Wood Science and Technology

, Volume 54, Issue 1, pp 31–61 | Cite as

Numerical modelling of timber and timber joints: computational aspects

  • Carmen SandhaasEmail author
  • Ani Khaloian Sarnaghi
  • Jan-Willem van de Kuilen


Timber joints with their simultaneous ductile and brittle failure modes still pose a major challenge when it comes to modelling. Wood is heterogeneous and highly anisotropic. It shows ductile behaviour in compression and brittle behaviour in tension and shear. A 3D constitutive model for wood based on continuum damage mechanics was developed and implemented via a subroutine into a standard FE framework. Embedment and joint tests using three different wood species (spruce, beech and azobé) were carried out, and the results were compared with modelling outcomes. The failure modes could be identified, and the general shape of the load–displacement curves agreed with the experimental outcomes.

List of symbols


Modulus of elasticity in longitudinal direction (parallel to the fibre direction)


Modulus of elasticity in radial direction (perpendicular to the fibre direction)


Modulus of elasticity in tangential direction (perpendicular to the fibre direction)


Shear modulus in LR-plane


Shear modulus in LT-plane


Shear modulus in RT-plane


Tensile strength parallel to the fibre direction


Compressive strength parallel to the fibre direction


Tensile strength perpendicular to the fibre direction


Compressive strength perpendicular to the fibre direction


Shear strength


Rolling shear strength


Fracture energy for tension parallel to the fibre direction


Fracture energy for tension perpendicular to the fibre direction


Fracture energy for shear


Fracture energy for rolling shear


Damage in tension parallel to the fibre direction


Damage in compression parallel to the fibre direction


Damage in tension perpendicular to the fibre direction (LT-plane)


Damage in compression perpendicular to the fibre direction (radial direction)


Damage in tension perpendicular to the fibre direction (LR-plane)


Damage in compression perpendicular to the fibre direction (tangential direction)


Damage in longitudinal shear (LT-plane)


Damage in longitudinal shear (LR-plane)


Damage in rolling shear (RT-plane)


Vector of stresses


Vector of strains


Modified stiffness matrix (including damage)


Poisson ratios


Viscosity parameter


Continuum damage mechanics


High-strength steel


Very high-strength steel


3D linear hexahedral elements with 8 nodes


3D quadratic hexahedral elements with 20 nodes


3D quadratic hexahedral elements with 20 nodes and reduced integration


State variable, showing the damage variable dc,0 in compression parallel to the fibre direction


State variable, showing the damage variable dt,90 in tension perpendicular to the fibre direction


