the stressstrain curve; 3. A line parallel to the linear portion is drawn at a distance from it equivalent to 0.2% strain. The intersection between the line and the stress-strain curve represents the experimental value of 𝜎0.2; 4. The first part of the stress-strain curve from the origin to 𝜎p is left unchanged. The second part of the curve from 𝜎p to 𝜎0.2 is determined by reducing the stress of a quantity in proportion with the strains at 𝜎p(𝜀p) and at 𝜎0.2(𝜀0.2) according to the following Equation: 𝜎s(𝑥)=𝜎d(𝑥)−∆0(𝜀x−𝜀p) 𝜀0.2−𝜀p (4.1) where ∆0 is the first static drop due to stress relaxation, and 𝜎s(𝑥) and 𝜎d(𝑥) are the static and dynamic stresses corresponding to the strain 𝜀x, respectively; 5. The part of the curve between the intermediate static drops is calculated according to Equation (4.2): 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 85 𝜎s(𝑥)=𝜎d(𝑥)−(∆i−1−∆i)∙(𝜀x−𝜀i−1) 𝜀i−𝜀i−1 (4.2) where ∆i and ∆i−1are the intermediate static drops due to stress relaxation, and 𝜀i and 𝜀i−1 are the strains corresponding to the intermediate static drops; 6. The last part of the curve is obtained by reducing the stress of the amount corresponding to the last static drop (∆u). (a) (b) Figure 4.2. Procedure to obtain static material properties: (a) proportional limit and 0.2% proof strength; (b) dynamic stress and static stress-strain curves. A lightweight floor system based on sandwich panel: characterization and development 86 The static engineering stress-strain curves (see Figure 4.3a) indicate that all sets of specimens present a pronounced yielding phenomenon [151]. A failed specimen exhibiting the typical necking phenomenon before fracture is shown in Figure 4.3b. (a) (b) Figure 4.3. Uniaxial coupon tensile test result: (a) stress-strain curves; (b) typical failure mode. The mechanical properties obtained from the tests are reported in Table 4.1, including the Young’s modulus (𝐸), the proof strength (𝜎0.2), the ultimate tensile strength (𝜎u), and the strain at fracture (𝜀f). 𝜎0.2 is the value of the yield strength (𝜎y) to be used in the structural safety verifications [80]. The three sets of specimens presented similar levels of ductility. Sets TC#1-1.0 and TC#2-1.0 satisfy the requirements of the steel grade S250GD+Z. A relatively small difference (less than 5%) in terms of 𝜎0.2 and 𝜎u is observed between the steel used in the production of Ferpainel steel insulated sandwich panels and that used to manufacture the final sandwich panel prototype. On the other hand, the set TC#1-1.5 does not meet the minimum requirements of the steel grade S220GD+Z in terms of yielding strength. Nevertheless, the values reported in the technical datasheet correspond to the static yielding strength. The mechanical properties reported in the technical datasheets usually refer to the dynamic values established according to the strain rates imposed by the standard [81]. A statistical analysis of the TC1.0 and TC-1.5 sets of data using the interquartile range (IQR) method was carried out to detect outliers [153]. The elastic modulus of specimen TC#1-1.5-1 was below the lower inner fence and, thus, identified as an outlier. This may be due to the presence of out-of-straightness defects in the specimen resulting in the introduction of bending moments and a non-uniform state of tensile stresses. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 87 Table 4.1. Summary of the mechanical properties of the CFS sheets. Specimen [-] 𝝈𝟎.𝟐 [MPa] 𝝈𝐮 [MPa] 𝜺𝐟 [%] 𝑬 [GPa] TC#1-1.0-1 261 351 30.87 190 TC#1-1.0-2 253 342 31.54 222 TC#1-1.0-3 253 340 30.97 201 TC#1-1.0-4 259 354 31.29 194 TC#1-1.0-5 258 346 31.75 209 TC#1-1.0-6 254 349 31.39 214 Avg. (CoV) 256 (1.3%) 347 (1.6%) 31.30 (1.1%) 205 (5.9%) TC#1-1.5-1 210 315 28.93 166* TC#1-1.5-2 210 319 31.69 198 TC#1-1.5-3 225 321 30.51 203 TC#1-1.5-4 215 321 33.19 190 TC#1-1.5-5 219 323 33.83 187 Avg. (CoV) 216 (3.0%) 320 (1.0%) 31.63 (6.3%) 194 (3.8%) TC#2-1.0-1 266.7 366.4 25.6 209 TC#2-1.0-2 271.4 362.9 28.4 208 TC#2-1.0-3 269.0 359.1 31.1 205 TC#2-1.0-4 270.8 360.0 27.3 209 TC#2-1.0-5 266.6 354.2 29.0 201 Avg. (CoV) 268.9 (0.8%) 360.5 (1.3%) 28.3 (7.1%) 206 (2.0%) Notes: Avg. – average value per set of specimens; CoV – coefficient of variation; the highlighted outliers with the symbol “*” were not considered in the average calculation. 4.1.2. Zinc coating thickness by microscopical examination Hot-dip galvanisation involves submerging a steel element in a bath of molten zinc. The procedure is most used to provide long-term corrosion protection to the steel for up to 50 years [154]. Since the datasheets provided by Ferpainel reported different coating masses of the zinc coating, a microscopical examination of the CFS sheet cross-section was carried out to estimate the coating thickness. Three specimens were taken from the same sheets described in Section 4.1.1. A thin layer of PUR foam was present on the inner side of the sheets (see Figure 4.4a). Therefore, before cutting the samples, the sheets were immersed in a Dimethylformamide bath, allowing the PUR foam to soak the compound. The PUR foam was then carefully removed from the sheet using a spatula. Finally, the sheets were first rinsed with light petroleum and, subsequently, distilled water. The dimensions of the specimens were 1 cm × 1 cm. The samples were mounted in epoxy resin, and the examined cross-section was polished with diamond paste particles of 1 μm according to the standard ASTM B487-85 (2013) [155] (see Figure 4.4b). A lightweight floor system based on sandwich panel: characterization and development 88 (a) (b) Figure 4.4. Specimen preparation for zinc coating thickness measurements: (a) CFS sheet cut out from the full-scale tested sandwich panel; (b) specimen cast in epoxy resin. Digital pictures were taken using an optical microscope to capture the cross-section regions of each sample (see Figure 4.5). The thickness of the coating (𝑡z) was measured by employing the ImageJ software. Five measurements were carried out on both coated surfaces of each specimen. The results of the zinc coating thickness by microscopical examination are summarised in Table 4.2. Figure 4.5. Optical microscope image from one of the samples. Table 4.2. Thickness of the zinc coating on the steel sheet of the full-scale sandwich panel. Specimen [-] 𝒕𝐳 [μm] Avg. (CoV) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 TZ01 17.1 15.2 17.1 16.0 19.3 18.7 17.0 15.4 14.7 17.5 17.7 (9.7%) TZ02 18.8 19.0 19.3 19.9 19.3 19.1 20.0 20.2 19.4 17.6 18.4 (8.4%) TZ03 18.8 20.3 18.8 15.6 18.0 17.2 18.4 16.0 17.6 17.6 17.6 (11.8%) Notes: Avg. – average value per set of specimens; CoV – coefficient of variation. