Computational Mechanical Modelling of Wood—From Microstructural Characteristics Over Wood-Based Products to Advanced Timber Structures

  • Josef FüsslEmail author
  • Markus Lukacevic
  • Stefan Pillwein
  • Helmut Pottmann
Part of the Lecture Notes in Civil Engineering book series (LNCE, volume 24)


Wood as structural bearing material is often encountered with skepticism and, therefore, it is not used as extensively as its very good material properties would suggest. Beside building physics and construction reasons, the main cause of this skepticism is its quite complex material behavior, which is the reason that design concepts for wood have so far not achieved a desirable prediction accuracy. Thus, for the prediction of effective mechanical properties of wood, advanced computational tools are required, which are able to predict as well as consider multidimensional strength information at different scales of observation. Within this chapter, three computational methods are presented: an extended finite element approach able to describe strong strain-softening and, thus, reproduce brittle failure modes accurately; a numerical limit analysis approach, exclusively describing ductile failure; and an elastic limit approach based on continuum micromechanics. Based on illustrative results, the performance of these methods is shown and discussed. Furthermore, a finite-element-based design procedure for an elastically-deformed wooden structure is outlined, showing how advanced mechanical information of the base material could be exploited within digital design of complex timber structures in future. Finally, geometric design concepts applicable within digital wood design are discussed, giving insights into possible future developments.


Computational mechanics Mechanical modelling Structural analysis Wooden microstructure Wood-based products 


  1. Aicher S, Gustafsson PJ, Haller P, Petersson H (2002) Fracture mechanics models for strength analysis of timber beams with a hole or a notch. A report of Rilem TC-133Google Scholar
  2. Andersen ED, Roos C, Terlaky T (2003) On implementing a primal-dual interior-point method for conic quadratic optimization. Math Program Ser B 95:249–277MathSciNetCrossRefGoogle Scholar
  3. Bader T, Hofstetter K, Hellmich C, Eberhardsteiner J (2010) Poromechanical scale transitions of failure stresses in wood: from the lignin to the spruce level. ZAMM - Zeitschrift für angewandte Mathematik und Mechanik 90:750–767MathSciNetCrossRefGoogle Scholar
  4. Bader T, Hofstetter K, Hellmich C, Eberhardsteiner J (2011) The poroelastic role of water in cell walls of the hierarchical composite softwood. Acta Mech 217:75–100CrossRefGoogle Scholar
  5. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Meth Eng 45(5):601–620CrossRefGoogle Scholar
  6. Benveniste Y (1987) A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech Mater 6:147–157CrossRefGoogle Scholar
  7. Blum Ch, Roli A (2003) Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Comput Surv (CSUR) 35(3):268–308CrossRefGoogle Scholar
  8. Böhm H (2004) A short introduction to continuum micro-mechanics. In: Böhm H (ed) Mechanics of microstructured materials. CISM Lecture notes 464. Springer, Wien/New York, pp 1–40CrossRefGoogle Scholar
  9. Chilton J, Tang G (2017) Timber gridshells: architecture, structure and craft. Routledge, New YorkGoogle Scholar
  10. Denton SR, Morley CT (2000) Limit analysis and strain-softening structures. Int J Mech Sci 42:503–522CrossRefGoogle Scholar
  11. Drucker DC, Greenberg HJ, Prager W (1951) The safety factor of an elastic-plastic body in plane strain. J Appl Mech 18:371–378MathSciNetzbMATHGoogle Scholar
  12. Drucker DC, Prager W, Greenberg HJ (1952) Extended limit design theorems for continuous media. Q Appl Math 9:381–389MathSciNetCrossRefGoogle Scholar
  13. Eberhardsteiner J (2002) Mechanisches Verhalten von Fichtenholz: Experimentelle Bestimmung der biaxialen Festigkeit- seigenschaften, Mechanical Behavior of Spruce Wood: Experimental Determination of Biaxial Strength (Properties), in German. Springer, Wien, New YorkCrossRefGoogle Scholar
  14. Eberhardsteiner J, Hofstetter K, Hellmich Ch (2005) Development and experimental validation of a continuum micromechanics model for the elasticity of wood. Eur J Mech A/Solids 24:1030–1053Google Scholar