State variable, showing the damage variable dvR in longitudinal shear


Coefficient of variation


Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. Ballerini M, Rizzi M (2005) A numerical investigation on the splitting strength of beams loaded perpendicular-to-grain by multiple dowel-type connections. Paper 38-7-1. CIB-W18 Meeting 38, Karlsruhe, GermanyGoogle Scholar
  2. Bažant ZP, Oh BH (1983) Crack band theory for fracture of concrete. Matériaux Constr 16(3):155–177. CrossRefGoogle Scholar
  3. Blaß HJ, Bejtka I (2008) Numerische Berechnung der Tragfähigkeit und der Steifigkeit von querzugverstärkten Verbindungen mit stiftförmigen Verbindungsmitteln (Numerical calculation of load-carrying capacity and stiffness of reinforced joints with dowel-type fasteners). Karlsruher Berichte zum Ingenieurholzbau, vol 10. Karlsruhe University of Technology, Germany (in German) Google Scholar
  4. Bocquet JF (1997) Modélisation des deformations locales du bois dans les assemblages brochés et boulonnés (Modelling of local deformations of the timber in dowelled and bolted joints). Dissertation, University Blaise Pascal Clermont-Ferrand (in French) Google Scholar
  5. Campilho RDSG, de Moura MFSF, Barreto AMJP, Morais JJL, Domingues JJMS (2009) Fracture behaviour of damaged wood beams repaired with an adhesively-bonded composite patch. Compos A 40(6-7):852–859CrossRefGoogle Scholar
  6. Cofer WF, Du Y, Hermanson JC (1999) Development of a simple three dimensional constitutive model for the analysis of wood, vol 231. American Society of Mechanical Engineers, Applied Mechanics Division (AMD), Mechanics of Cellulosic Materials, New York, pp 107–124Google Scholar
  7. Dias AMPG, Van de Kuilen JWG, Cruz HMP, Lopes SMR (2010) Numerical modelling of the load-deformation behavior of doweled softwood and hardwood joints. Wood Fiber Sci 42(4):480–489Google Scholar
  8. Dorn M (2012) Investigations on the serviceability limit state of dowel-type timber connections. Dissertation, Technische Universität WienGoogle Scholar
  9. Eberhardsteiner J (2002) Mechanisches Verhalten von Fichtenholz; experimentelle Bestimmung der biaxialen Festigkeitseigenschaften (Mechanical behaviour of spruce; experimental determination of biaxial strength properties). Springer, Vienna. ISBN: 978-3-211-83763-4. German) CrossRefGoogle Scholar
  10. EN 26891 (1991) Timber structures—joints made with mechanical fasteners—general principles for the determination of strength and deformation characteristics (ISO 6891). Comité Européen de Normalisation (CEN), BrusselsGoogle Scholar
  11. EN 338 (2009) Structural timber—strength classes. Comité Européen de Normalisation (CEN), BrusselsGoogle Scholar
  12. Federal Highway Administration (2007) Manual for LS-Dyna wood material model 143. Publication No. FHWA-HRT-04-097. U.S. Department of TransportationGoogle Scholar
  13. Fleischmann M (2005) Numerische Berechnung von Holzkonstruktionen unter Verwendung eines realitätsnahen orthotropen elasto-plastischen Werkstoffmodells (Numerical calculations of timber structures using a realistic orthotropic elasto-plastic material model). Dissertation, Technical University, Vienna (in German) Google Scholar
  14. Franke S (2008) Zur Beschreibung des Tragverhaltens von Holz unter Verwendung eines photogrammetrischen Messsystems (Description of load-carrying behaviour of timber using a photogrammetric measuring system). Dissertation, Bauhaus University Weimar (in German) Google Scholar
  15. Gharib M, Hassanieh A, Valipour H, Bradford MA (2017) Three-dimensional constitutive modelling of arbitrarily orientated timber based on continuum damage mechanics. Finite Elem Anal Des 135:79–90. CrossRefGoogle Scholar
  16. Grosse M (2005) Zur numerischen Simulation des physikalisch nichtlinearen Kurzzeittragverhaltens von Nadelholz am Beispiel von Holz-Beton-Verbundkonstruktionen (Numerical simulation of physical non-linear short-term behaviour of softwood using the example of timber-concrete composite structures). Dissertation, Bauhaus University Weimar (in German) Google Scholar
  17. Hambli R (2013) A quasi-brittle continuum damage finite element model of the human proximal femur based on element deletion. Med Biol Eng Comput 51(1–2):219–231. CrossRefPubMedGoogle Scholar
  18. Hashin Z (1980) Failure criteria for unidirectional fiber composites. J Appl Mech Trans ASME 47(2):329–334. CrossRefGoogle Scholar
  19. Hemmer K (1985) Versagensarten des Holzes der Weißtanne (Abies alba) unter mehrachsiger Beanspruchung (Failure modes of fir (Abies alba) subjected to multiaxial loading). Dissertation, University of Technology Karlsruhe (in German) Google Scholar
  20. Hofstetter K, Hellmich C, Eberhardsteiner J (2005) Development and experimental validation of a continuum micromechanics model for the elasticity of wood. Eur J Mech A Solids 24(6):1030–1053. CrossRefGoogle Scholar
  21. Jockwer R (2014) Structural behaviour of glued laminated timber beams with unreinforced and reinforced notches. Dissertation, ETH ZürichGoogle Scholar
  22. Juhasz TJ (2003) Ein neues physikalisch basiertes Versagenskriterium für schwach 3D-verstärkte Faserverbundlaminate (A new physical failure criterion for slightly 3D-reinforced composite materials). Dissertation, Technical University Braunschweig (in German) Google Scholar
  23. Keenan FJ (1973) Shear strength of glued-laminated timber beams. Dissertation, University of TorontoGoogle Scholar
  24. Khelifa M, Khennane A, El Ganaoui M, Celzard A (2016) Numerical damage prediction in dowel connections of wooden structures. Mater Struct 49:1829–1840. CrossRefGoogle Scholar
  25. Khennane A, Khelifa M, Bleron L, Viguier J (2014) Numerical modelling of ductile damage evolution in tensile and bending tests of timber structures. Mech Mater 68:228–236. CrossRefGoogle Scholar
  26. Lee H (2015) Damage modelling for composite structures. Dissertation, University of ManchesterGoogle Scholar