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 89 The average thickness of the zinc coating of the CFS sheets is 17.9 μm. The coating mass (𝑚z) may then be estimated according to the Equation provided in EN 10346 (2015) [151]: 𝑚z=2∙𝑑z∙𝑡z (4.3) where 𝑑z is the density of the zinc and 𝑡z is the thickness of the zinc coating. The estimated zinc coating mass is 247.8 g/m2 considering a zinc density of 7.1 g/cm3 [151]. The CFS sheet coating mass is conservatively designated as the coating mass Z225 whose typical thickness is 16 μm. The typical zinc coating thickness value is used to calculate the core thickness (𝑡cor) of the steel as prescribed by the Eurocode 3 Part 1-3 (2006) and discussed further ahead in Chapter 7. 4.2. POLYURETHANE FOAM AND ADHESIVE The objectives of the rigid closed-cell PUR foam employed as core material in the proposed sandwich panel are twofold: i) provide adequate thermal insulation; and ii) improve the resistance of the CFS sheets to local buckling phenomena. In this sense, it is interesting to characterize its compressive and tensile mechanical properties in the direction perpendicular to the face sheets. This is generally different from sandwich panel without webs, where the most relevant mechanical properties of the core materials concern its shear behaviour. The macroscopic mechanical properties of PUR foams depend on its microstructure. Indeed, PUR foams are cellular materials, the man’s attempt of reproducing materials that nature uses to build load-bearing structures, i.e. wood, bones, and coral [156]. Cellular materials are characterised by a porous microstructure comprising of solid and voids network. In the case of polymeric foams, the voids are called cells, and they are filled with gas. The solid are the faces surrounding the cells constituted by the polymer [157]. The microstructure of the PUR foam is a complicated process resulting from the interaction of different chemical reactions. The synthesis of polyurethane consists in the reaction of isocyanate groups and polyols to form urethane linkages. Simultaneously, physical (water) or chemical blowing agents (the infamous chlorofluorocarbon banned by the Montreal Protocol (1987) [158] and the slightly less harming hydrochlorofluorocarbons [159]) generate the gas to form the foamed structures. Urethane and urea linkages, resulting from the addition of diamines, react with the isocyanate groups to form allophanate and biuret linkages, respectively. These linkages are responsible for the branching and crosslinking of the A lightweight floor system based on sandwich panel: characterization and development 90 polymer. The gel point is the start of the crosslinking reaction which result in significantly increasing viscosity. The period after the gel point is defined as the curing period which generally involves heating. An imbalance between the polymerization-crosslinking and the blowing processes compromises the successful synthesis of the foamed product [160]. If gelation point is reached too slowly in foamed products, the foam structure may collapse or present imperfections which yield poor mechanical strength and performance. The mechanical properties of PUR foam are influenced, among others, by mixture components, substrates, and production technologies [161]. In the proposed sandwich panel, the PUR foam is produced by the sandwich foaming process [157]. The liquid prepolymer (an incompletely polymerized polymer) is poured onto the bottom face sheet. As polymerization and crosslinking occur, an adhesive bonding is provided between the face sheets and the expanding PUR foam. The PUR foam is constrained from expanding laterally by the webs and by the material behind and in front of it on the face sheet. Therefore, the foaming process causes the material to rise vertically, resulting in an anisotropic cell morphology [31]. The influence of cell structure on macroscopic properties has been investigated through visual inspection and mechanical testing. Scanning electron microscope pictures are used to correlate the size, shape, and orientation of foam cells to the foam's strength and stiffness. It was found that foams with cells with a larger aspect ratio in the rise direction, obtained from the preparation of a 100 kg/m3 PUR foam in narrow aluminium moulds, exhibited higher compressive stiffness and strength compared to the same foam prepared in wider moulds [162]. Additionally, cell orientation divergence from the rise direction was observed and attributed to the temperature gradient between the reacting polymer and the room temperature walls of the moulds. Furthermore, the tensile and compressive moduli of a 45 kg/m3 PUR foam used to fill CFS C-section were found to be 3.5 and 1.5 times higher, on average, in the rise direction, respectively [33]. Once again, the study confirmed that the superior properties resulted from the stretching and alignment of the cells in the rise direction. When different production technologies and testing methodologies are used, it may be more challenging to make a straightforward comparison. For example, when samples extracted from full-scale steel insulating sandwich panels were tested in flatwise compressive and tensile tests, they showed strength values 3.25 times higher in the longitudinal direction of the panel, which is perpendicular to the rise direction of the foam [163]. Additionally, the study indicates that 5.1% of the variation in the results of the 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 91 mechanical tests can be attributed to the heterogeneity resulting from the sandwich foam production technology. In the process of sandwich foaming, the polymerization and crosslinking reaction occurring in contact with the metal face sheets constitute an interface possessing its own mechanical characteristics. The mechanical properties of the interface are rarely well-characterised. To the best of the author’s knowledge, only one study in the literature attempted to characterise the galvanized steel-to-PUR foam interface in terms of critical strain release energy rate, maximum normal and tangential strengths, and normal and tangential displacements at maximum strength and failure for Mode I and II fracture [73]. Nevertheless, the estimated experimental properties present coefficient of variation of up to 90%. The difficulties arise both from the involved test setup required, i.e. a double cantilever beam test, as well as the variability of the properties of the PUR foam and the interface itself. In this brief literature review of rigid closed-cell PUR foam used as filler of CFS cross-section, it is interesting to highlight the significant number of factors that can influence the macroscopic mechanical properties of the core material. The complete characterisation would require the assessment of the chemical composition and cell morphology in relation to the mechanical properties. However, due to the timeframe of the LightSlab R&D Project, the envisaged experimental campaign aimed at retrieving the main mechanical properties of the PUR foam required by commercial finite element software to reproduce numerically the behaviour of the foam filled sandwich panel: i) flatwise compressive and tensile tests; and ii) shear tests. Additionally, TGAs were carried out to assess the decomposition processes of the foam due to temperature. 