  15. Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond Ser A 241:376–396MathSciNetCrossRefGoogle Scholar
  16. Foley Ch (2003) Modeling the effects of knots in structural timber. ISSN: 0349-4969Google Scholar
  17. Füssl J, Lackner R, Eberhardsteiner J, Mang HA (2008) Failure modes and effective strength of two-phase materials determined by means of numerical limit analysis. Acta Mech 195(1–4):185–202CrossRefGoogle Scholar
  18. Füssl J, Kandler G, Eberhardsteiner J (2016) Application of stochastic finite element approaches to wood-based products. Arch Appl Mech 86(1–2):89–110. Scholar
  19. Füssl J, Li M, Lukacevic M, Eberhardsteiner J, Martin CM (2017) Comparison of unit cell-based computational methods for predicting the strength of wood. Eng Struct 141:427–443. Scholar
  20. Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning, 1989. Addison-Wesley, ReadingGoogle Scholar
  21. Gravouil A, Moes N, Belytschko T (2002) Non-planar 3d crack growth by the extended finite element and level sets-part ii: level set update. Int J Numer Meth Eng 53(11):2569–2586CrossRefGoogle Scholar
  22. Guindos P (2011) Three-dimensional finite element models to simulate the behavior of wood with presence of knots, applying the flow-grain analogy and validation with close range photogrammetry. PhD thesis, University of Santiago de Compostela, Department of Agroforestry EngineeringGoogle Scholar
  23. Hochreiner G, Füssl J, Eberhardsteiner J (2014a) Cross-laminated timber plates subjected to concentrated loading: experimental identification of failure mechanisms. Strain 50(1):68–81CrossRefGoogle Scholar
  24. Hochreiner G, Füssl J, Serrano E, Eberhardsteiner J (2014b) Influence of wooden board strength class on the performance of cross-laminated timber plates investigated by means of full-field deformation measurements. Strain 50(2):161–173CrossRefGoogle Scholar
  25. Hofstetter K, Hellmich Ch, Eberhardsteiner J (2006) Micromechanical modeling of solid-type and plate-type deformation patterns within softwood materials. A review and an improved approach. Holzforschung 61:343–351Google Scholar
  26. Hofstetter K, Hellmich Ch, Eberhardsteiner J, Mang HA (2008) Micromechanical estimates for elastic limit states in wood, revealing nanostructural failure mechanisms. Mech Adv Mater Struct 15:474–484CrossRefGoogle Scholar
  27. Jiang C, Tang C, Vaxman A, Wonka P, Pottmann H (2015) Polyhedral patterns. ACM Trans Graph 34(6):#172, 1–12Google Scholar
  28. Johansson CJ (2003) Grading of timber with respect to mechanical properties. In: Timber engineering, pp 23–43. ISBN: 0-470-84469-8Google Scholar
  29. Kandler G, Füssl J (2017) A probabilistic approach for the linear behaviour of glued laminated timber. Eng Struct 148:673–685CrossRefGoogle Scholar
  30. Kandler G, Füssl J, Eberhardsteiner J (2015a) Stochastic finite element approaches for wood-based products: theoretical framework and review of methods. Wood Sci Technol 49(5):1055–1097CrossRefGoogle Scholar
  31. Kandler G, Füssl J, Serrano E, Eberhardsteiner J (2015b) Effective stiffness prediction of GLT beams based on stiffness distributions of individual lamellas. Wood Sci Technol 49(6):1101–1121CrossRefGoogle Scholar
  32. Kandler G, Lukacevic M, Füssl J (2016) An algorithm for the geometric reconstruction of knots within timber boards based on fibre angle measurements. Constr Build Mater 124:945–960CrossRefGoogle Scholar
  33. Kandler G, Lukacevic M, Füssl J (2018) Experimental study on glued laminated timber beams with well-known knot morphology. Eur J Wood Wood Prod 1–18Google Scholar
  34. Krieg O, Schwinn T, Menges A, Li JM, Knippers J, Schmitt A, Schwieger V (2014) Biomimetic lightweight timber plate shells: computational integration of robotic fabrication, architectural geometry and structural design. In: Block P et al (eds) Advances in architectural geometry 2014. Springer, pp 109–125Google Scholar
  35. Landelius J (1989) Finit area metoden. en bra metod f ̈or ber ̈akning av uppfl ̈akningsbrott? Rep No TVSM 5043:66 (in Swedish)Google Scholar
  36. Laws N (1977) The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material. J Elast 7(1):91–97MathSciNetCrossRefGoogle Scholar
  37. Li M, Füssl J, Lukacevic M, Eberhardsteiner J, Martin CM (2018a) Strength predictions of clear wood at multiple scales using numerical limit analysis approaches. Comput Struct 196:200–216CrossRefGoogle Scholar