  27. Li M, Füssl J, Lukacevic M, Eberhardsteiner J, Martin CM (2018) Strength predictions of clear wood at multiple scales using numerical limit analysis approaches. Comput Struct 196:200–216. CrossRefGoogle Scholar
  28. Lopes CS (2009) Damage and failure of non-conventional composite laminates. Dissertation, Delft University of TechnologyGoogle Scholar
  29. Maimí P (2006) Modelización constitutiva y computacional del daño y la fractura de materiales compuestos (Constitutive numerical modelling of damage and fracture of composite materials). Dissertation, University of Girona (in Spanish) Google Scholar
  30. Maimí P, Mayugo JA, Camanho PP (2008) A three-dimensional damage model for transversely isotropic composite laminates. J Compos Mater 42(25):2717–2745. CrossRefGoogle Scholar
  31. Maquer G, Schwiedrzik J, Zysset PK (2014) Embedding of human vertebral bodies leads to higher ultimate load and altered damage localisation under axial compression. Comput Methods Biomech Biomed Eng 17(12):1311–1322. CrossRefGoogle Scholar
  32. Matzenmiller A, Lubliner J, Taylor RL (1995) A constitutive model for anisotropic damage in fiber-composites. Mech Mater 20(2):125–152. CrossRefGoogle Scholar
  33. Moses DM, Prion HGL (2004) Stress and failure analysis of wood composites: a new model. Compos B Eng 35(3):251–261. CrossRefGoogle Scholar
  34. Nagy E, Landis EN, Davids WG (2010) Acoustic emission measurements and lattice simulations of microfracture events in spruce. Holzforschung 64(4):455–461. CrossRefGoogle Scholar
  35. Pinho ST, Dávila CG, Camanho PP, Iannucci L, Robinson P (2005) Failure modes and criteria for FRP under in-plane or three-dimensional stress states including shear non-linearity. Technical Report NASA/TM-2005-213530. NASA Langley Research Center, Hampton, VA, USAGoogle Scholar
  36. Pistoia W, van Rietbergen B, Lochmüller E-M, Lill CA, Eckstein E, Rüegsegger P (2002) Estimation of distal radius failure load with micro-finite element analysis models based on three-dimensional peripheral quantitative computed tomography images. Bone 30(6):842–848. CrossRefPubMedPubMedCentralGoogle Scholar
  37. Resch E, Kaliske M (2010) Three-dimensional numerical analyses of load-bearing behavior and failure of multiple double-shear dowel-type connections in timber engineering. Comput Struct 88(3–4):165–177. CrossRefGoogle Scholar
  38. Riks E (1972) The application of Newton’s method to the problem of elastic stability. J Appl Mech 39:1060–1066CrossRefGoogle Scholar
  39. Ruffoni D, van Lenthe GH (2017) 3.10 Finite element analysis in bone research: a computational method relating structure to mechanical function. In: Ducheyne P (ed) Comprehensive Biomaterials II, vol 3. Elsevier, Oxford, pp 169–196CrossRefGoogle Scholar
  40. Saavedra Flores EI, Saavedra K, Hinojosa J, Chandra Y, Das R (2016) Multi-scale modelling of rolling shear failure in cross-laminated timber structures by homogenisation and cohesive zone models. Int J Solids Struct 81:219–232. CrossRefGoogle Scholar
  41. Sandhaas C (2011) 3D material model for wood, based on continuum damage mechanics. Stevinrapport 6-11-4, Stevin II Laboratory. Delft University of Technology, The NetherlandsGoogle Scholar
  42. Sandhaas C (2012) Mechanical behaviour of timber joints with slotted-in steel plates. Dissertation, Delft University of TechnologyGoogle Scholar
  43. Sandhaas C, Ravenshorst G, Blaß HJ, Van de Kuilen JWG (2013) Embedment tests parallel-to-grain and ductility aspects using various wood species. Eur J Wood Prod 71(5):599–608. CrossRefGoogle Scholar
  44. Schlangen E, Qian Z (2009) 3D modeling of fracture in cement-based materials. J Multiscale Model 1(2):245–261. CrossRefGoogle Scholar
  45. Schmid M (2002) Anwendung der Bruchmechanik auf Verbindungen mit Holz (Application of fracture mechanics to timber joints). Dissertation, University of Technology Karlsruhe (in German) Google Scholar
  46. Schweigler M, Bader TK, Hochreiner G, Unger G, Eberhardsteiner J (2016) Load-to-grain angle dependence of the embedment behavior of dowel-type fasteners in laminated veneer lumber. Constr Build Mater 126:1020–1033. CrossRefGoogle Scholar
  47. Sirumbal-Zapata LF, Málaga-Chuquitaype C, Elghazouli AY (2018) A three-dimensional plasticity-damage constitutive model for timber under cyclic loads. Comput Struct 195:47–63. CrossRefGoogle Scholar
  48. Spengler R (1982) Festigkeitsverhalten von Brettschichtholz unter zweiachsiger Beanspruchung; Teil 1: Ermittlung des Festigkeitsverhaltens von Brettelementen aus Fichte durch Versuche (Load-carrying behaviour of glued laminated timber subjected to biaxial loading; part 1: experimental determination of load-carrying behaviour of spruce laminations). SFB96, Volume 62, reports on reliability theory of structures. Technical University Munich (in German) Google Scholar
  49. Toussaint P (2009) Application et modélisation du principe de la précontrainte sur des assemblages de structure bois. (Application and modelling of the principle of pre-stressing on timber joints). Dissertation, University Henri Poincaré ENSTIB Nancy (in French) Google Scholar
  50. Valipour H, Khorsandnia N, Crews K, Foster S (2014) A simple strategy for constitutive modelling of timber. Constr Build Mater 53:138–148. CrossRefGoogle Scholar
  51. van de Kuilen JWG, Gard W, Ravenshorst G, Antonelli V, Kovryga A (2017) Shear strength values for soft- and hardwoods. Paper 50-6-1. INTER Meeting 50, Kyoto, Japan, pp 49–64Google Scholar
  52. Xu BH, Taazount M, Bouchaïr A, Racher P (2009) Numerical 3D finite element modelling and experimental tests for dowel-type timber joints. Constr Build Mater 23(9):3043–3052. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Timber Structures and Building ConstructionKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Holzforschung MünchenTechnical University MunichMunichGermany
  3. 3.Biobased Structures and MaterialsDelft University of TechnologyDelftThe Netherlands

Personalised recommendations