4.2.1. Flatwise compressive and tensile tests Flatwise compressive and tensile specimens were carried out according to ASTM C297 (2004) [164] and ASTM C365 (2003) [165], respectively, on three sets of PUR foam specimens with a nominal density of 40 kg/m3 (FC-40 and FT-40), 50 kg/m3 (FC-50 and FT-50), and 60 kg/m3 (FC-60 and FT-60). The mechanical properties studied are those perpendicular to the face sheets of the sandwich panels, which are particularly relevant in the prediction of local instability phenomenon such as wrinkling. Furthermore, depending on the observed failure mode it is possible to assess either the bulk mechanical properties of the core materials or those of the PUR foam-to-face sheet interface. The tests were performed on cubic specimens with 60 mm sides. The specimens were cut out from larger sandwich panels with a core thickness of 60 mm and face sheet thickness of 1.0 mm. Before testing, the specimens’ dimensions A lightweight floor system based on sandwich panel: characterization and development 98 (a) 1 2 (b) Figure 4.11. DIC measurements of FT-40 under tension: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial tensile strain at the linear stage and at fracture. All units in [mε]. In Figure 4.12a, the stress-strain curves for the tracked area of one of the FC-50 specimens using DIC are displayed. The compressive axial strains obtained from the LVDTs and the DIC show good agreement. The three layers exhibit different stiffnesses, with BL remaining in a linear stage after yielding. Plastic deformations are mainly observed in ML and TL. In Figure 4.13a, the tensile stress-strain curves computed from the LVDTs and the DIC images show negligible discrepancies. The tensile elastic modulus of BL is the highest, as observed in the compressive behaviour. ML and TL exhibit a small hardening branch before the specimen fails. This may indicate the presence of a non-optimized microstructure with thick wall cells possibly exhibiting plastic deformation before ultimate failure. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 99 (a) 1 2 3 (b) Figure 4.12. DIC measurements of FC-50 under compression: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial compressive strain at the linear stage, at the onset of yielding and after yielding. All units in [mε]. A lightweight floor system based on sandwich panel: characterization and development 100 (a) 1 2 3 (b) Figure 4.13. DIC measurements of FT-50 under tension: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial tensile strain at the linear stage, at the onset of the hardening stage and fracture. All units in [mε]. The stress-strain curves of a typical specimen from the FC-60 set, which were computed using DIC tracking, are displayed in Figure 4.14a. The axial compressive strains measured by LVDTs agree well with the DIC recordings. TL and ML experience plastic deformation at yielding, while BL behaves elastically during the recorded time interval. However, there is a notable difference between the tensile curves of AVG-DIC and AVG-LVDTs (see Figure 4.15a). This could be due to non-uniform contrast in the speckle pattern, which causes the analysis software to fail in correlating the area of high-stress concentration. BL is stiffer than TL and ML, and ML exhibits a hardening branch before tensile failure, indicating the presence of imperfections in its microstructure (see Figure 4.13b). 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 101 (a) 1 2 3 (b) Figure 4.14. DIC measurements of FC-60 under compression: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial compressive strain at the linear stage, at the onset of yielding and after yielding. All units in [mε]. A lightweight floor system based on sandwich panel: characterization and development 102 (a) 1 2 3 (b) Figure 4.15. DIC measurements of FT-60 under tension: (a) stress-strain curve; (b) images extracted from the DIC recordings with the plot of the axial tensile strain at the linear stage, at the onset of the hardening stage and fracture. All units in [mε]. According to the analysis of the strain field using the DIC technique, there is a notable difference in the compressive and tensile mechanical properties between the 40 kg/m3 PUR foam and the 50 kg/m3 and 60 kg/m3 PUR foams. This difference is due to significantly different mechanical behaviour. The 40 kg/m3 PUR foam has weaker TL, while ML and BL have similar elastic properties. The 50 kg/m3 and 60 kg/m3 PUR foam have similar mechanical properties. There is an increasing gradient in the elastic properties from BL to TL that is easily recognizable. Additionally, the nonlinear tensile behaviour may suggest issues during the synthesis of the PUR foam. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 103 4.2.2. Shear test The difference in elastic properties between the CFS webs' shear modulus and the PUR foam core's shear modulus is significant. Therefore, the foam core material is not expected to have a significant influence on the overall shear performance of the sandwich panel. Nevertheless, it is worth noting that the mechanical properties of the bulk PUR foam material or the PUR foam-to-face sheet interface can be estimated based on the observed failure mode. This information can be utilised in the calibration of the constitutive models presented further ahead in Chapter 7. Five prismatic specimens with nominal dimensions of 360 mm × 115 mm × 30 mm were tested using the approach suggested in ASTM C273 (2016) [168]. The specimens were extracted from full-scale sandwich panels with a 1 mm thick steel face sheet in the longitudinal direction. The full-scale sandwich panel had a nominal PUR foam density of 40 kg/m3. As the flatwise compressive and tensile tests on the 50 kg/m3 and 60 kg/m3 PUR foam yielded unexpectedly low mechanical properties, and Ferpainel