  38. Li M, Füssl J, Lukacevic M, Eberhardsteiner J, Martin CM (2018b) A numerical upper bound formulation with sensibly-arranged velocity discontinuities and orthotropic material strength behaviour. J Theor Appl Mech 56Google Scholar
  39. Liu Y, Pottmann H, Wallner J, Yang YL, Wang W (2006) Geometric modelling with conical meshes and developable surfaces. ACM Trans Graph 25(3):681–689CrossRefGoogle Scholar
  40. Lubliner J (1990) Plasticity theory, Revised Pdf Edition 2006. University of California Berkeley, Previously published by Pearson Education, Inc.Google Scholar
  41. Lukacevic M, Füssl J (2014) Numerical simulation tool for wooden boards with a physically based approach to identify structural failure. Eur J Wood Wood Prod 72(4):497–508. Scholar
  42. Lukacevic M, Füssl J (2016) Application of a multisurface discrete crack model for clear wood taking into account the inherent microstructural characteristics of wood cells. Holzforschung 70(9):845–853CrossRefGoogle Scholar
  43. Lukacevic M, Füssl J, Griessner M, Eberhardsteiner J (2014) Performance assessment of a numerical simulation tool for wooden boards with knots by means of full-field deformation measurements. Strain 50(4):301–317. Scholar
  44. Lukacevic M, Füssl J, Lampert R (2015a) Failure mechanisms of clear wood identified at wood cell level by an approach based on the extended finite element method. Eng Fract Mech 144:158–175. Scholar
  45. Lukacevic M, Füssl J, Eberhardsteiner J (2015b) Discussion of common and new indicating properties for the strength grading of wooden boards. Wood Sci Technol 49(3):551–576CrossRefGoogle Scholar
  46. Lukacevic M, Lederer W, Füssl J (2017) A microstructure-based multisurface failure criterion for the description of brittle and ductile failure mechanisms of clear-wood. Eng Fract Mech 176:83–99CrossRefGoogle Scholar
  47. Mackenzie-Helnwein P, Eberhardsteiner J, Mang HA (2003) A multi-surface plasticity model for clear wood and its application to the finite element analysis of structural details. Comput Mech 31:204–218CrossRefGoogle Scholar
  48. Makrodimopoulos A, Martin CM (2005a) Limit analysis using large-scale socp optimization. In: Proceedings of 13th national conference of UK association for computational mechanics in engineering, Sheffield, pp 21–24Google Scholar
  49. Makrodimopoulos A, Martin CM (2005b) A novel formulation of upper bound limit analysis as a second-order cone programming problem. In: Proceedings of 8th international conference on computational plasticity, Barcelona, pp 1083–1086Google Scholar
  50. Makrodimopoulos A, Martin CM (2007) Upper bound limit analysis using simplex strain elements and second-order cone programming. Int J Numer Anal Meth Geomech 31(6):835–865CrossRefGoogle Scholar
  51. Masuda M (1988) Theoretical consideration on fracture criteria of wood—proposal of finite small area theory. In: Proceedings of the 1988 international conference on timber engineering, Seattle, vol 2, pp 584–595Google Scholar
  52. Melenk J, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1):289–314MathSciNetCrossRefGoogle Scholar
  53. Moes N, Gravouil A, Belytschko T (2002) Non-planar 3d crack growth by the extended finite element and level sets-part i: mechanical model. Int J Numer Meth Eng 53(11):2549–2568CrossRefGoogle Scholar
  54. MOSEK ApS (2006) The MOSEK optimization tools version 4.0 (revision 35). User’s Manual and Reference, available from
  55. Naicu D, Harris R, Williams C (2014) Timber gridshells: design methods and their application to a temporary pavilion. In: World conference on timber engineering, ViennaGoogle Scholar
  56. Nyström J (2003) Automatic measurement of fiber orientation in softwoods by using the tracheid effect. Comput Electron Agric 41(1):91–99CrossRefGoogle Scholar
  57. Olsson A, Oscarsson J (2014) Three dimensional fibre orientation models for wood based on laser scanning utilizing the tracheid effect. In: WCTE 2014, World conference on timber engineering, Quebec City, Canada, August 10–14Google Scholar
  58. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79(1):12–49MathSciNetCrossRefGoogle Scholar
  59. Pech S (2017) Metamodel assisted optimisation of glued laminated timber systems by reordering laminations using metaheuristic algorithms. Master thesis, TU, WienGoogle Scholar