experienced difficulties in synthesizing these foams, only the 40 kg/m3 PUR foam was tested as it had mechanical properties similar to those reported in the literature. Nevertheless, it is worth noting that the shear specimens showed visible manufacturing defects, namely large voids, concentrated near the interface, as shown in Figure 4.16. Figure 4.16. Shear specimens’ defects: voids along one of the interfaces. Before testing, the specimens’ dimensions were measured, and their weight was recorded with a scale. Based on that information, the estimated average density was 36.4 kg/m3, and its relative difference with the nominal density was -9.0%. The specimens were bonded to steel L-shaped profiles made of two UNP 120 steel profiles with SikaForce 7710 L100, a two-component polyurethane-based adhesive. Before applying the adhesive, the surfaces of the steel pieces and the face sheets of the specimens were polished with sandpaper and cleaned with acetone. The adhesive was then cured for 24 h at room temperature before the test. Two 3D bearing joints were screwed to the L-shaped profiles to avoid the transmission of the bending moment through the supports (see Figure 4.17). A tensile load is applied, resulting in the plates moving in opposite directions. Thus, a transverse displacement of the planes parallel to the bonded face sheets of the PUR foam is imposed [169]. A lightweight floor system based on sandwich panel: characterization and development 104 Figure 4.17. Test fixture for the assessment of PUR foam shear properties. The shear tests were performed in a UTM, and the load was recorded by a load cell with a capacity of 200 kN (±0.12 kN). The relative vertical displacement between the L-shaped profiles is measured by two LVDTs with a measuring stroke of 50 mm (±0.12 mm). The overall test setup is illustrated in Figure 4.18. The specimens were tested under quasi-static loading up to failure, under displacement control at a rate of 0.5 mm/min. Figure 4.18. Shear test experimental setup. The stress-strain curves for all the tested specimens tested are plotted in Figure 4.19. The shear strains are computed as the ratio between the relative displacement between the L-shaped profiles and the 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 105 thickness of the specimen. All the specimens presented an initial elastic stage followed by a hardening branch preceding the brittle failure. Two failure modes were identified based on tests performed on the five specimens: i) debonding at the PUR foam-to-face sheet interface (DI) (see Figure 4.20a); and ii) cohesive shear failure of the PUR foam (CS) with the formation of diagonal cracks (see Figure 4.20b). Figure 4.19. Stress-strain curves from the shear tests on PUR foam. (a) (b) Figure 4.20. Failure modes identified in the shear tests: (a) debonding failure at the interface PUR foam-to-face sheet interface; (b) cohesive shear failure of the PUR foam. A lightweight floor system based on sandwich panel: characterization and development 106 The material properties of the PUR foam are summarised in Table 4.5. The shear modulus (𝐺) is calculated as the slope of the initial part of the stress-strain curve (up to one-third of the shear yielding strength). The shear yield strength (𝜏y) is estimated as the intersection of the extrapolation of the linear and hardening region in the stress-strain curves following the method described in [170]. The shear yield strain is the strain corresponding to the shear yielding strength (𝛾y). Finally, in the table are included the ultimate shear strength (𝜏u) and the corresponding ultimate strain (𝛾u). Table 4.5. Shear properties of the PUR foam. Specimen 𝑮 [MPa] 𝝉𝐲 [MPa] 𝜸𝐲 [-] 𝝉𝐮 [MPa] 𝜸𝐮 [-] Failure mode [-] S01 2.53 0.14 0.055 0.17 0.155 CS S02 2.69 0.15 0.054 0.19 0.179 DI S03 3.77 0.14 0.037 0.16 0.067 DI S04 2.62 0.14 0.052 0.16 0.131 CS S05 3.11 0.16 0.050 0.19 0.115 DI Avg. (CoV) 2.94 (17.5%) 0.15 (5.4%) 0.049 (15.0%) 0.17 (8.7%) 0.129 (32.8%) All Avg. CS 2.57 0.14 0.053 0.16 0.143 CS Avg. DI (CoV) 3.19 (17.1%) 0.15 (6.1%) 0.047 (19.3%) 0.18 (10.5%) 0.120 (46.6%) DI Notes: Avg. – average results; CoV – coefficient of variation. Failure modes: DI – debonding at interface; CS – cohesive shear failure PUR foam. The stress-strain curves of the specimens failed due to debonding present a larger scatter in terms of ultimate stress and strain. This may be due to the unstable propagation of cracks at the PUR foam-toface sheet interface. It is also worth noting that the debonded specimens presented more noticeable defects, as depicted in Figure 4.16. The specimens that failed due to cohesive shear failure of the PUR foam presented lower mechanical strength than those that failed due to debonding. It may be argued that the specimens failed because the cohesive shear of the PUR foam presented a larger interface area, allowing for a uniform stress distribution through the thickness of the specimen. According to theory of elasticity of homogeneous and isotropic materials [171], the Poisson’s ratio (𝜈) of the PUR foam is given by Equation (4.4): 𝜈=𝐸−2∙𝐺 2∙𝐺 (4.4) According to the experimental data obtained from the flatwise compressive and tensile tests as well as the shear tests, 𝜈 of the PUR foam core material of the proposed sandwich panel present unusual values 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 107 of -0.02 and -0.04. Negative values of the Poisson’s ratio are also reported for the PUR foam core material of the steel insulated sandwich panel investigated in [163]. The authors of the above-mentioned study suggests that the assumption of isotropy for PUR foams may not be correct. Furthermore, in the case of the PUR foam tested in this work, the influence of the heterogeneity of the PUR foam on the value of ν should also be considered. The difference of the estimated densities between the sets of specimens is approximately 8%. Additionally, the specimens for the flatwise compressive and tensile tests were extracted from sandwich panels with larger total thickness than those produced for the production of the shear specimens. This may also contribute to different cell microstructure in the PUR foams. There is not a good agreement in the literature regarding the Poisson’s ratio of PUR foams found in civil engineering applications as can be seen in Table 4.6. The comparison is also made difficult by the different production technologies employed to produce sandwich panels. According to [172], the definition of the Poisson’s ratio of conventional PUR foams as a material constant is only valid for small strains. In their study, 𝜈 is