  60. Pech S, Kandler G, Lukacevic M, Füssl J (2018) Metamodel assisted optimization of glued laminated timber systems by reordering wooden lamellas using metaheuristic algorithms. Comput Struct (submitted)Google Scholar
  61. Persson K (2000) Micromechanical modelling of wood and fibre properties, vol 1013. Division of Structural Mechanics, Lund Institute of Technology. ISBN: 91-7874-094-0Google Scholar
  62. Pirazzi C, Weinand Y (2006) Geodesic lines on free-form surfaces—optimized grids for timber rib shells. In: World conference on timber engineering, vol 7Google Scholar
  63. Pottmann H, Asperl A, Hofer M, Kilian A (2007) Architectural geometry. Bentley Institute PressGoogle Scholar
  64. Pottmann H, Huang Q, Deng B, Schiftner A, Kilian M, Guibas L, Wallner J (2010) Geodesic patterns. ACM Trans Graph 29(4):#43, 1–10Google Scholar
  65. Pottmann H, Eigensatz M, Vaxman A, Wallner J (2015) Architectural geometry. Comput Graph 47:145–164CrossRefGoogle Scholar
  66. Puck A, Schuermann H (1998) Failure analysis of frp laminates by means of physically based phenomenological models. Compos Sci Technol 58(7):1045–1067CrossRefGoogle Scholar
  67. Schiftner A, Höbinger M, Wallner J, Pottmann H (2009) Packing circles and spheres on surfaces. ACM Trans Graph 28(5):#139, 1–8CrossRefGoogle Scholar
  68. Schling E, Barthel R (2017) Experimental studies on the construction of doubly curved structures. Detail Struct 01:52–56Google Scholar
  69. Schling E, Kilian M, Wang H, Schikore D, Pottmann H (2018) Design and construction of curved support structures with repetitive parameters. In: Advances in architectural geometry (submitted)Google Scholar
  70. Schmidt J, Kaliske M (2007) Simulation of cracks in wood using a coupled material model for interface elements. Holzforschung 61(4):382–389CrossRefGoogle Scholar
  71. Schmidt J, Kaliske M (2009) Models for numerical failure analysis of wooden structures. Eng Struct 31(2):571–579CrossRefGoogle Scholar
  72. Serrano E, Gustafsson J (2006) Fracture mechanics in timber engineering—strength analyses of components and joints. Mater Struct 40:87–96CrossRefGoogle Scholar
  73. Sjödin J, Serrano E (2008) A numerical study of methods to predict the capacity of multiple steel-timber dowel joints. Holz als Roh- und Werkstoff 66(6):447–454CrossRefGoogle Scholar
  74. Sjödin J, Serrano E, Enquist B (2008) An experimental and numerical study of the effect of friction in single dowel joints. Holz als Roh- und Werkstoff 66(5):363–372CrossRefGoogle Scholar
  75. Song JH, Areias PMA, Belytschko T (2006) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Meth Eng 67(6):868–893CrossRefGoogle Scholar
  76. Steffen A, Johansson CJ, Wormuth EW (1997) Study of the relationship between flatwise and edge-wise modull of elasticity of sawn timber as a means to improve mechanical strength grading technology. Holz als Roh- und Werkstoff 55(2–4):245–253CrossRefGoogle Scholar
  77. Stitic A, Weinand Y (2015) Timber folded plate structures—topological and structural considerations. Int J Space Struct 30:169–176CrossRefGoogle Scholar
  78. Suquet P (1987) Elements of homogenization for inelastic solid mechanics. In: Sanchez-Palencia E, Zaoui A (eds) Homogenization techniques for composite media. Lecture notes in physics 272. Springer, Wien, New York, pp 193–278Google Scholar
  79. Suquet P (1997) Continuum micromechanics. Springer, Wien, New YorkCrossRefGoogle Scholar
  80. Tsai SW, Wu EM (1971) A general theory of strength for anisotropic materials. J Compos Mater 5(1):5880CrossRefGoogle Scholar
  81. Zaoui A (2002) Continuum micromechanics: survey. ASCE J Eng Mech 128(8):808–816CrossRefGoogle Scholar
  82. Zimmermann T, Sell J, Eckstein D (1994) Rasterelektronenmikroskopische Untersuchungen an Zugbruchflächen von Fichtenholz. Holz als Roh- und Werkstoff 52(4):223–229CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Josef Füssl
    • 1
    Email author
  • Markus Lukacevic
    • 1
  • Stefan Pillwein
    • 1
  • Helmut Pottmann
    • 2
  1. 1.Institute for Mechanics of Materials and StructuresTU WienViennaAustria
  2. 2.Computer, Electrical and Mathematical Science and Engineering DivisionKing Abdullah University of Science and TechnologyThuwalSaudi Arabia

Personalised recommendations