estimated to be equal to 0.32 for small strains and approaching the value of 0.0 and 0.5 for large strain in compression and tension, respectively. In [173], the interval of Poisson’s ratio for polymeric foams is estimated to be between 0.1 and 0.4. Even considering less stringent definitions, more than half of the estimated values in Table 4.6 fall outside the typical range. This may suggest that the PUR foam has anisotropic material behaviour due to the preferential evolution of the foam in the rise direction [72]. Table 4.6. Poisson’s ratio of different PUR foams. Reference Production technology Nominal density [kg/m3] 𝑬𝐜 [MPa] 𝑬𝐭 [MPa] 𝑮 [MPa] 𝝂𝐜 [-] 𝝂𝐭 [-] [36] Slabstock foam 70 9.0 (±1.0) 14.58 (±1.0) 3.96* (±0.37) 0.14 0.84 [174] Slabstock foam 62 18.58 (±0.24) 18.1 (±0.457) 7.53 (±0.01) 0.23 0.20 [33] In-situ foam filling 45 12.44 (± 1.66) 19.0 (±5.41) 4.46 (± 0.45) 0.39 1.13 [17] Slabstock foam 100 26.8 (±1.4) 31.3 (±0.02) 8.7 (±1.0) 0.54 0.80 [163] Sandwich foaming 40 3.82 (±0.44) 3.59 (±0.17) 3.83 (±0.09) -0.50 -0.53 [18] Slabstock foam 48 6.30 (±0.57) 3.15 (±0.38) 0.00 -1.00 LightSlab Sandwich foaming 40 5.78 (±0.48) 5.63 (±1.39) 2.94 (±0.52) -0.02 -0.04 Notes: the values between parentheses are the corresponding standard deviation; 𝜈c – Poisson’s ratio based on 𝐸c; 𝜈t – Poisson’s ratio based on 𝐸t; the highlighted values the symbol “*” include debonding specimens in the average calculation. A lightweight floor system based on sandwich panel: characterization and development 114 automotive and aerospace applications, such a test is often used to evaluate the mechanical properties of the composite structure, and in particular the capacity of the core to transfer stresses from one face sheet to another [36], [40], [50], [51], [185]. The axial stiffness characterisation is particularly relevant in seismic-prone areas where the floor shall act as a diaphragm, equally redistributing the horizontal actions to the vertical structures. Moreover, an analytical study was conducted for predicting the performance of the panels. Edgewise compressive tests were carried out on two sets of specimens according to ASTM C364 (1999) [186]. The two sets of specimens had identical nominal dimensions and two different nominal thicknesses for the steel face sheet, namely 1.0 mm (EC-1.0) and 1.5 mm (EC-1.5). These specimens were extracted from larger sandwich panels with a core thickness of 100 mm and face sheet thicknesses of 1.0 mm and 1.5 mm, and had an area of 250 mm × 250 mm. Before testing, the specimens were measured and weighted, and an estimated density of 38.8 kg/m3 was calculated (relative difference from the nominal density of -3.1%). Visible defects were observed in the specimens upon delivery, which were caused by the extraction process from full-scale sandwich panels (see Figure 4.28). (a) (b) (c) (d) Figure 4.28. Sandwich panels specimens upon delivery: (a) face sheet defects; (b) milling machine treatment; (c) cohesive crack due to milling operation; (d) grinding wheel treatment. To achieve flat end surfaces perpendicular to the length of the specimens, different methods were used. Although the milling machine produced smooth end surfaces, it also induced stress that caused a cohesive crack in the PUR foam parallel to the face sheets (see Figure 4.28c). Thus, a grinding wheel was found to be more appropriate for the smoothing process. The quality of the end surfaces was slightly lower than that produced by the milling machine, but no cracks were observed in the PUR foam. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 115 The UTM used for testing had a load capacity of 200 kN. To ensure an even distribution of stress over the loaded area, a 50 mm thick solid steel plate was used to transfer the load. To prevent failure in the loading and support areas, 20 mm × 25 mm solid steel bars were attached to the plates on each side of the specimen. Four LVDTs were used to monitor vertical displacement on the bottom face of the loading plate (to remove the effect of possible rotations), and DIC was used to monitor one of the lateral faces of the specimen. The overall test setup is illustrated in Figure 4.29a. The tests were performed under displacement control at a speed of 0.5 mm/min. (a) (b) Figure 4.29. Edgewise compressive test: a) overall setup and DIC equipment; b) specimen. The load-displacement curves of the edgewise compressive tests conducted on the EC-1.0 and EC-1.5 sets of specimens are shown in Figure 4.30. Typically, both sets exhibit an initial nonlinear section of the curve, which can be attributed to the closing of gaps between the loading plates and the ends of the samples. Once the end cross-sections are entirely engaged, a linear elastic behaviour is observed until failure initiation. The EC-1.0 specimens exhibited various failure modes, which resulted in high scatter in peak load results. These modes included debonding between the core and face sheet (D), global buckling (GB), wrinkling of the face sheets (W), and local buckling near the loading plate (LB). Conversely, the EC1.5 specimens demonstrated a consistent failure mode due to global buckling. During the edgewise compressive test of the EC-1.0 specimen set, there was a significant deviation in the results, which could be attributed to several factors. These factors include the extraction process of the specimens from fullscale sandwich panels, resulting in irregular end surfaces and possible cracks in the PUR foam. A lightweight floor system based on sandwich panel: characterization and development 116 (a) (b) Figure 4.30. Edgewise compressive test load-displacement curves: a) EC-1.0 set; b) EC-1.5 set. The premature debonding of specimen EC-1.0-1 may have been caused by pre-existing cracks due to this process. Smooth end surfaces were difficult to achieve due to stress in the bulk of the PUR foam, leading to the concentration of stresses and localised end failure. The scatter in results could also be attributed to the heterogeneity of the PUR foam and the face sheet-to-core interface. This interface is crucial for the composite structure's monolithic behaviour [73]. In Figure 4.31, the compressive response of an EC-1.0 specimen is presented alongside its deformation history as recorded by the DIC. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 117 (a) 1 2 3 4 5 (b) Figure 4.31. Edgewise compressive test DIC results for EC-1.0 sample: a) load-displacement curve; b) deformation history. The load increases steadily until 0.3 mm, where a small load drop occurs due to local instability of the face sheet. When peak load is reached one of the face sheet wrinkles. The load drop after failure initiation is due to debonding at the top face sheet-to-core interface. The following constant load plateau corresponds to debonding propagation through the top face sheet-to-core interface. An average half wrinkling wavelength of 65.5 mm was estimated from the DIC measurements (see Figure 4.32). (a) (b) Figure 4.32. Wrinkling half wavelength estimation using DIC: a) EC-1.0-4; b) EC-1.0-5. All units in [mm]. A lightweight floor system based on sandwich panel: characterization and development 118 The typical deformation and collapse of an EC-1.5 sample is shown in Figure 4.33. Before failure initiation, two distinct linear branches are visible. Following the attainment of peak load due to global buckling, a softening branch appears, indicating the formation of a plastic hinge in the top face sheet located near the loading plate, and the simultaneous crushing of the PUR foam core. The out-of-plane bending of the top face sheet is transferred to the bottom face sheet through the core. However, the PUR foam core's tensile strength is insufficient to hold the face sheets together with the core. Ultimately, a significant drop in load is observed as the bottom face sheet separates from the core due to debonding. It is worth noting that most specimens failed due to the debonding of the top face sheet. After testing, the specimens were examined, revealing a thin layer of PUR foam remaining on the debonded face sheet. These findings are consistent with the results of the flatwise tensile test conducted on cubic sandwich panel specimens. The tensile tests highlighted the presence of a more flexible layer of PUR foam next to the top face sheet, where the cohesive tensile crack ultimately occurred. (a) 1 2 3 4 5 (b) Figure 4.33. Edgewise compressive test results for EC-1.5 sample: a) load-displacement curve; b) deformation history. In Table 4.9 a summary of the obtained results, including ultimate load (𝐹u), axial stiffness (𝐾), failure initiation, and failure propagation is presented. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 119 Table 4.9. Summary of the results of the edgewise compressive tests. Specimen [-] 𝑭𝐮 [kN] 𝑲 [kN/mm] Failure initiation [-] Failure propagation [-] EC-1.0-1 18.5 488.3 D (T) D (T) EC-1.0-2 48.0 395.1 LB + FC D (T) EC-1.0-3 43.5 - W (T) D (T) EC-1.0-4 52.4 520.7 W (T) + FC D (T) EC-1.0-5 36.1 592.0 W (T) D (T) Avg. (CoV) 39.7 (33.5%) 468.0 (17.5%) EC-1.5-1 56.0 533.4 GB D (T) EC-1.5-2 59.4 503.7 GB FC EC-1.5-3 59.4 - GB D (T) EC-1.5-4 67.0 571.2 GB FC and D (B) EC-1.5-5 45.6 555.7 GB FC and D (B) Avg. (CoV) 57.5 (13.5%) 541.0 (5.4%) Notes: Avg. – average value per set of specimens; CoV – coefficient of variation; the symbol “-“ indicates DIC measurements are unavailable for the specimen. The displacement of the cross-sections in Figure 4.34 was tracked by the DIC system. The sampling points were located in the PUR foam near the face sheet. It is assumed that the deformation of the face sheet and the PUR foam close to the interface is the same until debonding occurs. Figure 4.34. Location of the monitored points by the DIC in the edgewise compressive test. All units in [mm]. To validate the accuracy of the DIC system, the vertical displacement of the cross-section near the loading plate was compared with the results from LVDTs. The procedure is shown in Figure 4.35 for specimen EC-1.0-4. The DIC readings could not capture the rotations of the loading plate, causing a small difference (about 10%) from LDVTs measurement, as the DIC only monitors one of the vertical surfaces of the A lightweight floor system based on sandwich panel: characterization and development 120 specimen. Additionally, the reading of the loading plate cross-section was disrupted due to the crushing of the PUR foam, causing damage to the speckle pattern. Figure 4.35. Comparison between LVDTs and DIC measurements for specimen EC-1.0-4. The vertical displacement of the top and bottom cross-sections and their relative displacement (see Figure 4.36) showed that a significant component of the displacement occurred near the loading and supporting areas of the specimens due to the uneven end surfaces of the specimens. Figure 4.36. Comparison between LVDTs and DIC relative displacement between top and bottom cross-section for set EC-1.0. 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 121 The LVDTs measured the relative displacement between the loading and support plates, including the closing of gaps between the specimens and the test setup, as well as localised deformations. Therefore, to estimate the axial stiffness of the sandwich panels, load-displacement curves recorded by the DIC system were considered. However, the axial stiffness values for specimens EC-1.0-3 and EC-1.5-3 could not be estimated due to difficulties encountered during the image-matching process. These challenges were likely caused by varying lighting conditions in the laboratory facility due to changes in weather, resulting in insufficient pattern contrast for the algorithm to carry out the image correlation process [187]. 4.4. ANALYTICAL MODELLING OF EDGEWISE COMPRESSIVE TESTS To investigate the suitability of existing models to simulate the edgewise compressive behaviour of sandwich panels and to further understand the mechanical behaviour of the tested specimens, the analytical study on a sandwich panel with clamped ends is carried out. The panel has a length of 𝐿, width of 𝑏, and is subjected to compressive load 𝑃. It consists of face sheets with a thickness of 𝑡f and a core with a thickness of 𝑡c. The material properties of the face sheets and core materials are given in Table 4.1, Table 4.3, and Table 4.4. During the experiment, the sandwich panel's response up to the point of failure initiation was generally linear. The panel stiffness during this linear response is expressed as 𝐾. Assuming that the face sheet-to-core interface is perfectly bonded, the axial deflection of the sandwich panel is the same in the face sheet and the core, as given in Equation (4.5): 𝛿tot=𝛿f⇔𝑃∙𝐿 𝐸e∙𝐴tot=𝑃∙𝐿 𝐸f∙𝐴f (4.5) where 𝛿tot is the sandwich panel deformation, 𝛿f is the axial deformation of the face sheets, 𝐴tot is the total cross-section area of the sandwich panel, 𝐸e is the effective Young’s modulus of the sandwich panel [188], 𝐸f is the Young’s modulus of the face sheets, and 𝐴f is the cross-section area of the face sheets. Keeping in mind that the axial stiffness of a beam under compression is calculated by multiplying the Young's modulus of the material by the beam's area and dividing it by its length, the stiffness of the sandwich panel is determined according to Equation (4.6): 𝐾=𝐸e∙𝐴tot 𝐿 (4.6) When predicting the ultimate load of a sandwich panel, two failure modes are identified: global buckling and wrinkling. It is assumed that both the EC-1.0 and EC-1.5 sets consist of sandwich panels with thin A lightweight floor system based on sandwich panel: characterization and development 122 faces, where most of the bending stiffness (more than 99%) comes from the bending of the face sheets around the centroidal axis. The cores are weak, and their bending stiffness contributes to less than 1% of the total stiffness [189]. In this case, the global buckling load (𝑃cr) is a combination of Euler buckling (𝑃E) and buckling of the core due to shear (𝑃s) as shown in Equation (4.7): 1 𝑃cr=1 𝑃E+1 𝑃s (4.7) The Euler buckling and the core shear buckling are defined by the following expressions: 𝑃E=𝜋2∙(𝐸∙𝐼)eq (𝑘∙𝐿e)2 (4.8) 𝑃s=𝑏∙𝑡c∙𝐺c (4.9) where 𝐺c is the core shear modulus, and 𝐿e is the free length of the sandwich panel between the clamps. Wrinkling happens when periodic waves with a similar thickness as the core layer appear all over the face sheet. There are three types of wrinkling: i) rigid base (single-sided); ii) anti-symmetric; and iii) symmetric. In this study, symmetric wrinkling is the most relevant type since periodic waves can be seen on both face sheets. According to [190] the wrinkling half wavelength (𝑙w) and the wrinkling critical stress (𝜎w) in the face sheet are given by: 𝑙w=1.65∙𝑡f∙√𝐸f2 𝐸c∙𝐺c 6 (4.10) 𝜎w=0.91∙√𝐸f∙𝐸c∙𝐺c 3 (4.11) The critical wrinkling load (𝑃w) will be reached when the face sheets reach the critical wrinkling stress according to Equation (4.12): 𝑃w=2∙𝑏∙𝑡f∙𝜎w (4.12) A summary of the analytical results, along with the experimental values, is presented in Table 4.10. The axial stiffnesses for specimens EC-1.0 and EC-1.5 are 413.3 kN/mm and 581.0 kN/mm, respectively, according to Equation (4.6). The experimental results are within 10% of the analytical predictions, indicating the assumption of a perfectly bonded interface during the linear stage of axial response is reasonable. As mentioned in Section 4.2.2, the estimated values of 𝜈 based on the mechanical 4. MATERIAL CHARACTERISATION AND SMALL-SCALE TESTS 123 characterisation of the PUR foam yield unusual negative values and possible explanations have been provided. Based on that discussion, it was deemed adequate to estimate the value of 𝐺c according to [171] assuming that the ν of PUR foam for small strains may be taken as 0.3. Table 4.10. Summary of the geometrical properties and predicted and measured failure loads. Set 𝑳 [mm] 𝒕𝐜 [mm] 𝒕𝐟 [mm] 𝑲 [kN/mm] 𝑲𝐞𝐱𝐩 [mm] 𝑷𝐜𝐫 [kN] 𝒍𝐰 [mm] 𝑷𝐰𝐫 [kN] 𝑷𝐞𝐱𝐩 [kN] EC-1.0 250 100 1.0 413.3 468.0 55.6 64.1 62.9 44.0* EC-1.5 250 100 1.5 518.0 541.0 55.6 94.2 92.5 57.5 Notes: 𝐾exp – average experimental value of the axial stiffness of sandwich panels; 𝑃exp – average experimental value of the peak load of sandwich panels; highlighted with “*” is the average wrinkling load of specimens EC-1.0-3, EC-1.0-4, and EC-1.0-5. Equation (4.7) results in a value of 55.6 kN for both sets of specimens since the buckling of the core due to shear governs the overall failure mode. The estimated global buckling load agrees well with the average experimental ultimate load of EC-1.5 specimens (-3.4% relative difference). The critical wrinkling load is calculated using Equation (4.12), and the values of 62.9 kN and 92.5 kN are obtained for the EC-1.0 and EC-1.5 sets, respectively. The critical wrinkling load for EC-1.5 is higher than the global buckling load, indicating consistency in the failure mode. The predicted critical wrinkling load differs from the average experimental ultimate load of EC-1.0 specimens that failed due to wrinkling (44.0 kN). Nevertheless, several authors have proposed different values for the constant coefficient of Equation (4.11), which range from 0.5 to 0.91 [190], [191]. However, these coefficients were altered to fit the experimental results and may not be appropriate for the sandwich panel studied in this work. If a coefficient of 0.65 were used, the EC-1.0 and EC-1.5 specimens would have wrinkling loads of 44.9 kN and 66.1 kN, respectively. These results would suggest that global buckling occurred in the EC-1.5 specimens and would also decrease the difference between the experimental average peak load of the EC-1.0 specimens. The EC1.0 specimens had a wrinkling half wavelength of 64.1 mm, according to Equation (4.10), which matches the analytical prediction (-2.2% relative difference) despite the discrepancy in ultimate load. This difference may be due to the presence of geometrical imperfections, which affect the ultimate load instead of the instability mode's shape. A lightweight floor system based on sandwich panel: characterization and development 226 mechanical behaviour of the full-scale sandwich panel. To include such features, different modelling strategies are available, such as using failure criteria based on maximum principal stress or strain for the foam and explicitly modelling the interfaces by means of cohesive elements. Implementation of these strategies would require a new set of parameters that could be obtained either by calibrating the numerical models with the results of the flexural test of the sandwich panel or through experimental tests. In the latter option, tests such as the double cantilever beam and the single edge notch bending test should be carried out to obtain the traction-separation law of the interfaces and the fracture energy of the core material, respectively. Finally, the proposed sandwich panel floor system's applicability depends on future evaluations of its performance in terms of building physics functions, such as acoustic and thermal behaviour, as well as fire resistance. 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The 𝑖−𝑡ℎ individual in the OSPANEL code are represented by a structure which is a database containing variables of different types: i) the set of design variables (𝒙[𝑖]); ii) the cost (𝑐𝑠𝑡[𝑖]), weight 𝑤𝑡[𝑖], and environmental impact (𝑔𝑤𝑝[𝑖]); iii) the penalised fitness (𝑓𝑝[𝑖]), the ranking (𝑘[𝑖]), and the exponential ranking fitness (𝑓𝑒𝑥𝑝[𝑖]); iv) the mating probability (𝑝𝑚𝑎𝑡[𝑖]) and the cumulative mating probability (𝑝𝑐𝑢𝑚𝑢𝑙𝑚𝑎𝑡[𝑖]). The structure of each individual is stored in an array (𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖]) whose size (𝑁) may be chosen by the user. The user, through an input file, may also select the component 𝑤𝑖 of the weight vector (𝒘) of the objective function, the maximum number of generations (𝑡𝑚𝑎𝑥), and the selective pressure of the best individual (𝑠). The 𝑀𝑜𝑑𝑢𝑙𝑒 𝑆𝑜𝑟𝑡𝑀𝑖𝑛𝑀𝑎𝑥 is used to assign the ranking of each individual so that the individual with the smallest value of the penalised fitness is assigned a rank of 0 and the worst performing individual is assigned a rank of 𝑁−1 (see Figure A.1). Firstly, the penalised fitness of each individual (𝑓𝑝[𝑖]) is stored in an auxiliary float array (𝑃𝑒𝑛𝑎𝑙𝑖𝑧𝑒𝑑𝐹𝑖𝑡[𝑖]). The 𝑓𝑜𝑟 loop with the counter variable 𝑗 compares the value of the 𝑗−𝑡ℎ individual with the value of the variable 𝑚𝑖𝑛, which is the minimum value found so far of the penalised fitness. If the value of the current 𝑗−𝑡ℎ individual is less than 𝑚𝑖𝑛, its location in the array of individuals is stored in 𝑘𝑚𝑖𝑛. At the end of the 𝑓𝑜𝑟 loop with the counter variable 𝑗, the location of the best individual is store in an auxiliary integer array (𝑆𝑜𝑟𝑡𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖]) and its value in 𝑃𝑒𝑛𝑎𝑙𝑖𝑧𝑒𝑑𝐹𝑖𝑡[𝑖] is assigned a fictitious large value. This way, as the instruction are repeated in the 𝑓𝑜𝑟 loop with the counter variable 𝑖, the best performing individuals found in the 𝑖−1 loop is automatically discarded from the comparison. At the end of the 𝑓𝑜𝑟 loop with the counter variable 𝑖, the location of the individuals in the array are saved in 𝑆𝑜𝑟𝑡𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖] from the smallest to the largest. The last step consists in transferring this information from the 𝑆𝑜𝑟𝑡𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖] array to the 𝑘[𝑖] variable in the structure array 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖]. A lightweight floor system based on sandwich panel: characterization and development 244 Figure A.1. 𝑀𝑜𝑑𝑢𝑙𝑒 𝑆𝑜𝑟𝑡𝑀𝑖𝑛𝑀𝑎𝑥 of the algorithm OSPANEL. The 𝑀𝑜𝑑𝑢𝑙𝑒 𝐶𝑎𝑙𝑐𝑀𝑎𝑡𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 is used to determine the mating probability of each individual (𝑝𝑚𝑎𝑡[𝑖]). In the first 𝑓𝑜𝑟 loop the new fitness value of the 𝑖−𝑡ℎ individual (𝑓𝑒𝑥𝑝[𝑖]) is estimated (see Figure A.2). In the same loop the sum of the exponential ranking fitness of all the individuals is stored in the auxiliary float variable 𝑆𝑢𝑚. Finally, in the second for loop the mating probability (𝑝𝑚𝑎𝑡[𝑖]) of each individual is obtained as the ratio of the exponential ranking fitness of the single individual between the exponential ranking fitness of the entire population. ANNEX 245 Figure A.2. 𝑀𝑜𝑑𝑢𝑙𝑒 𝐶𝑎𝑙𝑐𝑀𝑎𝑡𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 of the algorithm OSPANEL. In the 𝑀𝑜𝑑𝑢𝑙𝑒 𝐶𝑎𝑙𝑐𝑀𝑎𝑡𝑃𝑜𝑜𝑙 the auxiliary integer array 𝑀𝑎𝑡𝑃𝑜𝑜𝑙[𝑖] is filled with the location of the individuals selected for reproduction in the structure array 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖] to form the mating pool 𝑀𝑡 (see Figure A.3). The selection is achieved by means of SUS, which can be visualised as the result of a spinning roulette wheel with the slots proportional in width to 𝑝𝑚𝑎𝑡[𝑖] with multiple, equally spaced pointers. The slots of the roulette wheel correspond to the cumulative probability (𝑝𝑐𝑢𝑚𝑢𝑙𝑚𝑎𝑡[𝑖]) of the individuals. The first 𝑓𝑜𝑟 loop consists in locating the position of the individual (𝑘𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙) through the auxiliary array 𝑆𝑜𝑟𝑡𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖] and constructing the roulette wheel. The cumulative probability is calculated by means of the auxiliary variable 𝑎𝑢𝑥, which stores the sum of the mating probability of the individuals at the end of each loop. Once the roulette is built, the auxiliary variables 𝑘𝑚𝑎𝑡𝑖𝑛𝑔𝑝𝑜𝑜𝑙 and 𝑃𝑜𝑖𝑛𝑡𝑒𝑟 are initialised. The cycle with the counter variable 𝑙 is used to locate the position of the pointers within the roulette slots. If the cycle is broken the last stored value of 𝑘𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 represent where the pointer landed on the roulette wheel. The position of the individual selected by the pointer is stored in the array 𝑀𝑎𝑡𝑃𝑜𝑜𝑙[𝑖], the 𝑃𝑜𝑖𝑛𝑡𝑒𝑟 location is updated and the 𝑓𝑜𝑟 loop with the counter variable 𝑗 is repeated until all the 𝑀𝑎𝑡𝑃𝑜𝑜𝑙[𝑖] array is filled. A lightweight floor system based on sandwich panel: characterization and development 246 Figure A.3. 𝑀𝑜𝑑𝑢𝑙𝑒 𝐶𝑎𝑙𝑐𝑀𝑎𝑡𝑃𝑜𝑜𝑙 of the algorithm OSPANEL. The 𝑀𝑜𝑑𝑢𝑙𝑒 𝐶𝑎𝑙𝑐𝑁𝑒𝑤𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 is utilised for the creation of the next generation of individuals 𝑃𝑡+1 (see Figure A.4). According to the elitist strategy the best genetic information of the previous generation is retained in the next one. Therefore, in the first 𝑓𝑜𝑟 loop, the best half (𝑁/2) of the parent generation is copied into the auxiliary array 𝑁𝑒𝑤𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖]. The best individuals are located ANNEX 247 through the integer array 𝑆𝑜𝑟𝑡𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖]. Furthermore, the auxiliary variable 𝐶𝑜𝑢𝑛𝑡𝑒𝑟 is used to keep track of the position of the individual in the structure array 𝑁𝑒𝑤𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖]. In the second nested 𝑓𝑜𝑟 loop the mating pool 𝑀𝑡 is shuffled. The 𝑓𝑜𝑟 loop with the counter variable 𝑖 starts from the value 𝑁. The location stored in 𝑀𝑎𝑡𝑃𝑜𝑜𝑙[𝑖] is swapped (↔) with the location in 𝑀𝑎𝑡𝑃𝑜𝑜𝑙[𝑗]. The 𝑗− 𝑡ℎ location is a random value ranging from 1 to 𝑖−1. The random location is obtained by using the 𝑟𝑎𝑛𝑑() operator which generates a pseudo random number and the modulus (%) operator which returns the remainder of the integer division between 𝑖 and 𝑟𝑎𝑛𝑑(). The shuffled 𝑀𝑎𝑡𝑃𝑜𝑜𝑙[𝑖] array is then used to generate the offspring population. Two consecutive individuals (𝑘𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙,𝐴 and 𝑘𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙,𝐵) are selected from the 𝑀𝑎𝑡𝑃𝑜𝑜𝑙[𝑖] and their genetic information is combined together through the cross-over operator (𝐶𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟). The offspring individual inherits a combination of the design variables of the parent individuals. A mutation operator (𝑀𝑢𝑡𝑎𝑡𝑖𝑜𝑛) with a predefined rate of 0.8 is applied to the set of design variables 𝒙[𝑖] of the offspring individual which is finally copied into the 𝑁𝑒𝑤𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖] at the location determined by the 𝐶𝑜𝑢𝑛𝑡𝑒𝑟 variable. The last step consists in copying the information from 𝑁𝑒𝑤𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖] to the original structure array 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙[𝑖]. A lightweight floor system based on sandwich panel: characterization and development 248 Figure A.4. 𝑀𝑜𝑑𝑢𝑙𝑒 𝐶𝑎𝑙𝑐𝑁𝑒𝑤𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 of the algorithm OSPANEL.